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# Norm inequalities for vector functions
B. A. Bhayo, V. Božin, D. Kalaj, M. Vuorinen
File: main.tex, printed: 2024-8-27, 21.22
Abstract. We study vector functions of ${\mathbb{R}}^{n}$ into itself, which
are of the form $x\mapsto g(|x|)x\,,$ where $g:(0,\infty)\to(0,\infty)$ is a
continuous function and call these radial functions. In the case when
$g(t)=t^{c}$ for some $c\in{\mathbb{R}}\,,$ we find upper bounds for the
distance of image points under such a radial function. Some of our results
refine recent results of L. Maligranda and S. Dragomir. In particular, we
study quasiconformal mappings of this simple type and obtain norm inequalities
for such mappings.
Mathematics Subject Classification (2000): 30C65, 26D15
Keywords and phrases: Quasiconformal map, normed linear space
## 1\. Introduction
In 2006 L. Maligranda [M] studied the following function
(1.1) $\alpha_{p}(x,y)=||x|^{p-1}x-|y|^{p-1}y|\,,p\in{\mathbb{R}}\,,$
for $x,y\in{\mathbb{R}}^{n}\setminus\\{0\\}\,,$ termed the $p-$angular
distance between $x$ and $y\,.$ It is clear that $\alpha_{p}$ satisfies the
triangle inequality and thus it defines a metric. Note that $\alpha_{0}(x,y)$
equals $2\sin(\omega/2)$ where $\omega\in[0,\pi]$ is the angle between the
segments $[0,x]$ and $[0,y]\,.$ He proved in [M, Theorem 2] the following
theorem in the context of normed spaces.
###### 1.2 Theorem.
$\alpha_{p}(x,y)\leq\left\\{\begin{array}[]{lll}(2-p)\displaystyle\frac{|x-y|\max\\{|x|^{p},|y|^{p}\\}}{(\max\\{|x|,|y|\\})}\quad{\rm
if}\,\,p\in(-\infty,0)\,\,{\rm and}\,x,y\neq 0;\\\ \\\
(2-p)\displaystyle\frac{|x-y|}{(\max\\{|x|,|y|\\})^{1-p}}\quad{\rm
if}\,\,p\in[0,1]\,\,{\rm and}\,\,x,y\neq 0;\\\ \\\
p\,(\max\\{|x|,|y|\\})^{p-1}|x-y|\quad{\rm
if}\,p\,\in(1,\infty).\end{array}\right.$
Soon thereafter, in 2009, S. Dragomir [D, Theorem 1] refined this result and
gave the following upper bound for the $p$-angular distance for nonzero
vectors $x,y\,.$
###### 1.3 Theorem.
$\alpha_{p}(x,y)\leq\left\\{\begin{array}[]{llllll}|x-y|(\max\\{|x|,|y|\\})^{p-1}+\left||x|^{p-1}-|y|^{p-1}\right|\min\\{|x|,|y|\\}\quad{\rm
if}\,\,p\in(1,\infty)\,;\\\ \\\
\displaystyle\frac{|x-y|}{(\min\\{|x|,|y|\\})^{1-p}}+\left||x|^{1-p}-|y|^{1-p}\right|\min\left\\{\frac{|x|^{p}}{|y|^{1-p}},\frac{|y|^{p}}{|x|^{1-p}}\right\\}\quad{\rm
if}\,\,p\in[0,1]\,;\\\ \\\
\displaystyle\frac{|x-y|}{(\min\\{|x|,|y|\\})^{1-p}}+\displaystyle\frac{||x|^{1-p}-|y|^{1-p}|}{(\max\\{|x|^{-p}|y|^{1-p},|y|^{-p}|x|^{1-p}\\})}\quad{\rm
if}\,\,p\in(-\infty,0)\,.\end{array}\right.$
Generalizations for operators were discussed very recently in [DFM]. For
general information about norm inequalities see [MPF, Chapter XVIII].
Studying sharp constants connected to the $p$-Laplace operator J. Byström [By,
Lemma 3.3] proved in 2005 the following result.
###### 1.4 Theorem.
For $p\in(0,1)$ and $x,y\in\mathbb{R}^{n}$, we have
$\alpha_{p}(x,y)\leq 2^{1-p}|x-y|^{p}$
with equality for $x=-y\,.$
In this paper we study a two exponent variant of the function
$x\mapsto|x|^{p-1}x$ defined for $a,b>0,x\in\mathbb{R}^{n}\,,$
(1.5) ${\mathcal{A}}_{a,b}(x)=\left\\{\begin{array}[]{ll}|x|^{a-1}x\quad
if\;|x|<1\\\ |x|^{b-1}x\quad if\;|x|\geq 1.\end{array}\right.$
This function, like its one exponent version (the special case $a=b$), defines
a quasiconformal mapping and it has been used in many examples to illuminate
various properties of these maps [Va, p.49]. For instance, if $a\in(0,1)$ the
function ${\mathcal{A}}_{a,b}$ is Hölder-continuous at the origin.
We prove that the change of distance under this function is maximal in the
radial direction, up to a constant, in the sense of the next theorem (observe
that the points $x$ and $z$ are on the same ray). Note that the result is
sharp for $a\to 1\,.$ This result is natural to expect, but the proof is
somewhat involved. For brevity we write ${\mathcal{A}}={\mathcal{A}}_{a,b}$ if
$0<a\leq 1\leq b\,.$
###### 1.6 Theorem.
Let $0<a\leq 1\leq b$ and
$C(a,b)=\sup_{|x|\leq|y|}Q(x,y),$
where
$Q(x,y)=\frac{|{\mathcal{A}}(x)-{\mathcal{A}}(y)|}{|{\mathcal{A}}(x)-{\mathcal{A}}(z)|}\,,\quad
x,y\in{\mathbb{R}}^{n}\setminus\\{0\\}\,\text{ with }\;x\neq y\,,$
and
$z=\frac{x}{|x|}(|x|+|x-y|).$
Then
$C(a,b)=\frac{2}{3^{a}-1}\text{ and }\lim_{a\to 1}C(a,b)=1.$
Because ${\mathcal{A}}_{a,b}$ agrees with $x\mapsto|x|^{a-1}x$ in
${\mathbb{B}}^{n}\,,$ we can compare Theorem 1.6 to Theorems 1.2, 1.3, and
1.4. We also have the following upper bound for $\alpha_{p}\,:$
###### 1.7 Theorem.
For all $x,y\in{\mathbb{R}}^{n}$ and $p\in(0,1)$
(1.8)
$\alpha_{p}(x,y)\leq|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(y)|\,,$
and furthermore, if $|x|\leq|y|$, we have also
(1.9)
$\alpha_{p}(x,y)\leq|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(y)|\leq\frac{2}{3^{p}-1}|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(z)|$
where $z$ is as in Theorem 1.6.
For a systematic comparison of the above results, see Section 5 where it is
shown that sometimes the bound in Theorem 1.7 is better than the other bounds
in Theorems 1.2, 1.3, 1.4.
We also discuss some properties of the distortion function $\varphi_{K}(r)$
associated with the quasiconformal Schwarz lemma, see [LV].
Acknowledgments. The first author is indebted to the Graduate School of
Mathematical Analysis and its Applications for support. He also wishes to
acknowledge the expert help of Dr. H. Ruskeepää in the use of the Mathematica®
software [Ru]. The fourth author was, in part, supported by the Academy of
Finland, Project 2600066611.
## 2\. Preliminary results
We prove here some inequalities for elementary functions that will be applied
in later sections. These inequalities deal with the logarithm and some of them
may be new results. Note also in the paper [KMV] some elementary Bernoulli
type inequalities were proved and used as a key tool. We use the notation sh,
ch, th, arsh, arch and arth to denote the hyperbolic sine, cosine, tangent and
their inverse functions, respectively.
As well-known, conformal invariants of geometric function theory are on one
hand closely linked with function theoretic extremal problems and on the other
hand with special functions such as complete elliptic integrals, elliptic
functions and hypergeometric functions. The connection between conformal
invariants and special functions is provided by conformal maps which can be
applied to express maps of quadrilaterals and ring domains onto canonical ring
domains such as a rectangle and an annulus.
For example, the quasiconformal version of the Schwarz lemma says that for a
$K$-quasiconformal map of the unit disk $\mathbb{B}^{2}$ onto itself keeping
$0$ fixed, we have for all $z\in\mathbb{B}^{2}$ the sharp bound [LV, p. 64]
(2.1) $|f(z)|\leq\varphi_{K}(|z|)\,,\quad\varphi_{K}(r)=\mu^{-1}(\mu(r)/K)$
where $\mu:(0,1)\longrightarrow(0,\infty)$ is a decreasing homeomorphism
defined by
(2.2) $\mu(r)=\frac{\pi}{2}\,\frac{{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}}(r^{\prime})}{{\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}}(r)},\quad{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp
K}}}(r)=\int_{0}^{1}\frac{dx}{\sqrt{(1-x^{2})(1-r^{2}x^{2})}}\,,$
and where ${\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}}(r)$ is Legendre’s complete elliptic integral of the
first kind and $r^{\prime}=\sqrt{1-r^{2}},$ for all $r\in(0,1)$. The function
$\varphi_{K}(r)$ has numerous applications to quasiconformal mapping theory,
see [LV, K, AVV2], which motivates the study of its properties. One of the
challenges is to find bounds, in the range $(0,1)$, and yet asymptotically
sharp when $K\to 1\,.$ For instance, the change of hyperbolic distances under
$K$-quasiconformal mappings of the unit disk onto itself can be estimated in
terms of the function $\varphi_{K}\,,$ see [AVV2, LV].
###### 2.3 Lemma.
The following functions are monotone increasing from $(0,\infty)$ onto
$(1,\infty)$;
$(1)\qquad f(x)=\displaystyle\frac{(1+x)\log(1+x)}{x}\,,\quad(2)\qquad
g(x)=\displaystyle\frac{x}{\log(1+x)}\,,$
$(3)$ For a fixed $t\in(0,1)$, the function $h(K)=K(1-t^{2/K})$ is monotone
increasing on $(1,\infty)$.
###### Proof.
For the proof of (1) see [KMV, p. 7]. For (2), we get
$g^{{}^{\prime}}(x)=\frac{1}{\log(1+x)}-\frac{x}{(1+x)(\log(1+x))^{2}}=\frac{(1+x)\log(1+x)-x}{(1+x)(\log(1+x))^{2}},$
and $g^{{}^{\prime}}(x)>0$ by (1). Moreover, $g$ tends to $1$ and $\infty$
when $x$ tends $0$ and $\infty$. Proof of $(3)$ follows easily because
$x\mapsto(1-a^{x})/x$ is decreasing on $(0,1)$ for each $a\in(0,1)$ [AVV2,
1.58(3)]. ∎
###### 2.4 Corollary.
For a fixed $x\in(0,1)$, the following functions, (1) $f(a)=(1+ax)^{1/a}$, (2)
$g(a)=(\log(1+x^{a}))^{1/a}$ are decreasing and increasing on $(1,\infty)$,
respectively. (3) The following inequality holds for $x\geq 0$ and
$a\in[0,1]$,
$\log(1+x^{a})\leq\max\\{\log(1+x),\log^{a}(1+x)\\}.$
###### 2.5 Lemma.
For $K>1\,,r\in(0,1),\,u={\rm arch}(1/r)/K$, the following functions
1. (1)
$f(K)=r\,{\rm arth}(1/{\rm ch(u)}){\rm sh}(u),$
2. (2)
$g(K)=rK\,{\rm arth}(1/{\rm ch(u)}){\rm sh}(u)$
are strictly decreasing and increasing, respectively. Moreover, both functions
tend to $\sqrt{1-r^{2}}\,{\rm arth}(r)$ when $K$ tends to $1$.
###### Proof.
Differentiating $f$ with respect to $K$ we get
$f^{{}^{\prime}}(K)=-\frac{r\,\textrm{arth}(1/r)}{K^{2}}\left(\textrm{arth}\left(\frac{1}{\textrm{ch}(u}\right){\rm
ch}(u)-1\right)\leq 0,$
$g^{{}^{\prime}}(K)=r\left(\textrm{ch}\left(\frac{1}{r}\right)\left(1-\textrm{arth}\left(\frac{1}{\textrm{ch}(u)}\right)+K\,\textrm{arth}\left(\frac{1}{\textrm{ch}(u)}\right)\textrm{sh}(u)\right)\right)\geq
0,$
respectively. We obtain
$f(1)=g(1)=r\,\textrm{arth}(r)\sqrt{({\rm ch}({\rm
arch}(1/r)))^{2}-1}=\sqrt{1-r^{2}}\,\textrm{arth}(r).$
∎
###### 2.6 Lemma.
(1) For a fixed $t>0$, the following function is monotone increasing in $K>1$.
Moreover, for $t=t_{0}=(e-1)/(e+1)$, the function is increasing from
$(1,\infty)$ onto $(m_{1},1)$,
$f(K)=\frac{K-\log(1/t)}{t^{2/K}(K+\log(1/t))},\;m_{1}=\frac{1+\log
t_{0}}{t^{2}_{0}(1-\log t_{0})}\approx 0.6027..$
(2) The following function is monotone increasing from $(1,\infty)$ onto
$(m_{2},1)$,
$g(K)=\frac{t^{1/K}_{0}\log(1/t^{2}_{0})}{K(1-t^{2/K}_{0})},\;m_{2}=\frac{2t_{0}\log
t_{0}}{t^{2}_{0}-1}\approx 0.9072..\;.$
###### Proof.
Differentiating $f$ with respect to $K$ we get
$\displaystyle f^{{}^{\prime}}(K)$ $\displaystyle=$
$\displaystyle\frac{t^{-2/K}}{K+\log(1/t)}-\frac{t^{-2/K}(K-\log(1/t))}{(K+\log(1/t))^{2}}+\frac{2t^{-2/K}(K-\log(1/t))\log
t}{K^{2}(K+\log(1/t))^{2}}$ $\displaystyle=$
$\displaystyle\frac{2t^{-2/K}(K^{2}\log(1/t)+K^{2}\log t-(\log(1/t))^{2}\log
t)}{K^{2}(K+\log(1/t))^{2}}$ $\displaystyle=$
$\displaystyle\frac{2t^{-2/K}(\log(1/t))^{3}}{K^{2}(K+\log(1/t))^{2}}>0.$
For $t=t_{0}$, $f$ tends to $m_{1}$ and $1$ when $K$ tends to $1$ and
$\infty$, respectively.
For the proof of (2), we differentiate $g$ with respect to $K$ and get,
$\displaystyle g^{{}^{\prime}}(K)$ $\displaystyle=$
$\displaystyle-\frac{t^{1/K}_{0}\log(1/t^{2}_{0})}{K^{2}(1-t^{2/K}_{0})}-\frac{2t^{3/K}_{0}\log(1/t^{2}_{0})\log
t_{0}}{K^{3}(1-t^{2/K}_{0})^{2}}-\frac{t^{1/K}_{0}\log(1/t^{2}_{0})\log
t_{0}}{K^{3}(1-t^{2/K}_{0})}$ $\displaystyle=$ $\displaystyle
t^{1/K}_{0}\log(1/t^{2}_{0})(-K(1-t^{2/K}_{0})-(1+t^{2/K}_{0})\log
t_{0})/(K^{3}(1-t^{2/K}_{0})^{2})$ $\displaystyle=$ $\displaystyle
t^{1/K}_{0}\log(1/t^{2}_{0})(t^{2/K}_{0}(1+\log(1/t_{0}))-(K-\log(1/t_{0})))/(K^{3}(1-t^{2/K}_{0})^{2})$
$\displaystyle=$
$\displaystyle\frac{t^{3/K}_{0}\log(1/t^{2}_{0})(1+\log(1/t_{0}))}{K^{3}(1-t^{2/K}_{0})^{2}}\left(1-\frac{K-\log(1/t_{0})}{t^{2/K}_{0}(1+\log(1/t_{0}))}\right)>0$
by (1). We can see that $g$ tends to $m_{2}$ and $1$ when $K$ tends to $1$ and
$\infty$, respectively. This completes the proof. ∎
###### 2.7 Lemma.
The following inequality holds for $K\geq 1$ and
$t\in[t_{0},1),t_{0}=(e-1)/(e+1)$
(2.8) $\log\left(\frac{1+t^{1/K}}{1-t^{1/K}}\right)\leq
K\log\left(\frac{1+t}{1-t}\right).$
###### Proof.
Write $h(t)=K\textrm{arth}(t)-\textrm{arth}(t^{1/K})$. Differentiating $h$
with respect to $t$ we get,
$\displaystyle h^{{}^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\frac{K}{1-t^{2}}-\frac{t^{1/K-1}}{K(1-t^{2/K})}=\frac{K^{2}t(1-t^{2/K})-t^{1/K}(1-t^{2})}{tK(1-t^{2})(1-t^{2/K})}$
$\displaystyle\geq$
$\displaystyle\frac{Kt(1-t^{2})-t^{1/K}(1-t^{2})}{tK(1-t^{2})(1-t^{2/K})}=\frac{Kt-t^{1/K}}{Kt(1-t^{2/K})}\geq
0.$
The first inequality holds by Lemma 2.3(3) and the second one holds when
$Kt\geq t^{1/K}\Leftrightarrow t\geq(1/K)^{K/(K-1)}=c_{1}(K)$. It is easy to
see by Lemma 2.3(1) that $c_{1}(K)$ is decreasing in $(1,\infty)$. We see that
$c_{1}(K)\to 1/e\approx 0.3679..$ and $0$ when $K\to 1$ and $\infty$
respectively, hence $h(t)$ is increasing in $t\geq 1/e$.
We can see that $h(t_{0})=K(1-2\,\textrm{arth}(t^{1/K}_{0})/K)/2$. Now it is
enough to prove that $f(K)=2\,\textrm{arth}(t^{1/K}_{0})/K<1$. Differentiating
$f$ with respect to $K$ we get
$\displaystyle f^{{}^{\prime}}(K)$ $\displaystyle=$
$\displaystyle\frac{-2\,\textrm{arth}(t^{1/K}_{0})}{K^{2}}-\frac{2t^{1/K}_{0}\log(t_{0})}{K^{3}(1-t^{2/K}_{0})}$
$\displaystyle=$ $\displaystyle
2(-K(1-t^{2/K}_{0})\,\textrm{arth}(t^{1/K}_{0})-t^{1/K}_{0}\log(t_{0}))/(K^{3}(1-t^{2/K}_{0}))$
$\displaystyle\leq$ $\displaystyle
2(-K(1-t^{2/K}_{0})\,\textrm{arth}(t_{0})+t^{1/K}_{0}\log(1/t_{0}))/(K^{3}(1-t^{2/K}_{0}))$
$\displaystyle=$ $\displaystyle
2(-(K(1-t^{2/K}_{0})/2)\log\left(\frac{1+t_{0}}{1-t_{0}}\right)+t^{1/K}_{0}\log(1/t_{0}))/(K^{3}(1-t^{2/K}_{0}))$
$\displaystyle=$
$\displaystyle(t^{1/K}_{0}\log(1/t^{2}_{0})-K(1-t^{2/K}_{0}))/(K^{3}(1-t^{2/K}_{0}))$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\left(\frac{t^{1/K}_{0}\log(1/t^{2}_{0})}{K(1-t^{2/K}_{0})}-1\right)<0$
by Lemma $\ref{2ll}(2)$, hence $f$ is a monotone decreasing function from
$(1,K)$ onto $(0,1/2)$. This implies the proof. ∎
###### 2.9 Lemma.
The following inequality holds for $K\geq 1$ and
$t\in(0,t_{0}],t_{0}=(e-1)/(e+1)$
(2.10) $\log\left(\frac{1+t^{1/K}}{1-t^{1/K}}\right)\leq
K\left(\log\left(\frac{1+t}{1-t}\right)\right)^{1/K}.$
###### Proof.
Write
$F(t)=K-\displaystyle\frac{\log((1+t^{1/K})/(1-t^{1/K}))}{\left(\log(1+t)/(1-t)\right)^{1/K}}\;.$
For the proof of (2.10) we show that $F(t)$ is decreasing in $t$ and
$F(t_{0})\geq 0$. Differentiating $F$ with respect to $t$ we get,
$F^{{}^{\prime}}(t)=\frac{\log\left(\frac{1+t}{1-t}\right)^{(K-1)/K}\left(2t^{1/K}(t^{2}-1)\log\left(\frac{1+t}{1-t}\right)-2t(t^{2/K}-1)\log\left(\frac{1+t^{1/K}}{1-t^{1/K}}\right)\right)}{Kt(t^{2}-1)(t^{2/K}-1)}\;.$
Now we show that
$t(t^{2/K}-1)\log\left(\frac{1+t^{1/K}}{1-t^{1/K}}\right)\geq
t^{1/K}(t^{2}-1)\log\left(\frac{1+t}{1-t}\right).\qquad(\ast)$
For the proof of $(\ast)$, it is enough to prove that $t(t^{2/K}-1)\geq
t^{1/K}(t^{2}-1).$ We get
$\displaystyle t(t^{2/K}-1)-t^{1/K}(t^{2}-1)$ $\displaystyle=$
$\displaystyle(t^{1/K+1}+t)(t^{1/K}-1)-(t^{1/K+1}+t^{1/K})(t-1)$
$\displaystyle=$ $\displaystyle t^{1/K+1/K+1}+t^{1/K}-t-t^{1/K+1+1}$
$\displaystyle=$ $\displaystyle t^{1/K}(t^{1/K+1}+1)-t(t^{1/K+1}+1)$
$\displaystyle=$ $\displaystyle(t^{1/K+1}+1)(t^{1/K}-t)\geq 0,$
this implies that $F(t)$ is decreasing in $t$. Now we prove that $F(t_{0})$ is
positive as a function of $K$. We write
$f(K)=K-\log\left(\frac{1+t^{1/K}_{0}}{1-t^{1/K}_{0}}\right)=K-2\,\textrm{arth}(t^{1/K}_{0})=F(t_{0})\;.$
Differentiating $f$ with respect to $K$ we get
$f^{{}^{\prime}}(K)=1+\frac{2t^{1/K}_{0}\log(t_{0})}{K^{2}(1-t^{2/K}_{0})}\geq
1-\frac{t^{1/K}_{0}\log(1/t^{2}_{0})}{K^{2}(1-t^{2/K}_{0})}>0$
by Lemma 2.6(2), hence $f$ is increasing in $K$. This implies the proof. ∎
###### 2.11 Corollary.
The following inequality holds for $K\geq 1$ and $t\in[0,1)$
(2.12) $\log\left(\frac{1+t^{1/K}}{1-t^{1/K}}\right)\leq
K\max\left\\{\left(\log\left(\frac{1+t}{1-t}\right)\right)^{1/K},\,\,\log\left(\frac{1+t}{1-t}\right)\right\\}.$
###### Proof.
The proof follows easily from inequalities (2.8) and (2.10). ∎
The next function tells us how the hyperbolic distances from the origin are
changed under the radial selfmapping of the the unit disk,
$z\mapsto|z|^{1/K-1}z,K>1,$ which is the restriction of
${\mathcal{A}}_{1/K,1/K}(z)$ to the unit disk. See also [BV].
###### 2.13 Theorem.
The following inequality holds for $K\geq 1$, $|z|<1$;
(2.14) $\rho(0,{\mathcal{A}}_{1/K,K}(z))\leq
K\max\\{\rho(0,|z|),\rho^{1/K}(0,|z|)\\}$
where $\rho$ is the hyperbolic metric [Vu, p. 19].
###### Proof.
Proof follows easily from inequality (2.12) and the formula
$\rho(0,r)=\log((1+r)/(1-r))\,.$ ∎
###### 2.15 Remark.
The constant $K$ can not be replaced by $K^{9/10}$ in (2.14), because for
$|z|=t_{0}$, the inequality (2.14) is equivalent to
$1-2\,\textrm{arth}(t^{1/K}_{0})/K^{9/10}\geq 0$. Write
$f(K)=1-2\,\textrm{arth}(t^{1/K}_{0})/K^{9/10}$, and we get
$f^{{}^{\prime}}(K)=\frac{9\,\textrm{arth}(t^{1/K}_{0})}{5K^{19/10}}+\frac{2t^{1/K}\log(t_{0})}{K^{29/10}(1-t^{2/K}_{0})},$
we see that $f^{{}^{\prime}}(1.005)=-0.004<0$, $f(K)$ is not increasing in
$K$.
###### 2.16 Lemma.
For $K>1$ the function
$F(r)=\frac{2{\rm arth}(1/{\rm ch}({\rm arch}(1/r)/K))}{\max\\{2{\rm
arth}(r),(2{\rm arth}(r))^{1/K}\\}}$
is monotone increasing in $(0,t_{0})$ and decreasing in $(t_{0},1)$.
###### Proof.
(1) Let $u={\rm arch}(1/r)/K$ and
$f(r)=\frac{{\rm arth}(1/{\rm ch}(u))}{{\rm arth}(r)}\,.$
Differentiating $f$ with respect to $r$ we get
$\displaystyle f^{{}^{\prime}}(r)$ $\displaystyle=$ $\displaystyle-\frac{{\rm
arth}(1/{\rm ch}(u))}{(1-r^{2})({\rm arth}(r))^{2}}+\frac{(1/{\rm ch}(u)){\rm
th}(u)}{K\sqrt{1/r-1}\sqrt{1+1/r}\,r^{2}\,{\rm arth}(r)(1-(1/{\rm
ch}(u))^{2})}$ $\displaystyle=$ $\displaystyle-\frac{Kr\,{\rm arth}(1/{\rm
ch}(u)){\rm sh}(u)-\sqrt{1-r^{2}}\,{\rm arth}(r)}{Kr(1-r^{2})({\rm
arth}(r))^{2}{\rm sh}(u)}\leq 0,$
by Lemma 2.5(2), hence $f$ is decreasing in $r\in(0,1)$.
(2) Let
$g(r)=\frac{2^{1-1/K}{\rm arth}(1/{\rm ch}(u))}{({\rm arth}(r))^{1/K}}\,.$
Differentiating $g$ with respect to $r$ we get
$g^{{}^{\prime}}(r)=\xi\left((1-r^{2}){\rm arth}(r)-r\sqrt{1-r^{2}}\,{\rm
arth}(1/{\rm ch}(u)){\rm sh}(u)\right)\geq 0$
by Lemma 2.5(1), here
$\xi=\frac{{2^{1-1/K}({\rm arth}(r))^{-(1+K)/K}}}{Kr(1-r^{2})^{3/2}\,{\rm
sh}(u)}\,.$
Hence $g$ is increasing in $r\in(0,1)$. We see that $f(t_{0})=g(t_{0})$. Thus
$F(r)$ increases in $r\in(0,t_{0})$ and decreases in $t\in(t_{0},1)$. ∎
For instance it is well-known that for all $K>1,r\in(0,1)$
(2.17) $\log\left(\frac{1+\varphi_{K}(r)}{1-\varphi_{K}(r)}\right)>\
K\log\left(\frac{1+r}{1-r}\right)\,$
[AVV1, (4.5)]. In the next theorem we study a function $p(r)$ which by [AVV2,
Thm 10.14] is a minorant of $\varphi_{K}(r)\,.$
###### 2.18 Theorem.
The following inequality holds for $K\geq 1$, $r\in(0,1),\,t_{0}=(e-1)/(e+1)$,
$\log\left(\frac{1+p\,(r)}{1-p\,(r)}\right)\leq
c_{3}(K)\max\left\\{\log\left(\frac{1+r}{1-r}\right),\left(\log\left(\frac{1+r}{1-r}\right)\right)^{1/K}\right\\}$
here $p\,(r)=1/{\rm ch}({\rm arch}(1/r)/K)$ and $c_{3}(K)=2\,{\rm
arth}(p\,(t_{0}))$. Moreover, $c_{3}(K)\to 1$ when $K\to 1\,.$
###### Proof.
The inequality follows easily from Lemma 2.16, because the maximum value of
the function given in Lemma 2.16 is $c_{3}(K)=1/{\rm ch}({\rm
arch}(1/t_{0})/K)$. ∎
We remark in passing that an inequality similar to (2.18) but with $p(r)$
replaced with $\varphi_{K}(r)$ and $c_{3}(K)$ replaced with a constant $c(K)$
was proved in [BV, Lemma 4.8].
## 3\. Quasiinvariance of the distance ratio metric
Our goal in this section is to study how the distances in the $j$-metric are
transformed under the function (1.5) following closely the paper [KMV]. The
main result here is Corollary 3.3.
###### 3.1 Lemma.
The following inequality holds for $K\geq 1$:
(3.2)
$\log\left(1+\frac{|{\mathcal{A}}_{1/K,K}(x)-{\mathcal{A}}_{1/K,K}(y)|}{\min\\{|{\mathcal{A}}_{1/K,K}(x)|,|{\mathcal{A}}_{1/K,K}(y)|\\}}\right)\leq
2^{1-1/K}\max\\{\log^{1/K}(t),\log(t)\\}$
here $t=1+\displaystyle\frac{|x-y|}{\min\\{|x|,|y|\\}}$, for all
$x,y\in\mathbb{B}^{n}$.
###### Proof.
By Theorem 1.4 and Corollary 2.4(1) we get
$1+\frac{|{\mathcal{A}}_{1/K,K}(x)-{\mathcal{A}}_{1/K,K}(y)|}{\min\\{|{\mathcal{A}}_{1/K,K}(x)|,|{\mathcal{A}}_{1/K,K}(y)|\\}}\leq
1+2^{1-1/K}\frac{|x-y|^{1/K}}{\min\\{|x|^{1/K},|y|^{1/K}\\}}$
$\leq\left(1+\left(\frac{|x-y|}{\min\\{|x|,|y|\\}}\right)^{1/K}\right)^{2^{1-1/K}}.\qquad(\ast)$
Taking $\log$ both sides to $(\ast)$ and by Corollary 2.4(3) we get
$\log\left(1+\frac{|{\mathcal{A}}_{1/K,K}(x)-{\mathcal{A}}_{1/K,K}(y)|}{\min\\{|{\mathcal{A}}_{1/K,K}(x)|,|{\mathcal{A}}_{1/K,K}(y)|\\}}\right)\leq\log\left(\left(1+\left(\frac{|x-y|}{\min\\{|x|,|y|\\}}\right)^{1/K}\right)^{2^{1-1/K}}\right)$
$\leq
2^{1-1/K}\max\left\\{\log\left(1+\frac{|x-y|}{\min\\{|x|,|y|\\}}\right),\left(\log\left(1+\frac{|x-y|}{\min\\{|x|,|y|\\}}\right)\right)^{1/K}\right\\}.$
∎
We denote by $\partial G$ the boundary of a domain $G$ and define
$d(z)=\min\\{|z-m|:m\in\partial G\\}.$
For a domain $G\subset\mathbb{R}^{n},G\neq\mathbb{R}^{n}$, the following
formula
$j(x,y)=\log\left(1+\frac{|x-y|}{\min\\{d(x),d(y)\\}}\right),\;x,y\in G$
defines $j$ as a metric in $G$ (see [Vu, p.28]).
###### 3.3 Corollary.
Let $D=\mathbb{R}^{n}\setminus\\{0\\}$, then we have
$j_{D}({\mathcal{A}}_{1/K,K}(x),{\mathcal{A}}_{1/K,K}(y))\leq
2^{1-1/K}\max\\{j_{D}(x,y),j^{1/K}_{D}(x,y)\\}$
for all $K\geq 1,x,y\in\mathbb{B}^{n}\cap D$.
###### Proof.
Follows from inequality (3.2). ∎
## 4\. Radial functions
###### 4.1 Definition.
Let $f:\overline{\mathbb{R}}^{n}\to\overline{\mathbb{R}}^{n}$ be a
homeomorphism. We say that $f$ is a _radial function_ if there exists a
homeomorphism $g:(0,\infty)\to(0,\infty)$ such that
$f(x)=g(|x|)x,\,x\in\mathbb{R}^{n}\setminus\\{0\\}$.
The following functions are examples of the radial functions:
1. (1)
$h(x)=\displaystyle\frac{x}{|x|^{2}}\,$, $x\in\mathbb{R}^{n}\setminus\\{0\\},$
$\,h(0)=\infty$, $\,h(\infty)=0\,.$
2. (2)
For $a,b>0,$
${\mathcal{A}}_{a,b}(x)=\left\\{\begin{array}[]{ll}|x|^{a-1}x\quad if\;|x|\leq
1\\\ |x|^{b-1}x\quad if\;|x|>1.\end{array}\right.$
###### 4.2 Remark.
Properties of ${\mathcal{A}}:$
1. (1)
For $|x|<1$ and $a,b,c,d>0$
$\displaystyle{\mathcal{A}}_{a,b}({\mathcal{A}}_{c,d}(x))$ $\displaystyle=$
$\displaystyle{\mathcal{A}}_{a,b}(|x|^{c-1}x)=||x|^{c-1}x|^{a-1}|x|^{c-1}x$
$\displaystyle=$ $\displaystyle|x|^{ac-c}|x|^{c-1}x=|x|^{ac-1}x.$
2. (2)
For $|x|>1$
$\displaystyle{\mathcal{A}}_{a,b}({\mathcal{A}}_{c,d}(x))$ $\displaystyle=$
$\displaystyle{\mathcal{A}}_{a,b}(|x|^{d-1}x)=||x|^{d-1}x|^{b-1}|x|^{d-1}x$
$\displaystyle=$ $\displaystyle|x|^{bd-d}|x|^{d-1}x=|x|^{bd-1}x.$
(1) and (2) imply that
${\mathcal{A}}_{a,b}({\mathcal{A}}_{c,d}(x))={\mathcal{A}}_{ac,bd}(x)$.
3. (3)
${\mathcal{A}}^{-1}_{a,b}(x)={\mathcal{A}}_{1/a,1/b}(x)$.
###### 4.3 Lemma.
[Vu, (1.5)] An inversion in $S^{n-1}(a,r)$ is defined as,
$h(x)=a+\frac{r^{2}(x-a)}{|x-a|^{2}},\;h(a)=\infty,\;h(\infty)=a.$
Moreover,
(4.4) $|h(x)-h(y)|=\frac{r^{2}|x-y|}{|x-a||y-a|}.$
One of the goals of this section is to find a partial counterpart of the
distance formula (4.4) for $\mathcal{A}$ and to prove Theorem 1.6.
###### 4.5 Lemma.
Let $h(w)=r^{2}w/|w|^{2},\,r>0,\,w\in\mathbb{R}^{n}\setminus\\{0\\}$ and let
$x,y\in\mathbb{R}^{n}\setminus\\{0\\}$ with $|x|\leq|y|$. Then with
$\lambda=(|x|+|x-y|)/|x|$ and $z=\lambda x$ we have
$\quad|h(x)-h(z)|\leq|h(x)-h(y)|\leq 3|h(x)-h(z)|.$
Equality holds in the upper bound for $x=-y$.
###### Proof.
For the proof of first inequality we observe that
$\displaystyle|h(x)-h(z)|$ $\displaystyle=$
$\displaystyle|h(x)-\frac{\lambda}{|\lambda|^{2}}h(x)|=\frac{|\lambda-1|}{\lambda}\frac{r^{2}}{|x|}$
$\displaystyle=$ $\displaystyle\frac{r^{2}|x-y|}{|x|(|x|+|x-y|)}$
$\displaystyle\leq$ $\displaystyle\frac{r^{2}|x-y|}{|x||y|}=|h(x)-h(y)|$
by triangle inequality.
For the second inequality, we have
$\displaystyle\frac{|h(x)-h(y)|}{|h(x)-h(z)|}$ $\displaystyle=$
$\displaystyle\frac{|x-y|}{|x||y|}\frac{|x|(|x|+|x-y|)}{|x-y|}$
$\displaystyle=$ $\displaystyle\frac{|x|}{|y|}+\frac{|x-y|}{|y|}\leq
1+\frac{|x|+|y|}{|y|}\leq 3.$
Note that here equality holds for $x=-y$. ∎
###### 4.6 Lemma.
The following inequality holds for $K\geq 1$:
$||x|^{K-1}x-|y|^{K-1}y|\leq
e^{\pi(K-1/K)}|x|^{K-1/K}\max\\{|x-y|^{1/K},|x-y|^{K}\\}$
for all $x,y\in\mathbb{C}\setminus\overline{\mathbb{B}}^{2}$.
###### Proof.
By [AVV2, Theorem 14.18, (14.4)] we get because $f:x\mapsto|x|^{K-1}x$ is
$K$-quasiconformal [Va, 16.2]
$||x|^{K-1}x,f(0),|y|^{K-1}y,f(\infty)|\leq\eta^{*}_{K,2}(|x,0,y,\infty|)=\eta_{K,2}\left(\frac{|x-y|}{|x|}\right).$
Finally by [AVV2, Theorem 10.24] and [Vu, Remark 10.31] we have
$\displaystyle||x|^{K-1}x-|y|^{K-1}y|$ $\displaystyle\leq$
$\displaystyle|x|^{K}\eta_{K,2}\left(\frac{|x-y|}{|x|}\right)$
$\displaystyle\leq$
$\displaystyle\lambda(K)|x|^{K}\max\\{\left(\frac{|x-y|}{x}\right)^{1/K},\left(\frac{|x-y|}{x}\right)^{K}\\}$
$\displaystyle\leq$ $\displaystyle
e^{\pi(K-1/K)}|x|^{K-1/K}\max\\{|x-y|^{1/K},|x-y|^{K}\\}.$
∎
###### 4.7 Lemma.
The following inequality holds for $K\geq 1$ and for all
$x,y\in\mathbb{R}^{n}\setminus\overline{\mathbb{B}}^{n}$:
$||x|^{\beta-1}x-|y|^{\beta-1}y|\leq
c(K)|x|^{\beta-\alpha}\max\\{|x-y|^{\alpha},|x-y|^{\beta}\\}$
here $c(K)=2^{K-1}K^{K}\exp(4K(K+1)\sqrt{K-1})$ and
$\alpha=K^{1/(1-n)}=1/\beta$.
###### Proof.
By [AVV2, Theorem 14.18] we get because $f:x\mapsto|x|^{K-1}x$ is
$K$-quasiconformal [Va, 16.2]
$||x|^{\beta-1}x,f(0),|y|^{\beta-1}y,f(\infty)|\leq\eta^{*}_{K,n}(|x,0,y,\infty|),$
and this is equivalent to
$||x|^{\beta-1}x-|y|^{\beta-1}y|\leq|x|^{\beta}\eta^{*}_{K,n}\left(\frac{|x-y|}{|x|}\right).$
By [AVV2, Theorem 14.6] we get
$\displaystyle||x|^{\beta-1}x-|y|^{\beta-1}y|$ $\displaystyle\leq$
$\displaystyle
c(K)|x|^{\beta}\max\left\\{\left(\frac{|x-y|}{x}\right)^{\alpha},\left(\frac{|x-y|}{x}\right)^{\beta}\right\\}$
$\displaystyle\leq$ $\displaystyle
c(K)|x|^{\beta-\alpha}\max\\{|x-y|^{\alpha},|x-y|^{\beta}\\}.$
∎
###### 4.8 Corollary.
The following inequalities hold for $K\geq 1$;
(4.9) $\left|\frac{x}{|x|^{1+1/K}}-\frac{y}{|y|^{1+1/K}}\right|\leq
2^{1-1/K}\frac{|x-y|^{1/K}}{(|x||y|)^{1/K}}$
for all $x,y\in\mathbb{R}^{n}\setminus\mathbb{B}^{n}$,
(4.10)
$\left|\frac{x}{|x|^{1+\beta}}-\frac{y}{|y|^{1+\beta}}\right|\leq\frac{c(K)}{{|x|^{\beta-\alpha}}}\max\left\\{\left(\frac{|x-y|}{|x||y|}\right)^{\alpha},\left(\frac{|x-y|}{|x||y|}\right)^{\beta}\right\\}$
for all $x,y\in\mathbb{B}^{n}$,
(4.11)
$\left|\frac{x}{|x|^{1+K}}-\frac{y}{|y|^{1+K}}\right|\leq\frac{e^{\pi(K-1/K)}}{|x|^{K-1/K}}\max\left\\{\left(\frac{|x-y|}{|x||y|}\right)^{1/K},\left(\frac{|x-y|}{|x||y|}\right)^{K}\right\\}$
for all $x,y\in\mathbb{B}^{2}$.
###### Proof.
For the proof of (4.9) we define
$g(z)={\mathcal{A}}_{1/K,K}(h(z))=\frac{z}{|z|^{1+1/K}},\;h(z)=\frac{z}{|z|^{2}},\;z\in\mathbb{R}^{n}\setminus\mathbb{B}^{n}.$
By Theorem 1.4 and (4.4) we get,
$|g(x)-g(y)|=\left|\frac{x}{|x|^{1+1/K}}-\frac{y}{|y|^{1+1/K}}\right|\\\ \leq
2^{1-1/K}|h(x)-h(y)|^{1/K}\leq 2^{1-1/K}\frac{|x-y|^{1/K}}{(|x||y|)^{1/K}}.$
Again for the proof of (4.10) we define
$g(z)={\mathcal{A}}_{\alpha,\beta}(h(z))=\frac{z}{|z|^{1+\beta}},\;h(z)=\frac{z}{|z|^{2}},\;z\in\mathbb{B}^{n}.$
By Lemma 4.7 and (4.4) we get,
$\displaystyle|g(x)-g(y)|$ $\displaystyle\leq$ $\displaystyle
c(K)|h(x)|^{\beta-\alpha}\max\\{|h(x)-h(y)|^{\alpha},|h(x)-h(y)|^{\beta}\\}$
$\displaystyle=$
$\displaystyle\frac{c(K)}{|x|^{\beta-\alpha}}\max\left\\{\left(\frac{|x-y|}{|x||y|}\right)^{\alpha},\left(\frac{|x-y|}{|x||y|}\right)^{\beta}\right\\}.$
Similarly, inequality (4.11) follows from Lemma 4.6 and (4.4). ∎
###### 4.12 Lemma.
For $0<a\leq 1\leq p<\infty$ and $0\leq s\leq 2\pi$ we have
$\frac{(1+p^{2a}-2p^{a}\cos
s)^{1/2}}{(-1+(1+X)^{a})}\leq\frac{1+p^{a}}{(-1+(2+p)^{a})}\,,\quad
X=\sqrt{1+p^{2}-2p\cos s}\,.$
###### Proof.
Let
$f_{p,a}(s)=\frac{(1+p^{2a}-2p^{a}\cos s)}{(-1+(1+X)^{a})^{2}}.$
Then
$f_{p,a}^{\prime}(s)=2\frac{(-a(p^{1-a}+p^{a+1}-2p\cos
s)/X+(1+X-(1+X)^{1-a}))\sin s}{p^{-a}(1+X)^{1-a}(-1+(1+X)^{a})^{3}}$
As
$p^{1-a}+p^{a+1}\leq 1+p^{2}$
because
$p^{1+a}(1-p^{1-a})\leq 1-p^{1-a}$
it follows that
$f^{\prime}_{p,a}(s)/\sin s\geq
2\frac{(-aX+(1+X-(1+X)^{1-a}))}{p^{-a}(1+X)^{1-a}(-1+(1+X)^{a})^{3}}.$
As
$(1+X)^{1-a}<1+(1-a)X,$
it follows that
$f^{\prime}_{p,a}(s)=0\text{ if and only $s=0$ or $s=\pi$.}$
For $s=0$, the function $f_{p,a}$ achieves its minimum
$f_{p,a}(0)=\left(\frac{-1+p^{a}}{-1+{p}^{a}}\right)^{2}=1$
and for $s=\pi$ its maximum
$f_{p,a}(\pi)=\left(\frac{1+p^{a}}{(-1+(2+p)^{a})}\right)^{2}.$
∎
###### 4.13 Lemma.
For $p\geq 1$, and $0<d\leq 1$ there holds
(4.14) $\frac{1+p^{d}}{(2+p)^{d}-1}\leq\frac{2}{3^{d}-1}.$
###### Proof.
Let
$h(p)=(3^{d}-1)(1+p^{d})-2((2+p)^{d}-1).$
We need to show that $h(p)\leq 0$. First of all
$h^{\prime}(p)=d\left((3^{d}-1)p^{d-1}-2(2+p)^{d-1}\right).$
Then
$h^{\prime}(p)\leq
0\Leftrightarrow\left(\frac{2}{p}+1\right)^{1-d}\leq\frac{2}{3^{d}-1}.$
Since
$\left(\frac{2}{p}+1\right)^{1-d}\leq 3^{1-d}$
we need to show that
$\left(3\right)^{1-d}\leq\frac{2}{3^{d}-1},$
but this is equivalent to
$3^{d}\leq 3$
which is obviously true. Thus $h^{\prime}(p)\leq 0$, and consequently
$h(p)\leq h(1)=0$ and this inequality coincides with (4.14). ∎
###### 4.15.
Proof of Theorem 1.6. The case $1\leq|x|\leq|y|$. Let us show that $Q(x,y)\leq
1$. Without loss of generality, we can assume that $x=r$ and $z$ are positive
real numbers, and $y=Re^{it}$. Then $z=r+|r-Re^{it}|$. Let
$p=\frac{R}{r}.$
Then $p\geq 1$. Next we have:
$\begin{split}\frac{|{\mathcal{A}}(x)-{\mathcal{A}}(y)|}{|{\mathcal{A}}(x)-{\mathcal{A}}(z)|}&=\frac{|1-p^{b}e^{it}|}{(1+|1-pe^{it}|)^{b}-1}\\\
&\leq\frac{|1-p^{b}e^{it}|}{(1+|1-p|)^{b-1}(1+|1-pe^{it}|)-1}\\\
&=\frac{|1-p^{b}e^{it}|}{p^{b-1}(1+|1-pe^{it}|)-1}\\\
&=\frac{|1-p^{b}e^{it}|}{p^{b-1}-1+|p^{b-1}-p^{b}e^{it}|}\\\
&=\frac{|1-p^{b-1}+p^{b-1}-p^{b}e^{it}|}{p^{b-1}-1+|p^{b-1}-p^{b}e^{it}|}\leq
1.\end{split}$
If $|x|\leq|y|\leq 1$ and $|z|\leq 1$, then by Lemmas 4.12 and 4.13 we get
$\displaystyle\frac{|{\mathcal{A}}(x)-{\mathcal{A}}(y)|}{|{\mathcal{A}}(x)-{\mathcal{A}}(z)|}$
$\displaystyle=$
$\displaystyle\frac{|r^{a}-R^{a}e^{it}|}{(r+|r-Re^{it}|)^{a}-r^{a}}$
$\displaystyle\leq$
$\displaystyle\frac{1+p^{a}}{(2+p)^{a}-1}\leq\frac{2}{3^{a}-1}.$
If $|x|\leq|y|\leq 1$ and $|z|\geq 1$, then it follows from Lemmas 4.12 and
4.13 and $|z|^{b}\geq|z|^{a}$ that
$\displaystyle\frac{|{\mathcal{A}}(x)-{\mathcal{A}}(y)|}{|{\mathcal{A}}(x)-{\mathcal{A}}(z)|}$
$\displaystyle\leq$
$\displaystyle\frac{|r^{a}-R^{a}e^{it}|}{(r+|r-Re^{it}|)^{a}-r^{a}}$
$\displaystyle\leq$
$\displaystyle\frac{1+p^{a}}{(2+p)^{a}-1}\leq\frac{2}{3^{a}-1}.$
The case $|x|\leq 1\leq|y|$ and $r^{a-1}>R^{b-1}$. Then there holds
$Q(x,y)\leq\frac{2}{3^{a}-1}.$
First of all
$\begin{split}\frac{|{\mathcal{A}}(x)-{\mathcal{A}}(y)|}{|{\mathcal{A}}(x)-{\mathcal{A}}(z)|}&=\frac{|r^{a}-R^{b}e^{it}|}{(r+|r-Re^{it}|)^{b}-r^{a}}\\\
&=\frac{|\alpha-e^{it}|}{(\beta+|\beta-e^{it}|)^{b}-\alpha}\end{split}$
where $\alpha=\frac{r^{a}}{R^{b}}$ and $\beta=\frac{r}{R}$. Take the
continuous function $k(q)=\beta^{q}$, $a\leq q\leq 1$. Since
$\beta=k(1)=\frac{r}{R}\leq\alpha=\frac{r^{a}}{R^{b}}\leq
k(a)=\frac{r^{a}}{R^{a}}$
it follows that there exists a constant $c$ with $a\leq c\leq 1$ such that
$k(c)=\beta^{c}=\alpha$. Then
$\displaystyle\frac{|\alpha-Re^{it}|}{(\beta+|\beta-e^{it}|)^{b}-\alpha}$
$\displaystyle=$
$\displaystyle\frac{|\beta^{c}-e^{it}|}{(\beta+|\beta-e^{it}|)^{b}-\beta^{c}}$
$\displaystyle\leq$
$\displaystyle\frac{|\beta^{c}-e^{it}|}{(\beta+|\beta-e^{it}|)^{c}-\beta^{c}}$
$\displaystyle\leq$ $\displaystyle\frac{1+\beta^{c}}{(2+\beta)^{c}-\beta^{c}}$
$\displaystyle\leq$ $\displaystyle\frac{2}{3^{c}-1}\leq\frac{2}{3^{a}-1},$
the second inequality follows from Lemma 4.12 and the third inequality follows
from Lemma 4.13 by taking $p=1/\beta$ and $c=d$.
Finally, let us show that $C(a,b)\geq 2/(3^{a}-1)\,.$ Suppose that
$x\in{\mathbb{R}}^{n}\setminus\\{0\\}$ is such that $3|x|<1\,,$ i.e.
$0<|x|<1/3\,$ and $y=-x\,.$ Then $z=x(|x|+|x-y|)/|x|=3x$ and
$Q(x,-x)=\frac{2|x|^{a}}{(3|x|)^{a}-|x|^{a}}=\frac{2}{3^{a}-1}\,,$
and hence $C(a,b)\geq 2/(3^{a}-1)\,.$ $\square$
## 5\. Conclusions
###### 5.1.
Proof of Theorem 1.7. For $|x|,|y|<1$ we have
$\alpha_{p}(x,y)=\left||x|^{p-1}x-|y|^{p-1}y\right|=\left|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(y)\right|.$
Consider the case $|x|<1<|y|\,$. It is obvious that
$\cos\theta\leq 1<\frac{|x|^{-p}(|y|^{1/p}+|y|^{p})}{2},$
this is equivalent to
$\cos\theta\leq
1<\frac{(|y|^{1/p}-|y|^{p})(|y|^{1/p}+|y|^{p})}{2|x|^{p}(|y|^{1/p}-|y|^{p})}$
$\Longleftrightarrow$
$\qquad
2|x|^{p}|y|^{1/p}\cos\theta-2|x|^{p}|y|^{p}\cos\theta<|y|^{2/p}-|y|^{2p}$
$\Longleftrightarrow$
$|y|^{2p}-2|x|^{p-1}|y|^{p-1}|x||y|\cos\theta<|y|^{2/p}-2|x|^{p-1}|y|^{1/p-1}|x||y|\cos\theta$
$\Longleftrightarrow$
$||x|^{p-1}x|^{2}+||y|^{p-1}y|^{2}-2|x|^{p-1}|y|^{p-1}x\,y<||x|^{p-1}x|^{2}+||y|^{1/p-1}y|^{2}-2|x|^{p-1}|y|^{1/p-1}x\,y$
$\Longleftrightarrow$
$\left||x|^{p-1}x-|y|^{p-1}y\right|^{2}<\left||x|^{p-1}x-|y|^{1/p-1}y\right|^{2}=\left|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(y)\right|^{2}.$
Consider now the case $1<|x|<|y|$. Starting with the observation that the
function $t\mapsto t^{1/p}-t^{p}$ is increasing for $t>1$ when $p\in(0,1)\,,$
we see that
$\frac{|x|^{1/p}}{|x|^{p}}\left(\left(\frac{|y|}{|x|}\right)^{1/p}-1\right)>\left(\left(\frac{|y|}{|x|}\right)^{p}-1\right)\Leftrightarrow(|y|^{1/p}-|x|^{1/p})^{2}>(|y|^{p}-|x|^{p})^{2}$
$\Longleftrightarrow$
$|x|^{2/p}-|y|^{2p}+|y|^{2/p}-|x|^{2p}>2|x|^{1/p}|y|^{1/p}-2|x|^{p}|y|^{p}.$
Now it is clear that
$\cos\theta\leq
1<\frac{|x|^{2/p}-|y|^{2p}+|y|^{2/p}-|x|^{2p}}{2|x|^{1/p}|y|^{1/p}-2|x|^{p}|y|^{p}}$
$\Longleftrightarrow$
$|x|^{2p}+|y|^{2p}-2|x|^{p}|y|^{p}\cos\theta<|x|^{2/p}+|y|^{2/p}-2|x|^{1/p}|y|^{1/p}\cos\theta$
$\Longleftrightarrow$
$||x|^{p-1}x|^{2}+||y|^{p-1}y|^{2}-2|x|^{p-1}|y|^{p-1}x\,y<||x|^{1/p-1}x|^{2}+||y|^{1/p-1}y|^{2}-2|x|^{1/p-1}|y|^{1/p-1}x\,y$
$\Longleftrightarrow$
$||x|^{p-1}x|^{2}+||y|^{p-1}y|^{2}-2|x|^{p-1}|y|^{p-1}x\,y<||x|^{p-1}x|^{2}+||y|^{1/p-1}y|^{2}-2|x|^{p-1}|y|^{1/p-1}x\,y$
$\Longleftrightarrow$
$\left||x|^{p-1}x-|y|^{p-1}y\right|^{2}<\left||x|^{1/p-1}x-|y|^{1/p-1}y\right|^{2}=\left|{\mathcal{A}}_{p,1/p}(x)-{\mathcal{A}}_{p,1/p}(y)\right|^{2}.\quad\square$
###### 5.2.
Comparison of the bounds. In what follows, we use the symbols $M,D,B,K$ for
the bounds given by Theorems 1.2, 1.3, 1.4, 1.6, respectively. In the case of
the complex plane, we will show by numerical examples that each of these four
bounds can occur as minimal. To this end, for each of the symbols $M,D,B,K$,
we give a table of four $x,y$ pairs and the corresponding upper bound values
associated with the four symbols $M,D,B,K$, such that the bound associated
with the symbol in question is the least one. For the computation of the $K$
bound it should be observed that in Theorem 1.7 we have the constraint
$|x|\leq|y|\,.$ If this is not the situation to begin with, we have swapped
the points for computation. In Tables 1-4 the parameter $p=0.5\,.$
Table 1. Sample points with $K<\min\\{B,D,M\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&B&D&M&K\\\ \hline\cr
1&-2.00-2.65i&2.65-2.65i&3.0496&143.4290&3.6030&2.6591\\\
2&2.25-0.75i&2.65+1.30i&2.0438&38.9860&1.8236&1.5158\\\
3&1.35+0.50i&1.95-0.65i&1.6107&14.8000&1.3571&1.2768\\\
4&1.10+2.30i&-2.40+2.10i&2.6479&82.4142&2.9447&2.3646\\\ \hline\cr\end{array}$
Table 2. Sample points with $D<\min\\{B,K,M\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&B&K&M&D\\\ \hline\cr
1&0.80-0.50i&-1.80+1.45i&3.6968&45.3884&3.2066&2.5495\\\
2&2.25-0.75i&0.00-0.05i&15.5147&32.3855&2.7931&2.6174\\\
3&2.55+1.50i&-1.10+1.70i&2.8148&76.9511&3.1879&2.7039\\\
4&-2.70+3.00i&1.50+0.60i&4.2727&106.6320&3.6118&3.1104\\\
\hline\cr\end{array}$ Table 3. Sample points with $B<\min\\{D,K,M\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&D&K&M&B\\\ \hline\cr
1&-2.45-2.205i&-1.2+0.55i&2.92&43.55&2.42&2.40\\\
2&-1.65+1.45i&2.15+2.75i&3.01&92.27&3.22&2.83\\\
3&-0.2-3i&-0.4+0.2i&5.21&34.64&2.77&2.53\\\
4&0.9-2.9i&-1.4+1.35i&3.74&115.15&4.16&3.11\\\ \hline\cr\end{array}$ Table 4.
Sample points with $M<\min\\{B,D,K\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&B&D&K&M\\\ \hline\cr
1&0.30+0.50i&-0.15+2.95i&2.23&3.69&23.73&2.17\\\
2&0.95+1.85i&0.55+1.55i&1.00&0.53&5.18&0.52\\\
3&1.60-0.25i&1.10-0.35i&1.01&0.64&3.93&0.60\\\
4&-0.60+0.30&-3.00+1.95i&2.41&4.02&32.84&2.31\\\ \hline\cr\end{array}$
Table 5. Sample points with $M<\min\\{(\ref{2j}),D\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&(\ref{2j})&D&M\\\
\hline\cr 1&2.25+2.45i&-0.01+2.95i&0.27&0.27&0.24\\\
2&-2.60+0.40i&-0.70-0.60i&1.23&3.30&1.19\\\
3&0.75-0.75i&-2.90-2.50i&1.32&4.53&1.23\\\
4&2.90+1.90i&1.20+0.85i&0.75&1.67&0.71\\\ \hline\cr\end{array}$ Table 6.
Sample points with $(\ref{2j})<\min\\{D,M\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&D&M&(\ref{2j})\\\
\hline\cr 1&-2.60-1.05i&-1.35-1.40i&0.70&0.65&0.56\\\
2&-0.45-1.05i&2.35+1.80i&3.95&1.83&1.46\\\
3&-1.15+2.30i&2.70+0.65i&0.99&2.12&0.96\\\
4&-0.10+1.25i&2.90+2.45i&0.71&0.94&0.60\\\ \hline\cr\end{array}$ Table 7.
Sample points with $D<\min\\{(\ref{2j}),M\\}\,.$
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr k&x_{k}&y_{k}&(\ref{2j})&M&D\\\
\hline\cr 1&1.35+2.95i&-1.35+2.90i&0.59&1.07&0.43\\\
2&-0.80+2.75i&-1.85+2.40i&0.38&0.49&0.25\\\
3&2.65+2.20i&-2.45+2.40i&0.49&0.64&0.49\\\
4&1.20-0.70i&1.30+0.70i&1.05&1.96&0.91\\\ \hline\cr\end{array}$
In conclusion, Tables 1-4 demonstrate that each of the above four bounds is
sometimes smaller than the minimum of the other three bounds. Some further
results, in addition to Theorems 1.2, 1.3, 1.4, 1.6 can be found in the papers
[M] and [D]. The tables were compiled with the help of the Mathematica
software package.
In Tables 5-7 we compare $(\ref{2j}),M$ and $D$, for
$x,y\in\mathbb{R}^{n}\setminus\mathbb{B}^{n},\,p=-0.6.$
## References
* [AVV1] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Dimension-free quasiconformal distortion in $n$-space. Trans. Amer. Math. Soc. 297 (1986), 687-706.
* [AVV2] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Conformal invariants, inequalities and quasiconformal maps. J. Wiley, 1997, 505 pp.
* [BV] B. A. Bhayo, and M. Vuorinen: On Mori’s theorem for quasiconformal maps in the $n$-space. Trans. Amer. Math. Soc. (to appear), arXiv:0906.2853[math.CV].
* [By] J. Byström: Sharp constants for some inequalities connected to the $p$-Laplace operator. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 2, Article 56, 8 pp. (electronic).
* [DFM] F. Dadipour, M. Fujii, and M. S. Moslehian: Dunkl–Williams inequality for operators associated with $p$-angular distance, arXiv:1006.1941[math.OA].
* [D] S. S. Dragomir: Inequalities for the $p$-angular distance in normed linear spaces. (English summary) Math. Inequal. Appl. 12 (2009), no. 2, 391–401.
* [KMV] R. Klén, V. Manojlović, and M. Vuorinen: Distortion of normalized quasiconformal mappings, arXiv:0808.1219[math.CV].
* [K] R. Kühnau, ed.: Handbook of complex analysis: geometric function theory. Vol. 1-2. North-Holland, Amsterdam, 2002 and 2005.
* [LV] O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane. Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer-Verlag, New York-Heidelberg, 1973. viii+258 pp.
* [M] L. Maligranda: Simple norm inequalities. Amer. Math. Monthly 113 (2006), no. 3, 256–260.
* [MPF] D. S. Mitrinović, J. E. Pecarić, and A. M. Fink: Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993. xviii+740 pp.
* [Ru] H. Ruskeepää: Mathematica® Navigator. 3rd ed. Academic Press, 2009.
* [Va] J. Väisälä: Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics 229, Springer-Verlag, Berlin, 1971.
* [Vu] M. Vuorinen: Conformal geometry and quasiregular mappings. Lecture Notes in Mathematics 1319, Springer, Berlin, 1988.
B. A. Bhayo and M. Vuorinen
Department of Mathematics
University of Turku
20014 Turku
Finland
barbha@utu.fi
V. Božin
Faculty of Mathematics
University of Belgrade
Studentski trg 16, Belgrade
Serbia
bozinv@turing.mi.sanu.ac.rs
D. Kalaj
University of Montenegro
Faculty of Mathematics
Dzordza Va ingtona b.b.
Podgorica
Montenegro
davidk@ac.me
|
arxiv-papers
| 2010-08-25T11:13:43 |
2024-09-04T02:49:12.462354
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Barkat A. Bhayo, Vladimir Bo\\v{z}in, David Kalaj, Matti Vuorinen",
"submitter": "Matti Vuorinen",
"url": "https://arxiv.org/abs/1008.4254"
}
|
1008.4385
|
# The Advection of Supergranules by the Sun’s Axisymmetric Flows
David H. Hathaway NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
david.hathaway@nasa.gov Peter E. Williams NASA Goddard Space Flight Center,
Greenbelt, MD 20771 USA peter.williams@nasa.gov Kevin Dela Rosa School of
Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213 USA
Manfred Cuntz Department of Physics, University of Texas at Arlington,
Arlington, TX 76019 USA cuntz@uta.edu
###### Abstract
We show that the motions of supergranules are consistent with a model in which
they are simply advected by the axisymmetric flows in the Sun’s surface shear
layer. We produce a 10-day series of simulated Doppler images at a 15-minute
cadence that reproduces most spatial and temporal characteristics seen in the
SOHO/MDI Doppler data. Our simulated data have a spectrum of cellular flows
with just two components – a granule component that peaks at spherical
wavenumbers of about 4000 and a supergranule component that peaks at
wavenumbers of about 110. We include the advection of these cellular
components by the axisymmetric flows – differential rotation and meridional
flow – whose variations with latitude and depth (wavenumber) are consistent
with observations. We mimic the evolution of the cellular pattern by
introducing random variations to the phases of the spectral components at
rates that reproduce the levels of cross-correlation as functions of time and
latitude. Our simulated data do not include any wave-like characteristics for
the supergranules yet can reproduce the rotation characteristics previously
attributed to wave-like behavior. We find rotation rates which appear faster
than the actual rotation rates and attribute this to the projection effects.
We find that the measured meridional flow does accurately represent the actual
flow and that the observations indicate poleward flow to $65\arcdeg-70\arcdeg$
latitude with equatorward counter cells in the polar regions.
Sun: granulation, Sun: rotation, Sun: photosphere, Sun: surface magnetism
## 1 Introduction
Supergranules are cellular flow structures observed in the solar photosphere
with typical diameters of about 30 Mm, lifetimes of about one day, and flow
velocities of 300 $\rm m\ s^{-1}$ (Rieutord & Rincon, 2010). They cover the
entire surface of the Sun except for the immediate surroundings of sunspots.
Supergranules were discovered by Hart (1954) but it was Leighton et al. (1962)
who suggested that these cellular flows were convective structures and coined
the term “supergranule.” However, Worden (1975), and others since, had
difficulty detecting any associated thermal features (i.e. hot cell centers)
consistent with a thermal convection origin. Recently however, Meunier et al.
(2007) have reported the detection of a small temperature excess at cell
centers of about 0.8-2.8 K.
Supergranules are intimately involved with the structure and evolution of the
magnetic field in the photosphere. Simon & Leighton (1964) found that the
magnetic structures of the chromospheric network are located at the boundaries
of these cells. Leighton (1964) quickly suggested that the random-walk of
magnetic elements by the evolving supergranules could transport following
polarity elements poleward to reverse the Sun’s polar field each solar cycle
while leading polarity elements would be transported across the equator where
they would meet and cancel their opposite polarity counterparts. While we now
recognize that Leighton over estimated the effective diffusivity implied by
this random walk (and that the Sun’s poleward meridional flow provides the
missing transport), supergranule diffusion is still not fully constrained or
understood. Leighton’s initial estimate of $770-1540\ \rm km^{2}\ s^{-1}$ for
the diffusivity was reduced to $200-400\ \rm km^{2}\ s^{-1}$ by Mosher (1977)
who studied the displacements of chromospheric network elements. Schrijver &
Martin (1990) measured the displacements of the magnetic elements themselves
using magnetograms and found a value of $250\ \rm km^{2}\ s^{-1}$ in plage
surrounding active regions but only $110\ \rm km^{2}\ s^{-1}$ in the active
regions themselves. Models for the surface magnetic flux transport typically
use values about twice this size – van Ballegooijen et al. (1998) used $450\
\rm km^{2}\ s^{-1}$ while Wang et al. (2002) used $500\ \rm km^{2}\ s^{-1}$.
Supergranules are embedded in a surface shear layer of their own making.
Foukal & Jokipii (1975) and Foukal (1979) suggested that the radial flows
within the supergranules would conserve angular momentum and produce a shear
layer with slower rotation at the surface and more rapid rotation below –
consistent with the earlier observations of Howard & Harvey (1970) that the
sunspots rotate about 5% faster than the surrounding surface plasma. The
production of this shear layer was succesfully modelled in spherical shells by
Gilman & Foukal (1979) and in plane-parallel layers by Hathaway (1982). The
theoretical arguements suggest a change in rotation frequency such that
$\delta\Omega\sim-2\Omega\delta r/r$. In addition, Hathaway (1982) found that
a meridional flow was also produced with poleward flow near the surface and
equatorward flow at the bottom.
The surface shear layer has been probed with helioseismic techniques which
confirm the existence of the layer but give somewhat inconsistent results for
the detailed structure in depth and latitude. Thompson et al. (1996) used the
global oscillation modes from GONG and found that the rotation rate increased
inward to a depth of about 35 Mm ($\delta r/r\sim 5\%$) and that the rate of
increase was larger at the equator than at $30\arcdeg$ latitude. Schou et al.
(1998) found similar results from global modes in MDI data but noted that the
shear appeared to disappear by $60\arcdeg$ latitude and perhaps reverse sign
at higher latitudes.
Basu et al. (1999) used the local helioseismology technique of ring-diagram
analysis. They found a thin (4 Mm) outer shear layer in which the shear did
not appear to reverse at high latitudes and a deeper shear layer in which it
did. They also found a poleward meridional flow which increased in amplitude
across the thin outer shear layer but then remained nearly constant across the
inner shear layer. Their meridional flow had peak velocities of about $50\ \rm
m\ s^{-1}$ at $50\arcdeg$ latitude. Giles et al. (1997) used a time-distance
local helioseismology technique and found a meridional flow with a peak
velocity of only about $24\ \rm m\ s^{-1}$ at $45\arcdeg$ latitude and
indicated that it was constant to a depth of about 26 Mm.
The observed rotation of the supergranules has added further mystery to their
nature. (The mechanism that selects the characteristic size of supergranules
is perhaps the biggest mystery.) Duvall (1980) cross-correlated the equatorial
Doppler velocity patterns and found that the supergranules rotated more
rapidly than the plasma at the photosphere and that even faster rotation rates
were obtained when there were longer (24-hour vs. 8-hour) time intervals
between Doppler images. He attributed this behavior to the surface shear layer
in which larger, longer-lived, cells extend deeper into the more rapidly
rotating layers. Snodgrass & Ulrich (1990) used data from Mount Wilson
Observatory to find the rotation rate at different latitudes and noted that
the rotation rates for the Doppler pattern were some 4% faster than the
spectroscopic rate and, mysteriously, some 2% faster than the small magnetic
features that are observed to outline their borders. More recently Beck &
Schou (2000) used a Fourier transform method to find that the larger features
do rotate more rapidly than the smaller features and that the low wavenumber
components of the Doppler pattern rotate more rapidly than the plasma at any
level within the surface shear layer. This led them to suggest that
supergranules have wave-like characteristics with a preference for prograde
propagation.
Hathaway et al. (2006) showed that this “super-rotation” of the Doppler
pattern could be attributed to projection effects associated with the Doppler
signal itself. As the velocity pattern rotates across the field of view the
observed line-of-sight component is modulated in a way that essentially adds
another half wave to the pattern and gives a higher rotation rate that
increases proportionally with decreasing wavenumber. They took a fixed
velocity pattern (which had spatial characteristics that matched the SOHO/MDI
data) and rotated it rigidly to show this “super-rotation” effect. While this
indicated that the Doppler projection effect should be accounted for, the
fixed pattern could not account for all the variations reported by Beck &
Schou (2000).
Gizon et al. (2003) used time-distance helioseismology to find the
supergranular flow (rather than direct Doppler measurements) and Schou (2003)
“divided-out” the line-of-sight modulation. Both studies found slower rotation
rates that matched that of the magnetic features but saw evidence for wave-
like prograde and retrograde moving components. Meunier & Roudier (2007)
compared rotation rates obtained by tracking Doppler features, magnetic
features, and divergence features (which were, in turn, derived from
correlation tracking of smaller intensity features). While they concluded that
projection effects do influence the rotation rate determined from the Doppler
features, they found that the magnetic features rotate more slowly than the
supergranules.
In this paper we report on our analyses of data from the SOHO/MDI instrument
[Scherrer et al. (1995)] and from simulated data in which the supergranules
are simply advected by differential rotation and meridional flow that vary
with latitude and depth. The simulated data are designed to faithfully mimic
the SOHO/MDI data [the same data that was analyzed in Beck & Schou (2000) and
Schou (2003)] with simple assumptions about the dynamical structure of the
surface shear layer. The analyses include reproductions of those done in
earlier studies. Through the simulations we can better determine the actual
differential rotation and meridional flow profiles consistent with the Doppler
observations (which are subject to line-of-sight projection effects).
## 2 The Data
The full-disk Doppler images from SOHO/MDI are obtained at a 1-minute cadence
to resolve the temporal variations associated with the p-mode oscillations. We
[cf. Hathaway et al. (2000) and Beck & Schou (2000)] have temporally filtered
the images to remove the p-mode signal by using a 31-minute long tapered
Gaussian with a FWHM of 16 minutes on sets of 31 images that were de-rotated
to register each to the central image. These filtered images were formed at
15-minute intervals over the 60-day MDI Dynamics Run in 1996. This filtering
process effectively removes the p-mode signal and leaves behind the Doppler
signal from flows with temporal variations longer than about 16 minutes.
Supergranules, with typical wavenumbers of about 110, are very well resolved
in this data (at disk center wavenumbers up to 1500 are resolved). While
granules are not well resolved, they do appear in the data as pixel-to-pixel
and image-to-image “noise,” as a convective blue shift (due to the correlation
between brightness and updrafts), and as resolved structures for the largest
members. These data are prepared for studying the cellular features by first
measuring and removing the Doppler signals due to: 1) observer motion, 2)
convective blue shift, 3) differential rotation, and 4) the
axisymmetricmeridional flow. The data are then mapped onto heliographic
coordinates with equal spacing in both longitude and latitude. This mapping
includes accounting for the position angle and tilt angle of the Sun’s
rotation axis as well as the orientation of the SOHO/MDI detector. Note that
the position angle and tilt angle of the Sun’s rotation axis determined by
Carrington (1863) have been found to be in error by about $0.1\arcdeg$ by Beck
& Giles (2005) and by Hathaway & Rightmire (2010). This correction is included
in this study.
The simulated data are constructed in the manner described previously
(Hathaway, 1988, 1992; Hathaway et al., 2000, 2002) from vector velocities
with
$V_{r}(\theta,\phi)=\sum_{l=0}^{lmax}\sum_{m=-l}^{l}R_{l}^{m}Y_{l}^{m}(\theta,\phi)$
(1)
$\displaystyle V_{\theta}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{lmax}\sum_{m=-l}^{l}S_{l}^{m}{\partial
Y_{l}^{m}(\theta,\phi)\over\partial\theta}+$ (2)
$\displaystyle\sum_{l=1}^{lmax}\sum_{m=-l}^{l}T_{l}^{m}{1\over\sin\theta}{\partial
Y_{l}^{m}(\theta,\phi)\over\partial\phi}$
$\displaystyle V_{\phi}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{lmax}\sum_{m=-l}^{l}S_{l}^{m}{1\over\sin\theta}{\partial
Y_{l}^{m}(\theta,\phi)\over\partial\phi}-$ (3)
$\displaystyle\sum_{l=1}^{lmax}\sum_{m=-l}^{l}T_{l}^{m}{\partial
Y_{l}^{m}(\theta,\phi)\over\partial\theta}$
where $Y_{\ell}^{m}(\theta,\phi)$ is the spherical harmonic function of degree
$\ell$ and azimuthal order $m$, $\theta$ is the colatitude measured southward
from the north pole, and $\phi$ is the longitude measured prograde from the
central meridian or some fixed longitude. The complex quantities $R_{l}^{m}$,
$S_{l}^{m}$, and $T_{l}^{m}$ are the spectral coefficients for the radial,
poloidal, and toroidal components, respectively.
To simulate the observed line-of-sight velocity, the spectral coefficients
from $\ell=0$ to $\ell=1500$ are specified and the three vector velocity
components are calculated using Eqns. 1-3 on a grid with 1500 points in
latitude and 4096 points in longitude. The Doppler velocity image is
constructed by determining the longitude and latitude at a point on the image,
finding the vector velocity at that point using bi-cubic interpolation, and
then projecting that vector velocity onto the line-of-sight using
$\displaystyle V_{los}(\theta,\phi)$ $\displaystyle=$ $\displaystyle
V_{r}(\theta,\phi)\sin B_{0}\ \cos\theta+$ (4) $\displaystyle
V_{r}(\theta,\phi)\cos B_{0}\ \sin\theta\ \cos\phi+$ $\displaystyle
V_{\theta}(\theta,\phi)\sin B_{0}\ \sin\theta-$ $\displaystyle
V_{\theta}(\theta,\phi)\cos B_{0}\ \cos\theta\ \cos\phi+$ $\displaystyle
V_{\phi}(\theta,\phi)\cos B_{0}\ \sin\phi$
where $B_{0}$ is the latitude at disk center (or equivalently the tilt of the
Sun’s north pole toward the observer) and velocities away from the observer
are taken to be positive. The line-of-sight velocity at an array of 49 points
within each pixel is determined and an average is taken to simulate the
integration over a pixel with the MDI instrument.
With the current simulations the instrumental blurring is treated in a more
realistic manner. Previously the Doppler velocity image itself was convolved
with an MDI point-spread-function. Here we make red and blue intensity images
from our Doppler velocity image and a simple limb darkened intensity image,
convolve those with an MDI point-spread-function, and construct the blurred
Doppler velocity image from the difference divided by the sum.
This process yields Doppler velocity images that are visually
indistinguishable from MDI Doppler velocity images. A velocity image
comparison is shown in Fig. 1.
Figure 1: Doppler velocity images (MDI on left, simulation on right) in which
the observer’s motion, the convective blue shift, and the Doppler signals due
to the axisymmetric flows — differential rotation and meridional circulation —
are removed to better reveal the cellular structures (supergranules). The
spatial characteristics are visually indistinguishable.
The velocity pattern is evolved in time by introducing changes to the spectral
coefficients based on two processes - the advection by the axisymmetric flows
(differential rotation and meridional flow) and random processes that lead to
the finite lifetimes of the cells.
The advection is governed by an advection equation
${\partial w\over\partial t}=-{V(\theta)\over r}{\partial
w\over\partial\theta}-{U(\theta)\over{r\sin\theta}}{\partial
w\over\partial\phi}$ (5)
where $w$ is a velocity component, $U(\theta)=r\sin\theta\Omega(\theta)$ gives
the differential rotation profile and $V(\theta)$ gives the meridional flow
velocity profile. Representing $w$ as a series of spherical harmonic
components (Eqs. 1-3) and projecting this advection equation onto a single
spherical harmonic gives a series of coupled equations for the evolution of
the spectral coefficients (Appendix A). Solid body rotation simply introduces
a constantly increasing phase for each coefficient. Differential rotation
couples the phase change in one spectral coefficient to spectral coefficients
with wavenumbers $\ell\pm 2$ and $\ell\pm 4$ for differential rotation of the
form
$\Omega(\theta)=\Omega_{0}+\Omega_{2}\cos^{2}\theta+\Omega_{4}\cos^{4}\theta$
(6)
while a simple but reasonable meridional flow profile with
$V(\theta)=V_{0}\cos\theta\sin\theta$ (7)
couples one spectral coefficient to spectral coefficients with wavenumbers
$\ell\pm 2$. (Spherical harmonics have fixed latitudinal structure. Spectral
power must pass from one spherical harmonic component to another in order to
move a feature in latitude.)
These cellular flows are embedded in the Sun’s surface shear layer. We
approximate the change in the rotation rate in the outermost 5% of the Sun as
reported by Howe et al. (2007) with
$\Omega(r,\theta)=\Omega(\theta)f(r)$ (8)
where
$f(r)=1+0.038\left[1-e^{-55(1-r/R_{\odot})}\right]$ (9)
and the latitude dependence is given by
${\Omega_{0}\over 2\pi}=452\ \rm nHz\ (14.07\arcdeg\ day^{-1})$ (10)
${\Omega_{2}\over 2\pi}=-55\ \rm nHz\ (-1.75\arcdeg\ day^{-1})$ (11)
${\Omega_{4}\over 2\pi}=-75\ \rm nHz\ (-2.30\arcdeg\ day^{-1})$ (12)
Assuming that the cells extend to depths similar to their horizontal
dimensions, and that they are advected at flow rates representative of that
depth, Eq. 9 is transformed into a function of $\ell$ with
$f(\ell)=1+0.038\left[1-e^{-90\pi/\ell}\right]$ (13)
This shear layer profile is illustrated in Fig. 2 along with the gradients
expected from theoretical arguements for flows that conserve angular momentum.
We assume a meridional flow which is constant with depth across this layer and
has a latitude dependence characterized by
$V_{0}=-30\ \rm m\ s^{-1}$ (14)
which gives a peak meridional flow velocity of $15\ \rm m\ s^{-1}$ at
$45\arcdeg$ latitude.
Figure 2: The assumed rotation rate as a function of radius in the surface
shear layer for three different latitudes. The surface rotation rate at each
latitude is indicated by the dashed lines. The variations in rotation rate for
flows that conserve angular momentum from the surface inward are indicated by
the dotted lines.
The finite lifetimes for the cells are simulated by introducing random
perturbations to the spectral coefficient phases. The size of these
perturbations increases with wavenumber to give shorter lifetimes to smaller
cells with
$\delta\Phi_{\ell}^{m}\propto\sqrt{\Delta t/\tau(\ell)}$ (15)
where $\delta\Phi_{\ell}^{m}$ is the change in phase for a complex spectral
coefficient of degree $\ell$ and order $m$, $\Delta t$ is the time interval
between simulated Doppler images, and $\tau(\ell)$ is proportional to the
lifetime for a spectral component of degree $\ell$. Lifetimes are well
approximated by a turn-over time for turbulent convective flows. The cellular
flow velocities are roughly proportional to $\ell$ while their diameters are
inversely proportional to $\ell$. The turn-over times should then be inversely
proportional to $\ell^{2}$. We find a reasonable fit to the data using
$\tau(\ell)=6.5{100^{2}\over\ell^{2}}\rm{hrs}$ (16)
## 3 The Analyses
Several anaylsis programs were applied to both the MDI data and the simulated
data. Power spectra were obtained to characterize and compare the distribution
of cell sizes and flow velocities. The rotation of the Doppler pattern was
determined using multiple techniques based on previous studies. The meridional
flow was measured based on the movement of the Doppler pattern and cell
lifetimes were estimated from cross-correlation analyses.
### 3.1 Convection Spectra
Convection spectra for individual images were obtained using the methods
described by Hathaway (1987) and Hathaway (1992) – the Doppler signal due to
the motion of the observer is removed, the convective blue shift signal is
identified and removed, the data is mapped to heliographic coordinates, the
axisymmetric flow signals due to differential rotation and meridional
circulation are identified and removed, and the remaining signal is projected
onto spherical harmonics. The averaged spectra from the 60-day 1996 MDI
Dynamics Run and from our 10-day simulated data run are shown in Fig. 3.
The match between these spectra is obtained by adjusting the input spectrum
for the simulated data. This spectrum contains two Lorentzian-like spectral
components – a supergranule component centered on $\ell\sim 110$ with a width
of about 100 and a granule component centered on $\ell\sim 4000$ with a width
of about 3000. The MDI spectrum is well matched with just these two components
without the addition of the mesogranule component suggested by November et al.
(1981). In fact, we have a distinct dip in the input spectrum at wavenumbers
$\ell\sim 500$ that should be representative of mesogranules. Hathaway et al.
(2000) showed that this dip is easily seen in the observed spectra from the
MDI high resolution data.
Figure 3: Power spectra from spherical harmonic analyses of MDI data (solid
line) and simulated data (dotted line). The spectral match indicates that both
datasets have very similar spatial structures. (The rapid drop in power for
wavenumbers beyond 600 is largely due to instrumental blurring.)
### 3.2 Rotation
Rotation information is obtained when additional analyses are applied to the
data after it has been mapped onto heliographic coordinates. Longitudinal
strips of this data, centered on latitudes from $75^{\circ}$ south to
$75^{\circ}$ north, were cross-correlated with corresponding strips from later
images as was done by Duvall (1980) and by Snodgrass & Ulrich (1990). The
shift in the location of the cross-correlation peak divided by the time
difference gives the rotation rate.
Fig. 4 shows the differential rotation from the cross-correlation analysis.
The profiles from the simulated data match those from the MDI data at all but
the highest latitudes. Both datasets show faster rotation rates for longer
time lags as noted by Duvall (1980) and by Snodgrass & Ulrich (1990). The
match between the MDI data and the simulated data indicates that the
latitudinal differential rotation profile is fairly well represented by Eq. 6
with the coefficients given by Eqns. 10-12 derived from the helioseismic
studies of Howe et al. (2007). (The simulated data would be a better match at
high latitudes for $\Omega_{4}=-85$ nHz.) The MDI data does show slightly
faster equatorial rotation. This indicates that either Eq. 13 should be
modified to give faster rotation at the lower wavenumbers or Eq. 16 should be
modified to give shorter lifetimes at the higher wavenumbers.
Figure 4: Differential rotation profiles from cross-correlation analyses of
MDI data (solid lines) and simulated data (dotted lines) for four different
time lags between Doppler images – 2, 4, 8, and 16 hours. All profiles are
well matched at all time lags and at all but the highest latitudes.
We reproduced the analysis of Beck & Schou (2000). Lines of data from the
heliographic maps were Fourier analyzed in longitude and those spectral
coefficents were then Fourier anaylzed in time over 10-day intervals. The
power spectra were averaged over all latitude lines between $\pm 9\arcdeg$.
The rotation rate for each wavenumber $\ell$ was determined by finding the
temporal frequency of the center of gravity of the power using a frequency
window of $\approx 58\mu\rm Hz$ (and iterating on the position of that window
four times) and then dividing the temporal frequency by the wavenumber.
Fig. 5 shows the equatorial rotation rates as functions of wavenumber for both
the MDI data and the simulated data. Here again we find a good match for all
but the lowest wavenumbers. This indicates that Eq. 13 gives a good wavenumber
dependence for the rotation rate. The drop in rotation rate for the MDI data
at wavenumbers below 30 is due to imaging artifacts that introduce signal at
low temporal frequencies which can overlap with the rotation signal at low
spatial frequencies. The effects of the line-of-sight projection on measured
rotation rates, as discussed by Hathaway et al. (2006), are evident in the
increase in the observed rotation rates above the input rates from Eq. 13 for
wavenumbers below $\sim 100$. Although the higher noise level from the single
10-day simulation makes precise comparisons difficult, it does appear that the
MDI data might be better fit if the rotation rates given by Eq. 13 were
somewhat higher at wavenumbers below $\sim 100$.
Figure 5: Equatorial rotation rates as functions of wavenumber from analyses
of 6 10-day sets of MDI data (solid line) and a single 10-day set of simulated
data (dotted line). The equatorial rotation rates used as input (Eq. 13) are
shown with the dashed line. The rotation rates match very well at all but the
lowest wavenumbers. Rotation rates faster than the input rates at wavenumbers
below about 100 are due to projection effects.
We have also reproduced one of the analyses of Schou (2003). Lines of data
from the heliographic maps are apodized and then multiplied by longitude
dependent weighting functions designed to remove the Doppler projection effect
and to isolate either longitudinal motions or latitudinal motions. The
weighting used to isolate longitudinal flows is based on the final term in Eq.
4 but with $B_{0}=0$ and a constant of 0.01 added to $\sin\phi$ to avoid
division by zero at the central meridian. The lines are shifted in longitude
according to the differential rotation rate and then Fourier analysed in space
and time to obtain “$k-\omega$” diagrams. Schou (2003) noted that these
diagrams show power for both prograde ($\omega/k<0$) and retrograde
($\omega/k>0$) motion but with excess power in the prograde components.
Fig. 6 shows the $k-\omega$ diagrams from 10 days of the MDI data and from the
simulated data for latitudes between $\pm 4.5\arcdeg$. Both show prograde and
retrograde components as similarly shaped wedges of enhanced power extending
from the origin to both positive and negative longitudinal wavenumbers.
However, the MDI data does show excess power for prograde motion (power excess
extending from the upper left to the lower right) that is not evident in the
simulated data. (Note that running this analysis on the full 60-day run of MDI
data shows a clear separation between prograde and retrograde motions.)
Figure 6: The $k-\omega$ diagrams from analyses of MDI data (left) and
simulated data (right). Darker shades indicate more power. Both datasets show
prograde and retrograde components with similar power and limits but the MDI
data shows excess power in the prograde components (power in upper left and
lower right quadrants). The thin line from upper right to lower left in the
MDI data is produced by imaging artifacts which appear to move retrograde at
the Sun’s rotation rate.
### 3.3 Meridional Flow
The meridional motion of the Doppler pattern can be measured by the cross-
correlation method used to find the differential rotation profiles illustrated
in Fig. 4 or by using the apodizing/weighting method described by Schou
(2003). Fig. 7 shows a comparison of the meridional flow profiles obtained
from the MDI data and the simulated data when we take $V_{0}=-30\ \rm m\
s^{-1}$, representing a poleward meridional flow with peak velocity of $15\
\rm m\ s^{-1}$. The meridional flow we find for the Doppler features in the
simulation very closely matches the input meridional flow (dashed lines in
Fig. 7). This indicates that the projection effects that produce the super-
rotation of the Doppler features do not impact the measurements of the
meridional flow.
Figure 7: Meridional flow profiles from cross-correlation analyses of MDI
data (solid lines) and simulated data (dotted lines) for four different time
lags between Doppler images – 2, 4, 8, and 16 hours. The meridional flow
profile input to the simulation is shown by the dashed lines.
It is clear that the actual meridional flow of the supergranules is more
complicated than that used in our simulation. The meridional flow profiles for
the supergranules shown in Fig. 7 for the MDI data agree very well with those
obtained by Gizon et al. (2003) and by Schou (2003) using somewhat different
methods. The MDI data indicates the presence of counter-cells in the polar
regions during the 1996 data run that would be better represented by
$V(\theta)=(50\cos\theta-60\cos^{3}\theta)\sin\theta$ (17)
In addition the flow appears to weaken for longer time differences between the
cross-correlated data. This suggests the the meridional flow in the surface
shear layer decreases in amplitude with depth.
### 3.4 Lifetimes
The lifetimes of the cells can be estimated by comparing the strength of the
cross-correlation coefficients for the Doppler features as functions of the
time interval between Doppler images. Fig. 8 shows the strength of the
correlations as functions of latitude and time lag for both the MDI data and
the simulated data. The profiles are similar in shape but slightly flatter
with the MDI data. The correlation coefficients are well matched at the
equator for time differences of 4-hours and 16-hours but the simulation shows
stronger correlations at 2-hours and weaker correlations at 8-hours. This
suggests that the formulae (Eqns. 15 and 16) for the random phase changes to
the spectral coefficients needs slight modification. In particular the
cellular features that dominate at $\Delta t=2$ hours need shorter lifetimes.
Figure 8: Cross-correlation peak profiles from analyses of MDI data (solid
lines) and simulated data (dotted lines). The profiles are well matched at
$\Delta t=4$ and $\Delta t=16$ hours but with slight differences at $\Delta
t=2$ and $\Delta t=8$ hours.
## 4 Conclusions
We have produced simulated data in which the cellular structures
(supergranules) are advected by differential rotation and meridional flow and
evolved by uncorrelated random changes. When we compare results from analyses
of these data with those from the MDI data we find that the simulated data
exhibit much of the same characteristics as the MDI data – the visual
structures, the power spectra, the rotation, the meridional flow, and the
evolution rates all match fairly closely. While some of the rotation
characteristics have been attributed to wave-like properties by Beck & Schou
(2000), Gizon et al. (2003), and Schou (2003), our simulated data is simply
advected by a zonal flow (differential rotation) with speeds that never exceed
those determined from helioseismology as reported by Howe et al. (2007).
While the similarities between the simulated data and the MDI data are strong,
the differences are important and interesting. The rotation variations with
depth and latitude were chosen to closely match those shown in Fig 1. of Howe
et al. (2007). The differences in rotation rate seen at high latitudes in our
Fig. 4 suggest that the magnitude of $\Omega_{4}$ should be increased from -75
nHz to -85 nHz. The differences in rotation rate seen at low wavenumbers in
our Fig. 5 suggest an increase in rotation across the surface shear layer of
about 2 nHz at the equator.
The lack of excess power in the prograde components represented in Fig 6 for
the simulation and the lack of evidence for a split between the prograde and
retrograde components indicate differences in the evolution of the pattern.
The manner in which we evolve the cells in the simulation is not fully
satisfactory. The phase (and amplitude) changes associated with the evolving
cells should replicate their advection by larger (nonaxysymmetric) flows
(Williams & Cuntz, 2009), their break-up into smaller cells, and their joining
to form larger cells. The random changes in phase that we introduce to evolve
the cells produces random offsets that are larger for larger wavenumbers and,
when modulated by the power spectrum shown in Fig 3, produces the wedge of
power shown in Fig 6 for the simulation. The split between prograde and
retrograde movement seen in the MDI data suggests that the motions are not
random but preferentially east-west. Lisle et al. (2004) found that
supergranules appear to have a persistent north-south alignment and suggested
that this was due to a north-south elongation of larger, giant cells that
advect the supergranules to their boundaries. The rotational constraints on
these giant cells [cf. Miesch et al. (2008) and references therein] should in-
fact give a preference for east-west flows near the equator and may produce an
added preference for prograde flows. Note that this interpretation of the
results does not involve wave-like properties for supergranules – but instead
invokes simple advection by larger flows.
The advection of the supergranules by the meridional flow shows considerable
promise. The fact that we fully and accurately recover the input velocity
profile indicates that the latitudinal movement of supergranules provides new
information on the meridional flow. The results with the MDI data indicate the
presence of counter-cells (equatorward flow) at latitudes above about
$65\arcdeg-70\arcdeg$. These results also indicate a meridional flow speed
that matches that found for the small magnetic elements as measured by Komm et
al. (1993) and by Hathaway & Rightmire (2010) but is significantly slower than
that reported in the helioseismic determination of Giles et al. (1997) and
Basu et al. (1999). The indication of a decrease in meridional flow speed with
depth is also at odds with those helioseismic results but in agreement with
the simulations of Hathaway (1982).
Magnetic element positions are known to coincide with the boundaries of
supergranules and move as the supergranule boundaries evolve. This was shown
fairly explicitly by Lisle et al. (2000). This suggests that the advection of
the supergranules by the axisymmetric flows should be directly related to the
advection of the magnetic elements. The differential rotation itself (relative
to the Carrington rotation) has a velocity range from $\sim 40\ \rm m\ s^{-1}$
prograde to $\sim 180\ \rm m\ s^{-1}$ retrograde while the meridional flow has
a peak velocity of only $\sim 11\ \rm m\ s^{-1}$. These weak flow velocities
are no match for the flows of several 100 $\rm m\ s^{-1}$ in the
supergranules. The magnetic elements should be quickly carried to the
boundaries and ultimately the interstices of the supergranules where the
direct action of the differential rotation and meridional flow are too weak to
dislodge them. The magnetic elements should experience differential rotation
and meridional flow only to the extent that the supergranules themselves are
advected by these flows at velocities representitive of deeper layers.
The variation in the differential rotation and meridional flow with depth and
latitude within the surface shear layer needs further examination. We have
already argued for some changes from what was assumed in our simulation. In
addition, we note that our association of different depths with different
wavenumbers $\ell$ as indicated in Eqns. 9 and 13 suggests that cells with
diameters $D=2\pi R_{\odot}/\ell$ are advected by flows at depth
$R_{\odot}-r\approx 0.82D$. This would make typical supergranules extend to
depths of 30 Mm. While this is deeper than suggested by the helioseismic
investigations of Duvall et al. (1997) and Woodard (2007), those
investigations were limited to layers not much deeper than 7 Mm. Shallower
cells would suggest that the surface shear layer is shallower as well.
We would like to thank: NASA for its support of this research through a grant
from the Heliophysics Guest Investigator Program to NASA Marshall Space Flight
Center; SOHO, which is a project of international cooperation between ESA and
NASA; and John Beck, who prepared the 31-minute filtered SOHO/MDI data used in
this paper.
## Appendix A Spectral Coefficient Changes Due to Advection
The changes in the spectral coefficients produced in the advection of the
pattern by the axisymmetric flows are best illustrated using the radial
component of the flow pattern as given by
$w(\theta,\phi,t)=\sum_{\ell=0}^{lmax}\sum_{m=-\ell}^{\ell}R_{\ell}^{m}(t)P_{\ell}^{m}(x)e^{im\phi}$
(A1)
where the time-varying spectral coefficient is $R_{\ell}^{m}(t)$,
$x=\cos\theta$, and $P_{\ell}^{m}$ is an Associated Legendre polynomial of
angular degree $\ell$ and azimuthal order $m$. The Associated Legendre
polynomials are normalized such that
$\int_{-1}^{1}P_{\ell}^{m}(x)P_{\ell^{\prime}}^{m^{\prime}}(x)dx=\delta_{\ell}^{\ell^{\prime}}\delta_{m}^{m^{\prime}}$
(A2)
The advection of this flow pattern by the axisymmetric flows is given by
${{\partial w}\over{\partial t}}=-{V\over r}{{\partial
w}\over{\partial\theta}}-{U\over{r\sin\theta}}{{\partial
w}\over{\partial\phi}}$ (A3)
The spherical harmonic representation gives explicit expressions for the
spatial derivatives with
${{\partial
w}\over{\partial\theta}}=\sum_{\ell=0}^{lmax}\sum_{m=-\ell}^{\ell}R_{\ell}^{m}(t){1\over\sin\theta}\left[{\ell\over
A_{\ell+1}^{m}}P_{\ell+1}^{m}(x)-{(\ell+1)\over
A_{\ell}^{m}}P_{\ell-1}^{m}(x)\right]e^{im\phi}$ (A4)
and
${{\partial
w}\over{\partial\phi}}=\sum_{\ell=0}^{lmax}\sum_{m=-\ell}^{\ell}R_{\ell}^{m}(t)imP_{\ell}^{m}(x)e^{im\phi}$
(A5)
where
$A_{\ell}^{m}=\left[{{(2\ell+1)(2\ell-1)}\over{(\ell+m)(\ell-m)}}\right]^{1\over
2}$ (A6)
Taking the meridional velocity as
$V(\theta)=V_{0}\cos\theta\sin\theta$ (A7)
and the azimuthal velocity as
$U(\theta)=r\sin\theta(\Omega_{0}+\Omega_{2}\cos^{2}\theta+\Omega_{4}\cos^{4}\theta)$
(A8)
and then projecting eq. A3 onto $Y_{\ell^{\prime}}^{m^{\prime}}$ gives
$\displaystyle{{\partial R_{\ell^{\prime}}^{m^{\prime}}}\over{\partial t}}$
$\displaystyle=$ $\displaystyle-{V_{0}\over
r}\sum_{\ell=m^{\prime}}^{lmax}R_{\ell}^{m^{\prime}}\int_{-1}^{1}\left[{\ell\over
A_{\ell+1}^{m^{\prime}}}xP_{\ell+1}^{m^{\prime}}(x)P_{\ell^{\prime}}^{m^{\prime}}(x)-{(\ell+1)\over
A_{\ell}^{m^{\prime}}}xP_{\ell-1}^{m^{\prime}}(x)P_{\ell^{\prime}}^{m^{\prime}}(x)\right]dx$
(A9) $\displaystyle-
im^{\prime}\Omega_{0}R_{\ell^{\prime}}^{m^{\prime}}-im^{\prime}\Omega_{2}\sum_{\ell=m^{\prime}}^{lmax}R_{\ell}^{m^{\prime}}\int_{-1}^{1}x^{2}P_{\ell}^{m^{\prime}}(x)P_{\ell^{\prime}}^{m^{\prime}}(x)dx$
$\displaystyle-
im^{\prime}\Omega_{4}\sum_{\ell=m^{\prime}}^{lmax}R_{\ell}^{m^{\prime}}\int_{-1}^{1}x^{4}P_{\ell}^{m^{\prime}}(x)P_{\ell^{\prime}}^{m^{\prime}}(x)dx$
The products of $x$, $x^{2}$, and $x^{4}$ with the Associated Legendre
polynomials can be replaced with expressions containing only Associated
Legendre polynomials of the same azimuthal order, $m$, using the recursion
relation
$P_{\ell}^{m}(x)=A_{\ell}^{m}xP_{\ell-1}^{m}(x)-B_{\ell}^{m}P_{\ell-2}^{m}(x)$
(A10)
where $A_{\ell}^{m}$ is given by eq. A6 and
$B_{\ell}^{m}=\left[{{(2\ell+1)(\ell+m-1)(\ell-m-1)}\over{(2\ell-3)(\ell+m)(\ell-m)}}\right]^{1\over
2}$ (A11)
Eq. A9 then reduces to (after dropping the primes)
$\displaystyle{{\partial R_{\ell}^{m}}\over{\partial t}}$ $\displaystyle=$
$\displaystyle-{V_{0}\over
r}\left[{{(\ell-2)}\over{A_{\ell}^{m}}A_{\ell-1}^{m}}R_{\ell-2}^{m}+\left[{{\ell
B_{\ell+2}^{m}}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}-{{(\ell+1)}\over{A_{\ell}^{m}A_{\ell}^{m}}}\right]R_{\ell}^{m}-{{(\ell+3)B_{\ell+2}^{m}}\over{A_{\ell+2}^{m}A_{\ell+2}^{m}}}R_{\ell+2}^{m}\right]-im\Omega_{0}R_{\ell}^{m}$
(A12) $\displaystyle-
im\Omega_{2}\left[{1\over{A_{\ell}^{m}A_{\ell-1}^{m}}}R_{\ell-2}^{m}+\left[{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}\right]R_{\ell}^{m}+{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}B_{\ell+2}^{m}R_{\ell+2}^{m}\right]$
$\displaystyle-
im\Omega_{4}\left[{1\over{A_{\ell}^{m}A_{\ell-1}^{m}}}{1\over{A_{\ell-2}^{m}A_{\ell-3}^{m}}}\right]R_{\ell-4}^{m}$
$\displaystyle-
im\Omega_{4}\left[{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}+{B_{\ell}^{m}\over{A_{\ell}^{m}A_{\ell-1}^{m}}}+{B_{\ell-1}^{m}\over{A_{\ell-1}^{m}A_{\ell-2}^{m}}}\right]{1\over{A_{\ell}^{m}A_{\ell-1}^{m}}}R_{\ell-2}^{m}$
$\displaystyle-
im\Omega_{4}\left[{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}+{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}\right]{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}R_{\ell}^{m}$
$\displaystyle-
im\Omega_{4}\left[{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}+{B_{\ell}^{m}\over{A_{\ell}^{m}A_{\ell-1}^{m}}}\right]{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}R_{\ell}^{m}$
$\displaystyle-
im\Omega_{4}\left[{B_{\ell+4}^{m}\over{A_{\ell+4}^{m}A_{\ell+3}^{m}}}+{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}+{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}\right]{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}B_{\ell+2}^{m}R_{\ell+2}^{m}$
$\displaystyle-
im\Omega_{4}\left[+{B_{\ell+5}^{m}\over{A_{\ell+5}^{m}A_{\ell+4}^{m}}}{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}B_{\ell+4}^{m}B_{\ell+2}^{m}\right]R_{\ell+4}^{m}$
A common factor in Eq. A12 is of the form
${B_{\ell}^{m}\over{A_{\ell}^{m}A_{\ell-1}^{m}}}={{(\ell+m-1)(\ell-m-1)}\over{(2\ell-1)(2\ell-3)}}$
(A13)
The dominant terms on the RHS of Eq. A12 are those for the direct rotation of
the $R_{\ell}^{m}$ coefficients. These terms are best handled analytically by
taking
$R_{\ell}^{m}=\mathcal{R}_{\ell}^{m}e^{im\Omega_{\ell}^{m}t}$ (A14)
where
$\displaystyle\Omega_{\ell}^{m}$ $\displaystyle=$
$\displaystyle\Omega_{0}+\Omega_{2}\left[{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}\right]+$
(A15)
$\displaystyle\Omega_{4}\left[{B_{\ell+3}^{m}\over{A_{\ell+3}^{m}A_{\ell+2}^{m}}}+{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}\right]{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+$
$\displaystyle\Omega_{4}\left[{B_{\ell+2}^{m}\over{A_{\ell+2}^{m}A_{\ell+1}^{m}}}+{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}+{B_{\ell}^{m}\over{A_{\ell}^{m}A_{\ell-1}^{m}}}\right]{B_{\ell+1}^{m}\over{A_{\ell+1}^{m}A_{\ell}^{m}}}$
With this substitution, Eq. A12 becomes an equation for the evolution of
$\mathcal{R}_{\ell}^{m}$ in which the meridional flow (terms multiplied by
$V_{0}$) couples the amplitude of the $\ell$ component to the $\ell\pm 2$
components while the differential rotation (terms multiplied by $\Omega_{2}$
and $\Omega_{4}$) couples the phase of the $\ell$ component to the $\ell\pm 2$
and $\ell\pm 4$ components. These spectral coefficients can be evolved in time
using Eq. A12 with a fourth-order Runge-Kutta scheme. The solutions are stable
and well behaved for quarter-hour time steps over at least 10-days for $\ell$
upto at least 1500.
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arxiv-papers
| 2010-08-25T21:17:46 |
2024-09-04T02:49:12.471243
|
{
"license": "Public Domain",
"authors": "David H. Hathaway, Peter E. Williams, Kevin Dela Rosa, and Manfred\n Cuntz",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1008.4385"
}
|
1008.4592
|
# Exact Solutions of the Two-Dimensional Discrete Nonlinear Schrödinger
Equation with Saturable Nonlinearity
Avinash Khare Institute of Physics, Bhubaneswar, Orissa 751005, India Kim Ø.
Rasmussen Theoretical Division and Center for Nonlinear Studies, Los Alamos
National Laboratory, Los Alamos, New Mexico, 87545, USA Mogens R. Samuelsen
Department of Physics, The Technical University of Denmark, DK-2800 Kgs.
Lyngby, Denmark Avadh Saxena Theoretical Division and Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
###### Abstract
We show that the two-dimensional, nonlinear Schrödinger lattice with a
saturable nonlinearity admits periodic and pulse-like exact solutions. We
establish the general formalism for the stability considerations of these
solutions and give examples of stability diagrams. Finally, we show that the
effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero.
## I Introduction.
The discrete nonlinear Schrödinger equation (DNLSE) finds widespread use in
physics due to its very general nonlinear character. It arises in the context
of the propagation of electromagnetic waves in optical waveguides emwave , and
it also appears in the study of Bose-Einstein condensates in optical lattices
bec . Recently we have studied the exact soliton solutions and their stability
for the one-dimensional DNLSE with a saturable nonlinearity krss1 . We were
also able to obtain staggered and short period solutions of this equation
krss2 as well as to generalize our results to arbitrarily higher-order
nonlinearities krsss .
Two-dimensional periodic lattices with a saturable nonlinearity in the
Schrödinger equation have been experimentally realized in photorefractive
materials efrem . Solitons have also been observed in these crystals fleischer
; martin . Localized traveling wave solutions that exist only for finite
velocities have been computed in this case melvin . The question of discrete
soliton mobility in these systems has been addressed as well magnus . Here we
show that it is also possible to obtain exact periodic and pulse-like soliton
solutions for the two-dimensional DNLSE with a saturable nonlinearity. We then
study the stability of these solutions as a function of the parameters of the
equation. We also show that similar to the one dimensional case, the effective
Peierls-Nabarro barrier (i.e. the discreteness barrier for soliton motion) for
the pulse-like soliton solutions is zero. In addition, we find several short
period solutions.
## II Two-dimensional discrete nonlinear Schrödinger equation with saturable
nonlinearity.
The equation we consider is the following asymmetric, DNLSE with a saturable
nonlinearity in two dimensions
$i\frac{d\phi_{n,m}}{dt}+[\zeta(\phi_{n+1,m}+\phi_{n-1,m})+\xi(\phi_{n,m+1}+\phi_{n,m-1})]+\frac{\nu|\phi_{n,m}|^{2}\phi_{n,m}}{1+|\phi_{n,m}|^{2}}=0\,.$
(1)
$|\zeta-\xi|$ is a measure of the spatial asymmetry and $\nu$ a measure of the
nonlinearity. This equation can be derived from the Hamiltonian
$\displaystyle
H=\sum_{n,m=1}^{N,M}\bigg{[}\zeta|\phi_{n+1,m}-\phi_{n,m}|^{2}+\xi|\phi_{n,m+1}-\phi_{n,m}|^{2}$
$\displaystyle-[2(\zeta+\xi)+\nu]|\phi_{n,m}|^{2}+\nu\ln(1+|\phi_{n,m}|^{2})\bigg{]}\,,$
(2)
and the equation of motion being derived from
$i\dot{\phi}_{n,m}=\frac{\partial H}{\partial\phi_{n,m}^{*}},$ (3)
considering $\phi_{n,m}$ and $i\phi_{n,m}^{*}$ as conjugate variables. There
are two conserved quantities for the field equation, Eq. (1), the Hamiltonian
$H$ and the power (norm) $P$ defined by
$P=\sum_{n,m=1}^{N,M}|\phi_{n,m}|^{2}\,.$ (4)
Note that the system is invariant under simultaneous interchange of $\zeta$
and $\xi$, and $n$ and $m$.
## III Exact solutions to the two-dimensional equation.
Exact stationary solutions can also be obtained in the case of this two-
dimensional discrete, asymmetric saturable nonlinear Schrödinger equation (1).
We are looking for stationary solutions using the ansatz
$\phi_{n,m}(t)=u_{n,m}e^{-i(\omega t+\delta)}\,,$ (5)
and obtain from Eq. (1), the following difference equation
$(1+u_{n.m}^{2})\omega
u_{n,m}+(1+u_{n,m}^{2})[\zeta(u_{n+1,m}+u_{n-1,m})+\xi(u_{n,m+1}+u_{n,m-1})]+\nu
u_{n,m}^{3}=0\,.$ (6)
Following Ref. krss1 we immediately find two different types of solutions.
One that is symmetric in $n$ and $m$ and one that only depends on $n$ (and by
symmetry one that only depends on $m$).
Symmetric case:
If one chooses
$\omega=-\nu\,,$ (7)
then
$u_{n,m}^{s}=\frac{{\rm sn}(\beta,k)}{{\rm cn}(\beta,k)}{\rm
dn}(\beta(n+m+\delta_{1}),k)\,,$ (8)
is a solution if $k$ is chosen to fulfill
$\frac{\nu}{\zeta+\xi}=2\frac{{\rm dn}(\beta,k)}{{\rm
cn}^{2}(\beta,k)}\,,~{}~{}\beta=\frac{2K(k)}{N_{p}}\,.$ (9)
Similarly
$u_{n,m}^{s}=k\frac{{\rm sn}(\beta,k)}{{\rm dn}(\beta,k)}{\rm
cn}(\beta(n+m+\delta_{1}),k)\,,$ (10)
is a solution provided
$\frac{\nu}{\zeta+\xi}=2\frac{{\rm cn}(\beta,k)}{{\rm
dn}^{2}(\beta,k)}\,,~{}~{}\beta=\frac{4K(k)}{N_{p}}\,.$ (11)
Here $k$ is the elliptic modulus (the elliptic parameter $m=k^{2}$ stegun ) of
the Jacobi elliptic functions ${\rm sn}(x,k)$, ${\rm cn}(x,k)$, and ${\rm
dn}(x,k)$ and $K(k)$ is the complete elliptic integral of the first kind
stegun ; Ryzhik . The integer $N_{p}$ denotes the spatial period of the
system. $N$ and $M$ in Eq. (II) must be chosen as multiples of $N_{p}$. The
two solutions have a common pulse-like limit for $k\rightarrow 1$ (and
$N_{p}\rightarrow\infty$),
$u_{n,m}^{s}=\sinh(\beta){\rm sech}[\beta(n+m+\delta_{1})]\,,$ (12)
which is a solution if $\beta$ fulfills
$\frac{\nu}{\zeta+\xi}=2\cosh(\beta)\,.$ (13)
By symmetry, a change of sign of $n$ (or $m$) in Eqs. (8), (10), and (12) will
give solutions with exactly the same properties.
Asymmetric case:
If one chooses for the frequency
$\omega=-\nu-2\xi\,,$ (14)
then
$u_{n,m}^{as}=\frac{{\rm sn}(\beta,k)}{{\rm cn}(\beta,k)}{\rm
dn}(\beta(n+\delta_{1}),k)\,,$ (15)
will be a solution for $k$ satisfying
$\frac{\nu}{\zeta}=2\frac{{\rm dn}(\beta,k)}{{\rm
cn}^{2}(\beta,k)}\,,~{}~{}\beta=\frac{2K(k)}{N_{p}}\,.$ (16)
Similarly, we also have a $cn$ solution.
$u_{n,m}^{as}=k\frac{{\rm sn}(\beta,k)}{{\rm dn}(\beta,k)}{\rm
cn}(\beta(n+\delta_{1}),k)\,,$ (17)
provided
$\frac{\nu}{\zeta}=2\frac{{\rm cn}(\beta,k)}{{\rm
dn}^{2}(\beta,k)}\,,~{}~{}\beta=\frac{4K(k)}{N_{p}}\,.$ (18)
Here $N=N_{p}$ and $M$ can be any integer.
Again both these solutions approach the pulse solution in the limit
$k\rightarrow 1$ and $N_{p}\rightarrow\infty$
$u_{n,m}^{as}=\sinh(\beta){\rm sech}[\beta(n+\delta_{1})]\,,$ (19)
provided
$\frac{\nu}{\zeta}=2\cosh(\beta)\,.$ (20)
Another asymmetric solution appears if we interchange $n$ and $m$ and $\zeta$
and $\xi$ ($M=N_{p}$ and $N$ any integer). We note that, in all cases,
changing the sign of $\xi$ is equivalent to staggering the solution in the $m$
direction ($(-1)^{m}$ as an amplitude factor) and changing the sign of $\zeta$
is equivalent to staggering the solution in the $n$ direction krss2 .
The described solutions are in some sense direct generalizations of our
earlier results for the one-dimensional version of the saturable nonlinear
Schrödinger equation, since they remain spatially uniform along a specific
direction in space. Such solutions are well-known for nonlinear partial
differential equations and are often, in such a continuum setting, referred to
as line solutions or line solitons. However, in continuum settings, such
solutions are generally not stable, because the extra dimension now allows for
an entire set of new instability modes to come into play. In our discrete
case, however, we shall demonstrate that these solutions indeed can be stable
in certain cases. The fact that these solutions have infinite extension along
one of their dimensions probably renders them less physically important.
However, their stability analysis does, as we will demonstrate, give detailed
insight into the intricate stability mechanisms of this nonlinear system.
Specifically, are the parameters $(\nu,\xi)$, which control the stability,
directly related to materials properties such as the change in refractive
index of the crystal.
## IV Stability of the solutions.
In order to study the linear stability of these exact solutions $u_{n,m}^{j}$
($j$ is “s” (symmetric) or “as” (asymmetric)) we introduce the following
expansion around the exact solution
$\phi_{n,m}(t)=u_{n,m}^{j}e^{-i(\omega t+\delta)}+\delta
u_{n,m}(t)e^{-i(\omega t+\delta)}$ (21)
applied in a frame rotating with frequency $\omega$ of the solution.
Substituting (21) into the field equation, Eq. (1), and retaining only terms
linear in the deviation, $\delta u_{n,m}$, we get
$i\delta\dot{u}_{n,m}+\zeta\big{(}\delta u_{n+1,m}+\delta
u_{n-1,m}\big{)}+\xi\big{(}\delta u_{n,m+1}+\delta
u_{n,m-1}\big{)}+\left(\omega+\frac{\nu|u_{n,m}^{j}|^{2}(2+|u_{n,m}^{j}|^{2})}{(1+|u_{n,m}^{j}|^{2})^{2}}\right)\delta
u_{n,m}+\frac{\nu|u_{n,m}^{j}|^{2}}{(1+|u_{n,m}^{j}|^{2})^{2}}\delta
u_{n,m}^{*}=0.$ (22)
We continue by splitting the deviations $\delta u_{n,m}$ into real parts
$\delta u_{n,m}^{(r)}$ and imaginary parts $\delta u_{n,m}^{(i)}$ ($\delta
u_{n,m}=\delta u_{n,m}^{(r)}+i\delta u_{n,m}^{(i)}$) and introducing the two
real vectors
$\displaystyle\delta\mbox{\boldmath$U^{r}$}=\\{\delta
u_{n,m}^{(r)}\\}=\\{\delta U_{J}^{(r)}\\},$ and
$\displaystyle\delta\mbox{\boldmath$U^{i}$}=\\{\delta
u_{n,m}^{(i)}\\}=\\{\delta U_{J}^{(i)}\\},$ (23)
where the pair of indices $m,n$ are replaced by a single index $J$ via:
$J=n+(m-1)N_{p}$. By introducing the real matrices
$\mbox{\boldmath$A$}=\\{A_{J,J^{\prime}}\\}$ and
$\mbox{\boldmath$B$}=\\{B_{J,J^{\prime}}\\}$ defined by
$\displaystyle
A_{J,J^{\prime}}=\zeta(\delta_{J,J^{\prime}+1}+\delta_{J,J^{\prime}-1})+\xi(\delta_{J,J^{\prime}+N_{p}}+\delta_{J,J^{\prime}-N_{p}})+\left(\omega+\frac{\nu|u_{n,m}^{j}|^{2}(3+|u_{n,m}^{j}|^{2})}{(1+|u_{n,m}^{j}|^{2})^{2}}\right)\delta_{J,J^{\prime}},$
(24) $\displaystyle
B_{J,J^{\prime}}=\zeta(\delta_{J,J^{\prime}+1}+\delta_{J,J^{\prime}-1})+\xi(\delta_{J,J^{\prime}+N_{p}}+\delta_{J,J^{\prime}-N_{p}})+\left(\omega+\frac{\nu|u_{n,m}^{j}|^{2}}{(1+|u_{n,m}^{j}|^{2})}\right)\delta_{J,J^{\prime}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(25)
where $J^{\prime}\pm 1$ and $J^{\prime}\pm N_{p}$ in the Kronecker $\delta$
means: $J^{\prime}\pm 1~{}mod~{}N_{p}$ and $J^{\prime}\pm N_{p}~{}mod~{}N_{p}$
to ensure periodic boundary conditions, Eq. (22) becomes
$\displaystyle-\delta\mbox{\boldmath$\dot{U^{i}}$}+\mbox{\boldmath$A$}\delta\mbox{\boldmath$U^{r}$}=\mbox{\boldmath$0$},$
and
$\displaystyle\delta\mbox{\boldmath$\dot{U^{r}}$}+\mbox{\boldmath$B$}\delta\mbox{\boldmath$U^{i}$}=\mbox{\boldmath$0$}.$
(26)
Combining these first order differential equations we get:
$\displaystyle\delta\mbox{\boldmath$\ddot{U^{i}}$}+\mbox{\boldmath$A$}\mbox{\boldmath$B$}\delta\mbox{\boldmath$U^{i}$}=\mbox{\boldmath$0$},$
and
$\displaystyle\delta\mbox{\boldmath$\ddot{U^{r}}$}+\mbox{\boldmath$B$}\mbox{\boldmath$A$}\delta\mbox{\boldmath$U^{r}$}=\mbox{\boldmath$0$}.$
(27)
The two matrices $A$ and $B$ are symmetric and have real elements. However,
since they do not commute $A$$B$ and
$\mbox{\boldmath$B$}\mbox{\boldmath$A$}=(\mbox{\boldmath$A$}\mbox{\boldmath$B$})^{tr}$
are not symmetric. $A$$B$ and $B$$A$ have the same eigenvalues, but different
eigenvectors. The eigenvectors for each of the two matrices need not be
orthogonal. The eigenvalue spectrum $\\{\gamma\\}$ of the matrix $A$$B$ (or
$B$$A$) determines the stability of the exact solutions. If $\\{\gamma\\}$
contains negative eigenvalues then the solution is unstable. The eigenvalue
spectrum always contains two eigenvalues which are zero. These eigenvalues
correspond to the translational invariance in space and time (represented by
$\delta_{1}$ and $\delta$). The given solutions are unstable for most of the
parameter space $(\zeta,\xi,\nu)$. From Ref. krss1 we know that $\xi=0$
generally leads to stable solutions. In determining the stability of the
solutions it is useful to note that the rescaling transformation
$(\zeta,\xi,\nu)\rightarrow\alpha(\zeta,\xi,\nu)$
($\omega\rightarrow\alpha\omega$) changes the eigenvalues by $\alpha^{2}$ and
therefore it does not affect the stability (i.e. the sign of eigenvalues) of
the solutions. Therefore the three-dimensional parameter space
$(\zeta,\xi,\nu)$ can be significantly reduced (into a two-dimensional
parameter space) as far as stability considerations are concerned. The
nonlinearity parameter $\nu\neq 0$, separates the three-dimensional parameter
space into two disconnected equivalent ones for $\nu>0$ and $\nu<0$,
respectively. The rescaling transformation with $\alpha=-1$ interchanges the
two equivalent half spaces. So we need only to consider positive $\nu$. From
here on we treat the symmetric and the asymmetric cases separately.
Stability of the symmetric case:
Since $\nu>0$, Eqs. (9), (11), and (13) require $\zeta+\xi>0$ and therefore
$\zeta>-\xi$. Further, we can always choose $\zeta>\xi$ due to the inter-
changeability of $\zeta$ and $\xi$. We therefore have $-\zeta<\xi<\zeta$ or
applying the scaling condition $-1<\xi/\zeta<1$. This means that the stability
of the entire parameter space can be mapped out onto the much smaller
parameter space $(1,\xi,\nu)$, where $-1<\xi<1$ (and $\nu>0$).
Figure 1: Stability analysis for the $dn$ solution, Eq. (8). Lowest
eigenvalues for $|\nu|=1.5$ (thick line, black online), $|\nu|=2$ (medium
thick line, blue online), and $|\nu|=3$ (thin line, red online). Dashed
(solid) lines indicate negative (positive) values of $\nu$. Stability occurs
when the lowest eigenvalue is zero. The entire existence interval is shown for
positive $\nu$. Negative $\nu$ results can be obtained by rescaling the
results for positive $\nu$. The remaining parameters are: $\zeta=1$ and
$N_{p}=8$.
Figure 2: General stability diagram for symmetric $dn$ (left) and $cn$ (right)
solutions of Eq. (8) and Eq. (10), respectively. In the grey region no
solution of the given kind exists. In the white regions solutions are
unstable, while the black area indicates stable solutions. The vertical line
$\xi=0$, which represents the one-dimensional stability result, is part of the
black area. The rough or jagged appearance of parts of the stability boundary
is due to limited numerical resolution rather than an intrinsic feature of the
problem. Parameters are: $\zeta=1$, and $N_{p}=8$.
In Fig. 1 we illustrate the stability analysis for the $dn$ solution, Eq. (8),
by showing the lowest eigenvalue as a function of $\xi$ for several values of
$\nu$. Stability occurs whenever the lowest eigenvalue is zero. The entire
existence regime is illustrated for positive $\nu$ and a few windows of
stability can be seen. It is important to note that the results for negative
values of $\nu$ ($\xi<-1$) are superfluous as they can be obtained by
rescaling the results for positive $\nu$. To demonstrate this we have for
$\zeta=1$:
$(1,-|\xi|,-|\nu|)\rightarrow(-1/|\xi|,1,|\nu|/|\xi|)\rightarrow(1,-1/|\xi|,|\nu|/|\xi|)$
where the last step follows by the inter-changeability of the two-coupling
parameters. This shows that the $\nu<0$ regime can be mapped onto the $\nu>0$
regime. Generally, stability is only observed for $\xi<0$. This means that
stability for the symmetric ${\rm dn}$ solutions can only be achieved when
$\xi$ and $\zeta$ have opposite signs. Recalling the equivalence between
staggered solutions and sign changes of $\xi$ or $\zeta$, another way to view
this result is that the symmetric ${\rm dn}$ solutions must be staggered in
one dimension when $\zeta$ and $\xi$ are both positive in order to be stable.
This is clearly a property arising from the discreteness that cannot be
achieved in a continuum system.
Assembling results like those shown in Fig. 1 for a range of $\nu$ and $\xi$
values we arrive at Fig. 2 where stability diagrams for both the $dn$ and the
$cn$ solutions to Eq. (8) and Eq. (10), respectively, are shown. The grey
region indicates that the solutions do not exist, whereas the black (white)
regions indicate the existence of stable (unstable) solutions. These two
stability diagrams have a very similar structure. However, the $cn$ solutions
are always stable in the proximity of the existence boundary marked by the
grey area. This property is related to the fact that the amplitude (which is
$\propto k$, see Eq.(10)) of this solution vanishes at the existence boundary
where $k=0$. Also, we note that the common pulse solution corresponds to the
corner close to $(\xi,\nu)=(-1,0)$.
We have looked at other values of $N_{p}$ and find that for larger $N_{p}$,
the stability diagram has similar features. The pulse-like solution only
exists in the limit $N_{p}\rightarrow\infty$, and here it has the same
stability properties as the $dn$ and $cn$ solutions for $k\rightarrow 1$.
Therefore, our analysis indicates that the symmetric pulse-like solutions are
stable for small $\nu$ and $\xi\sim-1$.
Stability of the asymmetric case:
Figure 3: General stability diagram for the asymmetric $dn$ (left) and $cn$
(right) solutions of Eq. (15) and Eq. (17), respectively. In the grey region
no solution of the given kind exists. In the white region solutions are
unstable, while the black area indicates stable solutions. The vertical line
$\xi=0$, which represents the one-dimensional stability result, is part of the
black area. Parameters are: $\zeta=1$, and $N_{p}=8$.
We proceed almost as in the symmetric case. We still have $\nu>0$, therefore
from Eqs. (16), (18), and (20) we have $\zeta>0$. So here the three-
dimensional parameter space can be reduced to $(1,\xi,\nu)$, where
$-\infty<\xi<\infty$ (and $\nu>0$). Illustrations of the stability diagrams
are given for the asymmetric $dn$ and $cn$ solutions, Eq. (15) and Eq. (17),
respectively in Fig. 3 for $\zeta=1$ and $N_{p}=8$. Again the two stability
diagrams have a very similar structure except that the parameter space for
stability of the asymmetric solution is much larger. As in the symmetric case,
even in the asymmetric case, the $cn$ solutions are always stable in the
proximity of the existence boundary marked by the grey area. Here we note that
the common pulse solution corresponds to large $\nu$ and $\xi$.
## V Peierls-Nabarro barrier for the pulse solution
We would now like to show the absence of Peierls-Nabarro barrier for the pulse
solution. However, we must remember that since both power $P$ and Hamiltonian
$H$ are constants of motion, one must compute the energy difference between
the solutions when $\delta_{1}=0$ and $\delta_{1}=1/2$ in such a way that the
power $P$ is same in both the cases.
For the pulse solution obtained above, the power $P$ is given by
$P=\sum_{n,m=-\infty}^{\infty}|\phi_{n,m}|^{2}=\sinh^{2}(\beta)\sum_{n,m=-\infty}^{\infty}{\rm
sech}^{2}[\beta(n+m+\delta_{1})]\,.$ (28)
This double sum can be evaluated using the single sum result
$\sum_{n=-\infty}^{\infty}{\rm
sech}^{2}[\beta(n+\delta_{2})]=\frac{2}{\beta}+\frac{2K(k)E(k)}{\beta^{2}}+\left(\frac{2K(k)}{\beta}\right)^{2}{\rm
dn}^{2}[2\delta_{2}K(k),k]\,,$ (29)
the above $P$ is given by
$P=\sum_{m=-\infty}^{\infty}\sinh^{2}(\beta)\left[\frac{2}{\beta}+\frac{2K(k)E(k)}{\beta^{2}}+\left(\frac{2K(k)}{\beta}\right)^{2}{\rm
dn}^{2}[2(m+\delta_{1})K(k),k]\right]\,.$ (30)
Note, only the last term on the rhs is $m$ dependent. In these equations
$E(k)$ is the complete elliptic integral of the second kind.
Let us now discuss the computation of the Hamiltonian $H$. Clearly, for the
pulse solution obtained above, $H$ as given by Eq. (II) takes the form
$\displaystyle H=\sum_{n,m=-\infty}^{\infty}\bigg{[}-\nu
P+\nu\ln[1+\sinh^{2}(\beta){\rm sech}^{2}(\beta[n+m+\delta_{1}])]$
$\displaystyle-2\sinh^{2}(\beta)[{\rm sech}(\beta[n+1+m+\delta_{1}]){\rm
sech}(\beta[n+m+\delta_{1}])(\zeta+\xi)]\bigg{]}\,.$ (31)
Again we use the single sum results to evaluate the double sum, i.e.
$\sum_{n=-\infty}^{\infty}\big{[}{\rm sech}[\beta(n+1+\delta_{2})]{\rm
sech}[\beta(n+\delta_{2})]\big{]}=\frac{2}{\sinh(\beta)}\,,$ (32)
$\sum_{n=-\infty}^{\infty}\ln[1+\sinh^{2}(\beta){\rm
sech}^{2}(\beta[n+\delta_{2}])]=2\beta\,,$ (33)
the above $H$ is given by
$H=\sum_{m=-\infty}^{\infty}\bigg{[}-\nu
P+2\nu\beta-4(\zeta+\xi)\sinh(\beta)\bigg{]}\,.$ (34)
We thus note that for a given power $P$ (which contains a sum over $m$), H is
indeed independent of $\delta_{1}$, i.e. the Peierls-Nabarro barrier is indeed
zero for the pulse solution. The same holds true for the asymmetric solution.
## VI Short period solutions
Recently we obtained short period solutions to the one-dimensional saturable
DNLSE krss2 . These short period ($N$) solutions, in the one-dimensional case,
can be written in the following compact form (coming from equally distributed
points on a circle so that projection on the $x$-axis should only be 0 or $\pm
a$):
$u_{N}(n)=\frac{a}{\cos(\varphi_{N})}\cos\left(\frac{2\pi
n}{N}+\varphi_{N}\right),$ (35)
where $\varphi_{1}$ =$\varphi_{2}$ =$\varphi_{4s}=0$, (4s is the stable period
4) and $\varphi_{4}=\frac{\pi}{4}$, $\varphi_{3}=\varphi_{6}=\frac{\pi}{6}$.
$\displaystyle
u_{N}(n+1)+u_{N}(n-1)=\frac{a}{\cos(\varphi_{N})}\bigg{[}\cos\left(\frac{2\pi(n+1)}{N}+\varphi_{N}\right)+\cos\left(\frac{2\pi(n-1)}{N}+\varphi_{N}\right)\bigg{]}$
(36) $\displaystyle=\frac{2a}{\cos(\varphi_{N})}\cos\left(\frac{2\pi
n}{N}+\varphi_{N}\right)\cos\frac{2\pi}{N}=2\psi_{N}(n)\cos\frac{2\pi}{N}.$
(37)
For $\omega$ we get
$\omega=-2\zeta\cos\frac{2\pi}{N}-\frac{\nu a^{2}}{1+a^{2}}.$ (38)
Assuming that in the two-dimensional case, the solution is a product of the
two one-dimensional solutions (with period $N$ and period $M$), i.e.
$u_{N,M}(n,m)=\frac{a}{\cos(\varphi_{N})\cos(\varphi_{M})}\cos\left(\frac{2\pi
n}{N}+\varphi_{N}\right)\cos\left(\frac{2\pi m}{M}+\varphi_{M}\right),$ (39)
we get:
$\omega=-2\zeta\cos\frac{2\pi}{N}-2\xi\cos\frac{2\pi}{M}-\frac{\nu
a^{2}}{1+a^{2}}.$ (40)
## VII Conclusions
We have given analytical expressions for the solutions to the two-dimensional
discrete nonlinear Schrödinger equation with saturable nonlinearity which
arises in photorefractive crystals efrem ; fleischer ; martin ; melvin ;
magnus . Due to their infinite extension along one of their dimensions, these
solutions are not very physically meaningful but it is very rare that
solutions to discrete nonlinear two-dimensional problems can be described in
closed form using standard mathematical functions as we have done here. This
feature of the solutions is physically significant because it provides an
opportunity for in-depth scrutiny and understanding that is not usually
available in a nonlinear physical system. These solutions are closely related
to the previously derived krss1 solutions to the corresponding one-
dimensional equation. However, in contrast to what one may expect based on
intuition derived from similar nonlinear partial differential equations, we
have shown that these solutions are linearly stable in certain regions of the
parameter space. Specifically, we have observed that the asymmetric versions
of these solutions lead to a very intricate stability diagram. We have shown
that the symmetric $dn$ solution is stable in certain regions of the parameter
space provided it is staggered in one dimension. However, the symmetric $cn$
solution as well as the asymmetric $cn$ and $dn$ solutions are stable in
certain regions of parameter space both when they are non-staggered or if they
are staggered in one dimension. The finding that nonlinear waveforms in two-
dimensional photorefractive materials best achieve stability in the presence
of phase asymmetry between the two spatial directions is crucial because the
photonic lattices that represent the physical realization of Eq.(1) tend to
naturally possess this property OL . Finally, we found that the Peierls-
Nabarro barrier for the pulse solution is zero. An understanding of the
mobility of these exact discrete two-dimensional solutions remains an
important issue magnus .
###### Acknowledgements.
This work was carried out under the auspices of the National Nuclear Security
Administration of the U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396.
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* (12) Table of Integrals, Series, and Products, I.S. Gradshteyn and I.M. Ryzhik, (Academic Press, San Diego, CA, 2007).
* (13) Khare A. and Sukhatme U. 2002 J. Math. Phys. 43, 3798;
Khare A., Lakshminarayan A. and Sukhatme U 2003 J. Math. Phys. 44, 1822;
math-ph/0306028; 2004 Pramana (Journal of Physics) 62, 1201.
* (14) Y.V. Kartashov, V.A. Visloukh, and L. Torner, Phys. Rev. E 68, 015603 (2003); A.S. Desyatnikov, D.N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jager, C. Denz, and Y. V. Kartashov, Opt. Letts. 30, 869 (2005).
|
arxiv-papers
| 2010-08-26T20:18:53 |
2024-09-04T02:49:12.483515
|
{
"license": "Public Domain",
"authors": "Avinash Khare, Kim \\O . Rasmussen, Mogens R. Samuelsen, and Avadh\n Saxena",
"submitter": "Kim Rasmussen",
"url": "https://arxiv.org/abs/1008.4592"
}
|
1008.4594
|
# Staggered and short period solutions of the Saturable Discrete Nonlinear
Schrödinger Equation
Avinash Khare1, Kim Ø. Rasmussen2, Mogens R. Samuelsen3, and Avadh Saxena2
1Institute of Physics, Bhubaneswar, Orissa 751005, India 2Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
3Department of Physics, The Technical University of Denmark, DK-2800 Kgs.
Lyngby, Denmark
###### Abstract
We point out that the nonlinear Schrödinger lattice with a saturable
nonlinearity also admits staggered periodic as well as localized pulse-like
solutions. Further, the same model also admits solutions with a short period.
We examine the stability of these solutions and find that the staggered as
well as the short period solutions are stable in most cases. We also show that
the effective Peierls-Nabarro barrier for the pulse-like soliton solutions is
zero.
###### pacs:
61.25.Hq, 64.60.Cn, 64.75.+g
The saturable discrete nonlinear Schrödinger equation is increasingly finding
applications in various physical situations. Most notably it serves as a model
for optical pulse propagation in optically modulated photorefractive media
[1], and in this context the pulse dynamics it describes have been intensely
studied [2, 3, 4]. In addition to its important role for such applications the
saturable discrete nonlinear Schrödinger equation is also of interest from a
purely nonlinear science viewpoint [5, 6, 7]. This interest arises because the
saturable discrete nonlinear Schrödinger equation has been demonstrated [8] to
admit onsite and intersite soliton solutions, which have the same energy. This
is contrasted by the standard cubic nonlinear Schrödinger lattice where the
onsite solution always has lower energy than the intersite solution. This
phenomenon have often been characterized in terms of a so-called Peierls-
Nabarro (PN) barrier, which is the energy difference between these two
distinct solutions. The particular feature of the saturable discrete nonlinear
Schrödinger equation is thus that it allows the PN barrier to change sign and
specifically vanish for certain solutions. The vanishing of the PN barrier
have been associated with the ability of these solutions to translate
undisturbed through the lattice, which is impossible in the cubic discrete
nonlinear Schrödinger equation. Here we derive analytical solutions to the
saturable discrete nonlinear Schrödinger equation and demonstrate that the
localized soliton solutions have a zero PN barrier.
Recently, we obtained [9] two different temporally and spatially periodic
solutions to the saturable equation[10]
$i\dot{\psi}_{n}+(\psi_{n+1}+\psi_{n-1}-2\psi_{n})+\frac{\nu|\psi_{n}|^{2}}{1+\mu|\psi_{n}|^{2}}\psi_{n}=0\,,$
(1)
where $\psi_{n}$ is a complex valued ‘wavefunction’ at site $n$, while $\nu$
and $\mu$ are real parameters. In particular, the first solution is
$\psi_{n}^{I}=\frac{1}{\sqrt{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{cn}(\beta,m)}\mbox{dn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right),$ (2)
where the modulus of the elliptic functions $m$ must be chosen such that
$2-\omega=\frac{\nu}{\mu}=\frac{2\mbox{dn}(\beta,m)}{\mbox{cn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{2K(m)}{N_{p}}\,,$
(3)
and $c$ and $\delta$ are arbitrary constants. We only need to consider c
between 0 and $\frac{1}{2}$ (half the lattice spacing). Here $K(m)$ denotes
the complete elliptic integral of the first kind [11]. The second solution is
$\psi_{n}^{II}=\sqrt{\frac{m}{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{dn}(\beta,m)}\mbox{cn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (4)
where the modulus $m$ is now determined such that
$2-\omega=\frac{\nu}{\mu}=\frac{2\mbox{cn}(\beta,m)}{\mbox{dn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{4K(m)}{N_{p}}\,.$
(5)
The integer $N_{p}$ denotes the spatial period of the solutions. In the limit
$N_{p}\rightarrow\infty$ ($m\rightarrow 1$), both the solutions $\psi_{n}^{I}$
and $\psi_{n}^{II}$ reduce to the same localized solution
$\psi_{n}^{III}=\frac{1}{\sqrt{\mu}}\frac{\sinh(\beta)}{\cosh([n+c]\beta)}e^{-i[\omega
t+\delta]},~{}~{}(N_{p}\rightarrow\infty),$ (6)
where $\beta$ is now given by
$2-\omega=\frac{\nu}{\mu}=2\mbox{cosh}\beta\,.$ (7)
In Ref. [9], we also developed the stability analysis and examined the linear
stability of these solutions to show that the solutions are linearly stable in
most cases.
The purpose of this note is to point out that the same model (1) also admits
the corresponding staggered solutions. In particular, using the identities for
the Jacobi elliptic functions [12], it is easily shown that the model admits
the following solutions
$\psi_{n}^{IS}=(-1)^{n}\frac{1}{\sqrt{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{cn}(\beta,m)}\mbox{dn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (8)
where the modulus $m$ must be chosen such that
$\omega-2=-\frac{\nu}{\mu}=\frac{2\mbox{dn}(\beta,m)}{\mbox{cn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{2K(m)}{N_{p}}\,.$
(9)
$\psi_{n}^{IIS}=(-1)^{n}\sqrt{\frac{m}{\mu}}\frac{\mbox{sn}(\beta,m)}{\mbox{dn}(\beta,m)}\mbox{cn}([n+c]\beta,m)\exp\left(-i[\omega
t+\delta]\right)\,,$ (10)
where the modulus $m$ is now determined such that
$\omega-2=-\frac{\nu}{\mu}=\frac{2\mbox{cn}(\beta,m)}{\mbox{dn}^{2}(\beta,m)},~{}~{}~{}\beta=\frac{4K(m)}{N_{p}}\,.$
(11)
In the limit $N_{p}\rightarrow\infty$ ($m\rightarrow 1$), both the solutions
$\psi_{n}^{IS}$ and $\psi_{n}^{IIS}$ reduce to the same localized staggered
solution
$\psi_{n}^{IIIS}=(-1)^{n}\frac{1}{\sqrt{\mu}}\frac{\sinh(\beta)}{\cosh([n+c]\beta)}e^{-i[\omega
t+\delta]},~{}~{}~{}(N_{p}\rightarrow\infty)\,,$ (12)
where $\beta$ is now given by
$\omega-2=-\frac{\nu}{\mu}=2\mbox{cosh}\beta\,.$ (13)
As an illustration we have plotted the exact solutions of the type IS and IIS
in Fig. 1. Here the period $N_{p}$ has to be even. We have shown two periods
for type IS and only one for type IIS.
There are, as expressed by Eqs. (9), (11), and (13), stringent conditions on
the parameters $\mu$ and $\nu$ for which these exact solutions exist. For
example, while the nonstaggered solutions are only valid for $\nu>0$ and hence
$\omega<2$, the staggered solutions are valid only if $\nu<0$ and hence
$\omega>2$. In the case IS the limitation is
$0~{}(m=1)<-\frac{2\mu}{\nu}<\cos^{2}\left(\frac{\pi}{N_{p}}\right)~{}(m=0)\,,$
(14)
while in the case IIS the limitation is
$0~{}(m=1)<-\frac{2\mu}{\nu}<\frac{1}{\cos(\frac{2\pi}{N_{p}})}~{}(m=0)\,.$
(15)
Similarly, the solution $\psi_{n}^{IIIS}$ exists only when $-\frac{2\mu}{\nu}$
is close to zero (m=1).
We have also examined the linear stability of these solutions and find that
the solutions are linearly stable in most cases. A single period $(N=N_{p}$i,
where $N$ is the lattice size) is always stable for both solutions IS and IIS.
A type IIS solution with more than one period ($N=jN_{p}$, where $j$ is an
integer larger than one) is also stable, while a type IS solution with more
than one period is always unstable. Thus, the first example in Fig. 1 is in
fact unstable.
Figure 1: Illustration of the exact solutions of two types. The parameters
are: $\nu=-1$, $\mu=0.4$, $\omega=4.5$, and $c=t=\delta=0$. $N_{p}=10$ for
$\psi_{n}^{IS}$ and 20 for $\psi_{n}^{IIS}$. The dashed curves represent the
solutions given by Eqs. (8) and (10) as if $n$ is a continuous variable. Lines
are guides to the eye.
For the solution IIIS, expressions for both the power and the Hamiltonian are
identical to those for the solution III and are given by Eqs. (13) and (14) of
Ref. [9]. Hence the PN barrier for the solutions III and IIIS is the same. We
would like to point out here that the calculation of PN barrier in I was not
quite correct. In particular, since both power P and the Hamiltonian H are
constants of motion, one must compute the energy difference between the
solutions when $c=0$ and $c=1/2$ in such a way that the power P is same in
both the cases. On using the expressions for P and H as given by Eqs. (13) and
(14) of I, we find that H for the solution III as well as IIIS is given by
$H=-\frac{4\sinh(\beta)}{\mu}+\frac{2\beta\nu}{\mu^{2}}+2\left(1-\frac{\nu}{2\mu}\right)P\,.$
(16)
Note that H is in fact independent of $c$, i.e. contrary to our claim in Ref.
[9], the PN barrier is in fact zero for our solution III (and hence also for
IIIS).
Before completing this note, we would like to mention that the model (1) also
admits a few short period solutions.
Using the ansatz
$\psi_{n}(t)=\phi_{n}e^{-i(\omega t+\delta)}\,,$ (17)
in Eq. (1) it is easily checked that the only possible short period solutions
are to Eq. (1) are:
1. 1.
Period 1 solution $\phi_{n}=(...,a,a,...)$ provided
$\omega=-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (18)
2. 2.
Period 2 solution $\phi_{n}=(...,a,-a,...)$ provided
$\omega=4-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (19)
3. 3.
Period 3 Solution $\phi_{n}=(...,a,0,-a,...)$ provided
$\omega=3-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (20)
4. 4.
Period 4 Solutions $\phi_{n}=(...,a,0,-a,0,...)$ and $(...,a,a,-a,-a,...)$
provided
$\omega=2-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (21)
5. 5.
Period 6 Solution $\phi_{n}=(...,a,a,0,-a,-a,0,...)$ provided
$\omega=1-\frac{\nu a^{2}}{1+\mu a^{2}}.$ (22)
Applying the stability analysis developed in Ref. [9] we have examined the
stability of these short period solutions and find that for a small
nonlinearity ($|\nu|<2\mu$) they are all stable. The period 4 solution
$(...,a,a,-a,-a,...)$ is always stable while all the other short period
solutions possess regions of instabilities at larger nonlinearity. For these
low period solutions the stability matrices given by Eqs. (20) and (21) of
Ref. [9] are simple and it is, for example, easy to see that the lowest non-
zero eigenvalue, $\lambda_{1}(a,\nu)$, of the stability problem for the period
1 solutions is given by ($\mu=1$)
$\lambda_{1}(a,\nu)=a^{4}+\left(2-\frac{2}{3}\nu\right)a^{2}+1.$ (23)
Similarly, we have for the period 2 solution
$\lambda_{2}(a,\nu)=a^{4}+\left(2+\frac{2}{3}\nu\right)a^{2}+1,$ (24)
and the period 4 solution
$\lambda_{4}(a,\nu)=a^{4}+(2-|\nu|)a^{2}+1.$ (25)
These expressions correspond to the relevant curves in Fig. 2.
It possible to derive similar expressions for the period 3 and period 6
solutions but the expressions are more complicated and will be omitted here.
Figure 2: Regions of stability for the short period solutions to Eq. (1) for
$\mu=1$. Period 1: thick full curve, period 2: dashed-dotted curve, period 3:
long-dashed curve, period 4: thin full curve, and period 6: short-dashed
curve. The instability occurs in the parameter region encompassed by the
respective curves.
Clearly the $p$ period solutions are unstable for the parameter values where
$\lambda_{p}(a,\nu)<0$, and we have illustrated these regions in Fig. 2.
Figure 2 shows the curves in the $(a,\nu)$-plane where $\lambda_{p}(a,\nu)=0$
so that the instability occurs in the regions that are encompassed by the
respective curves. A symmetry is apparent in this stability diagram and it is
easy to realize that this symmetry arises from the fact that the
transformation $(\nu,\phi_{n})\rightarrow(-\nu,(-1)^{n}\phi_{n})$ establishes
the following connection between the short period solutions: 1
$\leftrightarrow$ 2, 3 $\leftrightarrow$ 6, and 4 $\leftrightarrow$ 4.
In conclusion, we have obtained staggered as well as short period solutions of
the saturable discrete nonlinear Schrödinger equation. We also studied the
linear stability and found the solutions to be stable in certain parameter
ranges. Finally, we found that the Peierls-Nabarro barrier for the pulse
solutions is zero. Our results are relevant to optical soliton pulse
propagation in waveguides and photorefractive media [1].
Research at Los Alamos National Laboratory is carried out under the auspices
of the National Nuclear Security Administration of the U.S. Department of
Energy under Contract No. DE-AC52-06NA25396.
## References
* [1] Fleischer J.W. 2005 Opt. Express 13 1780\.
* [2] Fitrakis E.P., Kevrekidis P.G., Malomed B.A., and Frantzeskakis, D.J. 2006 Phys. Rev. E 74, 026605.
* [3] Vicencio R.A. and Johansson M 2006 Phys. Rev. E 73, 046602\.
* [4] Cuevas J and Eilbeck J.C. 2006 Phys. Lett. A 358, 15.
* [5] Melvin T.R.O., Champneys A.R., Kevrekidis P.G. 2006 Phys. Rev. Lett. 97, 124101.
* [6] Oxtoby O.F., Barashenkov I.V. 2007 Phys. Rev. E, 76, 036603.
* [7] Melvin T.R.O, Champneys A.R., Kevrekidis P.G. 2008 Physica D 237, 551.
* [8] L. Hadzievskii, A. Maluckov, M. Stepic, D. Kip 2004 Phys. Rev. Lett. 93, 033901.
* [9] Khare A., Rasmussen K.Ø, Samuelsen M.R., and Saxena A. 2005 J. Phys. A38, 807; hereafter we will refer to it as I.
* [10] Note that rewritting $\frac{\nu|\psi_{n}|^{2}}{1+\mu|\psi_{n}|^{2}}\psi_{n}=\frac{\nu}{\mu}(1-\frac{1}{1+\mu|\psi_{n}|^{2}})\psi_{n}$ and the notation change $\frac{\nu}{\mu}\rightarrow-\nu$ followed by the gauge transformation $\phi_{n}\rightarrow\phi_{n}\exp(-\nu t)$ render the equantion in a perhaps more often used form [7].
* [11] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun (U.S. GPO, Washington, D.C., 1964).
* [12] Khare A. and Sukhatme U. 2002 J. Math. Phys. 43, 3798; Khare A., Lakshminarayan A., Sukhatme U 2003 J. Math. Phys. 44, 1822; math-ph/0306028; 2004 Pramana (Journal of Physics) 62, 1201.
|
arxiv-papers
| 2010-08-26T20:31:35 |
2024-09-04T02:49:12.488491
|
{
"license": "Public Domain",
"authors": "Avinash Khare, Kim \\O . Rasmussen, Mogens R. Samuelsen, and Avadh\n Saxena",
"submitter": "Kim Rasmussen",
"url": "https://arxiv.org/abs/1008.4594"
}
|
1008.4630
|
# Dust Obscuration in Lyman Break Galaxies at ${z\sim 4}$
I-Ting Ho11affiliation: Institute of Astronomy & Astrophysics, Academia
Sinica, P.O. Box 23-141, Taipei 10617, Taiwan , Wei-Hao Wang11affiliation:
Institute of Astronomy & Astrophysics, Academia Sinica, P.O. Box 23-141,
Taipei 10617, Taiwan , Glenn E. Morrison22affiliation: Institute for
Astronomy, University of Hawaii, Honolulu, HI 96822, USA 33affiliation:
Canada-France-Hawaii Telescope, Kamuela, HI 96743, USA , and Neal A.
Miller44affiliation: Department of Astronomy, University of Maryland, College
Park, MD 20742, USA itho@ifa.hawaii.edu; itho@asiaa.sinica.edu.tw
###### Abstract
Measuring star formation rates (SFRs) in high-$z$ galaxies with their rest-
frame ultraviolet (UV) continuum can be uncertain because of dust obscuration.
Prior studies had used the submillimeter emission at $850~{}\mu\rm m$ to
determine the intrinsic SFRs of rest-frame UV selected galaxies, but the
results suffered from the low sensitivity and poor resolution ($\sim
15\arcsec$). Here, we use ultradeep Very Large Array 1.4 GHz images with $\sim
1\arcsec$–$2\arcsec$ resolutions to measure the intrinsic SFRs. We perform
stacking analyses in the radio images centered on $\sim 3500$ Lyman break
galaxies (LBGs) at $z\sim 4$ in the Great Observatories Origins Deep Survey-
North and South fields selected with Hubble Space Telescope/Advanced Camera
for Surveys data. The stacked radio flux is very low, $0.08\pm 0.15$ $\mu$Jy,
implying a mean SFR of $6\pm 11~{}{\rm M_{\sun}~{}yr^{-1}}$. This is
comparable to the uncorrected mean UV SFRs of $\sim 5~{}{\rm
M_{\sun}~{}yr^{-1}}$, implying that the $z\sim 4$ LBGs have little dust
extinction. The low SFR and dust extinction support the previous results that
$z\sim 4$ LBGs are in general not submillimeter galaxies. We further show that
there is no statistically significant excess of dust-hidden star-forming
components within $\sim 22$ kpc from the LBGs.
###### Subject headings:
dust, extinction — galaxies: evolution — galaxies: high-redshift — radio
continuum: galaxies
††slugcomment: ApJ accepted, August 2010
## 1\. INTRODUCTION
Understanding galaxy evolution in the early Universe requires large samples of
various kinds of high-$z$ galaxies. One important technique enabling
selections of $z>2.5$ star-forming galaxies is the Lyman break technique
(Cowie et al., 1988; Songaila et al., 1990; Lilly et al., 1991; Steidel &
Hamilton, 1993; Steidel et al., 1995; also see a review in Giavalisco, 2002).
The UltraViolet (UV) spectrum of a high-redshift star-forming galaxy usually
exhibits a Lyman continuum discontinuity at 912 Å, which is caused by
absorption of H I in the stellar atmosphere of massive stars, and interstellar
and intergalactic media. By using this feature, large samples of Lyman break
galaxies (LBGs) can be selected in various redshift ranges by searching for
sudden brightness dropouts between two adjacent broad-band images.
An important property of LBGs is their star formation rates (SFRs). However,
determining SFRs for $z>3$ systems is still challenging because most of the
diagnostics at low redshifts are unavailable when light becomes dimmer and
redshifts to wavelengths that are harder to observe. One of the most commonly
used method to determine SFRs at high redshift is to measure the rest-frame UV
continuum which shifts at $z>3$ to the optical and near-infrared (e.g.,
Kennicutt, 1998). Unfortunately, the UV continuum can be easily attenuated by
dust, which results in underestimation of the SFRs. Bouwens et al. (2009)
attempted to determine the correction factor by using the observed correlation
between the ratio of far-infrared (FIR) to UV flux and the UV spectral slope
(Meurer et al., 1999). They found that UV continuum of $z\sim 4$ LBGs in the
Great Observatories Origins Deep Survey-North (GOODS-N) and South (GOODS-S)
fields underestimates SFRs by factors of $\sim 3$–6. However, this result is
debatable since the correlation exhibits significant scatter in different
populations of galaxies (e.g., Cortese et al., 2006; Howell et al., 2010).
A more direct way to estimate the SFRs is to deduce them at longer wavelengths
where light is not affected by dust obscuration. Efforts had been made using
the submillimeter $850~{}\mu\rm m$ emission to determine the intrinsic SFRs in
LBGs (Peacock et al., 2000; Chapman et al., 2000; Webb et al., 2003). However,
these results have low signal-to-noise ratios (S/Ns), presumably because the
submillimeter single-dish maps were not deep enough, a consequence of
instrumental, sky, and confusion noises (i.e., uncertainties contributed by
nearby bright sources or faint undetected sources). In addition, an assumption
of dust temperature is required to estimate the total infrared (IR)
luminosity. These all make the estimate of SFRs based on submillimeter
measurements quite uncertain.
The radio wavelength is an alternative probe of SFRs. At 1.4 GHz, the radio
continuum is dominated by synchrotron radiation from relativistic electrons
produced by supernovae. By converting the 1.4 GHz luminosity to total IR
luminosity with the well-known radio—FIR correlation (Condon, 1992), it is
possible to estimate intrinsic SFRs (Kennicutt, 1998). The radio—FIR
correlation is fairly insensitive to dust temperature. The angular resolution
in the radio can be very high ($\sim 1\arcsec$–$2\arcsec$), meaning very
little confusion noise. Because of these great advantages, this radio-based
method has been used in various high-z studies (e.g., Reddy & Steidel, 2004;
Wang et al., 2006; Carilli et al., 2008; Pannella et al., 2009).
The Very Large Array (VLA) provides high resolution in the radio, but its
sensitivity is insufficient for directly detecting normal star-forming
galaxies at $z\gtrsim 1.5$. Therefore, stacking radio fluxes of large samples
of LBGs to determine their mean SFRs is a necessary approach. Reddy & Steidel
(2004) stacked $z\sim 2$ LBGs and found the dust correction to be $\sim 4.5$.
Using a similar method, Carilli et al. (2008) determined the dust correction
for $z\sim 3$ LBGs to be 1.8.
In order to determine the intrinsic SFRs for $z\sim 4$ LBGs, we performed
radio stacking analyses in the GOODS-N and GOODS-S fields. The GOODS Hubble
Space Telescope/Advance Camera for Surveys (HST/ACS) data (Giavalisco et al.,
2004) were used to locate $\sim 3500$ LBGs in the fields, and deep VLA 1.4 GHz
images (Miller et al., 2008; Morrison et al., 2010) were used to measure and
stack the radio fluxes. We note that although radio stacking of $z\sim 4$ LBGs
had also been attempted by Carilli et al. (2008), we used much deeper optical
images ($\sim 2$ mag deeper at V band ) and radio images ($\sim 15\%-40\%$
deeper). We are therefore probing much closer to the typical members in the
$z\sim 4$ LBG population, as in Bouwens et al. (2009).
In this paper, we first describe the observational data in Section 2. In
Section 3, we describe our methods of selecting LBGs, stacking analysis, and
estimations of SFRs using radio and UV. Finally, we compare the two different
SFRs and discuss the implications in Section 4, and give a summary in Section
5. We assume $\Omega_{0}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70~{}\rm
km~{}s^{-1}~{}Mpc^{-1}$. All magnitudes are in the AB magnitude system.
## 2\. DATA
### 2.1. HST/ACS
The HST/ACS multiband imaging in the GOODS fields (Giavalisco et al., 2004)
consists of four passbands: F435W, F606W, F775W, and F850LP, which are
referred to as $B_{435}$, $V_{606}$, $i_{775}$, and $z_{850}$, respectively.
We adopt the v2.0 source catalogs for selecting our LBG samples. Quantities
measured with SExtractor (Bertin & Arnouts, 1996) “automatic aperture”
(MAG_AUTO, FLUX_AUTO, etc.) are used to approximate total values.
### 2.2. VLA
The VLA 1.4 GHz images of the GOODS-N and GOODS-S fields are described in more
details in Morrison et al. (2010) and Miller et al. (2008), respectively. In
brief, the North field has an rms noise of $\sim 4~{}\mu\rm Jy~{}beam^{-1}$ at
the center, and $<6~{}\mu\rm Jy~{}beam^{-1}$ at the edges of the ACS fields.
The South field has an rms noise of $\sim 6~{}\mu\rm Jy~{}beam^{-1}$ at the
center, and $<7.5~{}\mu\rm Jy~{}beam^{-1}$ at the edges of the fields. These
sensitivities have been corrected for the primary beam response. The beam FWHM
of the North field is $1\farcs 7\times 1\farcs 6$, which corresponds to $\rm
11.8~{}kpc\times 11.1~{}kpc$ at $z=4$. The beam FWHM of the South field is
$2\farcs 8\times 1\farcs 6$, which corresponds to $\rm 19.5~{}kpc\times
11.1~{}kpc$ at $z=4$. Data Release 2 of the South field is used here.
## 3\. METHOD
### 3.1. Sample Selection
“B-dropout” galaxies at $z\sim 4$ are selected utilizing the redshifted Lyman
break located between $B_{435}$ and $V_{606}$. We adopt the well established
criteria (e.g., Beckwith et al., 2006; Bouwens et al., 2007):
$B_{435}-V_{606}>1.1,\\\ $ $B_{435}-V_{606}>(V_{606}-z_{850})+1.1,\\\ $
$V_{606}-z_{850}<1.6,\\\ $ $S/N(V_{606})>5,\ \ \ {\rm and}\ \ S/N(i_{775})>3.$
In addition, compact objects (SExtractor stellarity indices greater than 0.8)
with $i_{775}<$ 26.5 are rejected from our sample to prevent stellar
contamination. A color-color diagram illustrating the sample selection is
shown in Figure 1. In total, we selected 1778 and 1679 B-dropouts in the North
and South fields, respectively. These selection criteria efficiently select
LBGs between $z\sim 3$ and $z\sim 4.5$, with a mean redshift of 3.8 (see more
details in Bouwens et al., 2007). The number of LBGs selected by us is very
similar to that in Bouwens et al. (2007), who used similar GOODS ACS data.
Figure 1.— A color-color diagram showing the $B$-dropouts selection criteria.
Objects in the GOODS fields are plotted with dots. Intersection of the three
color selection criteria described in Section 3.1 are shaded in grey. The
selected LBGs are shown with red dots. Note that some of the objects inside
the grey region are not selected as LBGs because they do not meet the other
criteria (e.g., low S/N or too compact).
### 3.2. Stacking Analyses
We measured the radio fluxes of the $z\sim 4$ LBGs with aperture photometry at
their optical positions. Small offsets ($<0^{\prime\prime}.3$) between the
coordinates of the radio and optical images had been corrected utilizing their
source catalogs. Apertures of two different sizes were used here: small-
apertures with radii equal to the beam FWHMs ($\sim 11$ kpc at $z=4$); large-
apertures with radii equal to twice the beam FWHMs ($\sim 22$ kpc at $z=4$).
Here we first focus on the small aperture results and we will discuss the
large aperture results in Section 4.4. Elliptical apertures were used for the
South field to match its beam shape. Fluxes measured with our aperture
photometry were calibrated with the radio catalogs (Miller et al., 2008;
Morrison et al., 2010). For sources in the radio catalogs, we measured their
fluxes with our aperture photometry and computed the median of the ratios
between our fluxes and the catalog fluxes. The ratios, which were found to be
very close to unity for the aperture sizes we used, were applied back to our
radio fluxes.
Sources with radio fluxes $>100~{}\mu\rm Jy$ were rejected from our
measurements because their radio fluxes do not likely reflect their true SFRs.
There are eight such sources. If they are normal star-forming galaxies at
$z=4$, their IR luminosity would be $\gtrsim 5\times 10^{13}~{}L_{\sun}$ (or
SFR $\gtrsim 7000~{}{\rm M_{\sun}~{}yr^{-1}}$), which is unusually large. We
inspected these sources individually. They either are affected by nearby
bright radio sources that do not appear to be at the same redshifts of the
LBGs, or do not appear to be associated with known submillimeter galaxies
(SMGs) in the samples of Wang et al. (2004), Perera et al. (2008), Devlin et
al. (2009), and Weißet al. (2009). Therefore, they are highly unlikely high-
redshift ultraluminous starbursting galaxies. To avoid the bias from unrelated
nearby bright sources and from active galactic nuclei (AGNs), we did not
include these eight sources in our stacking analyses. For those sources with
radio fluxes $<100~{}\mu\rm Jy$, we then averaged their radio fluxes and
subtracted a background value (see below) from the means to get the final
stacked radio fluxes.
The uncertainties of our stacked fluxes were estimated with Monte Carlo
simulations, with an assumption that the galaxies are distributed randomly
over the map. We measured the mean radio fluxes at random positions, and the
number of random positions is the same as that of the $<100~{}\mu\rm Jy$
B-dropouts. The same apertures were used and the same rejection criterion of
$<100~{}\mu\rm Jy$ was adopted. This measurement is repeated 10,000 times, and
the mean radio fluxes have a fairly Gaussian-like distribution. We then
calculated the mean and the dispersion of these 10,000 measurements. The mean
was subtracted from the stacked radio flux of the LBG sample to form the final
stacked radio flux. This subtraction is to account for the effect of imperfect
clean and chance projection of random radio sources in our flux apertures.
Likewise, the dispersion can then represent the uncertainty of the stacked
radio flux.
There are two caveats in the above procedures. First, if $z\sim 4$ LBGs are
clustered at scales similar to the sizes of our apertures, our assumption that
LBGs are randomly distributed would break down. Our stacking method would then
overestimate the radio fluxes. We investigated clustering with the same method
used by Marsden et al. (2009). We found that the variance-to-mean ratio of
source numbers inside randomly placed $r<10\arcsec$ circles of our sample
exceeds that of random distribution by less than 0.1, and becomes smaller with
smaller radii. Therefore, we conclude that there is no significant small-scale
clustering in our LBG sample and the flux overestimate caused by small-scale
clustering can be neglected.
Second, the faulty estimate of uncertainties can be caused by concentration of
sources in high or low sensitivity regions (i.e., the slightly uneven
sensitivity distribution caused by the primary beam falloff in the GOODS-N or
by the mosaicking in the GOODS-S). Instead of stacking at random positions to
estimate the uncertainty, we also carried out the simulation based on the
source positions of the real LBG sample. We added an offset to the positions
of the LBGs and stacked their fluxes. We repeated this for 10,000 times with
different random offsets of $15\arcsec-60\arcsec$, and calculated the mean and
the dispersion. The results are within 30% to those measured from random
positions, suggesting that LBGs are not concentrated in specific regions.
With the above methods, we measured the final stacked fluxes and flux errors
of the LBGs in the two fields separately. For each aperture size, we combined
the two final stacked fluxes by weighting them with the inverse of the square-
errors to form the combined flux. The results are summarized in Table 1. The
stacked radio signal is very weak, consistent with zero within the noise.
### 3.3. Star Formation Rate
#### 3.3.1 Radio SFR
The stacked radio fluxes can be converted to SFRs. Assuming a universal
synchrotron emission spectral index of $\alpha=-0.8$, we convert the stacked
radio fluxes $S_{\rm stack}$ to the rest-frame 1.4 GHz luminosity densities
$L_{\rm 1.4GHz}$, i.e.,
$L_{\rm 1.4GHz}=4\pi d_{l}^{2}\sl S_{\rm stack}\sl(1+z)^{-(1+\alpha)},$ (1)
where $d_{l}$ is the luminosity distance and $z$ is the mean redshift of 3.8.
With the local radio—FIR correlation (Helou et al., 1985; Condon, 1992),
$L_{\rm 1.4GHz}$ can be converted to FIR luminosity $L_{\rm FIR(40-120)}$,
approximately the total luminosity between $\lambda=42.5~{}\mu\rm m$ and
$\lambda=122.5~{}\mu\rm m$, with the following relation:
$q=log{{L_{\rm FIR(40-120)}}\over{3.75\times 10^{12}\rm~{}W}}\sl-log{{{L_{\rm
1.4GHz}}}\over{\rm W~{}Hz^{-1}}},$ (2)
with $q=2.34\pm 0.01$(Yun et al., 2001). Although this value was derived
locally ($z\lesssim 0.15$), recent studies suggest weak evolution out to
$z\sim 2$ (Sargent et al., 2010; Ivison et al., 2010a, b). Extrapolating $q$
to $z=4$ using the relation $q\propto(1+z)^{\gamma}$ with $\gamma=-0.04\pm
0.03$ (Ivison et al., 2010b) implies that $L_{\rm FIR}$ (and therefore radio
SFRs) will be lowered by only $\sim 30\%\pm 20\%$, which is insignificant
compared to the uncertainties in our results.
SFRs can then be estimated with the conversion in Kennicutt (1998):
${{\rm SFR\over({\rm M_{\sun}~{}yr^{-1}})}}=4.5\times 10^{-44}{L_{\rm
IR(8-1000)}\over\rm erg~{}s^{-1}},$ (3)
where $L_{\rm IR(8-1000)}$ is approximately the total luminosity between
$8~{}\mu\rm m$ and $1000~{}\mu\rm m$. This estimation is within $\sim 30\%$ to
other published calibrations. We derived the ratio between $L_{\rm
FIR(40-120)}$ and $L_{\rm IR(8-1000)}$ by computing these two quantities on
model spectral energy distributions of six nearby normal and starburst
galaxies (Silva et al., 1998), using their original definitions over the
Infrared Astronomical Satellite(IRAS) bands (see the summary in Sanders &
Mirabel, 1996). The $L_{\rm IR(8-1000)}$ to $L_{\rm FIR(40-120)}$ ratio has a
range of $1.71-2.32$, with a weak anti-correlation between the ratio and IR
luminosity and a mean of 2.05. This mean ratio is slightly higher than that
used in Yun et al. (2001) of 1.5, which is based on measurements in luminous
IRAS galaxies and starburst galaxies (Sanders et al., 1991; Meurer et al.,
1999; Calzetti et al., 2000), but is closer to the values in nearby normal
spiral galaxies and low luminosity starbursts ($\sim 2$ for M 51 and M100, and
2.3 for M 82, based on the Silva et al. templates). Using this value, we
derived the SFRs listed in Table 2, which are consistent with normal galaxies
or low-luminosity starbursts.
Table 1Mean Radio Fluxes
Field | Small Apertureaafootnotemark: | Large Aperturebbfootnotemark:
---|---|---
| $F_{\rm LBG}$ | $F_{\rm random}$ | $F_{\rm LBG}$ | $F_{\rm random}$
| ($\mu\rm Jy$) | ($\mu\rm Jy$) | ($\mu\rm Jy$) | ($\mu\rm Jy$)
GOODS-N | $-0.05\pm 0.18$ | $0.36\pm 0.18$ | $0.29\pm 0.28$ | $1.10\pm 0.28$
GOODS-S | $0.36\pm 0.26$ | $0.10\pm 0.26$ | $0.54\pm 0.54$ | $-0.03\pm 0.54$
Combined | $0.08\pm 0.15$ | - | $0.34\pm 0.25$ | -
11footnotetext: Apertures with radii equal to the beam FWHM.
22footnotetext: Apertures with radii equal to twice the beam FWHM.
#### 3.3.2 UV SFR
The $i_{775}$ and $V_{606}$ bands correspond to rest-frame UV, and can be used
to calculate SFRs, uncorrected for extinction (Madau et al., 1998), with
${\rm SFR}({\rm M_{\sun}~{}yr^{-1}})={\sl L_{\rm UV}({\rm
ergs~{}s^{-1}~{}Hz^{-1}})\over C},$ (4)
where $L_{\rm UV}$ is the UV luminosity density at the same redshift of 3.8,
and $C$ is $8.0\times 10^{27}$ and $7.9\times 10^{27}$ at 1500Å and 2800Å,
respectively. We list in Table 2 the mean UV SFRs of the B-dropouts with radio
fluxes $<100~{}\mu\rm Jy$. The uncorrected SFRs derived from $i_{775}$ and
$V_{606}$ are remarkably similar to each other as well as to that derived from
the radio stacking analyses, indicating very little extinction in the observed
rest-frame UV emission.
Table 2Star Formation Rates Derived with Various Fluxes | Radio | Optical
---|---|---
| Small aperture | Large aperture | $i_{775}$ | $z_{850}$
SFRs(${\rm M_{\sun}~{}yr^{-1}}$) | $6.0\pm 11.0$ | $25.7\pm 18.8$ | 4.92 | 5.14
## 4\. RESULT AND DISCUSSION
### 4.1. Dust extinction in $z\sim 4$ LBGs
In Figure 2, we plot the $z_{850}$ magnitudes versus the small-aperture radio
fluxes and the corresponding SFRs of all the B-dropouts in the two fields,
with the red points indicating the mean ($<100~{}\mu\rm Jy$ sources). As can
be seen in the figure and Table 2, the average SFRs deduced from the radio and
the rest-frame UV are consistent within measurement uncertainties. The
relatively small SFR derived from the radio flux suggests little extinction in
the $z\sim 4$ LBGs. Unfortunately, although this is by far the deepest radio
stacking on $z\sim 4$ LBGs ($\sim 3\times$ deeper than that in Carilli et al.,
2008), the error in the stacked radio flux is still too large to pin down the
extinction correction. An extinction correction of $>6$ is ruled out at 2.2
$\sigma$ and an extinction correction of $>4$ is ruled out at 1.3 $\sigma$.
Bouwens et al. (2009) derived extinction corrections of $\sim 3-6$ for $z\sim
4$ LBGs based on their UV continuum slopes. Our result slightly favors a lower
value and is marginally ($\sim 2~{}\sigma$) consistent with the result of
Bouwens et al. (2009).
We note that an even stricter upper limit on the radio SFR can be placed if
any of the following is true: (1) the radio—FIR correlation $q$ factor
exhibits a decline with redshift (see Section 4.3); or (2) the conversion from
$L_{\rm FIR(40-120)}$ to $L_{\rm IR(8-1000)}$ is overestimated (see Section
3.3.1); or (3) there is AGN contribution in our radio SFR (see Section 4.2).
The combined effect of all the above may not be negligible and can make the
extinction correction for $z\sim 4$ LBGs even lower. However, this remains to
be tested by future observations.
Carilli et al. (2008) found a low extinction correction of 1.8 on $z\sim 3$
LBGs. On the other hand, on $z\sim 4$ LBGs, they had a 2 $\sigma$ detection of
$0.83\pm 0.42$ $\mu$Jy on 1447 $B$-dropouts, which also suggests a low
extinction correction. This radio flux is much higher than ours ($0.08\pm
0.15$ $\mu$Jy) but their ground-based $V$-band limiting magnitude is also much
shallower than ours (by $\sim 2$ mag). In other words, we are probing
different LBG luminosities. Comparing our $z\sim 4$ result with the $z\sim 4$
result in Carilli et al. (2008) is therefore not straightforward.
Figure 2.— HST $z_{850}$ magnitude vs. 1.4 GHz VLA flux of the $\sim 3500$
B-dropouts in the GOODS fields. Orange symbols indicate objects not included
in our stacking analyses. The radio fluxes are measured with small apertures
with radii equal to the beam FWHMs ($\sim 11~{}\rm kpc$ at $z=4$). The
$z_{850}$ uncertainty of each data point is indicated with vertical bars. The
radio uncertainties, which depend on the locations on the maps, are not shown
to avoid confusion. Instead, we plot the typical radio flux uncertainties for
galaxies close to the center of the 2 fields at the lower-right corner for
reference (upper: North, lower: South). The inferred SFR corresponding to the
measured flux density at 1.4 GHz (see text for details) is shown on the upper
x-axis. The average of all the black data points are shown with the red
symbols, with the radio uncertainties estimated from the MonteCarlo
simulation. The black curves indicate the locus of equal SFR deduced from the
$z_{850}$ and the radio. The color coded curves, from left to right, indicate
1/6, 1/4, 1/2, 2, 4, and 6 times dust corrections from the UV to radio SFR. A
blow up to the stacked data point is shown as an insert.
### 4.2. AGN contamination
To avoid AGN contamination, a flux cut of $<100~{}\mu\rm Jy$ was set and eight
sources were rejected (one in the north and seven in the south). If these
sources comprise all the AGNs in our sample, our sample would have a very low
AGN fraction of $\sim 0.2\%$. For comparison, the AGN fraction in LBGs at
$z\sim 3$ was estimated to be $\sim 3\%$, based on optical spectroscopy
(Shapley et al., 2003). This difference could be due to the fact that our LBGs
are much less luminous than those in the spectroscopic sample.
It is possible to roughly separate AGNs and star forming galaxies based on the
radio power. In the local radio luminosity function (e.g., Mauch & Sadler,
2007), the 1.4 GHz power that divides AGNs and starbursts is $\sim
10^{23}~{}\rm W~{}Hz^{-1}$. Cowie et al. (2004) show that this AGN/starburst
division increases to $\sim 10^{24}~{}\rm W~{}Hz^{-1}$ at $z\sim 1$. This
power corresponds to $\sim 10$ $\mu$Jy for $z=4$. However, the increasing
trend observed by Cowie et al. (2004) can continue to $z\gg 1$, moving the
AGN/starburst division to a radio flux greater than 10 $\mu$Jy. Furthermore,
high-redshift flat-spectrum radio sources (presumably AGNs) have a more
negative $K$-correction than that of steep-spectrum sources (starbursts). This
also increases the apparent flux division between AGNs and starbursts at
observed-frame 1.4 GHz (by roughly a factor of 2). We thus expect sources
below our 100 $\mu$Jy cut to be dominated by star forming galaxies.
It is interesting to test what happens if we assume that $>50$ $\mu$Jy sources
are also AGNs and remove them from the stacking analyses. To do this, we
rejected 13 sources that are brighter than 50 $\mu$Jy and introduced the same
50 $\mu$Jy cut in the Monte Carlo simulation. We found a stacked flux of
$0.05\pm 0.14$ $\mu$Jy, corresponding to a mean SFR of $3.8\pm 10.5$ ${\rm
M_{\sun}~{}yr^{-1}}$. The stacked flux decreases further if we keep lowering
the flux cut but the change is well within the noise. Nevertheless, it is
obvious that we will get a lower mean SFR if we assume more AGN contribution.
Although we are limited by noise here, our conclusions that $z\sim 4$ LBGs
have low SFRs and the mean extinction correction seems to be lower than 5 are
thus still valid.
### 4.3. The Radio—FIR Correlation
Carilli et al. (2008) discussed the use of the local radio—FIR correlation to
derive SFRs of high-redshift galaxies. They mentioned several possibilities
that this correlation may be different at high redshift. Indeed, a weak
evolution has been suggested by recent studies (Sargent et al., 2010; Ivison
et al., 2010a, b). However, as mentioned in Section 3.3.1, even if we
extrapolate the evolution based on a declining $q$ (Ivison et al., 2010b) and
adopt a $L_{\rm IR(8-1000)}$ to $L_{\rm FIR(40-120)}$ ratio that is 25% lower,
our main conclusions would not change and this would make the SFR of the LBGs
even lower. Furthermore, the local radio—FIR correlation is only used to
calibrate the relation between SFR and radio power. Even if there is an
evolution in the radio—FIR correlation at high redshift, we cannot be certain
that the evolution arises from the radio part of the correlation and affects
the radio—SFR relation. In short, we see neither evidence nor strong argument
that the radio SFR is significantly biased by the assumption of the local
radio–FIR correlation.
### 4.4. Implications for SMGs
The relatively low dust correction and SFRs imply that $z\sim 4$ LBGs in
general do not have extreme star formation activities enshrouded in dust,
unlike that in SMGs. If the total SFR of our $z\sim 4$ LBGs is only
contributed by SMGs with SFR of $1000~{}{\rm M_{\sun}~{}yr^{-1}}$ and if other
non-SMG LBGs have no star forming activities, then there can be no more than
$21\pm 39$ SMGs in our LBG sample. In comparison, the source counts at
$850~{}\mu\rm m$ (Coppin et al., 2006) suggest $\sim 350$ SMGs in a field of
the combined GOODS size, with $S_{850}\rm>1.4~{}mJy$ and a mean SFR of
$1000~{}{\rm M_{\sun}~{}yr^{-1}}$. This means that even if the total SFR of
our LBG sample is dominated by $z\sim 4$ SMGs, they can only account for
$<10\%$ of the entire SMG population at $S_{850}\rm>1.4~{}mJy$. This implies
that either LBGs and SMGs have little overlap at $z\sim 4$ or most SMGs are at
$z\ll 4$. We note that previous studies in the submillimeter showed that LBGs
are in general not SMGs (Chapman et al., 2000; Peacock et al., 2000; Webb et
al., 2003), and that most SMGs are at $z<3.5$ (Chapman et al., 2003, 2005).
Our results are consistent with these.
Another scenario related to SMGs worth investigating is dust-hidden companions
sitting close to the LBGs. An example of this is the $z=4.5$ SMG in Capak et
al. (2008). This galaxy is independently selected as an LBG, and has an
ultraluminous dusty component showing up at $>3.6\mu\rm m$ next to its rest-
frame UV emission. To know whether similar systems are common at $z\sim 4$, we
performed the same radio stacking using the large apertures with radii twice
the beam FWHM ($\sim 22$ kpc at $z=4$) to measure the radio fluxes. As shown
in Figure 3 and Table 2, the radio SFR is slightly higher but still consistent
with the UV SFR, implying that there is no statistically significant excess of
dust-hidden star-forming components close to LBGs.
Figure 3.— Same as Figure 2, but the radio fluxes are measured with apertures
of radii twice the beam FWHM ($\sim 22~{}\rm kpc$ at $z=4$).
## 5\. Summary
We employed radio stacking analyses on $\sim 3500$ LBGs at $z\sim 4$ in the
GOODS-N and GOODS-S fields. The mean radio flux was converted to FIR
luminosity via the well-known radio—FIR correlation, and then converted to an
intrinsic SFR, which is $6\pm 11{\rm M_{\sun}~{}yr^{-1}}$. By comparing with
SFRs estimated from rest-frame UV, we found a maximum UV extinction correction
of $\sim 6$. This is roughly consistent with that derived from UV continuum
slopes by Bouwens et al. (2009) with a very similar sample. The low extinction
correction confirms that $z\sim 4$ LBGs are in general not SMGs. We also
investigated the possibility of finding dust-hidden companions close to the
LBGs at $z\sim 4$. The stacked radio flux within $\sim 22\rm~{}kpc$ to the
LBGs implies a mean SFR that is still consistent with the UV inferred SFR,
which suggests no statistically significant excess of dust-hidden star-forming
components close to LBGs.
We thank the referee for providing comments that improve this paper. W.-H.W.
and I.-T.H. acknowledge a grant from the National Science Council of Taiwan
(98-2112-M-001-003-MY2).
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|
arxiv-papers
| 2010-08-27T01:56:24 |
2024-09-04T02:49:12.493545
|
{
"license": "Public Domain",
"authors": "I-Ting Ho, Wei-Hao Wang, Glenn E. Morrison and Neal A. Miller",
"submitter": "I-Ting Ho",
"url": "https://arxiv.org/abs/1008.4630"
}
|
1008.4724
|
arxiv-papers
| 2010-08-27T14:07:05 |
2024-09-04T02:49:12.500406
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hai-Jhun Wanng",
"submitter": "Hai-Jhun Wanng",
"url": "https://arxiv.org/abs/1008.4724"
}
|
|
1008.4794
|
# Generalized Thermalization in an Integrable Lattice System
Amy C. Cassidy Charles W. Clark Joint Quantum Institute, National Institute
of Standards and Technology, Gaithersburg, MD 20899, USA Marcos Rigol
Department of Physics, Georgetown University, Washington, DC 20057, USA
###### Abstract
After a quench, observables in an integrable system may not relax to the
standard thermal values, but can relax to the ones predicted by the
Generalized Gibbs Ensemble (GGE) [M. Rigol et al., PRL 98, 050405 (2007)]. The
GGE has been shown to accurately describe observables in various one-
dimensional integrable systems, but the origin of its success is not fully
understood. Here we introduce a microcanonical version of the GGE and provide
a justification of the GGE based on a generalized interpretation of the
eigenstate thermalization hypothesis, which was previously introduced to
explain thermalization of nonintegrable systems. We study relaxation after a
quench of one-dimensional hard-core bosons in an optical lattice. Exact
numerical calculations for up to 10 particles on 50 lattice sites ($\approx
10^{10}$ eigenstates) validate our approach.
###### pacs:
02.30.Ik,03.75.Kk,05.30.Jp,67.85.Hj
Once only of theoretical interest, integrable models of one-dimensional (1D)
quantum many-body systems can now be realized with ultracold atoms paredes04 ;
*kinoshita04. The possibility of controlling the effective dimensionality and
the degree of isolation have allowed access to the quasi-1D regime and to the
long coherence times necessary to realize integrable models. Additionally,
advances in the cooling and trapping of atoms have led to increased interest
in dynamics following quantum quenches, where a many-body system in
equilibrium is exposed to rapid changes in the confining potential or
interparticle interactions.
In general, in integrable quantum systems that are far from equilibrium,
observables cannot relax to the usual thermal state predictions because they
are constrained by the non-trivial set of conserved quantities that make the
system integrable sutherland04 . Relaxation to non-thermal values were
recently observed in a cold-atom system close to integrability kinoshita06 .
At integrability, it is natural to describe the observables after relaxation
by an updated statistical mechanical ensemble: the generalized Gibbs ensemble
(GGE) rigol07STATa , which is constructed by maximizing the entropy subject to
the integrability constraints jaynes57a ; *jaynes57b. In recent studies of
integrable systems rigol07STATa ; cazalilla06 ; *calabrese07a; *cramer08a;
*barthel08; *fioretto2010quantum; *rigol06STATb; kollar08 , the GGE has been
found to accurately describe various observables after relaxation, but a
microscopic understanding of its origin and applicability remains elusive. In
particular, an important question remains: how is it that expectation values
after relaxation can be described by an ensemble with exponentially fewer
parameters than the size of the Hilbert space? The full dynamics are
determined by as many parameters as the size of the latter. At a microscopic
level, thermalization for non-integrable systems can be understood in terms of
the eigenstate thermalization hypothesis (ETH) deutsch91 ; *srednicki94;
rigol08STATc , which, however, breaks down as one approaches integrability
rigol09STATa ; *rigol09STATb.
This paper is devoted to the study of how generalized thermalization, in the
sense of relaxation to the predictions of the GGE, takes place in integrable
systems. Answering this question is important not merely because of its
relevance to the foundations of statistical mechanics in integrable systems,
but also because it has become necessary to understand recent experiments with
ultracold gases in quasi-1D geometries. For integrable systems, we compare the
predictions of quantum mechanics with those of various statistical ensembles.
In particular, we introduce a microcanonical version of the GGE, which we use
to show that relaxation to the GGE can be understood in terms of a generalized
view of the ETH.
We study the dynamics following an instantaneous quench of 1D hard-core bosons
on a lattice, which is fully integrable. The Hamiltonian is given by
$\hat{H}=-J\sum_{i=1}^{L-1}\left(\hat{b}_{i}^{\dagger}\hat{b}_{i+1}+\textrm{H.c.}\right)+V(\tau)\sum_{i=1}^{L}(i-L/2)^{2}\hat{n}_{i}$
(1)
where $J$ is the hopping parameter; $V(\tau)$ gives the curvature of an
additional parabolic trapping potential for atoms on a lattice with lattice
constant $a$; $\hat{b}_{i}^{\dagger}\ (\hat{b}_{i})$ is the hard-core bosonic
creation (annihilation) operator; and
$\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}$ is the number operator. In
addition to the standard commutation relations for bosons, hard-core bosons
satisfy the constraint $\hat{b}_{i}^{\dagger 2}=\hat{b}_{i}^{2}=0$, which
forbids multiple occupancy of the lattice sites. This Hamiltonian can be
mapped onto non-interacting fermions through the Jordan-Wigner transformation
jordan28 , and the many-body eigenstates can be constructed as Slater
determinants of the single-particle fermionic eigenstates rigol05HCBc .
We will focus on the behavior of the momentum distribution function,
$\langle\hat{n}_{k}\rangle=\sum_{l,m}e^{-ik(l-m)}\langle\psi|\hat{b}_{m}^{\dagger}\hat{b}_{l}|\psi\rangle/L$,
for system sizes ranging from $N=5$ bosons on $L=25$ lattice sites to $N=10$
bosons on $L=50$ lattice sites ($\approx 10^{10}$ eigenstates). Initially, we
prepare the system in the ground state $|\psi_{0}\rangle$ of a 1D lattice with
hard-wall boundary conditions and an additional harmonic potential, with
trapping strength $V=V_{0}$. At time $\tau=0$, the harmonic trap is turned
off, $V(\tau\geq 0)=0$, and the state $|\psi(\tau)\rangle$ evolves under the
influence of the final Hamiltonian. Hereafter, we refer to this state as it is
immediately after the quench as the “quenched state”. Its time evolution is
given by
$|\psi(\tau)\rangle=\sum_{\alpha}c_{\alpha}e^{-iE_{\alpha}\tau/\hbar}|\alpha\rangle,$
where $|\alpha\rangle$ are the energy eigenstates of the final Hamiltonian
with energies $E_{\alpha}$, and
$c_{\alpha}=\langle\alpha\vphantom{\psi_{0}}|\psi_{0}\vphantom{\alpha}\rangle$
are the overlaps between the eigenstates of the final Hamiltonian and the
quenched state. After relaxation, assuming the degeneracies in energy levels
are irrelevant, the expectation value of an observable is expected to be given
by the so called diagonal ensemble (DE) rigol08STATc ; kollar08 ; rigol09STATa
; *rigol09STATb
$\langle\hat{A}\rangle_{\text{DE}}=\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_{0}^{\tau}d\tau^{\prime}\langle\psi(\tau^{\prime})|\hat{A}|\psi(\tau^{\prime})\rangle=\sum_{\alpha}|c_{\alpha}|^{2}\langle\alpha|\hat{A}|\alpha\rangle.$
We have checked numerically that, despite the integrability of our model,
$n_{k}$ relaxes to the DE prediction, with small fluctuations around this
result supp_mat .
Figure 1: (a),(b) Momentum distribution of the initial state (init), diagonal
(DE), generalized microcanonical (GME), generalized Gibbs (GGE), and the
microcanonical (ME) ensembles. (c),(d) Relative difference of the GME, GGE and
ME from the DE. (e),(f) Conserved quantities, $\langle\hat{I}_{n}\rangle$, in
the quenched state (identical to the DE), GME and ME. $\langle I_{n}\rangle$
are ordered in descending occupations in the quenched state. $L=50$, $N=10$,
$\delta_{\text{ME}}=0.05J$, $\delta_{\text{GME}}=0.8$. (a),(c),(e)
$\varepsilon=0.72J$, $V_{0}=0.029J$. (b),(d),(f) $\varepsilon=1.52J$,
$V_{0}=0.125J$.
Figure 1 shows the momentum distributions, $n_{k}$, before and after the
quench, for two different initial trap strengths, which correspond to
different energies per particle, $\varepsilon$, after the quench. These
results are compared with those of various ensembles of statistical mechanics.
The microcanonical ensemble (ME) is one in which all eigenstates in the
relevant energy window have identical weights. Within the microcanonical
ensemble, the expectation value of a generic observable $A$ is
$\langle\hat{A}\rangle_{\text{ME}}=N^{-1}_{\varepsilon,\delta_{\text{ME}}}\sum_{\alpha,|\varepsilon-\varepsilon_{\alpha}|<\delta_{\text{ME}}}\langle\alpha|\hat{A}|\alpha\rangle$,
where $\delta_{\text{ME}}$ is small, but still much greater than the mean
many-body level spacing. $N_{\varepsilon,\delta_{\text{ME}}}$ is the number of
eigenstates in the energy window
$|\varepsilon-\varepsilon_{\alpha}|<\delta_{\text{ME}}$. We have checked that
the results reported here are nearly independent of the specific value of
$\delta_{\text{ME}}$. The GGE is a grand-canonical ensemble that maximizes the
entropy subject to the constraints associated with non-trivial conserved
quantities of the quenched state. The density matrix takes the form
rigol07STATa
$\hat{\rho}_{\text{GGE}}=Z_{G}^{-1}e^{-\sum\lambda_{n}\hat{I}_{n}},\qquad
Z_{G}=\text{Tr}\left[e^{-\sum\lambda_{n}\hat{I}_{n}}\right],$ (2)
where {$\hat{I}_{n}$}, $n=1,\ldots,L$, are the conserved quantities. In our
systems, these correspond to the occupation of the single-particle eigenstates
of the underlying noninteracting fermions to which hard-core bosons are
mapped, and {$\lambda_{n}$} are Lagrange multipliers fixed by the initial
conditions,
$\lambda_{n}=\ln[(1-\langle\psi_{0}|\hat{I}_{n}|\psi_{0}\rangle)/\langle\psi_{0}|\hat{I}_{n}|\psi_{0}\rangle]$
rigol07STATa . Observables within this ensemble are then computed as
$\langle\hat{A}\rangle_{\text{GGE}}=\text{Tr}\left[\hat{A}\,\hat{\rho}_{\text{GGE}}\right]$
following Ref. rigol05HCBc .
As a step towards understanding the GGE as well as developing a more accurate
description of isolated integrable systems after relaxation, we introduce a
microcanonical version of the GGE, which we call the generalized
microcanonical ensemble (GME). Like the ME, where states within a small energy
window contribute with equal weight, within the GME we assign equal weight to
all eigenstates whose values of the conserved quantities are close to the
desired values. The expectation value of a generic observable within the
generalized microcanonical ensemble is given by
$\langle\hat{A}\rangle_{\text{GME}}=\mathcal{N}^{-1}_{\\{I_{n}\\},\delta_{\text{G
ME}}}\sum_{\alpha,\delta_{\alpha}<\delta_{\text{GME}}}\langle\alpha|\hat{A}|\alpha\rangle$,
where $\sum_{\alpha,\delta_{\alpha}<\delta_{\text{GME}}}$ is a sum over
eigenstates that are within the GME window and
$\mathcal{N}_{\\{I_{n}\\},\delta_{\text{GME}}}$ is the number of states within
that window and $\delta_{\alpha}$ is a measure of the distance of eigenstate
$\alpha$ from the target distribution.
In order to construct the GME, we include eigenstates of the Hamiltonian with
a similar distribution of conserved quantities which once averaged reproduce
the values of the conserved quantities in the quenched state. This approach is
characterized by three ingredients: (i) The ordered distribution (from largest
to smallest) of the conserved quantities in the DE, $\langle
I_{n}\rangle_{\textrm{DE}}\equiv\sum_{\alpha}|c_{\alpha}|^{2}I_{n,\alpha}$ [as
in Figs. 1(e) and 1(f)], (ii) a target distribution of the nonzero expectation
values of the conserved quantities $\\{I^{*}_{n^{*}_{i}}=1\\}$, where the
values of $n^{*}_{i}$ ($i=1,\ldots,N$) are chose to describe the distribution
$I_{n}$ in a coarse grained sense nstar , and (iii) for each individual many-
body eigenstate, the distance from the target state, $\delta_{\alpha}$, which
we define as
$\delta_{\alpha}=\left[\frac{1}{N}\sum_{i=1}^{N}I_{n^{*}_{i}}(n_{i,\alpha}-n^{*}_{i})^{2}\right]^{1/2}$.
Here $n_{i,\alpha}$ ($i=1,\ldots,N$) are the single-particle states occupied
in eigenstate $\alpha$, and $I_{n^{*}_{i}}$ are the interpolated values of
$\langle I_{n}\rangle_{\textrm{DE}}$, evaluated at $n^{*}_{i}$. The definition
of $\delta_{\alpha}$ is not unique and several variants that do not change our
conclusions were also considered supp_mat .
To better visualize the differences between the results of the various
ensembles in Figs. 1(a) and 1(b), we have plotted $\Delta\langle
n_{k}\rangle_{\text{stat}}=(\langle\hat{n}_{k}\rangle_{\text{DE}}-\langle\hat{n}_{k}\rangle_{\text{stat}})/\langle\hat{n}_{k}\rangle_{\text{DE}}$,
where “stat” stands for ME, GGE, or GME in Figs. 1(c) and 1(d). For weaker
initial confinements (smaller $\varepsilon$ \- Fig. 1(c)), the GME is
practically indistinguishable from the diagonal distribution. Both the GME and
the GGE accurately capture the tails of $n_{k}$, while the thermal ensemble
does not. For tighter initial traps (greater $\varepsilon$ \- Fig. 1(d)) all
four ensembles are very similar (note the scale), suggesting that $n_{k}$ in
the final steady state is indistinguishable from that of the thermal state.
The close agreement between DE and ME results in Fig. 1(b) raises the
question: how can an integrable system thermalize, given the constraints
imposed by the complete set of conserved quantities? We conjecture that if the
values of the conserved quantities in the quenched state are similar to those
of the ME, then the latter will accurately describe observables after
relaxation. This may occur for a variety of quenches.
In Figs. 1(e) and 1(f), we plot the values of the conserved quantities in the
quenched state and compare them with the expectation values of those
quantities in different statistical ensembles. (By definition, the
distribution of conserved quantities in the DE and GGE are identical to that
of the quenched state.) Figure 1(e) shows that the microcanonical values of
the conserved quantities are clearly different from the values in the quenched
state, while in Fig. 1(f) they are very similar. This supports our conjecture
above, and demonstrates that thermalization can occur in integrable systems
for special initial conditions. Additionally, the GME reproduces the correct
distribution of the conserved quantities supporting the validity of our method
for generating it.
To quantify the above observations, and to understand what happens in the
thermodynamic limit, we have studied the difference between the predictions of
the DE and the statistical ensembles for different system sizes. We compute
the integrated relative differences, $(\Delta
n_{k})_{\text{stat}}=\sum_{k}|\langle\hat{n}_{k}\rangle_{\text{DE}}-\langle\hat{n}_{k}\rangle_{\text{stat}}|/\sum_{k}\langle\hat{n}_{k}\rangle_{\text{DE}}$,
where again “stat” stands for ME, GGE, or GME.
Figure 2: (a) $(\Delta n_{k})_{\text{ME}}$ versus energy per particle of the
quenched state. $\delta_{\text{ME}}=0.05J$. (b) $(\Delta n_{k})_{\text{GME}}$
vs $\varepsilon$, $\delta_{\text{GME}}=0.8$. (c) Integrated difference between
the conserved quantities in the quenched state and the ME, $(\Delta
I_{n})_{\text{ME}}$. (d) $(\Delta n_{k})_{\text{GGE}}$ vs $\varepsilon$.
Inset: $(\Delta n_{k})_{\text{GGE}}$ vs $L^{-0.73}$ for $\varepsilon=1.07J$,
where a fit to $(\Delta n_{k})_{\text{GGE}}=zL^{-\gamma}$ gives
$\gamma=0.73\pm 0.02$.
In Fig. 2(a), we plot $(\Delta n_{k})_{\text{ME}}$ as a function of the final
energy per particle, $\varepsilon$, for different lattice sizes, $L$. To
perform finite-size scaling, $\varepsilon$ and the filling factor
($\nu=N/L=0.2$) are held constant as $L$ changes. Figure 2(a) shows that for
$\varepsilon\lesssim 1.3J$ the difference between the $n_{k}$ in the DE and
the ME increases with increasing $L$, indicating that the difference persists
in the thermodynamic limit. For $\varepsilon\gtrsim 1.3J$, the opposite
behavior is observed. From our previous discussion, one expects that $(\Delta
n_{k})_{\text{ME}}$ should closely follow the behavior of the integrated
differences between the conserved quantities in the quenched state and the ME,
$(\Delta I_{n})_{\text{ME}}={\sum_{n}|\langle I_{n}\rangle_{\text{DE}}-\langle
I_{n}\rangle_{\text{ME}}|}/{\sum_{n}\langle I_{n}\rangle_{\text{DE}}}$. This
is seen by comparing Figs. 2(a) and 2(c), which leads us to conclude that
$n_{k}$ need not relax to the standard thermal prediction, except when
$(\Delta I_{n})_{\text{ME}}$ becomes negligible. Qualitatively similar results
were obtained in the canonical ensemble supp_mat .
On the other hand, in Fig. 2(b) one can see that the differences between
$n_{k}$ in the diagonal and generalized microcanonical ensembles are very
small and decrease with increasing system size, so that the former
successfully describes this observable after relaxation. In the case of the
GGE [Fig. 2(d)], $(\Delta n_{k})_{\text{GGE}}$ is in general larger than
$(\Delta n_{k})_{\text{GME}}$, which is to be expected since the GGE is a
grand-canonical ensemble. As the system size increases $(\Delta
n_{k})_{\text{GGE}}\rightarrow 0$ as $L^{-\gamma}$, where $\gamma\approx 0.73$
[inset of Fig. 2(d)] and slightly depends on the energy supp_mat .
The question that remains to be answered is why the generalized Gibbs and the
generalized microcanonical ensemble are able to describe the $n_{k}$ after
relaxation, i.e., why
$\langle\hat{n}_{k}\rangle_{\text{GGE}}=\langle\hat{n}_{k}\rangle_{\text{GME}}=\langle\hat{n}_{k}\rangle_{\text{DE}}\equiv\sum_{\alpha}|c_{\alpha}|^{2}\langle\alpha|\hat{n}_{k}|\alpha\rangle$.
Note that whereas $\langle\hat{n}_{k}\rangle_{\text{GGE}}$ and
$\langle\hat{n}_{k}\rangle_{\text{GME}}$ are entirely determined by the $L$
independent values of the conserved quantities in the quenched state,
$\langle\hat{n}_{k}\rangle_{\text{DE}}$ is determined by the exponentially
larger $\binom{L}{N}$ values of the coefficients $c_{\alpha}$.
To address this question, we perform a spectral decomposition of
$\langle\hat{n}_{k}\rangle_{\text{DE}}$ and
$\langle\hat{n}_{k}\rangle_{\text{GME}}$. Figure 3 displays a coarse grained
view of the weight which eigenstates with a given zero momentum occupancy
$\langle\hat{n}_{k=0}\rangle_{\alpha}=\langle\alpha|\hat{n}_{k=0}|\alpha\rangle$
contribute to the DE [Fig. 3(a)] and the GME [Fig. 3(b)]. The correlation
between the results in both figures is apparent. However, it is not clear why
the details contained in the overlaps $c_{\alpha}$ are completely washed out
so that the DE and the GME results coincide, while they are different from
those in the ME. In the inset of Fig. 3(a), we plot a histogram of the values
of $n_{k=0}$ for the DE, GME and the ME. Clearly the histograms for the DE and
GME have a similar mean but different widths, while the ME has a different
mean and width supp_mat .
Figure 3: Density plot of the coarse-grained weights with which eigenstates
contribute to (a) the DE (sum over diagonal weights, $|c_{\alpha}|^{2})$ and
(b) the GME (fractional number of states = number of states/total number of
states) as a function of eigenstate energy and
$\langle\hat{n}_{k=0}\rangle_{\alpha}$. The sums are performed over window of
width $\delta n_{k}=0.0067$ , and $\delta\varepsilon=0.035J$. The horizontal
and vertical dotted lines are the expectation values of $\hat{n}_{k=0}$ and
$\varepsilon$ in each ensemble. $L=45$, $N=9$, $V_{0}=0.036J$,
$\varepsilon=0.72J$, $\delta_{\text{GME}}=0.85$. Inset in (a): Histogram of DE
weights (green), fractional number of GME states (blue) and fractional number
of ME states (black) summed over all energies. Bin width, $\delta
n_{k}=0.0067$. Vertical lines give the mean, $\langle\hat{n}_{k=0}\rangle$
within each ensemble. Inset in (b): Fluctuations of
$\langle\hat{n}_{ka=0}\rangle(\bullet)$ and
$\langle\hat{n}_{ka=2\pi/5}\rangle(\blacktriangle)$ within the DE (green) and
GME (blue) as a function of inverse system size. $\varepsilon=0.72J$.
Ultimately, one is interested in what happens in the thermodynamic limit. For
each $k$, we define the width of the distribution of
$\langle\hat{n}_{k}\rangle_{\alpha}$ for each ensemble as
$\sigma_{k}=\sqrt{\langle\hat{n}_{k}^{2}\rangle-\langle\hat{n}_{k}\rangle^{2}}$.
The inset of Fig. 3(b), shows $\sigma_{k}$ within the DE and the GME versus
$L^{-1}$. The scaling is depicted for two $k$ values and clearly shows that
the widths of both distributions vanish in the thermodynamic limit. This
demonstrates that the overwhelming majority of the states selected by the DE
as well as by the GME, which have similar values of the conserved quantities,
have identical expectation values of $n_{k}$. This is why details of the
distribution of $c_{\alpha}$ no longer matter as $L$ increases. We note that
with increasing $L$, the number of eigenstates contained in the generalized
microcanonical window increases exponentially, however, the ratio of the
number of states in the GME and the ME vanishes supp_mat .
The findings above provide a generalization of the ETH introduced previously
to understand thermalization in nonintegrable systems deutsch91 ;
*srednicki94; rigol08STATc . The ETH states that the expectation values of
few-body observables in generic systems do not fluctuate between eigenstates
that are close in energy. Thus all eigenstates within a microcanonical window
have essentially the same expectation values of the observables, and one can
say that thermalization occurs at the level of eigenstates. As seen in Fig. 3,
$\langle\hat{n}_{k}\rangle_{\alpha}$ exhibits large eigenstate-to-eigenstate
fluctuations in our integrable system, showing that ETH is invalid. However,
by selecting eigenstates with similar conserved quantities, ETH is restored,
although in a weaker sense: the overwhelming majority of eigenstates with
similar conserved quantities have similar values of $n_{k}$. These results
pave the way to a unified understanding of thermalization in generic
(nonintegrable systems) and its generalization in integrable systems. This
opens many new questions, such as whether the concepts of typicality tasaki98
; *goldstein06; *popescu06; *reimann08 and thermodynamics
polkovnikov08microscopic ; *polkovnikov08 can be generalized to isolated
integrable systems.
###### Acknowledgements.
This work was supported by NSF under Physics Frontier Grant No. PHY-0822671.
A.C.C. acknowledges support from NRC/NIST and M.R. acknowledges support from
the Office of Naval Research. We thank V. Dunjko, L. Mathey, and M. Olshanii
for helpful discussions.
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Supplementary material for EPAPS
Generalized Thermalization in an Integrable Lattice System
Amy C. Cassidy,1 Charles W. Clark,1 Marcos Rigol2
1Joint Quantum Institute, National Institute of Standards and Technology,
Gaithersburg, MD 20899, USA
2Department of Physics, Georgetown University, Washington, DC 20057, USA
Time Dynamics. In order to check that the DE accurately describes observables
after relaxation, we compare the time dynamics of the central momentum peak,
$\langle\hat{n}_{k=0}(\tau)\rangle=\langle\psi(\tau)|\hat{n}_{k=0}|\psi(\tau)\rangle$,
with the expectation value of $\hat{n}_{k=0}$ in the diagonal ensemble. Figure
1(a) shows that $\langle\hat{n}_{k=0}(\tau)\rangle$ relaxes to the diagonal
prediction, with small fluctuations around this result, indicating that the
diagonal ensemble correctly predicts the values of observables after
relaxation. Additionally, the expectation value of $\hat{n}_{k=0}$ in the GME
agrees with the DE, while the ME does not.
Figure 1: (a) Time evolution of $n_{k=0}$. Horizontal lines represent
$\langle\hat{n}_{k}\rangle_{\text{DE}}$,
$\langle\hat{n}_{k}\rangle_{\text{GME}}$,
$\langle\hat{n}_{k}\rangle_{\text{ME}}$. $L=50$, $N=10$, $\varepsilon=0.72J$,
$\delta_{\text{GME}}=0.8$, $\delta_{\text{ME}}=0.05J$. Main Panel: $\tau\leq
100$. Inset: $100\leq\tau\leq 1000$. (b) Integrated difference of the diagonal
and time-averaged momentum distribution,$(\Delta n_{k})_{\tau}$, versus
$L^{-0.98}$, where a fit to $\sigma_{\tau}=zL^{-\gamma}$ gives $\gamma=0.98\pm
0.25$. (c) Mean fluctuations of the momentum distribution, $\sigma_{\tau}$,
versus $L^{-0.4}$, where a fit to $(\Delta n_{k})_{\tau}=zL^{-\gamma}$ gives
$\gamma=0.40\pm 0.12$. (b),(c)
$\tau_{1}=100J,\tau_{2}=1000J,\varepsilon=0.72J$. $z$ is a generic
multiplicative constant.
In order to understand the relation between the DE prediction and the actual
time average over a finite time interval,
$\overline{n_{k}}=\frac{1}{\tau_{2}-\tau_{1}}\int_{\tau_{1}}^{\tau_{2}}d\tau\langle\hat{n}_{k}(\tau)\rangle$,
we study the integrated difference $(\Delta
n_{k})_{\tau}=\sum_{k}|\langle\hat{n}_{k}\rangle_{\text{DE}}-\overline{n_{k}}|/\sum_{k}\langle\hat{n}_{k}\rangle_{\text{DE}}$,
and the time fluctuations
$\sigma_{\tau}=\sum_{k}\sqrt{\overline{n_{k}^{2}}-\overline{n_{k}}^{2}}$ as a
function of system size. $(\Delta n_{k})_{\tau}$ is depicted in Fig. 1(b),
where we also show the results of a fit to $(\Delta
n_{k})_{\tau}=zL^{-\gamma}$ with $\gamma=0.98\pm 0.25$ ($z$ is a generic
multiplicative constant). The integrated difference, $(\Delta n_{k})_{\tau}$,
clearly decreases with increasing system size and is several orders of
magnitude smaller than similar comparisons with the statistical ensembles,
confirming that degeneracies are irrelevant and the diagonal distribution
accurately represents the long-time average. Figure 1(c) depicts the time
fluctuations of the momentum distribution, $\sigma_{\tau}$, where the exponent
was determined from fitting $\sigma_{\tau}=zL^{-\gamma}$. This plots makes
evident that the fluctuations also decrease with increasing system size
(likely $\propto 1/\sqrt{L}$) and are expected to vanish in the thermodynamic
limit.
Figure 2: (a) Distribution of conserved quantities in the diagonal ensemble,
$\langle I_{n}\rangle_{\textrm{DE}}$, ordered by energy of the single-particle
fermionic eigenstates. (b) $\langle I_{n}\rangle_{\textrm{DE}}$ sorted in
descending order. The target distribution of $n_{i}^{\ast}$ are labeled as
well as the corresponding $I_{n_{i}^{\ast}}$ for $i=1,2$. (c) Target
distribution of conserved quantities, $I_{n}^{\ast}$, and distribution of
eigenstate that minimized the distance $\delta_{\alpha}$ used to generate the
GME, $I_{n,\alpha}$. (d) $\langle I_{n}\rangle$ in the DE and GME using method
described in text as well as the standard metric (GME-std).
$L=20,N=4,\varepsilon=0.72J$.
Constructing the generalized microcanonical ensemble. We expand upon our
method for generating the generalized microcanonical ensemble with a series of
plots in Fig. 2. Constructing the generalized microcanonical ensemble for a
system of 1D hard-core bosons presents some unique challenges. In particular,
the set of conserved quantities are the single-particle occupations of the
underlying fermions used to build the many body eigenstates. Thus in any given
eigenstate, each conserved quantity is either $0$ or $1$ corresponding to
whether or not that fermionic state is occupied, such as $I_{n,\alpha}$ in
Fig. 2(c). On the other hand, the distribution in the diagonal ensemble is a
continuous variable, $0\leq\langle I_{n}\rangle_{\textrm{DE}}\leq 1$. For each
individual eigenstate, we must include or exclude it from the GME based on how
close the discrete distribution in the eigenstate is to the continuous
distribution of the quenched state. One possible method is a simple extension
of the microcanonical metric
$\delta^{\prime}_{\alpha}=\left[\sum_{n=1}^{L}(I_{n}^{\ast}-I_{n,\alpha})^{2}/\sum_{n}I_{n,\alpha}^{2}\right]^{1/2}.$
We found this was not guaranteed to accurately describe the distribution of
conserved quantities, particularly for low energies for the system sizes
studied (see Fig. 2(d)). Instead of using the standard metric, we perform a
weighted least-squares fit to a target distribution which is described as
follows.
The unsorted conserved quantities of the quenched state $\langle
I_{n}\rangle_{\textrm{DE}}$, ordered by energy of the single-particle
fermionic eigenstates, are plotted in Fig. 2(a). As a first step, the
distribution of $\langle I_{n}\rangle_{\textrm{DE}}$ is sorted in descending
order as shown in Fig 2(b). Next the values of $n_{i}^{\ast}$, which are not
limited to integer values, are chosen so that
$\int_{0.5}^{n_{1}^{\ast}}I(x)=0.5$, where $I(x)=\langle
I_{n}\rangle_{\text{DE}}$ for $x$ in the interval $(n-0.5,n+0.5]$.
Subsequently the $n_{i}^{\ast}$ are determined so
$\int_{n_{i-1}^{\ast}}^{n_{i}^{\ast}}I(x)=1$ as depicted in Fig. 2(b). This
set of $\\{n_{i}^{\ast}\\}$ determine the target distribution, labeled
$I_{n}^{\ast}$ in Fig. 2(c). Each $n_{i}^{\ast}$ is assigned a weight,
$I_{n_{i}^{\ast}}$, which is the interpolated value of $\langle
I_{n}\rangle_{\textrm{DE}}$ at $n_{i}^{\ast}$. The distance between the
distribution of conserved quantities in eigenstate $\alpha$ and the
distribution of the quenched state is then defined as
$\delta_{\alpha}=\left[\frac{1}{N}\sum_{i=1}^{N}I_{n^{*}_{i}}(n_{i,\alpha}-n^{*}_{i})^{2}\right]^{1/2},$
(1)
where the $n_{i,\alpha}$ are the single particle eigenstates that are occupied
in the many-body eigenstate $\alpha$.
Our method of constructing the GME is not unique. We tested various different
approaches, including an unweighted sum in Eq. (1), etc, which gave similar
results. We then employed the method which best reconstructed the distribution
of conserved quantities over the full range of parameters studied. Note that
for clarity Fig. 2 displays data for the case of $N=4$ particles on $L=20$
sites. There are only 4845 total eigenstates and 174 eigenstates used to
construct the GME for the data shown. Even for such a small system, the
distribution of conserved quantities within the GME is quite good.
## Lagrange multipliers and additivity.
In Fig. 3, we plot the expectation values of the conserved quantities in the
diagonal ensemble along with corresponding Lagrange multipliers. The Lagrange
multipliers are given by $\lambda_{n}=\ln\left(\frac{1-\langle
I_{n}\rangle_{\textrm{DE}}}{\langle I_{n}\rangle_{\textrm{DE}}}\right)$. As
can be seen, the Lagrange multipliers vary smoothly with the value of $I_{n}$
and with the index $n$ after the conserved quantities have been ordered in
descending order. Additionally, the distribution of conserved quantities is
very similar for two different lattice sizes with the same final energy-per-
particle, when the index is normalized by the total number of conserved
quantities. This is an important property of the Lagrange multipliers, which
shows that even though the conserved quantities in these integrable systems
are not additive in a strict sense, they can still be understood to be
additive in a coarse grained sense, because the values of $\lambda_{n}$ are a
smooth function of $I_{n}$.
Figure 3: Expectation values of the conserved quantities in the diagonal
ensemble and corresponding Lagrange multipliers for $N=5$, $L=25$ and $N=10$,
$L=50$. $\varepsilon=0.72t$.
Canonical Ensemble. In addition to the results reported for the microcanonical
distribution, we study the momentum distribution in the canonical ensemble
(CE). Within the canonical ensemble, observables are calculated as
$\langle\hat{A}\rangle_{\text{CE}}=Z_{\text{CE}}^{-1}\text{Tr}\left[\hat{A}e^{-\beta\hat{H}}\right]$,
where $Z_{\text{CE}}=\text{Tr}\left[e^{-\beta\hat{H}}\right]$ and $\beta$ is
the inverse temperature. The temperature is calculated numerically so that
$\langle E\rangle_{\text{CE}}=\langle\psi_{0}|\hat{H}(\tau\geq
0)|\psi_{0}\rangle$. In Fig. 4(a) we plot the integrated difference between
the momentum distribution in the DE and CE, $(\Delta n_{k})_{\text{CE}}$, for
different lattice sizes as a function of the energy per particle of the
quenched state. The comparison of the canonical and diagonal momentum
distributions are similar to the comparison between the microcanonical and
diagonal distributions, although there is some discrepancy between the
microcanonical and canonical distributions due to finite size effects. In
particular, the upturn in $(\Delta n_{k})_{\text{ME}}$ at $\varepsilon=1.4J$
is not present in $(\Delta n_{k})_{\text{CE}}$.
Figure 4: (a) Integrated difference between the diagonal and canonical
momentum distributions, $\left(\Delta n_{k}\right)_{\text{CE}}$, versus energy
per particle, $\varepsilon$. (b) Integrated difference of the momentum in the
microcanonical and canonical ensembles, $\left(\Delta n_{k}\right)_{\text{CE-
ME}}$, versus $L^{-1.19}$ for $\varepsilon=1.07J$, where a fit to
$\left(\Delta n_{k}\right)_{\text{CE-ME}}=zL^{-\gamma}$ gives $\gamma=1.0\pm
0.07$. Inset: Scaling exponent, $\gamma$ of $\left(\Delta
n_{k}\right)_{\text{CE-ME}}$ as a function of the energy per particle,
$\varepsilon$.
In general, the microcanonical distribution provides us with a better
description of the system because it is isolated, although it breaks down for
very small systems because of poor statistics due to an insufficient number of
states in the relevant energy window. Given the expectation that
$\langle\hat{n}_{k}\rangle_{\text{CE}}$ and
$\langle\hat{n}_{k}\rangle_{\text{ME}}$ are equal in the thermodynamic limit,
we calculate the scaling exponent of the integrated difference of the momentum
distribution, $\left(\Delta n_{k}\right)_{\text{CE-ME}}=zL^{-\gamma}$, where
$\left(\Delta n_{k}\right)_{\text{CE-
ME}}=\sum_{k}|\langle\hat{n}_{k}\rangle_{\text{CE}}-\langle\hat{n}_{k}\rangle_{\text{ME}}|/\sum_{k}\langle\hat{n}_{k}\rangle_{\text{CE}}$.
In Fig. 4(b) we plot $\left(\Delta n_{k}\right)_{\text{CE-ME}}$ versus
$L^{-\gamma}$ for energy $\varepsilon=1.31J$, where $\gamma=1.0$. In the
inset, the exponent $\gamma$ is plotted as a function of the energy per
particle. The mean value is $\gamma=1.03\pm 0.04$, which is consistent with
$L^{-1}$ scaling for the difference between the canonical and grand canonical
momentum distributions for hard-core bosons in a box found in Ref. rigol05HCBc
.
On general grounds, the canonical and microcanonical ensembles are expected to
equivalent in the thermodynamic limit, which is confirmed by our numerical
results. We also see increasing agreement between the results of the
generalized Gibbs and generalized microcanonical ensembles as system sizes
increase. There has been significant work in recent years on the equivalence
of classical ensembles and less in quantum systems. In the classical case,
Ellis et al. have demonstrated that the microcanonical and canonical ensembles
are equivalent if and only if the thermodynamic functions are equivalent,
which is the case if the microcanonical entropy is concave ellis2000large .
Furthermore, when the two ensemble are not equivalent, it may be possible to
construct a generalized canonical ensemble, which contains an additional
exponential which is a continuous function of the Hamiltonian, that is
equivalent to the microcanonical ensemble costeniuc2005generalized .
Scaling of the GGE. Given the trend observed in our results, and similar
calculations in rigol07STATa for much larger system sizes, we expect the
integrated difference between the momentum distribution in the DE and GGE,
$(\Delta n_{k})_{\text{GGE}}$, to vanish in the thermodynamic limit. We fitted
$(\Delta n_{k})_{\text{GGE}}=zL^{-\gamma}$ and report the results for
$\varepsilon=1.13J$ in the main text. In Fig. 5 we plot the exponent,
$\gamma$, as a function of the energy per particle of the quenched state for
all energies studied. The mean value of the exponent is $\gamma=0.731\pm
0.007$.
Figure 5: Exponent of power-law scaling of $(\Delta n_{k})_{\text{GGE}}$
versus inverse system size as a function of the energy per particle.
Number of States. The total number of states in our system, the number of
states in the ME, and the number of states in the GME all scale exponentially
with the system size as $z^{L}$. For the total number of states,
$L!/\left[N!(L-N)!\right]$, using Stirling’s approximation,
$z\approx(1-\nu)^{\nu-1}\nu^{-\nu}\approx 1.65$, where $\nu$ is the filling
factor. We confirm this scaling by numerically fitting to the total number of
states. For $\varepsilon=0.72J$, we find numerically that $z\approx 1.51$ in
the ME, ($\delta_{ME}=0.05J$) and $z\approx 1.37$ in the GME
($\delta_{GME}=0.8$). Thus the number of states in the ME and GME windows as a
the fraction of the total number of states vanishes in the thermodynamic
limit. Additionally the ratio of states in GME to ME vanishes. These values
are typical, although the precise exponents depend on the choice of the ME and
GME window sizes.
Figure 6: (a) Average density fluctuations, $\sigma_{\text{den}}$ versus
$L^{-0.5}$ for fixed energy per particle in the diagonal (DE), generalized
microcanonical (GME) and microcanonical (ME) ensembles. (b) Momentum
fluctuations, $\sigma_{k}$ for $ka=0$. $\varepsilon=0.72J$,
$\delta_{\text{ME}}=0.05J,\delta_{\text{GME}}=0.8.$
Fluctuations of Local Observables. For local observables, the fluctuation of
the eigenstate to eigenstate expectation values are expected to scale as
$L^{-1/2}$ biroli2009 . In Fig. 6(a) we plot the fluctuations of the site
occupations averaged over five sites,
$\sigma_{\text{den}}=1/5\sum_{i=1}^{5}\sqrt{\langle\hat{n}_{x=iN}^{2}\rangle-\langle\hat{n}_{x=iN}\rangle^{2}}$
vs. $L^{-0.5}$ in the DE, GME, and ME distributions along with a linear fit to
$\sigma_{\text{den}}$ versus $L^{-0.5}$. The data strongly suggests that the
fluctuations scale as $L^{-0.5}$ as predicted in Ref. biroli2009 and will
vanish in the thermodynamic limit for all three ensembles.
In the main text, we presented evidence that the fluctuations of the
eigenstate to eigenstate expectation value of the momenta occupations vanish
in the thermodynamic limit for the DE and GME. We have also studied how the
width of the distribution of the momenta eigenstate expectation values,
$\sigma_{k}$, scales for the ME. In Fig. 6(b), we plot the fluctuations of
$\hat{n}_{ka=0}$. For the microcanonical ensemble it is difficult to reach a
conclusion as to whether it vanishes or remains finite in the thermodynamic
limit. We do, however, find that for this observable the results are clearly
different from those for observables that only contain short range
correlations, and that the behavior of the width of its distribution within
the ME behaves quite differently from the one within the DE and the GME. In
the latter two the width decreases much rapidly with system size.
## References
* (1) M. Rigol, Phys. Rev. A 72, 063607 (2005).
* (2) R. Ellis, K. Haven, and B. Turkington, J. Stat. Phys. 101, 999 (2000).
* (3) M. Costeniuc, R. Ellis, H. Touchette, and B. Turkington, J. Stat. Phys. 119, 1283 (2005).
* (4) M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007).
* (5) G. Biroli, C. Kollath and A. Laeuchli, ArXiv/0907.3731.
|
arxiv-papers
| 2010-08-27T20:00:04 |
2024-09-04T02:49:12.504899
|
{
"license": "Public Domain",
"authors": "Amy C. Cassidy, Charles W. Clark, Marcos Rigol",
"submitter": "Marcos Rigol",
"url": "https://arxiv.org/abs/1008.4794"
}
|
1008.4814
|
Exploring connectivity of random
subgraphs of a graph.
Connectivity of random subgraphs of Cartesian products of $K_{2}$, $K_{3}$,
and $P_{3}$. A survey of uniformly most reliable networks.
BEHRANG MAHJANI
Supervisor: Jeffrey Steif
Department of Mathematical Sciences
CHALMERS UNIVERSITY OF TECHNOLOGY
Division of Engineering Mathematics
Göteborg, Sweden, 2010
Exploring connectivity of random subgraphs of a graph.
Connectivity of random subgraphs of Cartesian products of $K_{2}$, $K_{3}$,
and $P_{3}$. A survey of uniformly most reliable networks.
BEHRANG MAHJANI
©BEHRANG MAHJANI, 2010.
Department of Mathematical Sciences
Chalmers University of Technology
SE-412 96 Göteborg
Sweden
Telephone + 46 (0)31-772 1000
Exploring connectivity of random subgraphs of a graph
Connectivity of random subgraphs of Cartesian products of $K_{2}$, $K_{3}$,
and $P_{3}$. A survey of uniformly most reliable networks.
BEHRANG MAHJANI
Department of Mathematical Sciences
Chalmers University of Technology
Abstract
This work is divided into two main parts. The first part is devoted to
exploring the connectivity of random subgraphs of cartesian products of
$K_{1}$, $K_{2}$, and $P_{3}$. In the second part, the author presents a short
review of the results about network reliability.
The cartesian product of $K_{2}$, the complete graph with 2 vertices, is the
cube graph $Q^{n}$. A random subgraph of $Q^{n}$, $Q^{n}_{p_{n}}$, contains
all vertices of $Q^{n}$, and each edge of $Q^{n}$ independently with
probability $p_{n}$. One can call $p_{n}$ the percolation parameter. The
author explains in detail that for $p_{n}\geq 1-(1/2)(\log n)^{1/n}$,
$Q^{n}_{p_{n}}$ has no components with size larger than $1$ and smaller than
$2^{n}$, as $n\rightarrow\infty$. It is also explained that for
$p_{n}=1-(1/2)\lambda^{1/n}(1+o(1/n))$, the probability that $Q_{p_{n}}^{n}$
has no isolated point, as $n\rightarrow\infty$, tends to $e^{-\lambda}$;
hence, the probability that $Q_{p_{n}}^{n}$ is connected tends to
$e^{-\lambda}$. For constant percolation values larger than $1/2$, when $n$
tends to infinity, almost every random subgraph of $Q^{n}$ is connected; for
percolation values smaller than $1/2$, when $n$ tends to infinity, almost no
random subgraph of $Q^{n}$ is connected; and for percolation values equal
$1/2$, when $n$ tends to infinity, the probability that $Q^{n}_{1/2}$ is
connected tends to $e^{-1}$. At the end of this section, a comparison between
connectivity of a typical random graph with $M$ edges and $N$ vertices, $G\in
G(N=2^{n},M=n2^{n-1})$, and $Q^{n}$, after percolation with the parameter
$p_{n}$ is presented.
This work continues with exploring the threshold function for the cartesian
product of $K_{3}$, the complete graph with 3 vertices, denoted by
${}^{3}Q^{n}$. It is shown that for $p_{n}\geq 1-(1/\sqrt{3})(\log n)^{1/n}$,
${}^{3}Q_{p_{n}}^{n}$ has no components with size larger than $1$ and smaller
than $3^{n}$, as $n\rightarrow\infty$. Then, it is proved that for
$p_{n}=1-(1/\sqrt{3})\lambda^{1/2n}(1+o(1/n))$, the probability that
${}^{3}Q_{p_{n}}^{n}$ has no isolated point, as $n\rightarrow\infty$, tends to
$e^{-\lambda}$; hence, the probability that $Q_{p_{n}}^{n}$ is connected tends
to $e^{-\lambda}$. At last, the author suggests that the threshold value for
connectivity of the cartesian product of $P_{3}$, where $P_{3}$ is a path with
length $2$, denoted by $P_{3}^{n}$, is $2-\sqrt{2}$. One can show that for
percolation values smaller than $2-\sqrt{2}$, almost no random subgraph of
$P_{3}^{n}$ is connected, and for percolation values larger than $2-\sqrt{2}$,
almost every random subgraph of $P_{3}^{n}$ has no isolated point. The author
also shows that for percolation values larger than $0.68$ almost all random
subgraphs of $P_{3}^{n}$ are connected.
The last part of this work, sheds light on reliability of networks. The main
question in this part is: one is given 2 parameters, $n$ and $m$ where $n$ and
$m$ are positive integers. Among all graphs with $n$ vertices and $m$ edges,
which graph $G$, if any, maximizes the probability that when one does
percolation on $G$ with the parameter $p_{n}$, for all $p_{n}$ in $(0,1)$
there is one component? $G$ would be called the uniformly optimally reliable
graph (UOR graph) for the parameter $n$ and $m$. It is shown in this part, for
some $m$ and $n$ there is no UOR graph, since the graph which maximizes the
probability of connectivity depends on $p_{n}$ in that family of graphs. A
review of results about when the UOR graph exists is presented in this part.
Keywords: Random subgraphs, percolation, n-cube, path graph, reliable
networks.
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Professor Jeffrey Steif for
his supervision and guidance in this thesis. His invaluable comments and
suggestions were of enormous help during this research work. My special thanks
are due to my parents who supported me and made it possible for me to continue
my studies.
###### Contents
1. 1 Introduction
2. 2 Graph theory background
1. 2.1 Graph models and their matrix representation
2. 2.2 Connectivity
3. 3 Random graph theory background
1. 3.1 Evolution of Erdõs Rényi graphs
2. 3.2 Properties of almost every graphs
3. 3.3 Probabilistic methods
4. 4 Connected random subgraphs of the cube
1. 4.1 Isolated vertices
2. 4.2 Isoperimetric problem for the cube
3. 4.3 Isolated components of size larger than 2 and smaller than $2^{n}$
4. 4.4 Comparison between a typical random graph and $Q^{n}$
5. 5 Connected random subgraphs of the 3-cube
1. 5.1 Isolated vertices
2. 5.2 Isoperimetric problem for the 3-cube
3. 5.3 Isolated components of size larger than 2 and smaller than $3^{n}$
6. 6 Connected random subgraph of the $P_{3}$-product
1. 6.1 Isolated vertices
2. 6.2 Isolated components of size larger than 2 and smaller than $3^{n}$
7. 7 Reliable networks
1. 7.1 All terminal reliability
2. 7.2 Random accessibility
8. References
## Chapter 1 Introduction
Exploring the connectivity of random subgraphs of different families of graphs
is one of the most interesting topics in random graphs and percolation theory.
A random subgraph of a graph $G(V_{n},E_{m})$ is a graph which contains all
vertices of $G$, and each edge of $G$ independently with probability $p_{n}$.
$p_{n}$ is known as the percolation parameter. Connectivity of a random
subgraph of a graph can be investigated both for small and considerably large
(when $n$ tends to infinity) graphs. For considerably large graphs, the first
step in exploring the connectivity is to calculate $p_{c}$ which for all
constant $p=p_{n},p\in(0,1)$ and $p<p_{c}$, as $n$ tends to infinity, almost
all random subgraphs of $G(V_{n},E_{m})$ is connected; but for all $p\in(0,1)$
and $p>p_{c}$, as $n$ tends to infinity, almost no random subgraphs of
$G(V_{n},E_{m})$ is connected. The second step is to investigate what happens
when $p=p_{c}$. A more complete approach is to calculate $p_{c}$ when $p_{c}$
depends on $n$.
For small graphs, it is of interest to find the uniformly optimally reliable
graph (UOR graph). Consider $G(n,m)$ as the family of graphs with $n$ vertices
and $m$ edges. The UOR graph is the graph $G\in G(n,m)$ that maximizes the
probability that $G$ is connected after percolation with the parameter $p_{n}$
for fixed $n,m$ and all $p_{n}\in(0,1)$.
One of the interesting graphs for analyzing the connectivity of its random
subgraphs is the cube graph. The cube graph $Q^{n}$, is a graph with the
vertices labeling $1,2,3,4,...,2^{n}-1$. Two vertices in this graph are
adjacent if their binary representation differs only in one digit. Another way
to define $Q^{n}$ is using the cartesian products of $n$ copies of $K_{2}$,
where $K_{n}$ is the complete graph with $n$ vertices. It is shown by Paul
Erdös and Joel Spencer [4] that $p_{c}=1/2$ for $Q^{n}$. An extension of
$Q^{n}$ is the graph with the vertices labeling $1,2,3,4,...,3^{n}-1$, where
two vertices in this graph are adjacent if their ternary representation
differs only in one digit. We call this graph 3-cube denoted by ${}^{3}Q^{n}$.
One can show that ${}^{3}Q^{n}$ is the cartesian product of $K_{3}$.
Connectivity of random subgraphs of the cartesian product of $K_{i}$ is
investigated by Lane Clark [15]. Another extension of $Q^{n}$ is the cartesian
product of $n$ copies of a path with length 2 which we call it $P_{3}^{n}$.
This thesis is divided into two main parts. The first part (chapter 4,5,6) is
devoted to the connectivity of random subgraphs of some considerably large
graphs, and the second part is the connectivity of random subgraphs of small
graphs (chapter 7). Chapter 2 presents a very short review of definition and
results in graph theory. Chapter 3 is a short review of the definitions in
random graph theory; it is explained briefly in this chapter that how small
components construct a giant component and gradually a graph becomes connected
by adding more edges to it. In chapter $4$, the results by Bela Bollobás [1]
on finding $p_{c}$ for connectivity of random subgraphs of $Q^{n}$ is
explained in detail. In chapter $5$ the author calculates $p_{c}$ for
connectivity of random subgraphs of ${}^{3}Q^{n}$. After calculating the
threshold value for the connectivity of random subgraphs of ${}^{3}Q^{n}$, the
author found that this problem is solved for a general case of the random
subgraphs of cartesian product of $K_{i}$ [15]. Chapter $6$ is an approach to
find $p_{c}$ for connectivity of random subgraphs of $P_{3}^{n}$. This work
finishes with chapter $7$ which is a review of the results on finding the UOR
graph. In this chapter it is shown by the author that for some $m$ and $n$
there are no UOR graph.
## Chapter 2 Graph theory background
This chapter presents s short review of basic definitions in graph theory.
Most of the definitions in this chapter are extracted from [10].
### 2.1 Graph models and their matrix representation
#### Graphs
###### Definition 1.
Graph: A _graph_ $G(V,E)$ is an ordered pair consisting of the set of
_vertices_ $V$, and the set of _edges_ $E$. Each edge is associated with a set
of vertices which are called _endpoints_. Two vertices are _adjacent_ if they
are joined by an edge. Two edges are _adjacent_ if they have a common
endpoint. A vertex is _incident_ to an edge and viceversa, if that vertex is
an endpoint of the edge. A _self-loop_ is an edge which joins a vertex to
itself. A _multi-edge_ is a set of two or more edges having the same
endpoints. A _simple graph_ is a graph without self-loops and multi-edges.
#### Degrees
After defining a graph, it is of interest to get familiar with the
characteristics of different graphs in order to compare them. One of the basic
characteristics is the degree of each vertices.
###### Definition 2.
Degree: The _degree_ of a vertex, denoted by $deg(v)$, is the number of edges
incident on that vertex plus two times the number of its self-loops. The
_smallest degree_ in a graph is denoted by $\delta_{min}$ or $\delta$, and the
_largest degree_ in a graph is denoted by $\delta_{max}$ or $\Delta$. The
_degree sequence_ of a graph is the non-increasing sequence of vertice
degrees.
The first question that comes into mind, after defining the degree sequence of
a graph, is if there exists a degree sequence of a graph for each sequence of
positive integers.
###### Definition 3.
Graphic: A sequence of positive integers is _graphic_ if there is a
permutation of it that is the degree sequence of a simple graph. An explicit
sufficient and necessary condition for a sequence of positive integers to be
graphical is:
###### Theorem 1.
A sequence of non-negative integers $(d_{1},d_{2},...,d_{n})$ is graphical if
and only if
$\displaystyle\sum_{i=1}^{k}deg(i)\leq k(k-1)+\sum_{j=k+1}^{n}min(k,deg(i))$
(2.1)
for each $1\leq k\leq n$ [9].
#### Graph models
There are many types of graphs. Some of the most important types of simple
graphs are:
###### Definition 4.
Common families of graphs: A _complete graph_ $K_{n}$ is a simple graph with
$n$ vertices which every pair of vertices is connected by an edge. A
_bipartite graph_ $G$ is a graph with the set of vertices that can be
partitioned into two subsets $U$ and $W$, such that each edge in $G$ has one
endpoint in $U$ and one endpoint in $W$. A _regular graph_ is a graph where
all vertices have the same degree. A path graph is a simple graph with
$|V|=|E|+1$ that can be drawn, such that all vertices and edges are in a
single straight line. A _path graph_ with $|V|=n$ and $|E|=n-1$ is denoted by
$P_{n}$. A _hypercube graph (cube graph)_ is a simple n-regular graph with the
set of vertices labels from $0$ to $2^{n}-1$, in which two vertices are
adjacent if their binary representation differs only in one digit.
###### Definition 5.
Subgraphs: $H$ _subgraph_ of $G$ is a graph whose vertices and edges are in
$G$. If $V_{G}=V_{H}$ then the subgraph $H$ is said to _span_ the graph $G$.
The _induced subgraph_ on $U\subseteq V_{G}$ of $G$ is the graph whose set of
vertices is $U$, and set of edges is all edges of $G$ with two endpoints in
$U$. The _induced subgraph_ on $D\subseteq E_{G}$ of $G$ is the graph whose
its set of edges is $D$, and its set of vertices is all vertices which are
incident with an edge in $D$. A maximal connected subgraph of a graph $G$ is a
_component_ of $G$.
###### Definition 6.
Cartesian product of a graph: $G\times H$, the _Cartesian product_ of $G$ and
$H$ is the graph with the set of vertices $V_{G}\times V_{H}$ and the set of
edges $(V_{G}\times E_{H})\cup(E_{G}\times V_{H})$.
Defining a walk on a graph can help us to define some important characteristic
of a graph such as connectivity of a graph and spanning trees.
###### Definition 7.
Walk: In a graph $G$, a _walk_ from vertex $v_{0}$ to vertex $v_{n}$ is an
ordered sequence
$\displaystyle W=<v_{0},e_{1},v_{1},e_{2},...,v_{n-1},e_{n},v_{n}>$ (2.2)
of vertices and edges, such that the endpoints of $e_{i}$ is
$\\{v_{i-1},v_{i}\\}$ for $i=1,...,n$. For a simple graph one can abbreviate
the representation as a vertex sequence
$\displaystyle W=<v_{0},v_{1},...,v_{n}>$ (2.3)
###### Definition 8.
Tree, spanning tree: A _path_ is a walk with no repeated vertices (except the
initial and final vertices). A _cycle_ is a nontrivial closed path. A _tree_
is a connected graph without cycle. A spanning tree of a graph is a subgraph
of a graph which is a tree.
#### Matrix representations
The last important concept in this section is that each graph can be presented
as a matrix, as follows:
###### Definition 9.
Matrix representation of a graph: The _adjacency matrix_ of a simple graph $G$
is: $A_{G}[u,v]=\left\\{\begin{array}[]{ll}1,&\hbox{if $u$ and $v$ are
adjacent;}\\\ 0,&\hbox{otherwise.}\\\ \end{array}\right.$ for all pairs of
vertices $u$ and $v$ in $V_{G}$.
### 2.2 Connectivity
Connectivity of a graph is one of the most important property of a graph. A
graph is _connected_ if for every pair of vertices $u$ and $v$, there is walk
from $u$ to $v$. There are different types of connectivity for a graph:
###### Definition 10.
Vertex-connectivity: $\kappa_{v}(G)$, _vertex connectivity_ of a connected
graph $G$, is the minimum number of vertices which its removal will disconnect
$G$ or reduce it to a single vertex graph. A graph $G$ is _k-connected_ if
$\kappa_{v}(G)\geq k$.
###### Definition 11.
Edge-connectivity:$\kappa_{e}(G)$, _edge connectivity_ of a connected graph
$G$, is the minimum number of edges which its removal will disconnect $G$. A
graph $G$ is _k-edge-connected_ if $\kappa_{e}(G)\geq k$
###### Definition 12.
Algebraic connectivity: The _Laplacian matrix_ of a graph is $L:=D-A$, where
$A$ is the adjacency matrix and the $D$ is the diagonal matrix of vertex
outdegrees. The algebraic connectivity of an undirected graph with the
Laplacian matrix $L$ is the second smallest eigenvalue of $L$. If one arranges
the eigenvalues of $L$ as :
$\lambda_{1}(L)\leq\lambda_{2}(L)\leq...\lambda_{n}(L)$, then $\lambda_{2}(L)$
is the algebraic connectivity of a graph. The following theorem presents some
applications of algebraic connectivity:
###### Theorem 2.
For an undirected graph with minimum vertex degree $\delta$ and maximum vertex
degree $\Delta$, we have:
* •
$\lambda_{2}\geq 0$ with the inequality strict if and only if the graph is
connected.
* •
$\lambda_{2}\leq\frac{n}{n-1}\delta\leq\frac{n}{n-1}\triangle\leq\lambda_{n}$.
## Chapter 3 Random graph theory background
A _random graph_ is a graph with a specific number of vertices which adjacency
between two vertices are determined in a random way [10]. In this chapter, we
define Erdõs Rényi random graph, and then we explain briefly some properties
of it.
### 3.1 Evolution of Erdõs Rényi graphs
$G(n,M),0\leq M\leq\binom{n}{2}$, is the equiprobable space of all simple
graphs with the vertex set $V=\\{1,2,...,n\\}$ and $M$ edges. $G(n,p),0<p<1$,
is the collection of all graphs with the vertex set $V=\\{1,2,...,n\\}$ in
which two vertices are connected independently with the probability $p$ [10],
[1].
It is of interest to study the global structure of a random graph of order $n$
(with $n$ vertices) and size M(n) (with $M(n)$ edges). Let us define
$L_{j}(G)$ as the order of the jth largest component of a graph $G$, where if
$G$ has fewer than j components then $L_{j}(G)=0$. Consider the random graph
process $\widetilde{G}=(G_{t})_{t=0}^{N}$ where $G_{t}$ is getting larger by
adding more and more edges. When $t\sim\frac{1}{2}cn$ and $c<1$ then in a.e
$G_{t}$ the maximum of the order of its components is of order $\log n$. When
$c=1$, in a.e $G_{t}$ $L_{1}(G_{\lfloor n/2\rfloor})$ has order $n^{2/3}$.
When $t$ passes $n/2$, $L_{1}(G)$ begins to grow suddenly and the giant
component, which is a component whose order is much larger than other
components, appears. Eventually, small components join the giant component and
the graph becomes connected. Erdõs Rényi proved that $(n/2)\log n$ is the
sharp threshold for connectedness [1]. The following theorem illustrates this
fundamental result:
###### Theorem 3.
Let $c\in\mathbb{R}$ be fixed and let $M=(n/2)\\{\log
n+c+o(1)\\}\in\mathbb{N}$ and $p=\\{\log n+c+o(1)\\}/n$. Then [1]:
$\displaystyle\textbf{P}(G_{M}\text{ is connected})\rightarrow
e^{-e^{-c}}\text{ as }n\rightarrow\infty$ (3.1)
and
$\displaystyle\textbf{P}(G_{p}\text{ is connected})\rightarrow
e^{-e^{-c}}\text{ as }n\rightarrow\infty.$ (3.2)
For more information regarding random graphs one can check [1], [2], [13],
[14].
### 3.2 Properties of almost every graphs
A graph property $T$ is true for almost every (all) graph if for fixed
$p=p(n)$ [3]:
$\displaystyle\lim_{n\rightarrow\infty}\textbf{P}(G\in G(n,p)\text{ and
}G\text{ has property T})=1\text{ for }p>0$ (3.3)
Some of the important ”almost every graph properties” are:
###### Theorem 4.
For any integer $r\geq 1$ and all $p\in(0,1)$, almost every graph contains
$K_{r}$ [10].
###### Theorem 5.
Almost every graph is connected for all $p\in(0,1)$ [10].
###### Theorem 6.
For $k\in\mathbb{N}$ and all $p\in(0,1)$, almost every graph is k-connected
[10].
### 3.3 Probabilistic methods
Usually, the goal in the probabilistic method is to prove the existence of a
combinatorial structure with a certain property. The usual approach in these
methods is to first construct a suitable probability space, then show that
there exists a random object in that space with the desired properties [11].
Sometimes it is not easy to find the desired object, instead one proves that
there is an object which _almost_ satisfies the desired conditions [12].
Usually, it is possible to modify the almost close object in a deterministic
way so that one gets the desired object. Markov’s inequality, and Chebyshev
inequality are two important inequalities used for this purpose. An important
concept in probabilistic methods is the definition of threshold function,
which is:
###### Definition 13.
$r(n)$ is called a threshold function for a graph property $T$ for $G(n,M(n))$
if:
1\. When $\lim_{n\rightarrow\infty}\frac{M(n)}{r(n)}=0$ almost every graphs do
not satisfy $T$.
2\. When $\lim_{n\rightarrow\infty}\frac{M(n)}{r(n)}=1$ almost every graphs
satisfy $T$.
## Chapter 4 Connected random subgraphs of the cube
111The proof presented in this chapter is based on the proof presented by
B.Bollobás in [1] p.384-393.
The cube graph, $Q^{n}$, is a graph with $2^{n}$ vertices. If one labels each
vertex of $Q^{n}$ from $0$ to $2^{n}-1$, then two vertices are adjacent if
their binary representation differs only in one digit. Hence, one can conclude
that each vertex in $Q^{n}$ is connected to $n$ other vertices. In other
words, $Q^{n}$ has $n2^{n-1}$ edges. A random subgraph of $Q^{n}$ is denoted
by $Q_{p_{n}}^{n}$. $Q_{p_{n}}^{n}$ contains all vertices of $Q^{n}$, and each
edge of $Q^{n}$ independently with probability $p_{n}$.
It is of interest in this chapter to explore a critical value $p_{c}$, which
for fixed values of $p$ if $p<p_{c}$ then the probability that $Q_{p_{n}}^{n}$
is connected, as $n\rightarrow\infty$, tends to $0$; but if $p>p_{c}$ then the
probability that $Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends
to $1$. Burtin proved that this critical value is $1/2$ [6]. Later, P.Erdös
and J.Spencer proved that for $p=1/2$ the probability that $Q_{p_{n}}^{n}$ is
connected, as $n\rightarrow\infty$, tends to $e^{-1}$ [4].
In the first section of this chapter, first the probability that
$Q_{p_{n}}^{n}$ has no isolated point as $n\rightarrow\infty$, for fixed $p$,
is investigated. It is proved that for $p<1/2$ the probability that
$Q_{p_{n}}^{n}$ has no isolated point, as $n\rightarrow\infty$, tends to $0$.
Therefore, for $p<1/2$ the probability that $Q_{p_{n}}^{n}$ is connected, as
$n\rightarrow\infty$, tends to $0$. Then it is proved that, for $p>1/2$ the
probability that $Q_{p_{n}}^{n}$ has no isolated point, as
$n\rightarrow\infty$, tends to $1$. In the next step, the probability that
$Q_{p_{n}}^{n}$ has no isolated point, as $n\rightarrow\infty$, when $p$
depends on $n$ and it is close $1/2$, is explored. It is proved that for
$\lambda(n)=\lambda>0$ and $p_{n}=1-(1/2)\lambda^{1/n}(1+o(1/n))$, the
probability that $Q_{p_{n}}^{n}$ has no isolated point, as
$n\rightarrow\infty$, tends to $e^{-\lambda}$ [1]. Finally, it is proved that,
for fixed $p=1/2$ the probability that $Q_{p_{n}}^{n}$ has no isolated point,
as $n\rightarrow\infty$, tends to $e^{-1}$. These results are based on P.Erdös
and J.Spencer’s work [4].
In the second section, one sheds light on the Isoperimetric problem, which is
the problem of finding an inequality which relates the size of a subgraph to
the size of its boundary. The solution to this problem for $Q_{p_{n}}^{n}$ is
presented by S.Hart [5]. One needs such an inequality to explore the
probability that $Q_{p_{n}}^{n}$ has a component which is not the whole graph.
In the last section, the Isoperimetric inequality is applied to prove that
when $p$ depends on $n$ and $p_{n}\geq 1-(1/2)(\log n)^{1/n}$, then the
probability that there are no components with size larger than $1$ and smaller
than $2^{n}$ in $Q_{p_{n}}^{n}$ , as $n\rightarrow\infty$, tends to $1$.
Therefore, for $p_{n}=1-(1/2)\lambda^{1/n}(1+o(1/n))$, the probability that
$Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends to
$e^{-\lambda}$. Finally, as a special case, it is shown that, for fixed $p$ if
$p=1/2$, the probability that $Q_{p_{n}}^{n}$ is connected , as
$n\rightarrow\infty$, tend to $e^{-1}$; and if $p>1/2$, the probability that
$Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends to $1$.
### 4.1 Isolated vertices
For $p<0.5$:
Assume $p$ is fixed and $p<0.5$. First, consider the following definitions:
###### Definition 14.
$f_{n}(p_{n})$:=P($Q_{p_{n}}^{n}$ is connected)
###### Definition 15.
$g_{n}(p_{n}):=\textbf{P}(Q_{p_{n}}^{n}\text{contains an isolated point})$
###### Definition 16.
$X_{i}(n):=\left\\{\begin{array}[]{ll}1&\mbox{Vertex $i$ is isolated, $i\in
V(Q_{p_{n}}^{n}$)};\\\ 0&\mbox{Vertex $i$ is NOT isolated, $i\in
V(Q_{p_{n}}^{n}$)}.\end{array}\right.$ , and $X(n):=\displaystyle\sum_{i\in
V(Q_{p_{n}}^{n})}X_{i}(n)$.
Now, calculate $E[X(n)]$ and $Var[X(n)]$ as follows:
$\displaystyle\mu:=E[X(n)]=\sum_{i\in V(Q_{p_{n}}^{n})}E[X_{i}(n)]=\sum_{i\in
V(Q_{p_{n}}^{n})}(1-p)^{n}=2^{n}(1-p)^{n}$ (4.1) $\displaystyle
Var[X(n)]=\displaystyle\sum_{i\in
V(Q_{p_{n}}^{n})}Var[X_{i}(n)]+\displaystyle\sum_{i\neq j;i,j\in
V(Q_{p_{n}}^{n})}Cov[X_{i}(n),X_{j}(n)]$ (4.2)
where, $Var[X_{i}(n)]$ and $Cov[X_{i}(n),X_{j}(n)]$ are equal to:
$\displaystyle\displaystyle\sum_{i\in V(Q_{p_{n}}^{n})}Var[X_{i}(n)]$
$\displaystyle=2^{n}(1-p)^{n}-2^{n}(1-p)^{n}(1-p)^{n}=\mu-\mu(1-p)^{n}$ (4.3)
$\displaystyle Cov[X_{i}(n),X_{j}(n)]$
$\displaystyle=E[X_{i}(n)X_{j}(n)]-E[X_{i}(n)]E[X_{j}(n)]$ (4.4)
$\displaystyle=\left\\{\begin{array}[]{ll}0&\mbox{i,j not adjacent};\\\
(1-p)^{n}(1-p)^{n-1}-(1-p)^{n}(1-p)^{n}=\frac{\mu^{2}}{2^{2n}}(\frac{p}{1-p})&\mbox{i,j
adjacent}.\end{array}\right.$ (4.7)
and finally:
$\displaystyle
Var[X(n)]=\mu-\mu(1-p)^{n}+\frac{\mu^{2}}{2^{n}}(\frac{np}{1-p})=\mu+\mu(1-p)^{n}(\frac{np}{1-p}-1)$
(4.8)
Now, since we have $Var[X(n)]$, we can use Chebyshev’s inequality to estimate
$g_{n}(p)$. Chebyshev’s inequality states that:
$\displaystyle
1-g_{n}(p)=\textbf{P}[X(n)=0]\leq\textbf{P}[|X(n)-\mu|\geq\mu]\leq\frac{Var[X(n)]}{\mu^{2}}$
(4.9)
By applying Chebyshev’s inequality when $p<0.5$, one gets
$Var[X(n)]/\mu^{2}\rightarrow 0$, as $n\rightarrow\infty$. Therefore
$\lim_{n\rightarrow\infty}g_{n}(p)=1$. And finally, since $f_{n}(p)\leq
1-g_{n}(p)$, then for $p<0.5$ the probability that $Q_{p_{n}}^{n}$ is
connected for $p<0.5$, as $n\rightarrow\infty$, tends to $0$. $\blacksquare$
For $p>0.5$:
Assume $p$ is fixed and $p>0.5$. In order to calculate $g_{n}(p)$ when
$p>0.5$, as $n\rightarrow\infty$, one can use the following inequality:
$\displaystyle g_{n}(p)=\textbf{P}[X(n)>0]\leq E[X(n)]=\mu$ (4.10)
Since $E[X(n)]\rightarrow 0$ as $n\rightarrow\infty$, then
$\lim_{n\rightarrow\infty}g_{n}(p)=0$. This means that the probability that
there are no isolated points in $Q_{p_{n}}^{n}$ for $p>0.5$, as
$n\rightarrow\infty$, tends to $1$. $\blacksquare$
For $p_{n}=1-(1/2)\lambda^{1/n}(1+o(1/n))$:
One needs the following theorem from [1] to find the distribution of $X(n)$
(distribution of the number of isolated points).
###### Theorem 7.
Let $\lambda=\lambda(n)$ be a non-negative bounded function on N. Suppose the
non-negative integer valued random variables $X(1),X(2),...$ are such that:
$\displaystyle\lim_{n\rightarrow\infty}\\{E_{r}[X(n)]-\lambda^{r}\\}=0,\text{
}r=0,1,...$ (4.11)
where $E_{r}[X]$ is the $r$th factorial moment of $X$, i.e.
$E_{r}[X]=E[(X)_{r}]$. Then
$\displaystyle X(n)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\textbf{P}_{\lambda}$ (4.12)
Use the definition of $X(n)$ presented in definition 16. The goal is to
calculate $E[X(n)]$.
$\displaystyle E_{r}[X(n)]=E[X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)]$ (4.13)
Since $X(n):=\sum_{i\in V(Q_{p_{n}}^{n})}X_{i}(n)$ and $X_{i}$’s are indicator
functions, therefore:
$\displaystyle
X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)=\sum_{(i_{1},i_{2},...,i_{r})}X_{i_{1}}X_{i_{2}}...X_{i_{r}}$
(4.14)
where the sum is over all ordered sets of distinct vertices. Then:
$\displaystyle E_{r}[X(n)]$
$\displaystyle=E[X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)]$ (4.15)
$\displaystyle=E[\sum_{(i_{1},i_{2},...,i_{r})}X_{i_{1}}X_{i_{2}}...X_{i_{r}}]$
(4.16)
$\displaystyle=\sum_{(i_{1},i_{2},...,i_{r})}\textbf{P}[X_{i_{1}}=1,X_{i_{2}}=1,...,X_{i_{r}}=1]$
(4.17)
One knows that a set of $r$ vertices is incident with at most $rn$ edges.
There are $(r)_{r}\binom{2^{n}}{r}$ ways to choose such $r$ vertices. Hence:
$\displaystyle
E_{r}[X(n)]\geq(r)_{r}\binom{2^{n}}{r}(1-p_{n})^{rn}=(2^{n})_{r}(1-p_{n})^{rn}$
(4.18)
One the other hand, a set of $r$ vertices is incident with at least $r(n-r)$
edges. There are at most $(r-1)_{r-1}\binom{2^{n}}{r-1}(r-1)n$ ways to choose
a set of $r$ vertices in $Q_{p_{n}}^{n}$ where at least two vertices are
adjacent; since if we choose $r-1$ vertices independently, then the last
vertex must be connected to one of the chosen vertices. In other words, there
are at most $(r-1)_{r-1}\binom{2^{n}}{r-1}(r-1)n$ ways to choose $r$ vertices
which some of them are adjacent to each other. Hence:
$\displaystyle E_{r}[X(n)]$
$\displaystyle\leq(2^{n})_{r}(1-p_{n})^{rn}+(r-1)_{r-1}\binom{2^{n}}{r-1}(r-1)n(1-p_{n})^{r(n-r)}$
(4.19)
$\displaystyle\leq(2^{n})_{r}(1-p_{n})^{rn}+(2^{n})_{r}rn(1-p_{n})^{r(n-r)}$
(4.20)
$\displaystyle\leq(2^{n})_{r}(1-p_{n})^{rn}+2^{n(r-1)}rn(1-p_{n})^{r(n-r)}$
(4.21)
Finally from 4.18 and 4.21 one gets:
$\displaystyle(2^{n})_{r}(1-p_{n})^{rn}\leq
E_{r}[X(n)]\leq(2^{n})_{r}(1-p_{n})^{rn}+2^{n(r-1)}rn(1-p_{n})^{r(n-r)}$
(4.22)
which gives:
$\displaystyle(2(1-p_{n}))^{rn}(1-\frac{r}{2^{n}})^{r}\leq
E_{r}[X(n)]\leq(2(1-p_{n}))^{rn}\\{1+2^{-n}rn(1-p_{n})^{-r^{2}}\\}$ (4.24)
Since $r$ is fixed and $\lim_{n\rightarrow\infty}(2(1-p_{n}))^{n}=\lambda$,
then:
$\displaystyle\lim_{n\rightarrow\infty}(E_{r}[X(n)])=\lambda^{r}\text{ for
r=0,1,2,...}$ (4.25)
This shows that $X(n)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\textbf{P}_{\lambda}$. $\blacksquare$
For $p=0.5$:
In the calculation of $p_{n}=1-1/2\lambda^{1/n}(1+o(1/n))$, if we fix $p=1/2$
and let $\lambda=1$, then we get that the distribution of $X(n)$ , as
$n\rightarrow\infty$, tends to a Poisson distribution with mean $1$.
Therefore, one can conclude:
$\displaystyle\lim_{n\rightarrow\infty}(1-g_{n}(p))=\lim_{n\rightarrow\infty}(\textbf{P}(X(n)=0))=e^{-1}$
(4.26)
This shows that for $p=1/2$ the probability that $Q_{p_{n}}^{n}$ has no
isolated point, as $n\rightarrow\infty$, tends to $e^{-1}$. $\blacksquare$
### 4.2 Isoperimetric problem for the cube
One needs an inequality which relates the size of a subgraph of $Q^{n}$ to the
size of its boundary. This inequality will be applied to prove that for fixed
values of $p$ if $p\geq 0.5$, then the probability that subgraphs of $Q^{n}$
do not have a component of size larger than $2$ and smaller than $2^{n}$, as
$n\rightarrow\infty$, tends to $1$. The proof presented here is based on the
proof presented in [1].
###### Definition 17.
The edge boundary $b_{G}(H)$, where $H$ is an induced subgraph of G, is the
number of edges which joins vertices in $H$ to the vertices in $G\backslash
H$.
###### Definition 18.
$b_{G}(m):=\min\\{b_{G}(H),H\text{ is an induced subgraph of }G,|V(H)|=m\\}$.
The main task in this section is to calculate $b_{Q^{n}}(m)$. The answer,
loosely, is if $m=2^{k}$ for some $k<n$ then one should take a k-dimensional
sub-cube of $Q^{n}$ as $b_{Q^{n}}(H)$. If $2^{k}\leq m<2^{k+1}$, for some
$k<n$, then one should choose one side of a $(k+1)-$cube and $m-2^{k}$ more
vertices properly chosen in the other half. Since $Q^{n}$ is n-regular and $H$
is an induced subgraph of $G$ with $|V(H)|=m$, then:
$\displaystyle b_{Q^{n}}(H)$ $\displaystyle=mn-2e(H),\text{where e(H) is the
total number of edges in H.}$ (4.27) $\displaystyle b_{Q^{n}}(m)$
$\displaystyle=mn-2e_{n}(m),\text{where }e_{n}(m)=\max\\{e(H):H\text{ induced
subgraph of }Q^{n},|V(H)|=m\\}.$ (4.28)
###### Definition 19.
$h(i)$ := sum of digits in the binary expansion of $i$ and
$f(l,m):=\displaystyle\sum_{l\leq i<m}h(i)$
###### Lemma 1.
If $1\leq k\leq l$, then $f(l,l+k)\geq f(0,k)+k$
Proof:
Let look at the binary expansion of a few numbers:
$\begin{array}[]{ccccc}Column&3&2&1&0\\\ Bin\backslash
Dec&2^{3}&2^{2}&2^{1}&2^{0}\\\ 0&&&&0\\\ 1&&&&1\\\ 2&&&1&0\\\ 3&&&1&1\\\
4&&1&0&0\\\ 5&&1&0&1\\\ 6&&1&1&0\\\ 7&&1&1&1\\\ 8&1&0&0&0\\\ 9&1&0&0&1\\\
10&1&0&1&0\\\ 11&1&0&1&1\end{array}$
From this representation, one can observe that column $i$ starts with a block
of $2^{i}$ zeros. Therefore, sum of jth digits of $k$ consecutive numbers is
minimal if the first block of $0$’s is as long as possible. Hence, one can
conclude:
$\displaystyle f(l,l+k)\geq f(0,k)$ (4.29)
For every $i$ define $r$ such that $0\leq i\leq 2^{r}-1$. The binary
expansions of $i$ and $2^{r}-1-i$ are symmetric. This means that, if there is
a $1/0$ in an specific location of the binary expansion of $i$ then there is a
$0/1$ in the same location of the binary expansion of $2^{r}-1-i$. Therefore,
$\displaystyle h(i)+h(2^{r}-1-i)=r\text{ for }0\leq i\leq 2^{r}-1$ (4.30)
Consequently, since:
$\displaystyle\displaystyle\sum_{l\leq
i<l+k}h(i)+\displaystyle\sum_{2^{r}-l-k\leq i<2^{r}-l}h(i)=rk$ (4.31)
then:
$\displaystyle f(l,l+k)+f(2^{r}-l-k,2^{r}-l)=rk,\text{if }l+k\leq 2^{r}$
(4.32)
Let us prove lemma 1 with the assumption $k\leq 2^{r}\leq l$ by using
inequalities 4.29 and 4.32. This assumption means that the length of the
sequence in the binary expansion of $2^{r}+k$ and $2^{r}$ are equal.
With the same logic that one gets 4.29, one gets:
$\displaystyle f(l,l+k)\geq f(2^{r},2^{r}+k)\text{ when }2^{r}\leq l$ (4.33)
and then for $k\leq 2^{r}$ one can get:
$\displaystyle f(2^{r},2^{r}+k)=\displaystyle\sum_{2^{r}\leq i<2^{r}+k}h(i)$
(4.34)
$\sum_{2^{r}\leq i<2^{r}+k}h(i)$ is the sum over numbers with the same length
in their binary expansion’s sequence. When one removes the last digit in their
binary expansion, the remain is $f(0,k)$. Therefore, $\sum_{2^{r}\leq
i<2^{r}+k}h(i)$ is equal to $k$ $1$’s plus $f(0,k)$. Hence:
$\displaystyle f(2^{r},2^{r}+k)=k+f(0,k)\text{ when }k\leq 2^{r}\leq l$ (4.35)
and finally:
$\displaystyle f(l,l+k)\geq f(0,k)+k\text{ where }k\leq 2^{r}\leq l$ (4.36)
Now, one can prove lemma 1 by induction on $K$, without the assumption $k\leq
2^{r}\leq l$. We want to prove that for $1\leq K\leq l$, $f(l+K,l)\geq
K+f(0,K)$. Fix $k$ such that $1\leq k\leq l$ and $K<k$. For $K=1$ the
inequality in lemma 1 is trivial. Assume that the inequality is true for $K<k$
and $K>2$, which means:
$\displaystyle f(l,l+K)\geq K+f(0,K)\text{ when }1\leq k\leq l\text{ , and
}K<k$ (4.37)
Now, one should verify the inequality for $K=k$. Define $r\geq 1$ by
$2^{r-1}\leq k<2^{r}$. If $l\geq 2^{r}$, then $k\leq 2^{r}\leq l$ and the
lemma is implied by inequality 4.36. Hence, one may assume that
$2^{r-1}<l<2^{r}$. Now, one should apply inequality 4.32 and 4.37 in order to
get the final result:
$\displaystyle f(l+k)$ $\displaystyle=f(l,2^{r})+f(2^{r},l+k)\text{\emph{
(from definition of f and }}l\geq 2^{r})$ (4.38)
$\displaystyle=(2^{r}-l)r-f(0,2^{r}-l)+f(2^{r},l+k)\text{ \emph{ (from
\ref{1.5})}}$ (4.39)
$\displaystyle\geq(2^{r}-l)r-f(0,2^{r}-l)+f(0,l+k-2^{r})+l+k-2^{r}\text{\emph{
(from \ref{1.7})}}$ (4.40)
$\displaystyle\geq(2^{r}-l)r-f(2^{r}-k,2^{r}-k+2^{r}-l)+2^{r}-l+f(0,l+k-2^{r})+l+k-2^{r}\text{
\emph{(from \ref{1.7})}}$ (4.41)
$\displaystyle\geq(2^{r}-l)r-f(2^{r}-k,2^{r}-k+2^{r}-l)+f(0,l+k-2^{r})+k$
(4.42) $\displaystyle\geq f(l+k-2^{r},k)+f(0,l+k-2^{r})+k\text{ \emph{(from
\ref{1.5})}}$ (4.43) $\displaystyle\geq f(0,k)+k\text{ \emph{(from
characteristics of f)}}$ (4.44)
$\blacksquare$
###### Theorem 8.
For $2\leq m\leq 2^{n}$ we have $b_{Q^{n}}(m)=mn-2f(0,m)$. In other words,
$f(0,m)=e_{n}(m)\text{ where }e_{n}(m)=\max\\{e(H):H\text{ induced subgraph of
}Q^{n},|V(H)|=m\\}.$
Proof:
First, let us fix an $m$. As the first step one should prove that
$e_{n}(m)\geq f(0,m)$. Vertex $i$ is connected to $h(i)$ vertices $j$ with
$j<i$, since for each 1 in the binary expansion of $i$ there is exactly one j
$(j<i)$, which its binary expansion differs in the position of that $1$.
Therefore, one can conclude that $W=\\{0,1,2,...,m-1\\}$ contains $\sum_{0\leq
i<m}h(i)=f(0,m)$ edges. So, $e_{n}(m)\geq f(0,m)$.
As the second step, one should prove that $e_{n}(m)\leq f(0,m)$ by induction
on $n$. Fix $m$ and $n$ for $2\leq m\leq 2^{n}$. For $n=1$ the inequality is
trivially true. Assume that it is true for $N<n$, which means:
$\displaystyle e_{N}(m)\leq f(0,m),\text{ where }N<n\text{ and the fixed m is:
}2\leq m\leq 2^{n}$ (4.45)
Now, one should check the inequality 4.45 for $N=n$. This means that we should
find an $H$ induced subgraph of $Q^{n}$, $|V(H)|=m$, which maximize
$e_{n}(m)$. Let us split $Q^{n}$ into two (n-1)-dimensional cubes, the top
face with $2^{n-1}$ vertices and the bottom face with $2^{n-1}$ vertices. This
means, there are $(n-1)2^{n-2}$ edges in each face, and $2^{n-1}$ edges
between two faces. Now, one can construct $H$. Choose $m_{1}$ vertices for $H$
from the top face, and $m_{2}$ vertices from the bottom face, where
$m_{1}+m_{2}=m$ and $m_{1}\leq m_{2}$. In other words, $H$ is constructed from
two induced subgraphs, one from the top face, denoted by $H_{1}$, and the
other from the bottom face, denoted by $H_{2}$.
Each face is a (n-1)-dimensional cube, so inequality 4.45 holds for both
$H_{1}$ and $H_{2}$. Also, each vertex of the top face is connected to exactly
one vertex of the bottom face. Hence, the number of edges of $H$ is at most:
$\displaystyle e_{n}(m)\leq f(0,m_{1})+f(0,m_{2})+m_{1}\text{(from
\ref{1.21})}$ (4.46)
where $m_{1}$, in the right hand side of the inequality, is for the maximum
number of edges between $H_{1}$ and $H_{2}$, which one can choose here.
Finally, by applying lemma 1, one gets:
$\displaystyle e_{n}(m)$ $\displaystyle\leq f(0,m_{1})+f(0,m_{2})+m_{1}$
(4.47) $\displaystyle\leq f(m_{2},m_{2}+m_{1})+f(0,m_{2})\text{ (from lemma
\ref{1.1})}$ (4.48) $\displaystyle\leq f(0,m)\text{ (from definition of f)}$
(4.49)
$\blacksquare$
Theorem 8 shows that, if we want to choose an induced subgraph of $Q^{n}$,
with $m$ vertices, which has the smallest edge boundary, then we should choose
the induced subgraph of $Q^{n}$ with the set of vertices
$W=\\{0,1,2,...,m-1\\}$.
###### Corollary 1.
For all $k$ and $n$, $e_{n}(k)\leq\frac{k}{2}\lceil log_{2}k\rceil$, which is
equivalent to $b_{Q^{n}}(k)\geq k(n-\lceil log_{2}k\rceil)$.
Proof:
Let $r=\lceil log_{2}k\rceil$. Then
$\displaystyle 2f(0,k)$ $\displaystyle\leq f(0,k)+f(0,k)\leq
f(0,k)+f(2^{r}-k,2^{r})\text{ (from \ref{1.2}) }$ (4.50)
$\displaystyle=rk\text{ (from \ref{1.5})}$ (4.51)
Therefore:
$\displaystyle e_{n}(k)=f(0,k)\leq r\frac{k}{2}=\frac{k}{2}\lceil
log_{2}k\rceil$ (4.52)
$\blacksquare$
### 4.3 Isolated components of size larger than 2 and smaller than $2^{n}$
###### Definition 20.
$C_{s}$ is the family of s-subsets (subsets with size s) of $V=V(Q^{n})$ whose
induced graph is connected.
Remarks: $h(n):=o(g(n))$ means $\frac{h(n)}{g(n)}\rightarrow 0$ as
$n\rightarrow\infty$.
Remarks: The following inequality will be applied a lot in the rest of this
section:
$\displaystyle(\frac{n}{k})^{k}\leq\binom{n}{k}\leq\frac{n^{k}}{k!}\leq(\frac{ne}{k})^{k}$
(4.53)
###### Theorem 9.
If $p_{n}\geq 1-\frac{1}{2}(\log n)^{\frac{1}{n}}$, the probability that for
some $S\in C_{s}$, $2\leq s\leq 2^{n-1}$, no edges of $Q_{p_{n}}^{n}$ join $S$
to $V(Q^{n})\setminus S$, as $n\rightarrow\infty$, tends to 0.
Note: For $2^{n-1}<s<2^{n}$, if there exist a component of size smaller than
$2^{n}$ then there is at least one component of size smaller than $2^{n-1}$
which contradicts with the theorem.
Proof:
Consider $S\subset V=V(Q^{n})$ and set $b(S)=b_{Q^{n}}(H)$ for which $H$ is
the induced subgraph of $Q^{n}$ with the set of vertices S. One can observe
that:
$\displaystyle\textbf{P}(\text{No edges of }Q_{p_{n}}^{n}\text{ join S to
}V\setminus S)=(1-p_{n})^{b(S)}$ (4.54)
In order to prove the theorem, it is sufficient to show that:
$\displaystyle\sum_{s=2}^{2^{n-1}}\sum_{S\in C_{s}}(1-p_{n})^{b(S)}=o(1)$
(4.55)
From corollary 1, one knows that for $|S|=s$:
$\displaystyle b(S)\geq b(s)\geq s(n-\lceil\log_{2}s\rceil)$ (4.56)
and therefore:
$\displaystyle\sum_{S\in C_{s}}(1-p_{n})^{b(S)}\leq|C_{s}|(1-p_{n})^{b(s)}$
(4.57)
Now, one should partition s, $2\leq s\leq 2^{n-1}$, to different intervals in
order to find a bound for $|C_{s}|$ and $(1-p_{n})^{b(s)}$ for each interval.
First interval $2\leq s\leq
s_{1},s_{1}=\lfloor\frac{2^{\frac{n}{2}}}{n^{2}}\rfloor$:
First, one should find a bound for $|C_{s}|$. One has maximum $2^{n}$ choices
to choose the first element for $C_{s}$. The selected element is connected to
maximum $n$ vertices, therefore there are $n$ choices to choose the second
element. With the same logic there are at most $(s-1)n$ choices to choose the
last element for $C_{s}$. Therefore, one can show:
$\displaystyle|C_{s}|\leq 2^{n}(n)(2n)...((s-1)n))\leq(s-1)!(n)^{s-1}2^{n}$
(4.58)
Hence:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq(s-1)!(n)^{s-1}2^{n}(1-p_{n})^{s(n-\lceil\log_{2}s\rceil)}$
(4.59)
Since $p_{n}=1-\frac{1}{2}(\log n)^{\frac{1}{n}}$, so for large enough $n$:
$\displaystyle(1-p_{n})^{s(n-\lceil log_{2}s\rceil)}$
$\displaystyle\leq(2)^{-ns}(\log
n)^{s}(1-p_{n})^{-s(\log_{2}s)}\text{(neglecting some small terms)}$ (4.60)
$\displaystyle=(2)^{-ns}(\log n)^{s}2^{s\log_{2}s}(\log
n)^{\frac{-s\log_{2}s}{n}}$ (4.61) $\displaystyle(\text{ since for large
enough n: }(\log n)^{\frac{-s\log_{2}s}{n}}\leq 1)$ (4.62)
$\displaystyle\leq(2)^{-ns}(\log n)^{s}s^{s}$ (4.63)
From equations 4.59 and 4.63, one can show that:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq(s-1)!(n)^{s-1}2^{n}(2)^{-ns}(\log
n)^{s}s^{s}$ (4.64)
Assume that the right hand side of inequality 4.64 is equal to A. After
multiplying both sides of inequality 4.64 with $\frac{ns^{s+1}}{s!}$ and then
taking $\log_{2}$ from both sides, one gets:
$\displaystyle\log_{2}(|C_{s}|(1-p_{n})^{b(s)}\frac{ns^{s+1}}{s!})\leq\log_{2}(A\frac{ns^{s+1}}{s!})$
(4.65)
If $\log_{2}(A\frac{ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$
then $A\frac{ns^{s+1}}{s!}$ should tend to 0. This means that
$|C_{s}|(1-p_{n})^{b(s)}\frac{ns^{s+1}}{s!}$ tends to 0, as
$n\rightarrow\infty$. Therefore:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq\frac{s!}{ns^{s+1}}\text{ for large
enough n}$ (4.66)
which shows that:
$\displaystyle\sum_{s=2}^{s_{1}}\sum_{S\in C_{s}}(1-p_{n})^{b(S)}=o(1)$ (4.67)
Finally, it remains to prove
$\log_{2}(A\frac{ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$.
One can verify this for $s\leq n$ and $s>n$.
Second interval $s_{1}+1\leq s\leq 2^{n-1}$ and $S\in
C_{s}^{-},s_{1}=\lfloor\frac{2^{\frac{n}{2}}}{n^{2}}\rfloor$:
Let us define $C_{s}^{-}$ and $C_{s}^{+}$ as follows:
###### Definition 21.
$\displaystyle C_{s}^{-}:=\\{S\in C_{s}|b(s)\geq
s(n-\log_{2}s+\log_{2}n)\\}\text{, and }C_{s}^{+}:=C_{s}\backslash C_{s}^{-}$
(4.68)
One can bound $|C_{s}^{-}|$ for $s_{1}+1\leq s\leq 2^{n-1}$ as follows:
$\displaystyle|C_{s}^{-}|\leq|C_{s}|\leq\binom{2^{n}}{s}\leq\frac{2^{ns}}{s!}\leq(\frac{e2^{n}}{s})^{s}$
(4.69)
Hence:
$\displaystyle\displaystyle\sum_{s=s_{1}+1}^{2^{n-1}}\sum_{S\in
C_{s}^{-}}(1-p_{n})^{b(S)}$
$\displaystyle\leq\sum_{s=s_{1}+1}^{2^{n-1}}(\frac{e2^{n}}{s})^{s}(\frac{1}{2}(\log
n)^{\frac{1}{n}})^{s(n-\log_{2}s+\log_{2}n)}$ (4.70)
$\displaystyle\leq\sum_{s=s_{1}+1}^{2^{n-1}}(\frac{e2^{n}2^{-(n-\log_{2}s+\log_{2}n)}(\log
n)^{\frac{(n-\log_{2}s+\log_{2}n)}{n}}}{s})^{s}$ (4.71)
$\displaystyle\leq\sum_{s=s_{1}+1}^{2^{n-1}}(\frac{e2^{n}2^{-n}2^{\log_{2}s}2^{-\log_{2}n}\log
n}{s})^{s}(\log n)^{\frac{n(-\log_{2}s+\log_{2}n)}{s}}$ (4.72)
$\displaystyle(\text{ since for large enough n: }(\log
n)^{\frac{n(-\log_{2}s+\log_{2}n)}{s}}\leq 1)$ (4.73)
$\displaystyle\leq\sum_{s=s_{1}+1}^{2^{n-1}}(\frac{e\log n}{n})^{s}=o(1)$
(4.74)
$\blacksquare$
Third interval $s_{1}\leq s\leq s_{2}$,
$s_{1}=\lfloor\frac{2^{\frac{n}{2}}}{n^{2}}\rfloor,s_{2}=\lfloor\frac{2^{n}}{(\log
n)^{4}}\rfloor$ and $S\in C_{s}^{+}$:
For the 3rd and the 4th intervals one needs to know how to find a bound for
$|C_{s}^{+}|$. The following lemma, presented by B.Bollobas [1], helps us in
this matter:
###### Lemma 2.
Let G be a graph of order $v$ and suppose that $\Delta(G)\leq\Delta$,
$2e(G)=vd$ and $\Delta+1\leq u\leq v-\Delta-1$. Then, there is a u-set of U of
vertices with:
$\displaystyle|N(U)|=|U\cup\Gamma(U)|\geq
v\frac{d}{\Delta}\\{1-exp(\frac{-u(\Delta+1)}{v})\\}$ (4.75)
where, $\Delta(G):=$ Maximum degree in G, $d:=$ average degree in G and
$\Gamma(U)=\\{x\in V(G):xy\in E(G)\text{ for some y}\in U\\}$
Let $H=Q_{n}[S]$ (the induced subgraph of $Q_{n}$ with the set of vertices
$S$). From the definition of $C_{s}^{+}$ one knows that the average degree in
$H$ is at least:
$\displaystyle\log_{2}s-\log_{2}n$ (4.76)
The goal is to find $U\subset S$, where $|U|:=u:=\lfloor\frac{2s}{n}\rfloor$,
$\Delta=n$, $v=s$, $d\geq\log_{2}s-\log_{2}n$ and then use lemma 2 to
calculate $|N(U)|$. First, one should check the condition $\Delta+1\leq u\leq
v-\Delta-1$ for defined variables in order to use lemma 2. First, check if
$n+1\leq\lfloor\frac{2s}{n}\rfloor$, as $n\rightarrow\infty$:
$\displaystyle\frac{2s}{n}=\frac{2^{\frac{n}{2}+1}}{n^{3}}\text{for minimum s,
and trivially }n+1\leq\frac{2^{\frac{n}{2}+1}}{n^{3}}\text{ for large enough
n}$ (4.77)
and then check if $\lfloor\frac{2s}{n}\rfloor\leq s-(n+1)$. One should check
if $ns-n(n+1)\geq 2s$, which means one should check that whether:
$\displaystyle\frac{2^{\frac{n}{2}}(n-2)}{n^{3}(n+1)}\geq 1$ (4.79)
which is clearly true for large enough $n$. Now, one can apply lemma 2 on the
graphs generated by $S$ and get :
$\displaystyle\exists U\subset S:|N(U)|\geq
s\frac{\log_{2}s-\log_{2}n}{n}\\{1-exp(-\frac{u(n+1)}{s})\\}$ (4.80)
where:
$\displaystyle\frac{\log_{2}s-\log_{2}n}{n}\geq\frac{(\log_{2}(\frac{2^{\frac{n}{2}}}{n^{2}})-\log_{2}n)}{n}=\frac{n-6\log_{2}n}{2n}$
(4.81) $\displaystyle\text{and
}\lim_{n\rightarrow\infty}\frac{n-6\log_{2}n}{2n}=\frac{1}{2}$ (4.82)
on the other hand:
$\displaystyle\lim_{n\rightarrow\infty}(1-exp(-\frac{n+1}{s}(\frac{2s}{n}+1)))=1-e^{-2}$
(4.83)
Therefore, from 4.82 and 4.83 one gets:
$\displaystyle|N(U)|\geq\frac{1}{2}(1-e^{-2})s\geq\frac{s}{3}\text{ as
}n\rightarrow\infty$ (4.84)
Now that we have $|N(U)|$, we can estimate a bound for $|C_{s}^{+}|$ here. We
know from 4.84 that for each $S\in C_{s}^{+}$ there exist a $U\subseteq S$,
$|U|:=u:=\lfloor\frac{2s}{n}\rfloor$, such that $|N(U)|\geq s/3$. Therefore,
one can choose $S\in C_{s}^{+}$ as follows:
1\. Select $u$ vertices of $Q^{n}$; there are $\binom{2^{n}}{u}$ choices for
this $u$.
2\. Select $\lfloor\frac{s}{3}\rfloor-u$ neighbors of the selected vertices in
part $1$; there are maximum $(2^{n})^{u}$ choices, since there are at most
$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...\binom{n}{n}=2^{n}$ ways to find
neighbors of a vertex in $U$.
3\. Select $\lfloor\frac{2s}{3}\rfloor$ other vertices; there are at most
$\binom{2^{n}}{\lfloor\frac{2s}{3}\rfloor}$ choices.
Hence:
$\displaystyle|C_{s}^{+}|\leq\binom{2^{n}}{u}(2^{n})^{u}\binom{2^{n}}{\lfloor\frac{2s}{3}\rfloor}$
(4.85)
and:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\binom{2^{n}}{u}(2^{n})^{u}\binom{2^{n}}{\lfloor\frac{2s}{3}\rfloor}(1-p_{n})^{b(s)}$
(4.86)
where:
$\displaystyle(1-p_{n})^{b(s)}\leq 2^{-sn}s^{s}(\log n)^{s}$ (4.87)
consequently from 4.86, 4.87 and 4.53:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq(\frac{e2^{n}}{u})^{u}2^{un}(\frac{e2^{n}}{\lfloor\frac{2s}{3}\rfloor})^{\lfloor\frac{2s}{3}\rfloor}2^{-sn}s^{s}(\log
n)^{s}$ (4.88)
Write $s=2^{\beta n}$, ($\beta=\frac{log_{2}s}{n}$), so that:
$\displaystyle 2^{\beta n}\leq\frac{2^{n}}{(\log n)^{4}}\Rightarrow\beta\leq
1-\frac{4\log_{2}\log n}{n}$ (4.89)
Now, find a bound for the inequality 4.88. First calculate the first part of
the inequality:
$\displaystyle(\frac{e2^{n}}{u})^{u}2^{un}(\frac{e2^{n}}{\lfloor\frac{2s}{3}\rfloor})^{\lfloor\frac{2s}{3}\rfloor}$
$\displaystyle\leq(\frac{e2^{n}}{\frac{2s}{n}})^{\frac{2s}{n}}2^{2s}(\frac{e2^{n}}{\frac{2s}{3}})^{\frac{2s}{3}}=(2^{2}2^{2}(\frac{3}{2}e)^{\frac{2}{3}})^{s}\frac{2^{\frac{2s}{3n}}}{s^{\frac{2s}{3}}}(\frac{2^{n}}{s})^{\frac{2s}{3}}$
(4.90) $\displaystyle(\text{ since for large enough n and $s_{1}\leq s\leq
s_{2}$: }(\frac{e2^{n}}{\frac{2s}{3}})^{\frac{2s}{3}}\leq 1)$ (4.91)
$\displaystyle=(2^{2}2^{2}(\frac{3}{2}e)^{\frac{2}{3}})^{s}\frac{2^{\frac{2s}{3n}}}{s^{\frac{2s}{3}}}=c^{s}2^{\frac{2}{3}sn(1-\frac{\log_{2}s}{n})}=c^{s}2^{\frac{2}{3}sn(1-\beta)}$
(4.92)
where c is a positive constant. Now, by substituting 4.92 in 4.88 one gets:
$\displaystyle\sum_{S\in C_{s}^{+}}(1-p_{n})^{b(S)}$ $\displaystyle\leq
2^{-sn}s^{s}(\log n)^{s}c^{s}2^{\frac{2}{3}sn(1-\beta)}$ (4.93)
$\displaystyle=c^{s}(\log n)^{s}2^{-\frac{sn(1-\beta)}{3}}$ (4.94)
$\displaystyle\leq c^{s}(\log n)^{s}2^{-\frac{4s\log_{2}\log n}{3n}}\text{ ,
(from \ref{1.13})}$ (4.95) $\displaystyle=c^{s}(\log n)^{s}2^{\log_{2}(\log
n)^{\frac{-4s}{3}}}$ (4.96) $\displaystyle\leq c^{s}(\log n)^{s}(\log
n)^{\frac{-4s}{3}}$ (4.97) $\displaystyle=c^{s}(\log n)^{\frac{-s}{3}}$ (4.98)
and finally from 4.98:
$\displaystyle\sum_{s=s_{1}}^{s_{2}}\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\sum_{s=s_{1}}^{s_{2}}c^{s}(\log
n)^{\frac{-s}{3}}=o(1)$ (4.99)
$\blacksquare$
Fourth interval $s_{2}+1\leq s\leq 2^{n-1}$ and
$,s_{2}=\lfloor\frac{2^{n}}{(\log n)^{4}}\rfloor,S\in C_{s}^{+}$:
In $H=Q^{n}[S]$ (the induced subgraph of $Q^{n}$ with the set of vertices S),
the average degree is at least:
$\displaystyle\log_{2}s-\log_{2}n>n-2\log_{2}n$ (4.100)
since:
$\displaystyle s\geq\lceil\frac{2^{n}}{(\log
n)^{4}}\rceil\Rightarrow\log_{2}(\frac{2^{n}}{(\log n)^{4}})<\log_{2}s$
(4.101)
$\displaystyle\Rightarrow\log_{2}s-\log_{2}n\geq\log_{2}(\frac{2^{n}}{(\log
n)^{4}})-\log_{2}n\geq n-\log_{2}(\log n)^{4}-\log_{2}n$ (4.102)
$\displaystyle(\text{for large enough n one can get, }n>(\log n)^{4})$ (4.103)
$\displaystyle\geq n-2\log_{2}n$ (4.104)
First, look for a subgraph of H with large average degree. Let T be the set of
vertices of H with degree at least $n-(\log_{2}n)^{2}$, and set $t=|T|$. From
4.100 one can conclude that the sum of degrees in $H$ is at least
$s(n-2\log_{2}n)$.We also know that:
$\displaystyle\text{Sum of degrees in }S$ $\displaystyle\leq s(n-2\log_{2}n)$
(4.105) $\displaystyle\leq t\times(\text{Maximum degree of vertices in set $T$
of graph $H$ })$ (4.106) $\displaystyle+(s-t)\times(\text{Maximum degree of
vertices in set $S\setminus T$ of graph $H$ })$ (4.107) $\displaystyle\leq
tn+(s-t)(n-(\log_{2}n)^{2})$ (4.108) $\displaystyle\Rightarrow t\geq
s(1-\frac{2}{\log_{2}n})$ (4.109)
Define $H_{1}=Q^{n}[T]=H[T]$ as the induced subgraph spanned by $T$. We want
to calculate $|N_{H_{1}}(U)|$ for some $U$ in $H_{1}$, hence we should
estimate the size of $H_{1}$ and after that calculate the average degree in T.
Let us first calculate $e(H_{1})$, the total number of edges in $H_{1}$.
$\displaystyle e(H_{1})\geq
e(H)-(s-t)n\geq\frac{s}{2}(n-2\log_{2}n)-\frac{2s}{\log_{2}n}n\text{ (from
\ref{1.33} and \ref{1.34})}$ (4.110)
One knows that the average degree in $H_{1}$ is at least
$\frac{2e(H_{1})}{s}$, and:
$\displaystyle\frac{2e(H_{1})}{s}\geq n-2\log_{2}n-\frac{4n}{\log_{2}n}\geq
n-\frac{5}{log_{2}n}$ (4.111) $\displaystyle(\text{since:
}\log_{2}n^{2}<\frac{n}{\log_{2}n}\text{ for large enough n})$ (4.112)
Set $u=\lfloor\frac{2^{n}}{n^{\frac{1}{2}}}\rfloor$. One should check the
conditions of lemma 1 here. Let $v=t,\Delta=n,d\geq n-\frac{5}{\log_{2}n}$.
So, one should check if $n+1\leq\frac{2^{n}}{n^{\frac{1}{2}}}\leq t-(n+1)$ for
large enough $n$. Clearly, $n+1\leq\frac{2^{n}}{n^{\frac{1}{2}}}$, as
$n\rightarrow\infty$. It remains to prove $\frac{2^{n}}{n^{\frac{1}{2}}}\leq
t-(n+1)$, for large enough $n$. For minimum $s$ from 4.109 we can get:
$\displaystyle t\geq$ $\displaystyle\frac{2^{n}}{(\log
n)^{4}}(1-\frac{2}{\log_{2}n})\text{ (from \ref{1.34})}$ (4.113)
$\displaystyle\geq\frac{2^{n}}{n^{\frac{1}{2}}}+n+1\text{ (for large enough n)
}$ (4.114)
Now, one can use lemma 1 and estimate $|N_{H_{1}}(U)|$.
$\displaystyle|N_{H_{1}}(U)|$
$\displaystyle\geq\frac{t}{n}(n-\frac{5n}{\log_{2}n})\\{1-\exp(-\frac{n+1}{t}\frac{2^{n}}{n^{\frac{1}{2}}})\\}$
(4.115)
$\displaystyle\geq\frac{t}{2}(1-\frac{5}{\log_{2}n})\\{1-\exp(-\frac{n+1}{t}\frac{2^{n}}{n^{\frac{1}{2}}})\\}$
(4.116)
After that, let us estimate a bound for
$\exp(-\frac{n+1}{t}\frac{2^{n}}{n^{\frac{1}{2}}})$. One knows that $t\geq
s(1-\frac{2}{\log_{2}n})$. Since $\max(t)=s$ and $\max(s)=2^{n-1}$, then:
$\displaystyle\frac{2^{n}(n+1)}{n^{\frac{1}{2}}t}\geq\frac{2^{n}(n+1)}{n^{\frac{1}{2}}2^{n-1}}=\frac{2(n+1)}{n^{\frac{1}{2}}}\geq
n^{\frac{1}{4}}\text{( for large enough n)}$ (4.117)
$\displaystyle\Rightarrow\\{1-\exp(-\frac{n+1}{t}\frac{2^{n}}{n^{\frac{1}{2}}})\\}\geq\exp(-n^{\frac{1}{4}})\text{(
for large enough n)}$ (4.118)
By using the bound from 4.118 in 4.116, one gets:
$\displaystyle|N_{H}(U)|\geq|N_{H_{1}}(U)|$ $\displaystyle\geq
t(1-\frac{5}{\log_{2}n})\\{1-\exp(-n^{\frac{1}{4}})\\}$ (4.119)
$\displaystyle=t\\{1+\exp(-n^{\frac{1}{4}})\frac{5}{\log_{2}n}-\exp(-n^{\frac{1}{4}})-\frac{5}{\log_{2}n}\\}$
(4.120)
$\displaystyle(\lim_{n\rightarrow\infty}\exp(-n^{\frac{1}{4}})\frac{5}{\log_{2}n}=0\text{
and }\exp(-n^{\frac{1}{4}})<\frac{1}{\log_{2}n}\text{( for large enough n)})$
(4.121) $\displaystyle\geq
t\\{1-\frac{1}{\log_{2}n}-\frac{6}{\log_{2}n}\\}=t(1-\frac{6}{\log_{2}n})$
(4.122) $\displaystyle\geq
s(1-\frac{2}{\log_{2}n})(1-\frac{5}{\log_{2}n})=s(1+\frac{2}{\log_{2}n}\frac{6}{\log_{2}n}-\frac{8}{\log_{2}n})\text{(from
\ref{1.34})}$ (4.123) $\displaystyle\geq s(1-\frac{8}{\log_{2}n})$ (4.124)
Now that we have $|N_{H}(U)|$, we can estimate a bound for $|C_{s}^{+}|$ here.
We know from 4.124 that for each $S\in C_{s}^{+}$ there exist a $U\subseteq
S$, $|U|:=u:=\lfloor\frac{2^{n}}{n^{\frac{1}{2}}}\rfloor$, such that
$|N_{H}(U)|\geq s(1-\frac{8}{\log_{2}n})$. Therefore, one can choose $S\in
C_{s}^{+}$ as follows:
1\. Select $u$ vertices of $Q^{n}$; there are $\binom{2^{n}}{u}$ choices for
this $u$.
2\. Select $\lfloor s(1-\frac{8}{\log_{2}n})\rfloor-u$ neighbors of the
selected vertices in part 1. At most $(\log_{2}n)^{2}$ of the $n$ neighbors of
a vertex in $U$ do not belong to $N_{H}(U)$. Hence there are at most
$\sum_{(k_{j})}(\prod_{i=1}^{u}\binom{n}{j})$ ways to find neighbors of $u$
vertices in $U$, where the sum is over all
$(k_{1},k_{2},...,k_{u}),k_{i}\leq(\log_{2}n)^{2}$. We know that:
$\displaystyle\sum_{(k_{i})}\prod_{i=1}^{u}\binom{n}{k_{i}}$
$\displaystyle\leq\sum_{(k_{i})}\prod_{i=1}^{u}(\frac{n^{i}}{i!})$ (4.125)
$\displaystyle\leq\sum_{(k_{i})}\prod_{i=1}^{u}(\frac{n^{(\log_{2}n)^{2}}}{(\log_{2}n)^{2}!})$
(4.126)
$\displaystyle=\sum_{(k_{i})}\frac{n^{u(\log_{2}n)^{2}}}{((\log_{2}n)^{2}!)^{u}}$
(4.127)
$\displaystyle={((\log_{2}n)^{2})}^{u}\frac{n^{u(\log_{2}n)^{2}}}{((\log_{2}n)^{2}!)^{u}}$
(4.128) $\displaystyle\leq n^{u(\log_{2}n)^{2}}$ (4.129)
3\. Select $\lfloor\frac{8s}{\log_{2}n}\rfloor$ other vertices; there are at
most $\binom{2^{n}}{\lfloor\frac{8s}{\log_{2}n}\rfloor}$ choices.
Hence:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\binom{2^{n}}{u}n^{u(\log_{2}n)^{2}}\binom{2^{n}}{\lfloor\frac{8s}{\log_{2}n}\rfloor}2^{-s(n-\log_{2}s)}(\log
n)^{s(1-\frac{\log_{2}s}{n})}$ (4.131)
where:
$\displaystyle\binom{2^{n}}{u}n^{u(\log_{2}n)^{2}}\binom{2^{n}}{\lfloor\frac{8s}{\log_{2}n}\rfloor}\leq
2^{o(s)}$ (4.133)
Therefore:
$\displaystyle\sum_{S\in C_{s}^{+}}(1-p_{n})^{b(S)}\leq 2^{\varepsilon(s)}$
(4.134)
where:
$\displaystyle\varepsilon(s)=o(s)-s\\{n-\log_{2}s-\log_{2}\log
n+\frac{\log_{2}s}{n}\log_{2}\log n\\}$ (4.135)
Since $s\leq 2^{n-1}$, hence one can get:
$\displaystyle\varepsilon(s)$ $\displaystyle\leq o(s)-s\\{n-(n-1)-\log_{2}\log
n+\frac{n-1}{n}\log_{2}\log n\\}$ (4.136)
$\displaystyle=o(s)-s\\{1-\frac{1}{n}\log_{2}\log n\\}\leq\frac{-s}{2}$
(4.137)
Therefore, for large enough n, one can get:
$\displaystyle\sum_{s=s_{2}+1}^{2^{n-1}}\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\sum_{s=s_{2}+1}^{2^{n-1}}2^{-\frac{s}{2}}=o(1)$
(4.138)
$\blacksquare$
### 4.4 Comparison between a typical random graph and $Q^{n}$
Consider $Per(G(V,E),p_{n})$ as the subgraph of $G(V,E)$ after percolation
with the parameter $p_{n}$. The goal in this section is to compare
$\textbf{P}(Q^{n}_{p_{n}}\text{ is connected})$ with P($Per(G\in
G(N=2^{n},M=n2^{n-1}),p_{n}$) is connected) as $n\rightarrow\infty$.
One knows, the probability that $G_{p}\in G(n,p)$ is connected, for $p=c\log
n/n$ as $n\rightarrow\infty$, tends to [1] :
$\displaystyle\lim_{n\rightarrow\infty}\textbf{P}(G_{p}\in G(n,p)\text{ is
connected})=\left\\{\begin{array}[]{ll}1&\mbox{if $c>1$};\\\ 1-e^{-1}&\mbox{if
$c=1$};\\\ 0&\mbox{if $c<1$}.\end{array}\right.$ (4.142)
###### Theorem 10.
If $Q$ is a convex property and $pq\binom{n}{2}\rightarrow\infty$, then almost
every graph in $G(n,p)$ has $Q$ iff for every fixed $x$ a.e. graph in $G(n,M)$
has $Q$, when $M=\lfloor p\binom{n}{2}+x(pq\binom{n}{2}^{0.5})\rfloor$ [1].
###### Definition 22.
$Q$ is a convex property if $F\subset G\subset H$ and $F\in Q$ and $H\in Q$
then $G\in Q$.
We want to calculate $\textbf{P}(Per(G_{m}\in G(n,M=m),p_{n})\text{ is
connected})$ as $n\rightarrow\infty$. From theorem 10, one can conclude that
$G(n,p)$ and $G(n,M=p\binom{n}{2})$ have almost the same behavior for
connectivity, as $n\rightarrow\infty$. Hence:
$\displaystyle\textbf{P}(Per(G_{M}\in G(n,M=p\binom{n}{2}),p_{n})\text{ is
connected})$ (4.143) $\displaystyle\approx\textbf{P}(G_{p}\in
G(n,pp_{n}))\text{ is connected}),\text{ as }n\rightarrow\infty$ (4.144)
We can calculate $\textbf{P}(G_{p}\in G(n,pp_{n}))\text{ is connected})$ from
theorem 10 for $pp_{n}=c\log n/n$, as $n\rightarrow\infty$. Therefore, we
should consider $p_{n}=c(n-1)\log n/(2m)$. Finally, one can calculate the
probability that $G_{M}\in G(n,M=m)$ is connected after percolation with the
parameter $p_{n}=c(n-1)\log n/(2m)$ as:
$\displaystyle\lim_{n\rightarrow\infty}\textbf{P}(Per(G_{M}\in
G(n,M),p_{n})\text{ is connected})=\left\\{\begin{array}[]{ll}1&\mbox{for
$p_{n}$'s such that $c>1$};\\\ 1-e^{-1}&\mbox{for $p_{n}$'s such that
$c=1$};\\\ 0&\mbox{for $p_{n}$'s such that $c<1$}.\end{array}\right.$ (4.148)
Now, let us calculate $\textbf{P}(Per(G_{M}\in G(2^{n},n2^{n-1}),p_{n})\text{
is connected)}$. From (4.148), for $p_{n}=c(1-\frac{1}{2^{n}})$, one gets:
$\displaystyle\lim_{n\rightarrow\infty}\textbf{P}(Per(G_{M}\in
G(2^{n},n2^{n-1}),p_{n})\text{ is
connected})=\left\\{\begin{array}[]{ll}1&\mbox{for $p_{n}$'s such that $c>\log
2$};\\\ 1-e^{-1}&\mbox{for $p_{n}$'s such that $c=\log 2$};\\\ 0&\mbox{for
$p_{n}$'s such that $c<\log 2$}.\end{array}\right.$ (4.152)
Finally, it can be concluded that,
$\lim_{n\rightarrow\infty}\textbf{P}(Per(G_{M}\in
G(2^{n},n2^{n-1}),p^{\prime})\text{ is connected})=1$ for $0.3\approx\log
2<p^{\prime}<0.5$, while
$\lim_{n\rightarrow\infty}\textbf{P}(Q^{n}_{p^{\prime}}\text{ is
connected})=0$ for $\log 2<p^{\prime}<0.5$. Hence, a typical graph with
$2^{n}$ vertices and $n2^{n-1}$ edges is ”more connected” than $Q^{n}$.
## Chapter 5 Connected random subgraphs of the 3-cube
111The proof presented for the 3-cube, in this chapter, is independent of the
proof presented in the previous chapter; therefore, some parts of the proofs
overlap each other.
The 3-cube graph, ${}^{3}Q^{n}$, is a simple graph with $3^{n}$ vertices. If
one labels each vertex of ${}^{3}Q^{n}$ from $0$ to $3^{n}-1$, then two
vertices are adjacent if their ternary representation differs only in one
digit. The number of edges in ${}^{3}Q_{n}$ is $n3^{n}$, since each vertex is
connected to $2n$ vertices. A random subgraph of ${}^{3}Q^{n}$ contains all
vertices of ${}^{3}Q^{n}$, and each edge independently with probability
$p_{n}$. ${}^{3}Q_{p_{n}}^{n}$ stands for a random subgraph of ${}^{3}Q^{n}$.
The main goal in this chapter is to explore a critical value $p_{c}$, which
for fixed values of $p$ if $p<p_{c}$ then the probability that
${}^{3}Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends to $0$; but
if $p>p_{c}$ then the probability that ${}^{3}Q_{p_{n}}^{n}$ is connected, as
$n\rightarrow\infty$, tends to $1$. We prove that this critical value is
$(\sqrt{3}-1)/(\sqrt{3})$.
In the first section of this chapter, first the probability that
${}^{3}Q_{p_{n}}^{n}$ has no isolated point as $n\rightarrow\infty$, for fixed
$p$, is investigated. It is proved that for $p<(\sqrt{3}-1)/(\sqrt{3})$ the
probability that ${}^{3}Q_{p_{n}}^{n}$ has no isolated point, as
$n\rightarrow\infty$, tends to $0$. Therefore, for $p<(\sqrt{3}-1)/(\sqrt{3})$
the probability that ${}^{3}Q_{p_{n}}^{n}$ is connected, as
$n\rightarrow\infty$, tends to $0$. Then it is proved that, for
$p>(\sqrt{3}-1)/(\sqrt{3})$ the probability that ${}^{3}Q_{p_{n}}^{n}$ has no
isolated point, as $n\rightarrow\infty$, tends to $1$. In the next step, the
probability that ${}^{3}Q_{p_{n}}^{n}$ has no isolated point, as
$n\rightarrow\infty$, when $p$ depends on $n$ and it is close
$(\sqrt{3}-1)/(\sqrt{3})$, is explored. It is proved that for
$\lambda(n)=\lambda>0$ and $p_{n}=1-(1/\sqrt{3})\lambda^{1/2n}(1+o(1/n)$, the
probability that ${}^{3}Q_{p_{n}}^{n}$ has no isolated point, as
$n\rightarrow\infty$, tends to $e^{-\lambda}$. Finally, it is proved that, for
fixed $p=(\sqrt{3}-1)/(\sqrt{3})$ the probability that ${}^{3}Q_{p_{n}}^{n}$
has no isolated point, as $n\rightarrow\infty$, tends to $e^{-1}$.
In the second section, one sheds light on the Isoperimetric problem, which is
the problem of finding an inequality which relates the size of a subgraph to
the size of its boundary. One needs such an inequality to explore the
probability that ${}^{3}Q_{p_{n}}^{n}$ has a component which is not the whole
graph.
In the last section, the Isoperimetric inequality is applied to prove that
when $p$ depends on $n$ and $p_{n}\geq 1-(1/\sqrt{3})(\log n)^{1/n}$, then the
probability that there are no components with size larger than $1$ and smaller
than $3^{n}$ in ${}^{3}Q_{p_{n}}^{n}$ , as $n\rightarrow\infty$, tends to $1$.
Therefore, for $p_{n}=1-(1/\sqrt{3})\lambda^{1/2n}(1+o(1/n))$, the probability
that ${}^{3}Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends to
$e^{-\lambda}$. Finally, as a special case, it is shown that, for fixed $p$ if
$p=(\sqrt{3}-1)/(\sqrt{3})$, the probability that ${}^{3}Q_{p_{n}}^{n}$ is
connected , as $n\rightarrow\infty$, tend to $e^{-1}$; and if
$p>(\sqrt{3}-1)/(\sqrt{3})$, the probability that ${}^{3}Q_{p_{n}}^{n}$ is
connected, as $n\rightarrow\infty$, tends to $1$.
### 5.1 Isolated vertices
For $p_{n}<\frac{\sqrt{3}-1}{\sqrt{3}}$:
First, one should consider the following definitions:
###### Definition 23.
$f_{n}(p_{n})$:=P(${}^{3}Q_{p_{n}}^{n}$ is connected)
###### Definition 24.
$g_{n}(p_{n}):=\textbf{P}(^{3}Q_{p_{n}}^{n}\text{contains an isolated point})$
###### Definition 25.
$X_{i}(n):=\left\\{\begin{array}[]{ll}1&\mbox{Vertex $i$ is isolated, $i\in
V(^{3}Q_{p_{n}}^{n}$)};\\\ 0&\mbox{Vertex $i$ is NOT isolated, $i\in
V(^{3}Q_{p_{n}}^{n}$)}.\end{array}\right.$ , and
$X(n):=\displaystyle\sum_{i\in V(^{3}Q_{p_{n}}^{n})}X_{i}(n)$.
Now, calculate $E[X(n)]$ and $Var[X(n)]$ as follows:
$\displaystyle\mu:=E[X(n)]=\sum_{i\in
V(^{3}Q_{p_{n}}^{n})}E[X_{i}(n)]=\sum_{i\in
V(^{3}Q_{p_{n}}^{n})}(1-p)^{2n}=3^{n}(1-p)^{2n}$ (5.1) $\displaystyle
Var[X(n)]=\displaystyle\sum_{i\in
V(^{3}Q_{p_{n}}^{n})}Var[X_{i}(n)]+\displaystyle\sum_{i\neq j;i,j\in
V(^{3}Q_{p_{n}}^{n})}Cov[X_{i}(n),X_{j}(n)]$ (5.2)
where, $Var[X_{i}(n)]$ and $Cov[X_{i}(n),X_{j}(n)]$ are equal to:
$\displaystyle\displaystyle\sum_{i\in V(^{3}Q_{p_{n}}^{n})}Var[X_{i}(n)]$
$\displaystyle=3^{n}(1-p)^{2n}-3^{n}(1-p)^{2n}(1-p)^{2n}=\mu-\mu(1-p)^{2n}$
(5.3) $\displaystyle Cov[X_{i}(n),X_{j}(n)]$
$\displaystyle=E[X_{i}(n)X_{j}(n)]-E[X_{i}(n)]E[X_{j}(n)]$ (5.4)
$\displaystyle=\left\\{\begin{array}[]{ll}0&\mbox{i,j not adjacent};\\\
(1-p)^{2n}(1-p)^{2n-1}-(1-p)^{2n}(1-p)^{2n}=\frac{\mu^{2}}{3^{2n}}(\frac{p}{1-p})&\mbox{i,j
adjacent}.\end{array}\right.$ (5.7)
and finally:
$\displaystyle
Var[X(n)]=\mu-\mu(1-p)^{2n}+\frac{\mu^{2}}{3^{n}}(\frac{2np}{1-p})=\mu+\mu(1-p)^{2n}(\frac{2np}{1-p}-1)$
(5.8)
Now, since we have $Var[X(n)]$, we can use Chebyshev’s inequality to estimate
$g_{n}(p)$. Chebyshev’s inequality states that:
$\displaystyle
1-g_{n}(p)=\textbf{P}[X(n)=0]\leq\textbf{P}[|X(n)-\mu|\geq\mu]\leq\frac{Var[X(n)]}{\mu^{2}}$
(5.9)
By applying Chebyshev’s inequality when $p<(\sqrt{3}-1)/(\sqrt{3})$, one gets
$Var[X(n)]/\mu^{2}\rightarrow 0$, as $n\rightarrow\infty$. Therefore
$\lim_{n\rightarrow\infty}g_{n}(p)=1$. And finally, since $f_{n}(p)\leq
1-g_{n}(p)$, then for $p<(\sqrt{3}-1)/(\sqrt{3})$ the probability that
${}^{3}Q_{p_{n}}^{n}$ is connected, as $n\rightarrow\infty$, tends to $0$.
$\blacksquare$
For $p_{n}>\frac{\sqrt{3}-1}{\sqrt{3}}$:
Assume $p$ is fixed and $p>(\sqrt{3}-1)/(\sqrt{3})$. In order to calculate
$g_{n}(p)$ when $p>(\sqrt{3}-1)/(\sqrt{3})$, as $n\rightarrow\infty$, one can
use the following inequality:
$\displaystyle g_{n}(p)=\textbf{P}[X(n)>0]\leq E[X(n)]=\mu$ (5.10)
Since $E[X(n)]\rightarrow 0$ as $n\rightarrow\infty$, then
$\lim_{n\rightarrow\infty}g_{n}(p)=0$. This means that the probability that
there are no isolated points in ${}^{3}Q_{p_{n}}^{n}$ for
$p>(\sqrt{3}-1)/(\sqrt{3})$, as $n\rightarrow\infty$, tends to $1$.
$\blacksquare$
For $p_{n}=1-(1/\sqrt{3})\lambda^{1/2n}(1+o(1/n))$:
One needs the following theorem from [1] to find the distribution of $X(n)$
(distribution of the number of isolated points).
###### Theorem 11.
Let $\lambda=\lambda(n)$ be a non-negative bounded function on N. Suppose the
non-negative integer valued random variables $X(1),X(2),...$ are such that:
$\displaystyle\lim_{n\rightarrow\infty}\\{E_{r}[X(n)]-\lambda^{r}\\}=0,\text{
}r=0,1,...$ (5.11)
where $E_{r}[X]$ is the $r$th factorial moment of $X$, i.e.
$E_{r}[X]=E[(X)_{r}]$. Then
$\displaystyle X(n)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\textbf{P}_{\lambda}$ (5.12)
Use the definition of $X(n)$ presented in definition 25. The goal is to
calculate $E[X(n)]$.
$\displaystyle E_{r}[X(n)]=E[X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)]$ (5.13)
Since $X(n):=\sum_{i\in V(Q_{p}^{n})}X_{i}(n)$ and $X_{i}$’s are indicator
functions, therefore:
$\displaystyle
X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)=\sum_{(i_{1},i_{2},...,i_{r})}X_{i_{1}}X_{i_{2}}...X_{i_{r}}$
(5.14)
where the sum is over all ordered sets of distinct vertices. Then:
$\displaystyle E_{r}[X(n)]$
$\displaystyle=E[X(n)(X(n)-1)(X(n)-2)...(X(n)-r+1)]$ (5.15)
$\displaystyle=E[\sum_{(i_{1},i_{2},...,i_{r})}X_{i_{1}}X_{i_{2}}...X_{i_{r}}]$
(5.16)
$\displaystyle=\sum_{(i_{1},i_{2},...,i_{r})}\textbf{P}[X_{i_{1}}=1,X_{i_{2}}=1,...,X_{i_{r}}=1]$
(5.17)
One knows that a set of $r$ vertices is incident with at most $2rn$ edges.
There are $(r)_{r}\binom{3^{n}}{r}$ ways to choose such $r$ vertices. Hence:
$\displaystyle
E_{r}[X(n)]\geq(r)_{r}\binom{3^{n}}{r}(1-p_{n})^{2rn}=(3^{n})_{r}(1-p_{n})^{2rn}$
(5.18)
One the other hand, a set of $r$ vertices is incident with at least $r(2n-r)$
edges. There are at most $(r-1)_{r-1}\binom{3^{n}}{r-1}(r-1)2n$ ways to choose
a set of $r$ vertices in ${}^{3}Q_{p_{n}}^{n}$ where at least two vertices are
adjacent; since if we choose $r-1$ vertices independently, then the last
vertex must be connected to one of the chosen vertices. In other words, there
are at most $(r-1)_{r-1}\binom{3^{n}}{r-1}(r-1)2n$ ways to choose $r$ vertices
which some of them are adjacent to each other. Hence:
$\displaystyle E_{r}[X(n)]$
$\displaystyle\leq(3^{n})_{r}(1-p_{n})^{2rn}+(r-1)_{r-1}\binom{3^{n}}{r-1}2(r-1)n(1-p_{n})^{r(2n-r)}$
(5.19)
$\displaystyle\leq(3^{n})_{r}(1-p_{n})^{2rn}+(3^{n})_{r}2rn(1-p_{n})^{r(2n-r)}$
(5.20)
$\displaystyle\leq(3^{n})_{r}(1-p_{n})^{2rn}+3^{n(r-1)}2rn(1-p_{n})^{r(2n-r)}$
(5.21)
Finally from 5.18 and 5.21 one gets:
$\displaystyle(3^{n})_{r}(1-p_{n})^{2rn}\leq
E_{r}[X(n)]\leq(3^{n})_{r}(1-p_{n})^{2rn}+3^{n(r-1)}2rn(1-p_{n})^{r(2n-r)}$
(5.22)
which gives:
$\displaystyle(3(1-p_{n})^{2})^{rn}(1-\frac{r}{3^{n}})^{r}\leq
E_{r}[X(n)]\leq(3(1-p_{n})^{2})^{rn}\\{1+3^{-n}2rn(1-p_{n})^{-r^{2}}\\}$
(5.24)
Since $r$ is fixed and
$\lim_{n\rightarrow\infty}(3(1-p_{n})^{2})^{n}=\lambda$, then:
$\displaystyle\lim_{n\rightarrow\infty}(E_{r}[X(n)])=\lambda^{r}\text{ for
r=0,1,2,...}$ (5.25)
This shows that $X(n)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\textbf{P}_{\lambda}$. $\blacksquare$
For $p=0.5$:
In the calculation of $p_{n}=1-1/\sqrt{3}\lambda^{1/2n}(1+o(1/n))$, if we fix
$p=(\sqrt{3}-1)/(\sqrt{3})$ and let $\lambda=1$, then we get that the
distribution of $X(n)$ , as $n\rightarrow\infty$, tends to a Poisson
distribution with mean $1$. Therefore, one can conclude:
$\displaystyle\lim_{n\rightarrow\infty}(1-g_{n}(p))=\lim_{n\rightarrow\infty}(\textbf{P}(X(n)=0))=e^{-1}$
(5.26)
This shows that for $p=(\sqrt{3}-1)/(\sqrt{3})$ the probability that
${}^{3}Q_{p_{n}}^{n}$ has no isolated point, as $n\rightarrow\infty$, tends to
$e^{-1}$. $\blacksquare$
### 5.2 Isoperimetric problem for the 3-cube
One needs an inequality which relates the size of a subgraph of ${}^{3}Q^{n}$
to the size of its boundary. This inequality will be applied to prove that for
$p\geq(\sqrt{3}-1)/(\sqrt{3})$, the probability that subgraphs of
${}^{3}Q^{n}$ do not have a component of size larger than $2$ and smaller than
$3^{n}$, as $n\rightarrow\infty$, tends to $1$.
###### Definition 26.
The edge boundary $b_{G}(H)$, where $H$ is an induced subgraph of G, is the
number of edges which joins vertices in $H$ to the vertices in $G\backslash
H$.
The main task in this section is to calculate $b_{{}^{3}Q^{n}}(k)$. Since
${}^{3}Q^{n}$ is 2n-regular and $H$ is an induced subgraph of $G$ with
$|V(H)|=m$, then:
$\displaystyle b_{{}^{3}Q^{n}}(H)$ $\displaystyle=mn-2e(H),\text{where e(H) is
the total number of edges in H.}$ (5.27) $\displaystyle b_{{}^{3}Q^{n}}(k)$
$\displaystyle=mn-2e_{n}(k),\text{where }e_{n}(k)=max\\{e(H):H\text{ induced
subgraph of }^{3}Q^{n},|V(H)|=m\\}.$ (5.28)
###### Definition 27.
$h(i):=$ sum of digits in the ternary expansion of $i$ and
$f(l,m):=\displaystyle\sum_{l\leq i<m}h(i)$
###### Lemma 3.
If $1\leq k\leq l$, then $f(l,l+k)\geq f(0,k)+k$
Proof:
Look at the ternary expansion of a few numbers:
$\begin{array}[]{ccccc}Column&2&1&0\\\ Bin\backslash Dec&3^{2}&3^{1}&3^{0}\\\
0&&&0\\\ 1&&&1\\\ 2&&&2\\\ 3&&1&0\\\ 4&&1&1\\\ 5&&1&2\\\ 6&&2&0\\\ 7&&2&1\\\
8&&2&2\\\ 9&1&0&0\\\ 10&1&0&1\\\ 11&1&0&2\\\ 12&1&1&0\\\ 13&1&1&1\\\
14&1&1&2\\\ 15&1&2&0\\\ 16&1&2&1\\\ 17&1&2&2\\\ 18&2&0&0\\\ 19&2&0&1\\\
20&2&0&2\\\ 21&2&1&0\end{array}$
From this representation, one can observe that column $i$ starts with a block
of $3^{i}$ zeros. Therefore, sum of jth digits of $k$ numbers, one after the
other, is minimal if the first block of $0$’s is as long as possible. Hence,
one can conclude:
$\displaystyle f(l,l+k)\geq f(0,k)$ (5.29)
For every $i$ define $r$ such that $0\leq i\leq 3^{r}-1$. The ternary
expansions of $i$ and $3^{r}-1-i$ are symmetric. This means that, if there is
a $2/0$ in an specific location of the ternary expansion of $i$ then there is
a $0/2$ in the same location of the ternary expansion of $3^{r}-1-i$; and if
there is a $1$ in an specific location of the ternary expansion of $i$ then
there is a $1$ in the same location of the ternary expansion of $3^{r}-1-i$.
Therefore,
$\displaystyle h(i)+h(3^{r}-1-i)=2r\text{ for }0\leq i\leq 3^{r}-1$ (5.30)
Consequently, since:
$\displaystyle\displaystyle\sum_{l\leq
i<l+k}h(i)+\displaystyle\sum_{3^{r}-l-k\leq i<3^{r}-l}h(i)=2rk$ (5.31)
then:
$\displaystyle f(l,l+k)+f(3^{r}-l-k,3^{r}-l)=2rk,\text{where }l+k\leq 3^{r}$
(5.32)
One should first prove lemma 3 with the assumption $k\leq 3^{r}\leq l$ by
using 5.29 and 5.32. This assumption means that the length of the sequence of
0’s, 1’s and 2’s in the ternary expansion of $3^{r}+k$ and $3^{r}$ are equal.
With the same logic that one gets 5.29, one gets:
$\displaystyle f(l,l+k)\geq f(3^{r},3^{r}+k)\text{, when }3^{r}\leq l$ (5.33)
and then for $k\leq 3^{r}$ one can get:
$\displaystyle f(3^{r},3^{r}+k)=\displaystyle\sum_{3^{r}\leq i<3^{r}+k}h(i)$
(5.34)
$\sum_{3^{r}\leq i<3^{r}+k}h(i)$ is the sum over numbers with the same length
in their ternary expansion’s sequence. When one removes the last digit in
their ternary expansion, what remains is $f(0,k)$. Therefore, $\sum_{3^{r}\leq
i<3^{r}+k}h(i)$ is equal to sum of the last digits plus $f(0,k)$. Hence:
$\displaystyle f(l,l+k)\geq f(3^{r},3^{r}+k)=k+f(0,k)\text{ when }k\leq
3^{r}\leq l$ (5.35)
and finally:
$\displaystyle f(l,l+k)\geq f(0,k)+k\text{ where }k\leq 3^{r}\leq l$ (5.36)
Lemma 3 is proved with the assumption $k\leq 3^{r}\leq l$. Now, one should
prove lemma 3 without this assumption. The proof is based on induction on $K$.
We want to prove that for $1\leq K\leq l$, $f(l+K,l)\geq K+f(0,K)$. Fix $k$
such that $1\leq k\leq l$ and $K<k$. For $K=1$ the inequality is trivial.
Assume that the inequality is true for $K<k$ and $K>2$, which means:
$\displaystyle f(l,l+K)\geq K+f(0,K)\text{ when }1\leq k\leq l\text{ , and
}K<k$ (5.37)
Now, one should prove the inequality for $K=k$. Define $r\geq 1$ by
$3^{r-1}\leq k<3^{r}$. If $l\geq 3^{r}$, then $k\leq 3^{r}\leq l$ and the
lemma is implied by inequality 5.36. One may assume that $3^{r-1}<l<3^{r}$.
Finally, one should apply inequality 5.32 and 5.37 in order to get the final
result:
$\displaystyle f(l+k)$ $\displaystyle=f(l,3^{r})+f(3^{r},l+k)\text{\emph{
(from definition of f and }}l\geq 3^{r})$ (5.38)
$\displaystyle=(3^{r}-l)2r-f(0,3^{r}-l)+f(3^{r},l+k)\text{ \emph{ (from
\ref{2.5})}}$ (5.39)
$\displaystyle\geq(3^{r}-l)2r-f(0,3^{r}-l)+f(0,l+k-3^{r})+l+k-3^{r}\text{\emph{
(from \ref{2.7})}}$ (5.40)
$\displaystyle\geq(3^{r}-l)2r-f(3^{r}-k,3^{r}-k+3^{r}-l)+3^{r}-l+f(0,l+k-3^{r})+l+k-3^{r}\text{
\emph{(from \ref{2.7})}}$ (5.41)
$\displaystyle\geq(3^{r}-l)2r-f(3^{r}-k,3^{r}-k+3^{r}-l)+f(0,l+k-3^{r})+k$
(5.42) $\displaystyle\geq f(l+k-3^{r},k)+f(0,l+k-3^{r})+k\text{ \emph{(from
\ref{2.5})}}$ (5.43) $\displaystyle\geq f(0,k)+k\text{ \emph{(from
characteristics of f)}}$ (5.44)
$\blacksquare$
###### Theorem 12.
For $2\leq m\leq 3^{n}$ we have $b_{{}^{3}Q^{n}}(m)=mn-2f(0,m)$. In other
words, $f(0,m)=e_{n}(m)\text{ where }e_{n}(m)=\max\\{e(H):H\text{ induced
subgraph of ${}^{3}Q^{n}$ }|V(H)|=m\\}.$
Proof:
First, let us fix an $m$. As the first step, one should prove that
$e_{n}(m)\geq f(0,m)$. Vertex $i$ is connected to $h(i)$ vertices $j$ with
$j<i$, since for each $1$ ($2$) in the ternary expansion of $i$ there is
exactly one j $(j<i)$, which its ternary expansion differs in the position of
that $1$ ($2$). Therefore, on can conclude that $W=\\{0,1,2,...,m-1\\}$
contains $\sum_{0\leq i<m}h(i)=f(0,m)$ edges. So, $e_{n}(m)\geq f(0,m)$.
As the second step, one should prove that $e_{n}(m)\leq f(0,m)$ by induction
on $n$ for fixed $m$. For $n=1$ the inequality is trivially true. Assume that
it is true for $N<n$, which means:
$\displaystyle e_{N}(m)\leq f(0,m),\text{ where }N<n\text{ and the fixed m is
}3\leq m\leq 3^{n}$ (5.45)
Now, one should check the inequality 5.45 for $N=n$. This means that we should
find an $H$ induced subgraph of ${}^{3}Q^{n}$, $|V(H)|=m$, which maximize
$e_{n}(m)$. Let us split ${}^{3}Q^{n}$ into three (n-1)-dimensional cubes,
face-1, face-2 and face-3 each with $3^{n-1}$ vertices and $(n-1)3^{n-1}$
edges. Now, one can construct $H$. Choose $m_{1}$ vertices for $H$ from
face-1, $m_{2}$ vertices from face-2, and $m_{3}$ vertices from face-3 where
$m_{1}+m_{2}+m_{3}=m$ and $m_{1}\leq m_{2}\leq m_{3}$ and $m_{1}+m_{2}\leq
m_{3}$. In other words, $H$ is constructed from three induced subgraphs,
denoted by $H_{1}$, $H_{2}$, and $H_{3}$.
Each face is a (n-1)-dimensional cube, therefore inequality 5.45 holds for
$H_{1}$, $H_{2}$ and $H_{3}$. Also, each vertex of each face is connected to
exactly on vertex from one face and one vertex from the other face. Hence, the
number of vertices of $H$ is at most:
$\displaystyle e_{n}(m)\leq f(0,m_{1})+f(0,m_{2})+f(0,m_{3})+2m_{1}+m_{2}$
(5.46)
$2m_{1}+m_{2}$ is the maximum number of edges between three faces that one can
choose here. $2m_{1}$ is the maximum number of edges between chosen vertices
in $H_{1}$ and $H_{2}$ plus the maximum number of edges between chosen
vertices in $H_{1}$ and $H_{3}$. Consequently, $m_{2}$ is the maximum number
of edges between chosen vertices in $H_{2}$ and $H_{3}$. Therefore:
$\displaystyle e_{n}(m)$ $\displaystyle\leq
f(0,m_{1})+f(0,m_{2})+f(0,m_{3})+2m_{1}+m_{2}\text{ (from \ref{2.21})}$ (5.47)
$\displaystyle\leq f(m_{2},m_{2}+m_{1})+f(0,m_{2})+m_{1}+m_{2}\text{ (from
lemma \ref{2.1})}$ (5.48) $\displaystyle\leq
f(m_{3},m_{3}+m_{2}+m_{1})+f(0,m_{3})\text{ (from lemma \ref{2.1})}$ (5.49)
$\displaystyle\leq f(0,m)\text{ (from definition of f)}$ (5.50)
$\blacksquare$
Theorem 12 shows that, if we want to choose an induced subgraph of
${}^{3}Q^{n}$, with $m$ vertices, which has the smallest edge boundary, then
we should choose the induced subgraph of ${}^{3}Q^{n}$ with the set of
vertices $W=\\{0,1,2,...,m-1\\}$.
###### Corollary 2.
For all $k$ and $n$, $e_{n}(k)\leq k\lceil log_{3}k\rceil$, which is
equivalent to $b_{{}^{3}Q^{n}}(k)\geq 2k(n-\lceil log_{3}k\rceil)$.
Proof:
Let $r=\lceil\log_{3}k\rceil$. Then
$\displaystyle 2f(0,k)$ $\displaystyle=f(0,k)+f(0,k)\leq
f(0,k)+f(3^{r}-k,3^{r})\text{ (from \ref{2.2})}$ (5.51)
$\displaystyle=2rk\text{ (from \ref{2.5})}$ (5.52)
Therefore:
$\displaystyle e_{n}(k)=f(0,k)\leq rk=k\lceil\log_{3}k\rceil$ (5.53)
$\blacksquare$
### 5.3 Isolated components of size larger than 2 and smaller than $3^{n}$
###### Definition 28.
$C_{s}$ is the family of s-subsets (subsets with size s) of $V=V(^{3}Q^{n})$
whose induced graph is connected.
Remarks: $h(n):=o(g(n))$ means $\frac{h(n)}{g(n)}\rightarrow 0$ as
$n\rightarrow\infty$.
Remarks: The following inequality will be applied a lot in the rest of this
section:
$\displaystyle(\frac{n}{k})^{k}\leq\binom{n}{k}\leq\frac{n^{k}}{k!}\leq(\frac{ne}{k})^{k}$
(5.54)
###### Theorem 13.
If $p_{n}\geq 1-\frac{1}{\sqrt{3}}(\log n)^{\frac{1}{n}}$, the probability
that for some $S\in C_{s},2\leq s\leq\frac{3^{n}}{2}$, no edges of
${}^{3}Q_{p_{n}}^{n}$ join $S$ to $V(^{3}Q^{n})\setminus S$, as
$n\rightarrow\infty$, tends to 0.
Note: For $\frac{3^{n}}{2}<s<3^{n}$, if there exist a component of size
smaller than $3^{n}$ then there is at least one component of size smaller than
$\frac{3^{n}}{2}$ which contradicts with the theorem.
Proof:
Consider $S\subset V=V(^{3}Q^{n})$ and set $b(S)=b_{Q^{n}}(H)$ where $H$ is
the induced subgraph of ${}^{3}Q^{n}$ with the set of vertices S. One can
observe that:
$\displaystyle\textbf{P}(\text{No edges of }^{3}Q_{p_{n}}^{n}\text{ join S to
}V\setminus S)=(1-p_{n})^{b(S)}$ (5.55)
In order to prove the theorem, it is sufficient to show:
$\displaystyle\sum_{s=2}^{3^{\frac{n}{2}}}\sum_{S\in
C_{s}}(1-p_{n})^{b(S)}=o(1)$ (5.56)
From corollary 2, one knows that for $|S|=s$:
$\displaystyle b(S)\geq b(s)\geq 2s(n-\lceil\log_{3}s\rceil)$ (5.57)
and therefore:
$\displaystyle\sum_{S\in C_{s}}(1-p_{n})^{b(S)}\leq|C_{s}|(1-p_{n})^{b(s)}$
(5.58)
One may partition $s$, $2\leq s\leq\frac{3^{n}}{2}$, to different intervals in
order to find a small enough bound for $|C_{s}|$ and $(1-p_{n})^{b(s)}$.
First interval $2\leq s\leq
s_{1},s_{1}=\lfloor\frac{3^{\frac{n}{2}}}{n^{2}}\rfloor$:
First, one should find a bound for $|C_{s}|$. One has maximum $3^{n}$ choices
to choose the first element for $C_{s}$. The selected element is connected to
maximum $2n$ vertices, therefore, there are maximum $2n$ choices to choose the
second element. With the same logic, there are at most $2n(s-1)$ choices to
choose the last element for $|C_{s}|$. Hence, one can show:
$\displaystyle|C_{s}|\leq
3^{n}(2n)(2n(2))...(2n(s-1))\leq(s-1)!(2n)^{s-1}3^{n}$ (5.59)
and:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq(s-1)!(2n)^{s-1}3^{n}(1-p_{n})^{2s(n-\lceil\log_{3}s\rceil)}$
(5.60)
Since $p_{n}=1-\frac{1}{\sqrt{3}}(\log n)^{\frac{1}{n}}$, so for large enough
$n$:
$\displaystyle(1-p_{n})^{2s(n-\lceil log_{3}s\rceil)}$
$\displaystyle\leq(3)^{-ns}(\log
n)^{2s}(1-p_{n})^{-2s(\log_{3}s)}\text{(neglecting some small terms)}$ (5.61)
$\displaystyle\text{( since: }(\log n)^{\frac{-2s\log_{3}s}{n}}\leq 1\text{
for large enough }n)$ (5.62) $\displaystyle=(3)^{-ns}(\log
n)^{2s}3^{s\log_{3}s}(\log n)^{\frac{-2s\log_{3}s}{n}}$ (5.63)
$\displaystyle\leq(3)^{-ns}(\log n)^{2s}s^{s}$ (5.64)
From equations 5.60 and 5.64, one gets:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq(s-1)!(2n)^{s-1}3^{n}(3)^{-ns}(\log
n)^{2s}s^{s}$ (5.65)
Assume that the right hand sides of inequality 5.65 is equal to A. After
multiplying both side of inequality 5.65 with $\frac{2ns^{s+1}}{s!}$ and then
getting $\log_{3}$ from both sides, one gets:
$\displaystyle\log_{3}(|C_{s}|(1-p_{n})^{b(s)}\frac{2ns^{s+1}}{s!})\leq\log_{3}(A\frac{2ns^{s+1}}{s!})$
(5.66)
If $\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$
then $A\frac{2ns^{s+1}}{s!}$ should tend to 0\. This means that
$|C_{s}|(1-p_{n})^{b(s)}\frac{2ns^{s+1}}{s!}$ tends to 0, as
$n\rightarrow\infty$. Therefore:
$\displaystyle|C_{s}|(1-p_{n})^{b(s)}\leq\frac{s!}{2ns^{s+1}}\text{ for large
values of n}$ (5.67)
which shows that:
$\displaystyle\sum_{s=2}^{s_{1}}\sum_{S\in C_{s}}(1-p_{n})^{b(S)}=o(1)$ (5.68)
Finally, it remains to prove
$\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$.
One can verify this for $s\leq n$ and $s>n$. $\blacksquare$
Second interval $s_{1}+1\leq s\leq\frac{3^{n}}{2}$ and $S\in
C_{s}^{-},s_{1}=\lfloor\frac{3^{\frac{n}{2}}}{n^{2}}\rfloor$:
Define $C_{s}^{-}$ and $C_{s}^{+}$ as follows:
###### Definition 29.
$\displaystyle C_{s}^{-}:=\\{S\in C_{s}|b(s)\geq
2s(n-\log_{3}s+\log_{3}n)\\}\text{, and }C_{s}^{+}:=C_{s}\backslash C_{s}^{-}$
(5.69)
One can bound $|C_{s}^{-}|$ for $s_{1}+1\leq s\leq 3^{\frac{n}{2}}$ as
follows:
$\displaystyle|C_{s}^{-}|\leq|C_{s}|\leq\binom{3^{n}}{s}\leq\frac{3^{ns}}{s!}\leq(\frac{e3^{n}}{s})^{s}$
(5.70)
Hence:
$\displaystyle\displaystyle\sum_{s=s_{1}+1}^{3^{\frac{n}{2}}}\sum_{S\in
C_{s}^{-}}(1-p_{n})^{b(S)}$
$\displaystyle\leq\sum_{s=s_{1}+1}^{3^{\frac{n}{2}}}(\frac{e3^{n}}{s})^{s}(\sqrt{3}^{-1}(\log
n)^{\frac{1}{n}})^{2s(n-\log_{3}s+\log_{3}n)}$ (5.71)
$\displaystyle\leq\sum_{s=s_{1}+1}^{3^{\frac{n}{2}}}(\frac{e3^{n}3^{-(n-\log_{3}s+\log_{3}n)}(\log
n)^{\frac{2(n-\log_{3}s+\log_{3}n)}{n}}}{s})^{s}$ (5.72)
$\displaystyle\leq\sum_{s=s_{1}+1}^{3^{\frac{n}{2}}}(\frac{e3^{n}3^{-n}3^{\log_{3}s}3^{-\log_{3}n}(\log
n)^{2}}{s})^{s}(\log n)^{\frac{2n(-\log_{3}s+\log_{3}n)}{s}}$ (5.73)
$\displaystyle\text{ (since for large enough n: }(\log
n)^{\frac{2n(-\log_{3}s+\log_{3}n)}{s}}\leq 1)$ (5.74)
$\displaystyle\leq\sum_{s=s_{1}+1}^{3^{\frac{n}{2}}}(\frac{e(\log
n)^{2}}{n})^{s}=o(1)$ (5.75)
$\blacksquare$
Third interval $s_{1}\leq s\leq
s_{2},s_{1}=\lfloor\frac{3^{\frac{n}{2}}}{n^{2}}\rfloor,s_{2}=\lfloor\frac{3^{n}}{(\log
n)^{7}}\rfloor$ and $S\in C_{s}^{+}$:
For the 3rd and the 4th intervals one needs to find a bound for $|C_{s}^{+}|$.
The following lemma, presented by B.Bollobas [1], helps us in this matter:
###### Lemma 4.
Let G be a graph of order $v$ and suppose that $\Delta(G)\leq\Delta$,
$2e(G)=vd$ and $\Delta+1\leq u\leq v-\Delta-1$. Then, there is a u-set of U of
vertices with [1]:
$\displaystyle|N(U)|=|U\cup\Gamma(U)|\geq
v\frac{d}{\Delta}\\{1-\exp(\frac{-u(\Delta+1)}{v})\\}$ (5.76)
where, $\Delta(G):=$ Maximum degree in G, $d:=$ average degree in G and
$\Gamma(U)=\\{x\in V(G):xy\in E(G)\text{ for some y}\in U\\}$
Let $H=$${}^{3}Q^{n}[S]$ (the induced subgraph of ${}^{3}Q_{n}$ with the set
of vertices $S$). From the definition of $C_{s}^{+}$ one knows that the
average degree in $H$ is at least:
$\displaystyle 2(\log_{3}s-\log_{3}n)$ (5.77)
The goal is to find $U\subset S$, where $|U|:=u:=\lfloor\frac{2s}{n}\rfloor$,
$\Delta=2n$, $v=s$, $d\geq\log_{3}s-\log_{3}n$ and then use lemma 4 to
calculate the boundary size of $U,|N(U)|$. First, check if
$2n+1\leq\lfloor\frac{2s}{n}\rfloor$, as $n\rightarrow\infty$:
$\displaystyle\frac{2s}{n}=\frac{23^{n/2}}{n^{3}}\text{for minimum s, and
trivially }2n+1\leq\frac{23^{n/2}}{n^{3}},\text{for large enough }n$ (5.78)
and then check if $\lfloor\frac{2s}{n}\rfloor\leq s-(2n+1)$. One should check
if $ns-n(2n+1)\geq 2s$ which means one should check that whether:
$\displaystyle\frac{3^{\frac{n}{2}}(n-2)}{n^{3}(2n+1)}\geq 1$ (5.80)
which is clearly true for large enough $n$. Now, one can apply lemma 4 on the
graph generated by $S$ and get:
$\displaystyle\exists U\subset S:|N(U)|\geq
s\frac{2(\log_{3}s-\log_{3}n)}{2n}\\{1-\exp(-\frac{u(2n+1)}{s})\\}$ (5.81)
where:
$\displaystyle\frac{2(\log_{3}s-\log_{3}n)}{2n}\geq\frac{(\log_{3}(\frac{3^{\frac{n}{2}}}{n^{2}})-\log_{3}n)}{n}=\frac{\frac{n}{2}-3\log_{3}n}{n}$
(5.82) $\displaystyle\text{which
}\lim_{n\rightarrow\infty}\frac{\frac{n}{2}-3\log_{3}n}{n}=\frac{1}{2}$ (5.83)
on the other hand:
$\displaystyle\lim_{n\rightarrow\infty}(1-\exp(-\frac{2n+1}{s}(\frac{2s}{n}+1)))=1-e^{-4}$
(5.84)
Therefore, from 5.83 and 5.84 one gets:
$\displaystyle|N(U)|\geq\frac{1}{2}(1-e^{-4})s\geq\frac{s}{3}\text{ as
}n\rightarrow\infty$ (5.85)
Now that we have $|N(U)|$, we can estimate a bound for $|C_{s}^{+}|$ here. We
know from 5.85 that for each $S\in C_{s}^{+}$ there exist a $U\subseteq S$,
$|U|:=u:=\lfloor\frac{2s}{n}\rfloor$, such that $|N(U)|\geq s/3$. Therefore,
one can choose $S\in C_{s}^{+}$ as follows:
1\. Select u vertices of ${}^{3}Q^{n}$; there are $\binom{3^{n}}{u}$ choices
for this u.
2\. Select $\lfloor\frac{s}{3}\rfloor-u$ neighbors of the selected vertices of
u in part 1; there are maximum $(2^{n})^{u}$ choices, since there are at most
$\binom{2n}{0}+\binom{2n}{1}+\binom{2n}{2}+...\binom{2n}{2n}=2^{2n}$ ways to
find neighbors of a vertex in $U$.
3\. Select $\lfloor\frac{2s}{3}\rfloor$ other vertices; there are at most
$\binom{3^{n}}{\lfloor\frac{2s}{3}\rfloor}$ choices.
Hence:
$\displaystyle|C_{s}^{+}|\leq\binom{3^{n}}{u}(2^{2n})^{u}\binom{3^{n}}{\lfloor\frac{2s}{3}\rfloor}$
(5.86)
and:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\binom{3^{n}}{u}(2^{2n})^{u}\binom{3^{n}}{\lfloor\frac{2s}{3}\rfloor}(1-p_{n})^{b(s)}$
(5.87)
where:
$\displaystyle(1-p_{n})^{b(s)}\leq 3^{-sn}s^{s}(\log n)^{2s}$ (5.88)
consequently from 5.87, 5.88 and 5.54:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq(\frac{e3^{n}}{u})^{u}2^{2un}(\frac{e3^{n}}{\lfloor\frac{2s}{3}\rfloor})^{\lfloor\frac{2s}{3}\rfloor}3^{-sn}s^{s}(\log
n)^{2s}$ (5.89)
Write $s=3^{\beta n}$, ($\beta=\frac{log_{3}s}{n}$), so that:
$\displaystyle 3^{\beta n}\leq\frac{3^{n}}{(\log n)^{7}}\Rightarrow\beta\leq
1-\frac{7\log_{3}\log n}{n}$ (5.90)
Now, find a bound for the inequality 5.89. First calculate the first part of
the inequality:
$\displaystyle(\frac{e3^{n}}{u})^{u}2^{2un}(\frac{e3^{n}}{\lfloor\frac{2s}{3}\rfloor})^{\lfloor\frac{2s}{3}\rfloor}$
$\displaystyle\leq(\frac{e3^{n}}{\frac{2s}{n}})^{\frac{2s}{n}}3^{2s}2^{4s}(\frac{e3^{n}}{\frac{2s}{3}})^{\frac{2s}{3}}$
(5.91) $\displaystyle\text{ (since for large enough n and $s_{1}\leq s\leq
s_{2}$: }(\frac{e3^{n}}{\frac{2s}{n}})^{\frac{2s}{n}}\leq 1)$ (5.92)
$\displaystyle\leq(3^{2}2^{4}(\frac{3}{2}e)^{\frac{2}{3}})^{s}\frac{3^{\frac{2s}{3n}}}{s^{\frac{2s}{3}}}=c^{s}3^{\frac{2}{3}sn(1-\frac{\log_{3}s}{n})}=c^{s}3^{\frac{2}{3}sn(1-\beta)}$
(5.93)
where c is a positive constant. Now, by substituting 5.93 in 5.89 one gets:
$\displaystyle\sum_{S\in C_{s}^{+}}(1-p_{n})^{b(S)}$ $\displaystyle\leq
3^{-sn}s^{s}(\log n)^{2s}c^{s}3^{\frac{2}{3}sn(1-\beta)}$ (5.94)
$\displaystyle=c^{s}(\log n)^{2s}3^{-\frac{sn(1-\beta)}{3}}$ (5.95)
$\displaystyle\leq c^{s}(\log n)^{2s}3^{-\frac{7s\log_{3}\log n}{3n}}\text{ ,
(from \ref{2.13})}$ (5.96) $\displaystyle=c^{s}(\log n)^{2s}3^{\log_{3}(\log
n)^{\frac{-7s}{3}}}$ (5.97) $\displaystyle\leq c^{s}(\log n)^{2s}(\log
n)^{\frac{-7s}{3}}$ (5.98) $\displaystyle=c^{s}(\log n)^{\frac{-s}{3}}$ (5.99)
and finally from 5.99:
$\displaystyle\sum_{s=s_{1}}^{s_{2}}\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\sum_{s=s_{1}}^{s_{2}}c^{s}(\log
n)^{\frac{-s}{3}}=o(1)$ (5.100)
$\blacksquare$
Fourth interval $s_{2}+1\leq s\leq\frac{3^{n}}{2}$ and
$,s_{2}=\lfloor\frac{3^{n}}{(\log n)^{9}}\rfloor,S\in C_{s}^{+}$:
In $H=$${}^{3}Q^{n}[S]$ (the induced subgraph ${}^{3}Q^{n}$ with the set of
vertices S), the average degree is at least:
$\displaystyle 2(\log_{3}s-\log_{3}n)>2(n-2\log_{3}n)$ (5.101)
since:
$\displaystyle s\geq\lceil\frac{3^{n}}{(\log
n)^{9}}\rceil\Rightarrow\log_{3}(\frac{3^{n}}{(\log n)^{9}})<\log_{3}s$
(5.102)
$\displaystyle\Rightarrow\log_{3}s-\log_{3}n\geq\log_{3}(\frac{3^{n}}{(\log
n)^{9}})-\log_{3}n\geq n-\log_{3}(\log n)^{9}-\log_{3}n$ (5.103)
$\displaystyle(\text{for large enough n one can get }n>(\log n)^{9})$ (5.104)
$\displaystyle\geq n-2\log_{3}n$ (5.105)
First, look for a subgraph of H with large average degree. Let T be the set of
vertices of H with degree at least $2(n-(\log_{3}n)^{2})$, and set $t=|T|$.
From 5.101 one can conclude that the sum of degrees in $H$ is at least
$s(n-2\log_{3}n)$. We also know that:
$\displaystyle\text{Sum of degrees in }S$ $\displaystyle\leq 2s(n-2\log_{3}n)$
(5.106) $\displaystyle\leq t\times(\text{Maximum degree of vertices in set $T$
of graph $H$ })$ (5.107) $\displaystyle+(s-t)\times(\text{Maximum degree of
vertices in set $S\setminus T$ of graph $H$ })$ (5.108) $\displaystyle\leq
2tn+2(s-t)(n-(\log_{3}n)^{2}$ (5.109) $\displaystyle\Rightarrow t\geq
s(1-\frac{4}{\log_{3}n})$ (5.110)
Define $H_{1}=$${}^{3}Q^{n}[T]=H[T]$ as the induced subgraph spanned by $T$.
We want to calculate $|N_{H_{1}}(U)|$ in $H_{1}$, hence we should estimate the
size of $H_{1}$ and after that calculate the average degree in T. Let us first
calculate $e(H_{1})$, the total number of edges in $H_{1}$.
$\displaystyle e(H_{1})\geq e(H)-2(s-t)n\geq
2\frac{s}{2}(n-2\log_{3}n)-\frac{4s}{\log_{3}n}n\text{ (from \ref{2.33.1} and
\ref{2.34.1})}$ (5.111)
One knows that the average degree in $H_{1}$ is at least
$\frac{2e(H_{1})}{s}$, and:
$\displaystyle\frac{2e(H_{1})}{s}\geq 2(n-2\log_{3}n)-\frac{8n}{\log_{3}n}\geq
2n-\frac{9}{\log_{3}n}$ (5.112) $\displaystyle\text{(since:
}\log_{3}n^{2}<\frac{n}{\log_{3}n}\text{ for large enough n)}$ (5.113)
Set $u=\lfloor\frac{3^{n}}{n^{\frac{1}{2}}}\rfloor$. One should check the
conditions of lemma 3 here. Let $v=t,\Delta=2n,d\geq 2n-\frac{9}{\log_{3}n}$.
So, one should check if $2n+1\leq\frac{3^{n}}{n^{\frac{1}{2}}}\leq t-2(n+1)$,
for large enough $n$. Clearly, $2n+1\leq\frac{3^{n}}{n^{\frac{1}{2}}}$, as
$n\rightarrow\infty$. It remains to prove $\frac{3^{n}}{n^{\frac{1}{2}}}\leq
t-2(n+1)$, for large enough $n$. For minimum $s$ from 5.110 we can get:
$\displaystyle t\geq$ $\displaystyle\frac{3^{n}}{(\log
n)^{7}}(1-\frac{4}{\log_{3}n})\text{ (from \ref{2.34.1})}$ (5.114)
$\displaystyle\geq\frac{3^{n}}{n^{\frac{1}{2}}}+2(n+1)\text{ (for large enough
n) }$ (5.115)
Now, one can use lemma 3 and estimate $|N_{H_{1}}(U)|$.
$\displaystyle|N_{H_{1}}(U)|$
$\displaystyle\geq\frac{t}{2n}(2n-\frac{9n}{\log_{3}n})\\{1-\exp(-\frac{2n+1}{t}\frac{3^{n}}{n^{\frac{1}{2}}})\\}$
(5.116)
$\displaystyle\geq\frac{t}{2}(n-\frac{9}{\log_{3}n})\\{1-\exp(-\frac{2n+1}{t}\frac{3^{n}}{n^{\frac{1}{2}}})\\}$
(5.117)
Let us estimate a bound for
$\exp(-\frac{2n+1}{t}\frac{3^{n}}{n^{\frac{1}{2}}})$. One knows that $t\geq
s(1-\frac{4}{\log_{3}n})$. Since $\max(t)=s$ and $\max(s)=\frac{3^{n}}{2}$,
then:
$\displaystyle\frac{3^{n}(2n+1)}{n^{\frac{1}{2}}t}\geq\frac{3^{n}(2n+1)}{n^{\frac{1}{2}}3^{n}}=\frac{(2n+1)}{n^{\frac{1}{2}}}\geq
n^{\frac{1}{4}}\text{ (for large enough n) }$ (5.118)
$\displaystyle\Rightarrow\\{1-\exp(-\frac{2n+1}{t}\frac{3^{n}}{n^{\frac{1}{2}}})\\}\geq\exp(-n^{\frac{1}{4}})\text{
(for large enough n) }$ (5.119)
By using the bound from 5.119 in 5.117, one gets:
$\displaystyle|N_{H}(U)|\geq|N(H_{1})|$
$\displaystyle\geq\frac{t}{2}(2-\frac{9}{\log_{3}n})\\{1-\exp(-n^{\frac{1}{4}})\\}$
(5.120)
$\displaystyle=\frac{t}{2}\\{2+\exp(-n^{\frac{1}{4}})\frac{9}{\log_{3}n}-2\exp(-n^{\frac{1}{4}})-\frac{9}{\log_{3}n}\\}$
(5.121)
$\displaystyle(\lim_{n\rightarrow\infty}\exp(-n^{\frac{1}{4}})\frac{9}{\log_{3}n}=0\text{
and }\exp(-n^{\frac{1}{4}})<\frac{1}{\log_{3}n}\text{ (for large enough n) })$
(5.122)
$\displaystyle\geq\frac{t}{2}\\{2-\frac{2}{\log_{3}n}-\frac{9}{\log_{3}n}\\}=\frac{t}{2}(2-\frac{11}{\log_{3}n})$
(5.123)
$\displaystyle\geq\frac{s}{2}(1-\frac{2}{\log_{3}n})(2-\frac{11}{\log_{3}n})=\frac{s}{2}(2+\frac{2}{\log_{3}n}\frac{11}{\log_{3}n}-\frac{15}{\log_{3}n})\text{
(for large enough n) }$ (5.124)
$\displaystyle\geq\frac{s}{2}(2-\frac{15}{\log_{3}n})=s(1-\frac{7.5}{\log_{3}n})$
(5.125)
Now that we have $|N_{H}(U)|$, we can estimate a bound for $|C_{s}^{+}|$ here.
We know from 5.125 that for each $S\in C_{s}^{+}$ there exist a $U\subseteq
S$, $|U|:=u:=\lfloor\frac{3^{n}}{n^{\frac{1}{2}}}\rfloor$, such that
$|N_{H}(U)|\geq s(1-\frac{7.5}{\log_{3}n})$. Therefore, one can choose $S\in
C_{s}^{+}$ as follows:
1\. Select u vertices of ${}^{3}Q^{n}$; there are $\binom{3^{n}}{u}$ choices
for this u.
2\. Select $\lfloor s(1-\frac{7.5}{\log_{n}3})\rfloor-u$ neighbors of the
selected vertices in part 1. At most $2(\log_{3}n)^{2}$ of the $2n$ neighbors
of a vertex in $U$ do not belong to $N_{H}(U)$. Hence there are at most
$\sum_{(k_{j})}(\prod_{i=1}^{u}\binom{2n}{j})$ ways to find neighbors of $u$
vertices in $U$, where the sum is over all $(k_{1},k_{2},...,k_{u}),k_{i}\leq
2(\log_{3}n)^{2}$. We know that:
$\displaystyle\sum_{(k_{i})}\prod_{i=1}^{u}\binom{2n}{k_{i}}$
$\displaystyle\leq\sum_{(k_{i})}\prod_{i=1}^{u}(\frac{(2n)^{i}}{i!})$ (5.126)
$\displaystyle\leq\sum_{(k_{i})}\prod_{i=1}^{u}(\frac{(2n)^{(2(\log_{3}n)^{2})}}{(2(\log_{3}n)^{2})!})$
(5.127)
$\displaystyle=\sum_{(k_{i})}\frac{(2n)^{2u(\log_{3}n)^{2}}}{((2(\log_{3}n)^{2})!)^{u}}$
(5.128)
$\displaystyle={(2(\log_{3}n)^{2})}^{u}\frac{(2n)^{2u(\log_{3}n)^{2}}}{((2(\log_{3}n)^{2})!)^{u}}$
(5.129) $\displaystyle\leq(2n)^{2u(\log_{3}n)^{2}}$ (5.130)
3\. Select $\lfloor\frac{7.5}{\log_{3}n}\rfloor$ other vertices; there are
$\binom{3^{n}}{\lfloor\frac{7.5}{\log_{3}n}\rfloor}$ choices.
Hence:
$\displaystyle\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\binom{3^{n}}{u}(2n)^{2u(\log_{3}n)^{2}}\binom{3^{n}}{\lfloor\frac{7.5}{\log_{3}n}\rfloor}3^{-2s(n-\log_{3}s)}(\log
n)^{2s(1-\frac{\log_{3}s}{n})}$ (5.132)
where:
$\displaystyle\binom{3^{n}}{u}(2n)^{2u(\log_{3}n)^{2}}\binom{3^{n}}{\lfloor\frac{7.5}{\log_{3}n}\rfloor}=3^{o(s)}$
(5.134)
Therefore:
$\displaystyle\sum_{S\in C_{s}^{+}}(1-p_{n})^{b(S)}\leq 3^{\varepsilon(s)}$
(5.135)
where:
$\displaystyle\varepsilon(s)=o(s)-2s\\{n-\log_{3}s-\log_{3}\log
n+\frac{\log_{3}s}{n}\log_{3}\log n\\}$ (5.136)
Since $s\leq\frac{3^{n}}{2}$, hence one can get:
$\displaystyle\varepsilon(s)$ $\displaystyle\leq
o(s)-2s\\{n-(n-1)-\log_{3}\log n+\frac{n-1}{n}\log_{3}\log n\\}$ (5.137)
$\displaystyle=o(s)-2s\\{1-\frac{1}{n}\log_{3}\log n\\}\leq-s$ (5.138)
Therefore, for large enough n, one can get:
$\displaystyle\sum_{s=s_{2}+1}^{\frac{3^{n}}{2}}\sum_{S\in
C_{s}^{+}}(1-p_{n})^{b(S)}\leq\sum_{s=s_{2}+1}^{\frac{3^{n}}{2}}3^{-s}=o(1)$
(5.139)
$\blacksquare$
## Chapter 6 Connected random subgraph of the $P_{3}$-product
The $P_{3}^{n}$ graph is the cartesian products of $n$ copies of $P_{3}$,
where $P_{3}$ stands for a path with length 2. If one labels each vertex of
$P_{3}^{n}$ from $0$ to $3^{n}-1$, then two vertices are adjacent if the
difference between their ternary representation is 1, in other words, vertex
$x=(x_{1},x_{2},...,x_{n})$ is connected to the vertex
$y=(y_{1},y_{2},...,y_{n})$ if for some $i$ we have $|x_{i}-y_{i}|=1$ and
$x_{j}=x_{j}$ for $j\neq i$. Vertex $x=(x_{1},x_{2},...,x_{n})$ is connected
to $n+i$ vertices where $i=|\\{j|x_{j}=1,j\in\\{1,..,n\\}\\}|$. A random
subgraph of $P_{3}^{n}$ contains all vertices of $P_{3}^{n}$, and each edge
independently with probability $p$. $p$ is called the percolation parameter
and $P_{3,p}^{n}$ stands for the random subgraph of $P_{3}^{n}$
The main goal in this chapter is to explore a critical value $p_{c}$, which
for fixed values of $p$ if $p<p_{c}$ then almost no random subgraphs of
$P_{3}^{n}$ is connected, as $n\rightarrow\infty$; but if $p>p_{c}$ then
almost all random subgraphs of $P_{3}^{n}$ are connected, as
$n\rightarrow\infty$. We suggest that this critical value is $2-\sqrt{2}$. The
proof in this chapter is not complete and there is a place for further work.
In the first section of this chapter, it is proved that for $p<2-\sqrt{2}$
almost no random subgraphs of $P_{3}^{n}$ are connected, as
$n\rightarrow\infty$. Then, it is proved that, for $p>2-\sqrt{2}$ almost all
random subgraphs of $P_{3}^{n}$ have no isolated point, as
$n\rightarrow\infty$. In the second section, the probability that random
subgraphs of $P_{3}^{n}$, with the percolation parameter $p>2-\sqrt{2}$, have
no components with size larger than $1$ and smaller than $3^{n}$, as
$n\rightarrow\infty$, is explored.
### 6.1 Isolated vertices
For $p<2-\sqrt{2}:$
Let us define $X_{i}$ and $X$ for a graph $G$ as follows:
###### Definition 30.
$X_{i}(n):=\left\\{\begin{array}[]{ll}1&\mbox{Vertex $i$ is isolated, $i\in
V(G)$};\\\ 0&\mbox{Vertex $i$ is NOT isolated, $i\in
V(G)$}.\end{array}\right.$ , and $X(n):=\displaystyle\sum_{i\in
V(G)}X_{i}(n)$.
As the first step we calculate $E[X(n)]$. One can categorize the set of
vertices $V$ into $n+1$ subsets $V_{i}$’s where $x=(x_{1},x_{2},...,x_{n})\in
V_{i}$ if $|\\{j|x_{j}=1,j\in\\{1,..,n\\}\\}|=i$. Hence:
$\displaystyle\mu:=E[X(n)]=\sum_{j\in
V(P^{n}_{3,p})}E[X_{j}(n)]=\sum_{i=0}^{n}\sum_{k\in
V_{i}}E[X_{k}(n)]=\sum_{i=0}^{n}\binom{n}{i}2^{n-i}(1-p)^{n+i}=(1-p)^{n}(3-p)^{n}$
(6.1)
From (6.1), the threshold value for $E[X]$ is $p_{c}=2-\sqrt{2}$ which is the
solution to the equation $(1-p)^{n}(3-p)^{n}=1$. This means that
$E[X]\rightarrow\infty$ for $p<p_{c}$, but $E[X]\rightarrow 0$ for $p>p_{c}$.
Now, one should calculate $Var[X]$.
$\displaystyle Var[X(n)]$ $\displaystyle=\sum_{i\in
V(P^{n}_{3,p})}Var[X_{i}(n)]+\sum_{i,j\in V(P^{n}_{3,p}),i\neq
j}Cov[X_{i}(n),X_{j}(n)]$ (6.2) $\displaystyle=\sum_{i=0}^{n}\sum_{k\in
V_{i}}Var[X_{k}(n)]+\sum_{i,j\in V(P^{n}_{3,p}),i\neq
j}Cov[X_{i}(n),X_{j}(n)]$ (6.3)
where:
$\displaystyle\sum_{i=0}^{n}\sum_{k\in V_{i}}Var[X_{k}(n)]$
$\displaystyle=\sum_{i=0}^{n}\binom{n}{i}2^{n-i}((1-p)^{n+i}-(1-p)^{2n+2i})$
(6.4) $\displaystyle=(1-p)^{n}(3-p)^{n}-(1-p)^{2n}(2+(1-p)^{2})^{n}$ (6.5)
In order to calculate $\sum_{i,j\in V(P^{n}_{3,p}),i\neq
j}Cov[X_{i}(n),X_{j}(n)]$ one should know that if $x\in V_{i}$ then $x$ is
incident to $2i$ vertices in $V_{i-1}$ and $n-i$ vertices in $V_{i+1}$. Also,
one knows:
$\displaystyle Cov[X_{i}(n),X_{j}(n)]$
$\displaystyle=E[X_{i}(n)X_{j}(n)]-E[X_{i}(n)]E[X_{j}(n)]=0\text{ if i,j not
adjacent}$ (6.6)
Hence:
$\displaystyle\sum_{i,j\in V(P^{n}_{3,p}),i\neq j}$ $\displaystyle
Cov[X_{i}(n),X_{j}(n)]$ (6.7)
$\displaystyle=\sum_{i=0}^{n}\binom{n}{i}2^{n-i}\\{(n-i)(1-p)^{n+i-1}(1-p)^{n+i+1}+2i(1-p)^{n+i-1}(1-p)^{n+i-1}\\}$
(6.8)
$\displaystyle=\sum_{i=0}^{n}\binom{n}{i}2^{n-i}(1-p)^{2n+2i-1}\\{(n-i)(1-p)+\frac{2i}{1-p}\\}$
(6.9)
$\displaystyle=(1-p)^{2n}\sum_{i=0}^{n}\binom{n}{i}2^{n-1}(1-p)^{2i-1}\\{n(1-p)+i(\frac{2}{1-p}-(1-p))\\}$
(6.10)
$\displaystyle=n(1-p)^{2n}\sum_{i=0}^{n}\binom{n}{i}2^{n-i}(1-p)^{2i}+(1-p)^{2n-2}(2-(1-p)^{2})\sum_{i=0}^{n}\binom{n}{i}2^{n-i}(1-p)^{2i}i$
(6.11)
$\displaystyle=n(1-p)^{2n}(2+(1-p)^{2})^{n}+(1-p)^{2n-2}(1-(1-p)^{2})2^{n}n\frac{(1-p)^{2}}{2}(1+\frac{(1-p)^{2}}{2})^{n-1}$
(6.12)
$\displaystyle=n(1-p)^{2n}(2+(1-p)^{2})^{n}+n(2-(1-p)^{2})(2+(1-p)^{2})^{n-1}(1-p)^{2n}$
(6.13) $\displaystyle=4n(1-p)^{2n}(2+(1-p)^{2})^{n-1}$ (6.14)
and finally, from (6.5) and (6.14) one gets:
$\displaystyle Var[X(n)]$
$\displaystyle=(1-p)^{n}(3-p)^{n}-(1-p)^{2n}(2+(1-p)^{2})^{n}+4n(1-p)^{2n}(2+(1-p)^{2})^{n-1}$
(6.15)
$\displaystyle=\mu-\mu^{2}\frac{(2+(1-p)^{2})^{n}}{(3-p)^{2n}}+\mu^{2}\frac{4n(2+(1-p)^{2})^{n}}{(3-p)^{2n}}$
(6.16)
Now, since we have $Var[X(n)]$, we can use Chebyshev’s inequality to estimate
$g_{n}(p)$. Chebyshev’s inequality states that:
$\displaystyle 1-\textbf{P}[P_{3,p}^{n}\text{ contains an isolated
point}]=\textbf{P}[X(n)=0]\leq\textbf{P}[|X(n)-\mu|\geq\mu]\leq\frac{Var[X(n)]}{\mu^{2}}$
(6.17)
From 6.16 when $p<2-\sqrt{2}$, one gets $Var[X(n)]/\mu^{2}\rightarrow 0$, as
$n\rightarrow\infty$. Therefore, from (6.17) one gets
$\lim_{n\rightarrow\infty}\textbf{P}[P_{3,p}^{n}\text{ contains an isolated
point}]=1$. Finally, since
$\textbf{P}[P_{3,p}^{n}\text{ is connected}]\leq\textbf{P}[P_{3,p}^{n}\text{
does contains an isolated point}]$
, then for $p<2-\sqrt{2}$ the probability that $P_{3,p}^{n}$ is connected, as
$n\rightarrow\infty$, tends to $0$. $\blacksquare$
For $p>2-\sqrt{2}$:
In order to calculate $\textbf{P}[P_{3,p}^{n}\text{ contains an isolated
point}]$ when $p>2-\sqrt{2}$, as $n\rightarrow\infty$, one can use the
following inequality:
$\displaystyle\textbf{P}[P_{3,p}^{n}\text{ contains an isolated
point}]=\textbf{P}[X(n)>0]\leq E[X(n)]=\mu$ (6.18)
Since $E[X(n)]\rightarrow 0$ as $n\rightarrow\infty$, then
$\lim_{n\rightarrow\infty}\textbf{P}[P_{3,p}^{n}\text{ contains an isolated
point}]=0$. This means that the probability that there are no isolated points
in $P_{3,p}^{n}$ for $p>2-\sqrt{2}$, as $n\rightarrow\infty$, tends to $1$.
$\blacksquare$
### 6.2 Isolated components of size larger than 2 and smaller than $3^{n}$
###### Definition 31.
$C_{s}$ is the family of subsets of $V(P_{3}^{n})$ with size $s$ whose their
induced graph is connected.
###### Definition 32.
The edge boundary $b_{G}(H)$, where $H$ is an induced subgraph of G, is the
number of edges which joins vertices in $H$ to the vertices in $G\backslash
H$. Then $b_{G}(k)=min\\{b_{G}(H):H\subset G,|H|=k\\}$
###### Theorem 14.
If $p\geq 2-\sqrt{2}$, the probability that for some $S\in C_{s},2\leq s\leq
3^{0.44n}$, no edges of $P_{3,p}^{n}$ join $S$ to $V(P_{3}^{n})\setminus S$,
as $n\rightarrow\infty$, tends to 0.
Proof:
Consider $S\subset V=V(P_{3}^{n})$ and set $b(S)=b_{P_{3}^{n}}(H)$ where $H$
is the induced subgraph of $P_{3}^{n}$ with the set of vertices S. One can
observe that:
$\displaystyle\textbf{P}(\text{No edges of }P_{3,}^{n}\text{ join S to
}V\setminus S)=(1-p)^{b(S)}$ (6.19)
In order to prove the theorem, it is sufficient to show:
$\displaystyle\sum_{s=2}^{3^{0.44n}}\sum_{S\in C_{s}}(1-p)^{b(S)}=o(1)$ (6.20)
From [16] one knows that for $|S|=s$:
$\displaystyle b(S)\geq b(s)\geq\frac{e}{3}s\ln\frac{3^{n}}{s}=\frac{e\ln
3}{3}s(n-\log_{3}s)$ (6.21)
one knows that:
$\displaystyle\sum_{S\in C_{s}}(1-p)^{b(S)}\leq|C_{s}|(1-p)^{b(s)}$ (6.22)
First, one should find a bound for $|C_{s}|$. One has maximum $3^{n}$ choices
to choose the first element for $C_{s}$. The selected element is connected to
maximum $2n$ vertices, therefore, there are maximum $2n$ choices to choose the
second element. With the same logic, there are at most $2n(s-1)$ choices to
choose the last element for $|C_{s}|$. Hence, one can show:
$\displaystyle|C_{s}|\leq
3^{n}(2n)(2n(2))...(2n(s-1))\leq(s-1)!(2n)^{s-1}3^{n}$ (6.23)
and:
$\displaystyle|C_{s}|(1-p)^{b(s)}\leq(s-1)!(2n)^{s-1}3^{n}(1-p)^{\frac{e\ln
3}{3}s(n-\log_{3}s)}$ (6.24)
Set $a:=\frac{e\ln 3}{3}$. Assume that the right hand sides of inequality
(6.24) is equal to A. After multiplying both sides of inequality (6.24) with
$\frac{2ns^{s+1}}{s!}$ and then taking $\log_{3}$ from both sides, one gets:
$\displaystyle\log_{3}(|C_{s}|(1-p)^{b(s)}\frac{2ns^{s+1}}{s!})\leq\log_{3}(A\frac{2ns^{s+1}}{s!})$
(6.25)
If $\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$
then $A\frac{2ns^{s+1}}{s!}$ should tend to 0 for $2\leq s\leq 3^{0.44n}$.
This means that $|C_{s}|(1-p)^{b(s)}\frac{2ns^{s+1}}{s!}$ tends to 0 for
$2\leq s\leq 3^{0.44n}$, as $n\rightarrow\infty$. Therefore:
$\displaystyle|C_{s}|(1-p)^{b(s)}\leq\frac{s!}{2ns^{s+1}}\text{ for large
values of n}$ (6.26)
which shows that:
$\displaystyle\sum_{s=2}^{3^{0.44n}}\sum_{S\in C_{s}}(1-p)^{b(S)}=o(1)$ (6.27)
Finally, it remains to prove
$\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as $n\rightarrow\infty$.
One can verify this for $s\leq n$ and $s>n$. After multiplying both sides of
inequality (6.24) with $\frac{2ns^{s+1}}{s!}$ and then getting $\log_{3}$ from
both sides, one gets:
$\displaystyle\log_{3}(|C_{s}|(1-p)^{b(s)}\frac{2ns^{s+1}}{s!})\leq
s(\log_{3}2n+\log_{3}s+an\log_{3}(1-p)-a(\log_{3}s)\log_{3}(1-p))+n$ (6.28)
For $s\leq n$ the largest factor in equation (6.28) is
$n(a(\log_{3}(1-p))s+1)$ which is negative for $p=p_{c}$ and $2\leq s\leq n$.
Therefore, $\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as
$n\rightarrow\infty$ for $s\leq n$. For $s>n$ the largest factor in equation
(6.28) is $s(an(\log_{3}(1-p))+(\log_{3}s)(1-a\log_{3}(1-p))$ which is
negative if $s<3^{\frac{-a\log_{3}(1-p)}{1-a\log_{3}(1-p)}n}\approx
3^{0.44n}$. Therefore, $\log_{3}(A\frac{2ns^{s+1}}{s!})\rightarrow-\infty$ as
$n\rightarrow\infty$ for $s>n$. $\blacksquare$
For $p\geq 0.67$:
###### Theorem 15.
If $p\geq 0.67$, the probability that for some $S\in C_{s},2\leq s\leq 3^{n}$,
no edges of $P_{3,p}^{n}$ join $S$ to $V(P_{3}^{n})\setminus S$, as
$n\rightarrow\infty$, tends to 0. Hence, the probability that $P^{n}_{3,p}$ is
connected tends to $1$, as $n\rightarrow\infty$.
Proof:
We know that:
$\displaystyle|C_{s}|\leq\binom{3^{n}}{s}<(\frac{e3^{n}}{s})^{s}$ (6.29)
hence:
$\displaystyle|C_{s}|(1-p)^{b(s)}\leq\binom{3^{n}}{s}<(\frac{e3^{n}}{s})^{s}(1-p)^{\frac{e\ln
3}{3}s(n-\log_{3}s)}$ (6.30)
By using Mathematica 111Mathematica is a computational software program
developed by Wolfram Research of Champaign, Illinois one can show the right
hand side of equation (6.30) tends to zero as $n\rightarrow\infty$.
$\blacksquare$
## Chapter 7 Reliable networks
In many engineering applications, it is of interest to construct a graph
(network), with a specific number of edges and vertices, which is the most
reliable one in that family of graph with n vertices and m edges. One of the
measures of reliability is all-terminal reliability.
### 7.1 All terminal reliability
###### Definition 33.
Random subgraph of a graph, Percolation on a graph: Random subgraph of
$G(V_{n},E_{m})$ is the graph $G_{p_{n}}$ which contains all vertices of $G$,
and each edge of $G$ independently with probability $p_{n}$. Doing percolation
on a graph with the parameter $p_{n}$ is the same as finding a random subgraph
of a graph with the parameter $p_{n}$. $p_{n}$ is called percolation
parameter.
###### Definition 34.
Uniformly optimally reliable graph (UOR): The UOR graph, if it exists, is the
graph $G_{n}\in G(n,m)$ which maximizes the probability that $G_{n}$ is
connected after percolation with the parameter $p_{n}$ for fixed $n,m$ and all
$p_{n}\in(0,1)$.
###### Definition 35.
Reliability polynomial: Let $s_{k}$ be the number of spanning connected
subgraphs of $G_{n}\in G(n,m)$ having exactly $k$ edges. Let $R(G_{n},p_{n})$
be the probability that $G_{n}$ is connected after percolation with the
parameter $p_{n}$. One can formulate $R(G_{n},p_{n})$ as follows:
$\displaystyle R(G_{n},p_{n}):=\sum_{i=0}^{m}s_{i}p_{n}^{i}(1-p_{n})^{m-i}$
(7.1)
$R(G_{n},p_{n})$ is called _reliability polynomial_ of graph $G_{n}$ or _all-
terminal reliability_. In this definition, $s_{i}=0$ for $i<n-1$ and
$s_{m}=1$. Also, $s_{m-1}$ is $m-$number of cuts in $G_{n}$, and $s_{n-1}$ is
the number of spanning tress of $G_{n}$.
The UOR graph, if it exists, is the graph $G_{n}\in G(n,m)$ which maximizes
$R(G_{n},p_{n})$ for all $p_{n}\in(0,1)$. From the definition of reliability
polynomial one can see that the value of reliability polynomial depends on the
structure of a graph as well as percolation value. Trivially, for fixed values
of $p_{n}$ there always exists an optimal solution for $R(G,p_{n})$. In other
words, for fixed values of $p_{n}$, there always exists a $G_{n}\in G(n,m)$
which maximizes $R(G_{n},p_{n})$. The following theorem and corollary can be
helpful to find the UOR graph. This theorem and corollary are extracted from
[17].
###### Theorem 16.
Let G and H be two undirected simple graphs both having n nodes and m edges
and $s_{k}(G),s_{k}(H)$ denote the number of spanning connected subgraphs of G
and H, respectively, with exactly k edges [17].
1. 1.
If there exists an integer $0\leq k\leq m-1$ such that $s_{i}(G)=s_{i}(H)$ for
$i=0,1,...,k$ and $s_{k+1}(G)>s_{k+1}(H)$, then there exists a $\rho>0$ such
that for all $0<p<\rho$ we have $R(G,p)>R(H,p)$.
2. 2.
If there exists an integer $0\leq k\leq m$ such that $s_{i}(G)=s_{i}(H)$ for
$i=m,m-1,...,m-k$ and $s_{m-k-1}(G)>s_{m-k-1}(H)$, then there exists a
$\rho<1$ such that for all $\rho<p<1$ we have $R(G,p)>R(H,p)$.
###### Corollary 3.
If G is UOR, then [17]:
1. 1.
G has the maximum number of spanning trees among all simple graphs having n
nodes and m edges, and
2. 2.
G is $max-\lambda$ , i.e. has the maximum possible value of $\lambda$ among
all simple graphs having n nodes and m edges, namely $\lambda(G)=\lfloor
2m/n\rfloor$, and the minimum number of cutsets of size $\lambda$ among all
such $max-\lambda$ graphs.
where $\lambda(G)$ is the edge connectivity of G, i.e., the minimum number of
edges whose its removal will disconnect $G$.
Important coefficients for large n: Let $n$ be large and $m$ sufficiently
larger than $n$. When $p_{n}$ is close to $0$ then $s_{n-1}$, the number of
spanning trees of $G_{n}$, has the most significant contribution in
$R(G_{n},p_{n})$ since $(1-p_{n})$ is almost $1$ and $p_{n}^{n-1}$ is much
larger than $p_{n}^{m}$. Similarly, when $p_{n}$ is close to $1$, then
$s_{m-1}$, $m-$number of cuts, has the most significant contribution in
$R(G_{n},p_{n})$.
#### Laplacian
The _Laplacian matrix_ of a graph is described briefly in chapter 1. Here, we
present a few results on Laplacian. The author calculated the algebraic
connectivity for all graphs with $n=5,6,7$ and $n-1<m<\binom{n}{2}$ and could
not find a direct relation between algebraic connectivity and all-terminal
reliability. There is a room for further work in this part.
The Laplacian matrix is $L:=(l_{i,j})_{n\times n}$ where
$l:=\left\\{\begin{array}[]{ll}deg(v_{i}),&\hbox{if $i=j$;}\\\ -1,&\hbox{if
$i\neq j$ and $v_{i}$ adjacent to $v_{j}$;}\\\ 0,&\hbox{o.w.}\\\
\end{array}\right.$. Arrange the eigenvalues of $L$ as :
$\lambda_{1}(L)\leq\lambda_{2}(L)\leq...\lambda_{n}(L)$. This set of
$\lambda_{i}$’s are called the spectrum of $L$ and $\lambda_{2}(L)$ is called
the algebraic connectivity of a graph. The following interesting lemma sheds
light on some applications of Laplacian matrix (this lemma is extracted from
[24]):
###### Lemma 5.
For a graph $G$ on $n$ vertices, we have [24]
(i):
$\sum_{i}\lambda_{i}\leq n$
with equality holding if and only if $G$ has no isolated vertices.
(ii): For $n\geq 2$
$\lambda_{1}\leq\frac{n}{n-1}$
with equality holding if and only if $G$ is the complete graph on $n$
vertices. Also, for a graph $G$ without isolated vertices, we have:
$\lambda_{n-1}\geq\frac{n}{n-1}.$
(iii:) For a graph which is not a complete graph, we have
$\lambda_{1}\leq 1.$
(iv:) If $G$ is connected, then $\lambda_{1}>0$. If $\lambda_{i}=0$ and
$\lambda_{i+1}\neq 0$, then $G$ has exactly $i+1$ connected components.
(v): For all $i\leq n-1$, we have:
$\lambda_{i}\leq 2$
with
$\lambda_{n-1}=2$
if and only of a connected component of $G$ is bipartite and nontrivial.
(vi:) The spectrum of a graph is the union of the spectrum of its connected
components.
#### Reliability for $n=5,6,7$
The author calculated $R(G,p)$ for $n=5,6,7$ and $n-1<m<\binom{n}{2}$. The
coefficients of the reliability polynomials are presented in tables 7.1, 7.2
and 7.3. As table 7.1 illustrates, for $n=5$ there is always a UOR graph. For
$n=6,7$ there are always a UOR graph except two cases. If, $(n,m)=(6,11)$, the
optimal solution for $R(G_{n},p)$ depends on the value of $p$. For $p<0.29$
(approximately 0.29) graph 7.1(a) is optimal while for $p>0.29$ graph 7.1(b)
optimal. Also, if $(n,m)=(7,15)$, the optimal solution for $R(G_{n},p)$
depends on the value of $p$. For $p<0.81$ (approximately 0.81) graph 7.2(a) is
optimal while for $p>0.81$ graph 7.2(b) is optimal. From these observation one
can conclude that the UOR does not exist for all values of $n$ and $m$.
n=5 | $s_{m-1}$ | $s_{m-2}$ | … | … | … | $s_{n-1}$
---|---|---|---|---|---|---
m=5 | 5 | | | | |
m=6 | 6 | 12 | | | |
m=7 | 7 | 20 | 24 | | |
m=8 | 8 | 28 | 52 | 45 | |
m=9 | 9 | 36 | 82 | 111 | 75 |
m=10 | 10 | 45 | 120 | 205 | 222 | 125
Table 7.1: The coefficients of the reliability polynomials of the UOR graph for $n=6$ and $n-1<m<\binom{n}{2}$. n=6 | $s_{m-1}$ | $s_{m-2}$ | … | … | … | … | … | … | … | $s_{n-1}$
---|---|---|---|---|---|---|---|---|---|---
m=6 | 6 | | | | | | | | |
m=7 | 7 | 16 | | | | | | | |
m=8 | 8 | 26 | 36 | | | | | | |
m=9 | 9 | 36 | 78 | 81 | | | | | |
m=10 | 10 | 45 | 116 | 177 | 135 | | | | |
_$m_{a}$ =11_ | 11 | 55 | 163 | 309 | 368 | 225 | | | |
_$m_{b}$ =11_ | 11 | 55 | 163 | 310 | 370 | 224 | | | |
m=12 | 12 | 66 | 220 | 489 | 744 | 740 | 384 | | |
m=13 | 13 | 78 | 286 | 771 | 1249 | 1552 | 1292 | 576 | |
m=14 | 14 | 91 | 364 | 999 | 1978 | 2877 | 3040 | 2196 | 864 |
m=15 | 15 | 105 | 455 | 1365 | 2997 | 4945 | 6165 | 5700 | 3660 | 1296
Table 7.2: The coefficients of the reliability polynomials of the UOR graph for $n=6$ and $n-1<m<\binom{n}{2}$. For $m=11$, and $p<0.29$ (approximately 0.29) the row with $m_{a}$ is the UOR graph and for $p>0.29$ the row with $m_{b}$ is the UOR graph. n=7 | $s_{m-1}$ | $s_{m-2}$ | … | … | … | … | … | … | … | … | … | … | … | … | $s_{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
m=7 | 7 | | | | | | | | | | | | | |
m=8 | 8 | 21 | | | | | | | | | | | | |
m=9 | 9 | 33 | 51 | | | | | | | | | | | |
m=10 | 10 | 44 | 104 | 117 | | | | | | | | | | |
m=11 | 11 | 55 | 159 | 273 | 231 | | | | | | | | | |
m=12 | 12 | 66 | 216 | 456 | 612 | 432 | | | | | | | | |
m=13 | 13 | 78 | 284 | 690 | 1146 | 1248 | 720 | | | | | | | |
m=14 | 14 | 91 | 364 | 994 | 1932 | 2668 | 2460 | 1200 | | | | | | |
_$m_{a}$ =15_ | 15 | 105 | 455 | 1360 | 2946 | 4704 | 5464 | 4320 | 1840 | | | | | |
_$m_{b}$ =15_ | 15 | 105 | 455 | 1360 | 2946 | 4705 | 5465 | 4305 | 1805 | | | | | |
m=16 | 16 | 120 | 560 | 1817 | 4328 | 7766 | 10548 | 10628 | 7396 | 2800 | | | | |
m=17 | 17 | 136 | 680 | 2379 | 6169 | 1226 | 18762 | 22226 | 19808 | 12320 | 4200 | | | |
m=18 | 18 | 153 | 816 | 3060 | 8562 | 18485 | 31344 | 41964 | 44000 | 35094 | 19716 | 6125 | | |
m=19 | 19 | 171 | 969 | 3876 | 11624 | 27073 | 49985 | 73888 | 87468 | 81976 | 58958 | 30109 | 8575 | |
m=20 | 20 | 190 | 1140 | 4845 | 15502 | 38725 | 77240 | 124605 | 163400 | 173646 | 147500 | 96915 | 45530 | 12005 |
m=21 | 21 | 210 | 1330 | 5985 | 20349 | 54257 | 116175 | 202755 | 290745 | 343140 | 331506 | 258125 | 156555 | 68295 | 16807
Table 7.3: The coefficients of the reliability polynomials of the UOR graph
for $n=7$ and $n-1<m<\binom{n}{2}$. For $m=15$, and $p<0.81$ (approximately
0.81) the row with $m_{a}$ is the UOR graph and for $p>0.81$ the row with
$m_{b}$ is the UOR graph.
(a)
(b)
Figure 7.1: $(n,m)=(6,11)$, For $p<0.29$ (approximately 0.29) graph $(a)$ is
optimal while for $p>0.29$ graph $(b)$ is optimal.
(a)
(b)
Figure 7.2: $(n,m)=(7,15)$, For $p<0.81$ (approximately 0.81) graph $(a)$ is
optimal while for $p>0.81$ graph $(b)$ is optimal.
#### Reliability for $m=n-1,n,n+1,n+2,n+3$
For $m=n-1,n,n+1,n+2$ there always exists a UOR graph. For $m=n-1$, any tree
is the UOR graph. For $m=n$, $C_{n}$, single cycle with n vertices, is the UOR
graph. The first non-trivial case is $m=n+1$, which is solved by F. Boesch
[21], [20]. The UOR graph in this case is: for $n\geq 5$, start with a
multigraph with 2 vertices and 3 edges. Then add total of $n-2$ vertices of
degree 2 in each lines of the graph so that the number of vertices in each
line differs by at most one [20]. For $m=n+2$ the problem is also solved by F.
Boesch. The UOR graph in this case is: start with $K_{4}$, then add total of
$n-2$ vertices of degree 2 in each lines of the graph so that the number of
vertices in each line differs by at most one [20]. For $m=n+3$, the UOR graph
is found by G. Wang [22]. The UOR graph in this case is: start with $K_{3,3}$,
a complete bipartite graph with 3 vertices in each part, and then add the
remanning vertices as before.
#### Family of counterexamples
Kelmans [18] and Myrvold et al. [19] found infinite families of counter
examples which the UOR graph does not exist. As an example, for $n$ even and
$n\geq 6$ and $m=n(n-1)/2-(n+2)/2$, or for $n$ odd and $n\geq 7$ and
$m=n(n-1)/2-(n+5)/2$ there always exists a graph in which it maximizes
$R(G,p_{n})$ for $p$ close to $1$, but do not have the maximum number of
spanning trees. Therefore, from theorem 16 and corollary 3 the UOR graph does
not exists for these families.
### 7.2 Random accessibility
M. Ebneshahrashoob, T. Gao and M. Sobel introduced the concept of random
accessibility for simple graphs [23]. They believe that finding the UOR graph
is related to the concept of random accessibility. In random accessibility,
the goal is to find the expectation and the variance of the number of
transitions $X_{j}$ needed to visit $j$ new vertices in $G\in G(V_{n},E_{m})$.
In this approach the starting point is not considered as a new vertex. The
result of the expectation and the variance can depend on starting point.
Hence, one should change the starting point depending on the degree of it as a
weighing factor. If one considers a graph with enough symmetry, the result
does not depend on starting point. From analyzing numerical results, they make
the following interesting conjuncture:
* •
If the family of graphs contains both regular and non-regular graphs, then the
UOR graph is among the regular graphs. Also, the expectation for random
accessibility of graphs is equal or greater than the corresponding result of
the UOR graph for each value of $j$ close to $m-1$ (with the same ordering of
the graph as for all-terminal reliability).
## References
* [1] Bollobás, Bela, (2001), Random graphs. Springer.
* [2] Bollobás, Bela, (2002), Modern Graph Theory. Cambridge University Press; 2nd edition.
* [3] Diestel, Reinhard (2006), Graph Theory. Springer; 3rd edition.
* [4] Erdös, Paul and Spencer, Joel, (1979), Evolution of the n-cube. Comp. $\&$ Maths. with Appls., Vol.5, pp. 33-39
* [5] Hart, Sergiu (1976), A note on the edges of the n-cube. Discrete Mathematics, Vol.14, pp. 157-163
* [6] Burtin, Yu. D. (1977), On the probability of connectedness of a random subgraph of the n-cube. Problemy Pered. Inf., 13, 90-95
* [7] Spencer, Joel (1993), Nine lectures on random graphs. Springer Berlin / Heidelberg; Volume 1541/1993
* [8] Spencer, Joel (1987), Ten Lectures on the Probabilistic Method. Society for Industrial Mathematics; Second edition
* [9] Chai Wah Wu (2007), Synchronization in Complex Networks of Nonlinear Dynamical Systems. World Scientific Publishing Company
* [10] L. Gross, Jonathan (2005), Graph Theory and Its Applications. Chapman and Hall/CRC; Second edition
* [11] Alon, Noga and Spencer, Joel (2008), The probabilistics method. Wiley-Interscience; Third edition
* [12] Matoušek, Jiřá and , Vondrák (2008), The probabilistics method. Lectures notes; Department of Applied Mathematics, Charles University
* [13] Kolchin, V.F. (2009), Random graphs. Cambridge University Press; First edition
* [14] Janson, Svante and Luczak, Tomasz and Rucinski, Andrzej (2000), Random graphs. Wiley-Interscience; First edition
* [15] Clark, Lane (2002), Random subgraphs of cerain graph powers. IJMMS 32:5, 285 292
* [16] Tillich, Jean-Pierre (2000), Edge isoperimetric inequalities for product graphs. Discrete Mathematics, V.213, Issue 1-3, 291-320
* [17] Boesch, F T and Satyanarayana A. and Suffel, C.L. (2009), A survey of some network reliability analysis and synthesis results . Networks, V.54, Issue 2, 99-107
* [18] Kelmans, A.K (1981), On graph with randomly deleted edges. Acta Math Acad Sci Hung, V.37, 77-88
* [19] Myrvold, W and Cheung, L.Page and Perry J.E. (1991), Uniformly most reliable networks do not always exist. Networks, V.21, 417-419
* [20] Boesch, F.T and Li, X. and Suffel, C. (1991), On the existence of uniformly optimally reliable networks. Networks, V.21, 181-194
* [21] Bauer, D. and Boesch F, and Suffel, C. and Tindell, R. (1991), Combinatorial optimization problems in the analysis and design of probabilistic networks. Networks, V.15, 257-271
* [22] Wang, G. (1994), A proof of Boesch’s conjecture. Networks, V.24, 277-284
* [23] Ebneshahrashoob. Morteza and Gao, Tangan and Sobel, Milton (2005), Random Accessibility as a Parallelism to Reliability Studies on Simple Graphs. Communications in Statistics - Theory and Methods, V.34, 1423 - 1436
* [24] Chung, Fan R. K. (1997), Spectral Graph Theory. American Mathematical Society
|
arxiv-papers
| 2010-08-27T22:15:12 |
2024-09-04T02:49:12.512671
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Behrang Mahjani",
"submitter": "Behrang Mahjani",
"url": "https://arxiv.org/abs/1008.4814"
}
|
1008.4846
|
# Laguerre-Gaussian modes: entangled state representation and generalized
Wigner transform in quantum optics
Li-yun Hu1 and Hong-yi Fan2
1College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Corresponding author. _E-mail addresses_ : hlyun2008@126.com,
hlyun2008@gmail.com)
###### Abstract
By introducing a new entangled state representation, we show that the
Laguerre-Gaussian (LG) mode is just the wave function of the common
eigenvector of the orbital angular momentum and the total photon number
operators of 2-d oscillator, which can be generated by 50:50 beam splitter
with the phase difference $\phi=\pi/2$ between the reflected and transmitted
fields. Based on this and using the Weyl ordering invariance under similar
transforms, the Wigner representation of LG is directly obtained, which can be
considered as the generalized Wigner transform of Hermite Gaussian modes.
PACS: 03.65.-w-Quantum mechanics
PACS: 42.50.-p-Quantum optics
## 1 Introduction
It has been known that a Laguerre-Gaussian (LG) beam of paraxial light has a
well-defined orbital angular momentum [1, 2, 3, 4, 5], which is useful in
studying quantum entanglement [6]. In Ref. [3] Nienhuis and Allen employed
operator algebra to describe the Laguerre-Gaussian beam, and noticed that
Laguerre-Gaussian modes are laser mode analog of the angular momentum
eigenstates of the isotropic 2-d harmonic oscillator. In Ref. [7] Simon and
Agarwal presented a phase-space description (the Wigner function) of the LG
mode by exploiting the underlying phase-space symmetry. In this Letter we
shall go a step further to show that LG mode is just the wave function of the
common eigenvector $\left|n,l\right\rangle$ of the orbital angular momentum
operator and the total photon number operator of 2-d oscillator in the
entangled state representation (ESR). The ESR was constructed [8, 9] based on
the Einstein-Podolsky-Rosen quantum entanglement [10]. It is shown that
$\left|n,l\right\rangle$ can be generated by 50:50 beam splitter with the
phase difference $\phi=\pi/2$ between the reflected and transmitted fields.
Then we use the Weyl ordering form of the Wigner operator and the Weyl
ordering’s covariance under similar transformations to directly derive the
Wigner representation of LG beams, which seems economical. The marginal
distributions of Wigner function (WF) are also obtained by the entangled state
representation. It is found that the amplitude of marginal distribution is
just the eigenfunction of the fractional Fourier transform (FrFT). In
addition, LG mode can also be considered as the generalized Wigner transform
of Hermite Gaussian modes by using the Schmidt decomposition of the ESR.
## 2 Eigenvector corresponding to Laguerre-Guassian mode
In Ref. [3] the Bosonic operator algebra of the quantum harmonic oscillator is
applied to the description of Gaussian modes of a laser beam, i.e., a paraxial
beam of light is described by operators’ eigenvector equations
$\displaystyle N\left|n,l\right\rangle$
$\displaystyle=n\left|n,l\right\rangle,\text{\ \
}N\equiv\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right),$
$\displaystyle L\left|n,l\right\rangle$
$\displaystyle=l\left|n,l\right\rangle,\text{ \ }L\equiv
X_{1}P_{2}-X_{2}P_{1},$ (1)
since $\left[N,L\right]=0$, where $a_{i}^{\dagger}$ and $a_{i}$ ($i=1,2$) are
Bose creation operator and annihilation operator; $L$ and $N$ are the orbital
angular momentum operator and the total photon number operator of a paraxial
beam of light, respectively. Using
$X_{i}=\left(a_{i}+a_{i}^{\dagger}\right)/\sqrt{2}$ and
$P_{i}=\left(a_{i}-a_{i}^{\dagger}\right)/(\mathtt{i}\sqrt{2})$, and
$[a_{i},a_{j}^{\dagger}]=\delta_{ij}$, then
$L=\mathtt{i}(a_{2}^{\dagger}a_{1}-a_{1}^{\dagger}a_{2}).$ (2)
Here we search for the common eigenvector of $\left(N,L\right)$ in the
entangled state representation. By introducing
$A_{+}=\frac{1}{\sqrt{2}}(a_{1}-ia_{2}),\text{
}A_{-}=\frac{1}{\sqrt{2}i}(a_{1}+ia_{2}),$ (3)
which obey the commutative relation
$\displaystyle[A_{+},A_{+}^{\dagger}]$ $\displaystyle=1,\text{\
}[A_{-},A_{-}^{\dagger}]=1,$ (4) $\displaystyle[A_{+},A_{-}^{\dagger}]$
$\displaystyle=0,\text{ }[A_{-},A_{+}^{\dagger}]=0,$
one can see
$N=A_{+}^{\dagger}A_{+}+A_{-}^{\dagger}A_{-},\text{
}L=A_{+}^{\dagger}A_{+}-A_{-}^{\dagger}A_{-}.$ (5)
Now we introduce the entangled state representation in Fock space,
$\left|\eta\right\rangle=\exp\left\\{-\frac{1}{2}\left|\eta\right|^{2}+\eta
A_{+}^{\dagger}-\eta^{\ast}A_{-}^{\dagger}+A_{+}^{\dagger}A_{-}^{\dagger}\right\\}\left|00\right\rangle,$
(6)
here $\eta=\left|\eta\right|e^{i\varphi}=\eta_{1}+i\eta_{2},$
$\left|00\right\rangle$ is annihilated by $A_{+}$ and $A_{-}.$ It is not
difficult to see that $\left|\eta\right\rangle$ is the common eigenvector of
operators $\left(X_{1}-X_{2}-P_{1}+P_{2},P_{1}+P_{2}-X_{1}-X_{2}\right)$
$\displaystyle\left(X_{1}-X_{2}-P_{1}+P_{2}\right)\left|\eta\right\rangle$
$\displaystyle=2\eta_{1}\left|\eta\right\rangle,$
$\displaystyle\left(P_{1}+P_{2}-X_{1}-X_{2}\right)\left|\eta\right\rangle$
$\displaystyle=2\eta_{2}\left|\eta\right\rangle.$ (7)
Using the normal ordering form of vacuum projector
$\left|00\right\rangle\left\langle
00\right|=\colon\exp\left(-A_{+}^{\dagger}A_{+}-A_{-}^{\dagger}A_{-}\right)\colon,$
(8)
(where : : denotes normal ordering) and the technique of integral within an
ordered product (IWOP) of operators [11, 12] we can prove the completeness
relation and the orthonormal property of $\left|\eta\right\rangle,$
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1,\text{
}\left\langle\eta\right.\left|\eta^{\prime}\right\rangle=\pi\delta\left(\eta-\eta^{\prime\ast}\right)\delta\left(\eta^{\ast}-\eta^{\prime}\right).$
(9)
Thus $\left|\eta\right\rangle$ is qualified to make up a new representation.
It follows from (6) and (4) that
$A_{+}\left|\eta\right\rangle=(\eta+A_{-}^{\dagger})\left|\eta\right\rangle,\text{\
}A_{+}^{\dagger}\left|\eta\right\rangle=\left(\frac{\partial}{\partial\eta}+\frac{\eta^{\ast}}{2}\right)\left|\eta\right\rangle,$
(10)
$A_{-}\left|\eta\right\rangle=(A_{+}^{\dagger}-\eta^{\ast})\left|\eta\right\rangle,A_{-}^{\dagger}\left|\eta\right\rangle=\left(-\frac{\partial}{\partial\eta^{\ast}}-\frac{\eta}{2}\right)\left|\eta\right\rangle.$
(11)
which lead to (denote $r=\left|\eta\right|$ for simplicity)
$\displaystyle(A_{+}^{\dagger}A_{+}+A_{-}^{\dagger}A_{-})\left|\eta\right\rangle$
$\displaystyle=\left(\frac{1}{2}r^{2}-1-2\frac{\partial^{2}}{\partial\eta\partial\eta^{\ast}}\right)\left|\eta\right\rangle$
$\displaystyle=\left[\frac{r^{2}}{2}-1-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{{}^{r^{2}}}\frac{\partial^{2}}{\partial\varphi^{2}}\right)\right]\left|\eta\right\rangle$
$\displaystyle(A_{+}^{\dagger}A_{+}-A_{-}^{\dagger}A_{-})\left|\eta\right\rangle$
$\displaystyle=\left(\eta\frac{\partial}{\partial\eta}-\eta^{\ast}\frac{\partial}{\partial\eta^{\ast}}\right)\left|\eta\right\rangle=-\mathtt{i}\frac{\partial}{\partial\varphi}\left|\eta\right\rangle.$
(12)
Projecting Eqs.(1) onto the $\left\langle\eta\right|$ representation and using
(5), (10)-([20]), one can obtain the following equations
$l\left\langle\eta\right|\left.n,l\right\rangle=i\frac{\partial}{\partial\varphi}\left\langle\eta\right|\left.n,l\right\rangle,$
(13)
and
$n\left\langle\eta\right|\left.n,l\right\rangle=\left[\frac{r^{2}}{2}-1-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{{}^{r^{2}}}\frac{\partial^{2}}{\partial\varphi^{2}}\right)\right]\left\langle\eta\right|\left.n,l\right\rangle.$
(14)
Eq.(13) indicates that $\left\langle\eta\right|\left.n,l\right\rangle\propto
e^{-il\varphi}.$ From the uniqueness of wave function,
$e^{-il\varphi}|_{\varphi=0}=e^{-il\varphi}|_{\varphi=2\pi},$ we know $l=0,$
$\pm 1,$ $\pm 2\cdots.$ So letting
$\left\langle\eta\right|\left.n,l\right\rangle=R(r)e^{-il\varphi}$ and
substituting it into (14) yields
$\frac{d^{2}R}{dr^{2}}+\frac{1}{r}\frac{dR}{dr}+\left(-r^{2}+2\left(n+1\right)-\frac{l^{2}}{{}^{r^{2}}}\right)R=0.$
(15)
Introducing $\xi=r^{2}$ such that
$\frac{dR}{dr}=2\sqrt{\xi}\frac{dR}{d\xi},\text{
}\frac{d^{2}R}{dr^{2}}=2\frac{dR}{d\xi}+4\xi\frac{d^{2}R}{d\xi^{2}},$ (16)
Eq. (15) becomes
$\frac{d^{2}R}{d\xi^{2}}+\frac{1}{\xi}\frac{dR}{d\xi}+\left(-\frac{1}{4}+\frac{n+1}{2\xi}-\frac{l^{2}}{4\xi^{2}}\right)R=0.$
(17)
Then make the variable transform in (17)
$R(\xi)=e^{-\xi/2}\xi^{\left|l\right|/2}u(\xi),$ (18)
one can obtain the equation for $u(\xi),$
$\xi\frac{d^{2}u}{d\xi^{2}}+(\left|l\right|+1-\xi)\frac{du}{d\xi}+\frac{n-\left|l\right|}{2}u=0.$
(19)
Eq.(19) is just a confluent hypergeometric equation whose solution is the
associate Laguerre polynomials, $L_{n_{\rho}}^{\left|l\right|}(\xi)$, where
$n_{\rho}=\frac{n-\left|l\right|}{2},$ $(n_{\rho}=0,$ $1,$ $2,\cdots)$[13].
Thus the wave function of $\left|n,l\right\rangle$ in
$\left\langle\eta\right|$ representation is given by
$\left\langle\eta\right|\left.n,l\right\rangle=C_{1}e^{-il\varphi}e^{-\frac{1}{2}r^{2}}r^{\left|l\right|}L_{n_{\rho}}^{\left|l\right|}(r^{2}),$
(20)
where $C_{1}$ is an integral constant. The right-hand side of Eq.(20) is just
the LG mode, so we reach the conclusion that the wave function of
$\left|n,l\right\rangle$ in the entangled state representation is just the LG
mode, i.e., the LG mode gets its new physical meaning in quantum optics.
Next, we further derive the explicit expression of $\left|n,l\right\rangle.$
Using the completeness relation of $\left\langle\eta\right|$ (9) and (20), we
have
$\displaystyle\left|n,l\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|n,l\rangle$
$\displaystyle=C_{1}\int\frac{d^{2}\eta}{\pi}e^{-\frac{1}{2}r^{2}}\left|\eta\right\rangle
e^{-il\varphi}r^{\left|l\right|}L_{n_{\rho}}^{\left|l\right|}(r^{2}).$ (21)
Then noticing the relation between two-variable Hermite polynomial [14, 15]
and Laguerre polynomial,
$H_{m,n}\left(\eta,\eta^{\ast}\right)=m!\left(-1\right)^{m}\eta^{\ast}{}^{n-m}L_{m}^{n-m}\left(\eta\eta^{\ast}\right),$
(22)
where $m<n,\ $and the generating function of
$H_{m,n}\left(\eta,\eta^{\ast}\right)$ is
$H_{m,n}\left(x,y\right)=\left.\frac{\partial^{m+n}}{\partial t^{m}\partial
t^{\prime
n}}\exp\left[-tt^{\prime}+tx+t^{\prime}y\right]\right|_{t=t^{\prime}=0},$ (23)
as well as using the integral formula [16]
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{
Re}\left(\xi\right)<0,$ (24)
we can reform Eq.(21) as (without loss of the generality, setting $l>0$ and
$m_{\rho}=[n+\left|l\right|]/2$)
$\displaystyle\left|n,l\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{n_{\rho}}C_{1}}{n_{\rho}!}\int\frac{d^{2}\eta}{\pi}H_{n_{\rho},m_{\rho}}\left(\eta,\eta^{\ast}\right)e^{-\frac{1}{2}\left|\eta\right|^{2}}\left|\eta\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{n_{\rho}}C_{1}}{n_{\rho}!}\frac{\partial^{n_{\rho}+m_{\rho}}}{\partial
t^{n_{\rho}}\partial t^{\prime
m_{\rho}}}\exp\left[-tt^{\prime}+A_{+}^{\dagger}A_{-}^{\dagger}\right]$
$\displaystyle\times\int\frac{d^{2}\eta}{\pi}\exp\left[-\left|\eta\right|^{2}+\left(A_{+}^{\dagger}+t\right)\eta+\left(t^{\prime}-A_{-}^{\dagger}\right)\eta^{\ast}\right]_{t=t^{\prime}=0}\left|00\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{n_{\rho}}C_{1}}{n_{\rho}!}\frac{\partial^{n_{\rho}+m_{\rho}}}{\partial
t^{n_{\rho}}\partial t^{\prime
m_{\rho}}}\exp\left[A_{+}^{\dagger}t^{\prime}-tA_{-}^{\dagger}\right]_{t=t^{\prime}=0}\left|00\right\rangle$
$\displaystyle=\frac{C_{1}}{n_{\rho}!}\left(A_{+}^{\dagger}\right)^{m_{\rho}}\left(A_{-}^{\dagger}\right)^{n_{\rho}}\left|00\right\rangle.$
(25)
## 3 Generation of $\left|n,l\right\rangle$ by Beam Splitter
Note Eq.(3) and
$\displaystyle A_{+}^{{\dagger}}$
$\displaystyle=e^{i\frac{\pi}{2}J_{x}}a_{1}^{{\dagger}}e^{-i\frac{\pi}{2}J_{x}},\text{
\
}A_{-}^{{\dagger}}=e^{i\frac{\pi}{2}J_{x}}a_{2}^{{\dagger}}e^{-i\frac{\pi}{2}J_{x}},\text{
}$ $\displaystyle J_{x}$
$\displaystyle=\frac{1}{2}\left(a_{1}^{{\dagger}}a_{2}+a_{2}^{{\dagger}}a_{1}\right),\text{
}\left|00\right\rangle=e^{i\frac{\pi}{2}J_{x}}\left|00\right\rangle,$ (26)
thus Eq.(25) can be further put into the following form
$\displaystyle\left|n,l\right\rangle$
$\displaystyle=\frac{C_{1}}{n_{\rho}!}e^{i\frac{\pi}{2}J_{x}}\left(a_{1}^{{\dagger}}\right)^{m_{\rho}}\left(a_{2}^{{\dagger}}\right)^{n_{\rho}}\left|00\right\rangle$
$\displaystyle=C_{1}\sqrt{\frac{m_{\rho}!}{n_{\rho}!}}e^{i\frac{\pi}{2}J_{x}}\left|m_{\rho},n_{\rho}\right\rangle.$
(27)
It is easy to see that the normalized constant can be chosen as
$C_{1}=\sqrt{n_{\rho}!/m_{\rho}!},$ which further leads to
$\left|n,l\right\rangle=e^{i\frac{\pi}{2}J_{x}}\left|m_{\rho},n_{\rho}\right\rangle,$
(28)
where $J_{x}$ can be expressed by angular momentum operators
$J_{+}=a_{1}^{{\dagger}}a_{2}$ and $J_{-}=a_{1}a_{2}^{{\dagger}}$,
$J_{x}=\frac{1}{2}\left(J_{+}+J_{-}\right).$ $J_{+},$ $J_{-}$ and
$J_{z}=\frac{1}{2}\left(a_{1}^{{\dagger}}a_{1}-a_{2}^{{\dagger}}a_{2}\right)$
make up a close SU(2) Lie algebra.
On the other hand, the beam splitter is one of the few experimentally
accessible devices that may act as an entangler. In fact, the role of a beam
splitter operator [17, 18] is expressed by
$B\left(\theta,\phi\right)=\exp\left[\frac{\theta}{2}\left(a_{1}^{{\dagger}}a_{2}e^{i\phi}-a_{1}a_{2}^{{\dagger}}e^{-i\phi}\right)\right],$
(29)
with the amplitude reflection and transmission coefficients
$T=\cos\frac{\theta}{2},$ $R=\sin\frac{\theta}{2}.$ The beam splitter gives
the phase difference $\phi$ between the reflected and transmitted fields.
Comparing Eq.(29) with $e^{i\frac{\pi}{2}J_{x}}$ leads us to choose
$\theta=\pi/2$ (corresponding to 50:50 beam splitter) and $\phi=\pi/2,$ thus
$B\left(\pi/2,\pi/2\right)$ is just equivalent to $e^{i\frac{\pi}{2}J_{x}}$ in
form. This indicates that $\left|n,l\right\rangle$ can be generated by acting
a symmetric beam splitter with $\phi=\pi/2$ on two independent input Fock
states
$\left|m_{\rho},n_{\rho}\right\rangle=\left|m_{\rho}\right\rangle_{1}\left|n_{\rho}\right\rangle_{2}.$
In addition, note that $n=m_{\rho}+n_{\rho},$ i.e., when the total number of
input photons is $n,$ so the output state becomes an ($n+1$)-dimensional
entangled state [19].
## 4 The Wigner representation
As is well-known, the Wigner quasidistribution provides with a definite phase
space distribution of quantum states and is very useful in quantum statistics
and quantum optics. In this section, we evaluate the Wigner representation of
$\left|n,l\right\rangle.$ According Ref.[20], the Wigner representation of
$\left|n,l\right\rangle$ is given by
$\displaystyle W_{\left|n,l\right\rangle}$ $\displaystyle=\left\langle
n,l\right|\Delta_{1}\left(x_{1},p_{1}\right)\Delta_{2}\left(x_{2},p_{2}\right)\left|n,l\right\rangle$
$\displaystyle=\left\langle
m_{\rho},n_{\rho}\right|e^{-i\frac{\pi}{2}J_{x}}\Delta_{1}\left(x_{1},p_{1}\right)$
$\displaystyle\times\Delta_{2}\left(x_{2},p_{2}\right)e^{i\frac{\pi}{2}J_{x}}\left|m_{\rho},n_{\rho}\right\rangle,$
(30)
where $\Delta_{1}\left(x_{1},p_{1}\right)$ is the single-mode Wigner operator,
whose Weyl ordering form [21] is
$\Delta_{1}\left(x_{1},p_{1}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(p_{1}-P_{1}\right)\delta\left(x_{1}-X_{1}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(31)
where the
symbol$\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$denotes Weyl
ordering [22]. Note that the order of Bose operators $a$ and $a^{{\dagger}}$
within $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ can be
permitted. That is to say, even though $[a$,$a^{{\dagger}}]$ $=1$, we can have
$\genfrac{}{}{0.0pt}{}{:}{:}aa^{{\dagger}}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}a^{{\dagger}}a\genfrac{}{}{0.0pt}{}{:}{:}$.
According to the covariance property of Weyl ordering under similar
transformations [21] and
$\displaystyle e^{-i\frac{\pi}{2}J_{x}}X_{1}e^{i\frac{\pi}{2}J_{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(X_{1}-P_{2}\right),\text{ }$
$\displaystyle e^{-i\frac{\pi}{2}J_{x}}P_{1}e^{i\frac{\pi}{2}J_{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(P_{1}+X_{2}\right),$ $\displaystyle
e^{-i\frac{\pi}{2}J_{x}}X_{2}e^{i\frac{\pi}{2}J_{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(X_{2}-P_{1}\right),\text{ }$
$\displaystyle e^{-i\frac{\pi}{2}J_{x}}P_{2}e^{i\frac{\pi}{2}J_{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(P_{2}+X_{1}\right),$ (32)
we have
$\displaystyle
e^{-i\frac{\pi}{2}J_{x}}\Delta_{1}\left(x_{1},p_{1}\right)\Delta_{2}\left(x_{2},p_{2}\right)e^{i\frac{\pi}{2}J_{x}}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(p_{1}-\frac{P_{1}+X_{2}}{\sqrt{2}}\right)\delta\left(x_{1}-\frac{X_{1}-P_{2}}{\sqrt{2}}\right)$
$\displaystyle\times\delta\left(p_{2}-\frac{P_{2}+X_{1}}{\sqrt{2}}\right)\delta\left(x_{2}-\frac{X_{2}-P_{1}}{\sqrt{2}}\right)\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\Delta_{1}\left(\frac{x_{1}+p_{2}}{\sqrt{2}},\frac{p_{1}-x_{2}}{\sqrt{2}}\right)\Delta_{2}\left(\frac{x_{2}+p_{1}}{\sqrt{2}},\frac{p_{2}-x_{1}}{\sqrt{2}}\right).$
(33)
Since the Wigner representation of number state $\left|m\right\rangle$ is well
known [23],
$\displaystyle W_{\left|m\right\rangle}$ $\displaystyle=\left\langle
m\right|\Delta_{1}\left(x_{1},p_{1}\right)\left|m\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{m}}{\pi}e^{-\left(x_{1}^{2}+p_{1}^{2}\right)}L_{m}\left[2\left(x_{1}^{2}+p_{1}^{2}\right)\right],$
(34)
so we directly obtain the Wigner representation of L-G mode,
$\displaystyle W_{\left|n,l\right\rangle}$ $\displaystyle=\left\langle
m_{\rho}\right|\Delta_{1}\left(\frac{x_{1}+p_{2}}{\sqrt{2}},\frac{p_{1}-x_{2}}{\sqrt{2}}\right)\left|m_{\rho}\right\rangle$
$\displaystyle\times\left\langle
n_{\rho}\right|\Delta_{2}\left(\frac{x_{2}+p_{1}}{\sqrt{2}},\frac{p_{2}-x_{1}}{\sqrt{2}}\right)\left|n_{\rho}\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{m_{\rho}+n_{\rho}}}{\pi^{2}}e^{-Q_{0}}L_{m_{\rho}}\left(Q_{0}+Q_{2}\right)L_{n_{\rho}}\left(Q_{0}-Q_{2}\right),$
(35)
where $Q_{0}=\allowbreak p_{1}^{2}+p_{2}^{2}+x_{1}^{2}+x_{2}^{2}$ and
$Q_{2}=2p_{2}x_{1}-2p_{1}x_{2}$. Eq.(35) is in agreement with the result of
Ref. [5, 7]. Our derivation seems economical.
## 5 The marginal distributions and fractional Fourier transform of
$\left|n,l\right\rangle$
The fractional Fourier transform (FrFT) has been paid more and more attention
within different contexts of both mathematics and physics. It is also very
useful tool in Fourier optics and information optics. In this section, we
examine the relation between the FrFT and the marginal distributions of
$W_{\left|n,l\right\rangle}$.
For this purpose, we recall that the two-mode Wigner operator
$\Delta_{1}\left(x_{1},p_{1}\right)\Delta_{2}\left(x_{2},p_{2}\right)\equiv\Delta_{1}\left(\alpha\right)\Delta_{2}\left(\beta\right)$
($\alpha=(x_{1}+ip_{1})/\sqrt{2}$, $\beta=(x_{1}+ip_{1})/\sqrt{2}$) in
entangled state representation $\left\langle\tau\right|.$ Using the IWOP
technique we have shown in [22] that $\Delta_{1,2}\left(\sigma,\gamma\right)$
is just the product of two independent single-mode Wigner operators
$\Delta_{1}\left(\alpha\right)\Delta_{2}\left(\beta\right)=\Delta_{1,2}\left(\sigma,\gamma\right)$
i.e.,
$\Delta_{1,2}\left(\sigma,\gamma\right)=\int\frac{d^{2}\tau}{\pi^{3}}\left|\sigma-\tau\right\rangle\left\langle\sigma+\tau\right|e^{\tau\gamma^{\ast}-\tau^{\ast}\gamma},$
(36)
where $\sigma=\alpha-\beta^{\ast},$ $\gamma=\alpha+\beta^{\ast}$ and
$\left|\tau=\tau_{1}+i\tau_{2}\right\rangle$ can be expressed in two-mode Fock
space as [12, 13]
$\left|\tau\right\rangle=\exp\left\\{-\frac{1}{2}\left|\tau\right|^{2}+\tau
a_{1}^{\dagger}-\tau^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$
(37)
which is the common eigenvector of $X_{1}-X_{2}$ and $P_{1}+P_{2}$, which
obeys the eigenvector equations
$\left(X_{1}-X_{2}\right)\left|\tau\right\rangle=\sqrt{2}\tau_{1}\left|\tau\right\rangle,$
$\left(P_{1}+P_{2}\right)\left|\tau\right\rangle=\sqrt{2}\tau_{2}\left|\tau\right\rangle.$
Performing the integration of $\Delta_{1,2}\left(\sigma,\gamma\right)$ over
$d^{2}\gamma$ ($d^{2}\sigma$) leads to the projection operator of the
entangled state $\left|\tau\right\rangle$ ($\left|\xi\right\rangle$)
$\displaystyle\int d^{2}\gamma\Delta_{1,2}(\sigma,\gamma)$
$\displaystyle=\frac{1}{\pi}\left|\tau\right\rangle\left\langle\tau\right||_{\tau=\sigma},$
$\displaystyle\int d^{2}\sigma\Delta_{1,2}(\sigma,\gamma)$
$\displaystyle=\frac{1}{\pi}\left|\xi\right\rangle\left\langle\xi\right||_{\xi=\gamma},$
(38)
where $\left|\xi\right\rangle$ is the conjugate state of
$\left|\tau\right\rangle$. Thus the marginal distributions for quantum states
$\rho$ in ($\tau_{1},\tau_{2}$) and ($\xi_{1},\xi_{2}$) phase space are given
by
$\displaystyle\int d^{2}\sigma W\left(\sigma,\gamma\right)$
$\displaystyle=\frac{1}{\pi}\left\langle\xi\right|\rho\left|\xi\right\rangle|_{\xi=\gamma},\text{
}$ $\displaystyle\int d^{2}\gamma W\left(\sigma,\gamma\right)$
$\displaystyle=\frac{1}{\pi}\left\langle\tau\right|\rho\left|\tau\right\rangle|_{\tau=\sigma},$
(39)
respectively. Eq.(39) shows that, for bipartite system, the marginal
distributions can be calculated by evaluating the quantum average of $\rho$ in
$\left\langle\xi\right|$, $\left\langle\tau\right|$ representations.
Now we calculate the inner-product
$\left\langle\tau\right.\left|n,l\right\rangle.$ Note that
$a_{1}^{{\dagger}}a_{2}$, $a_{1}a_{2}^{{\dagger}}$, and
$J_{z}=\frac{1}{2}\left(a_{1}^{{\dagger}}a_{1}-a_{2}^{{\dagger}}a_{2}\right)$
make up a close SU(2) Lie algebra, thus $e^{i\frac{\pi}{2}J_{x}}$ can be
decomposed as
$e^{i\frac{\pi}{2}J_{x}}=e^{ia_{1}^{{\dagger}}a_{2}}\exp\left[\frac{1}{2}\left(a_{1}^{{\dagger}}a_{1}-a_{2}^{{\dagger}}a_{2}\right)\ln
2\right]e^{ia_{2}^{{\dagger}}a_{1}},$ (40)
then we have
$\displaystyle\left\langle\tau\right.\left|n,l\right\rangle$
$\displaystyle=\sqrt{\frac{m_{\rho}!}{n_{\rho}!}}\sum_{k=0}^{m_{\rho}}\left(\sqrt{2}\right)^{m_{\rho}-n_{\rho}-2k}$
$\displaystyle\times\frac{\left(n_{\rho}+k\right)!}{k!\left(m_{\rho}-k\right)!}\sum_{j=0}^{n_{\rho}+k}\frac{i^{k+j}}{j!}\sqrt{\frac{\left(m_{\rho}-k+j\right)!}{\left(n_{\rho}+k-j\right)!}}$
$\displaystyle\times\left\langle\tau\right.\left|m_{\rho}-k+j,n_{\rho}+k-j\right\rangle.$
(41)
Using the generating function of $H_{m,n},$ we have
$\left\langle\tau^{\prime}\right.\left|m,n\right\rangle=\frac{\left(-1\right)^{n}}{\sqrt{m!n!}}H_{m,n}\left(\tau^{\prime\ast},\tau^{\prime}\right)e^{-\left|\tau^{\prime}\right|^{2}/2}.$
(42)
Substituting Eq.(42) into Eq.(41) leads to
$\displaystyle\left\langle\tau\right.\left|n,l\right\rangle$
$\displaystyle=\left(-\right)^{n_{\rho}}2^{\left(m_{\rho}-n_{\rho}\right)/2}e^{-\left|\tau\right|^{2}/2}$
$\displaystyle\sqrt{\frac{m_{\rho}!}{n_{\rho}!}}\sum_{k=0}^{m_{\rho}}\frac{\left(n_{\rho}+k\right)!}{2^{k}k!\left(m_{\rho}-k\right)!}$
$\displaystyle\times\sum_{j=0}^{n_{\rho}+k}\frac{\left(-i\right)^{k+j}}{j!\left(n_{\rho}+k-j\right)!}H_{m_{\rho}-k+j,n_{\rho}+k-j}\left(\tau^{\ast},\tau\right).$
(43)
Thus the marginal distribution is
$\displaystyle\int d^{2}\gamma W\left(\sigma,\gamma\right)$
$\displaystyle=\frac{e^{-\left|\sigma\right|^{2}}}{\pi}2^{m_{\rho}-n_{\rho}}\frac{m_{\rho}!}{n_{\rho}!}\left|\sum_{k=0}^{m_{\rho}}\frac{\left(-i\right)^{k}\left(n_{\rho}+k\right)!}{2^{k}k!\left(m_{\rho}-k\right)!}\right.$
$\displaystyle\times\left.\sum_{j=0}^{n_{\rho}+k}\left(-i\right)^{j}\frac{H_{m_{\rho}-k+j,n_{\rho}+k-j}\left(\sigma^{\ast},\sigma\right)}{j!\left(n_{\rho}+k-j\right)!}\right|^{2}.$
(44)
Due to the presence of sum polynomial, the marginal distribution is not
Gaussian. In a similar way, one can obtain the other marginal distribution in
$\gamma$ direction.
Before the end of this section, we mention the relation between the FrFT and
marginal distribution. In Ref.[24] we have proved that, in the context of
quantum optics, the FrFT can be described as the matrix element of fractional
operator $\exp[-i\alpha(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})]$ between
$\left\langle\tau^{\prime}\right|$ and $\left|f\right\rangle$, i.e.,
$\mathcal{F}_{\alpha}\left[f\left(\tau^{\prime}\right)\right]=\frac{e^{i(\alpha-\frac{\pi}{2})}}{2\sin\alpha}\int\frac{d^{2}\tau^{\prime}}{\pi}\exp\left[\frac{i(\left|\tau^{\prime}\right|^{2}+\left|\tau\right|^{2})}{2\tan\alpha}-\frac{i\left(\tau^{\ast}\tau^{\prime}+\tau^{\prime\ast}\tau\right)}{2\sin\alpha}\right]f\left(\tau^{\prime}\right)=\left\langle\tau\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\left|f\right\rangle,$
where
$f\left(\tau^{\prime}\right)=\left\langle\tau^{\prime}\right|\left.f\right\rangle.$
When
$\left|f\right\rangle=e^{i\frac{\pi}{2}J_{x}}\left|m_{\rho},n_{\rho}\right\rangle,$
the corresponding FrFT is
$\displaystyle\mathcal{F}_{\alpha}\left[\left\langle\tau^{\prime}\right.\left|n,l\right\rangle\right]$
$\displaystyle=\left\langle\tau\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]e^{i\frac{\pi}{2}J_{x}}\left|m_{\rho},n_{\rho}\right\rangle$
$\displaystyle=e^{-i\alpha\left(m_{\rho}+n_{\rho}\right)}\left\langle\tau\right.\left|n,l\right\rangle,$
(45)
where we have used $e^{-i\alpha a_{1}^{\dagger}a_{1}}a_{1}$ $e^{i\alpha
a_{1}^{\dagger}a_{1}}=a_{1}e^{i\alpha}.$ Eq.(45) implies that the
eigenequations of FrFT can also be
$\left\langle\tau\right.\left|n,l\right\rangle$ with the eigenvalue being
$e^{-i\alpha\left(m_{\rho}+n_{\rho}\right)}$, which is the superposition (41)
of two-variable Hermite polynomials. In Ref.[25], we have proved that the two-
variable Hermite polynomials (TVHP) is just the eigenfunction of the FrFT in
complex form by using the IWOP technique and the bipartite entangled state
representations. Here, we should emphasize that for any unitary two-mode
operators $U$ obeying the relation
$\exp[-i\alpha(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})]U\exp[i\alpha(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})]=U,$
the wave function
$\left\langle\tau\right|U\left|m_{\rho},n_{\rho}\right\rangle$ is the
eigenfunction of FrFT with the eigenvalue being
$e^{-i\alpha\left(m_{\rho}+n_{\rho}\right)}$ [2].
Combing Eqs.(45) and (39), one can obtain a simple formula connecting the FrFT
and the marginal distribution of $W_{\left|n,l\right\rangle},$
$\int d^{2}\gamma
W\left(\sigma,\gamma\right)=\frac{1}{\pi}\left|\mathcal{F}_{\alpha}\left[\left\langle\tau=\sigma\right.\left|n,l\right\rangle\right]\right|^{2},$
(46)
which is $\alpha-$independent. Thus we can also obtain the marginal
distribution by the FrFT.
## 6 L-G mode as generalized Wigner transform of Hermit-Gaussian modes
In this section we shall reveal the relation between the L-G mode and the
single variable Hermit-Gaussian(H-G) modes. Note that by taking the Fourier
transformation of $\left|\tau\right\rangle$ with regard to $\tau_{2}$ followed
by the inverse Fourier transformation, we can recover the entangled state
$\left|\tau\right\rangle.$ In another word,
$\left|\tau=\tau_{1}+i\tau_{2}\right\rangle$ can be decomposed into
$\displaystyle\left|\tau\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dxe^{ix\sqrt{2}\tau_{2}}\left|x+\frac{\tau_{1}}{\sqrt{2}}\right\rangle_{1}\otimes\left|x-\frac{\tau_{1}}{\sqrt{2}}\right\rangle_{2}$
$\displaystyle=e^{-i\tau_{1}\tau_{2}}\int_{-\infty}^{\infty}dxe^{ix\sqrt{2}\tau_{2}}\left|x\right\rangle_{1}\otimes\left|x-\sqrt{2}\tau_{1}\right\rangle_{2},$
(47)
in which $\left|x\right\rangle_{i}$ ($i=1,2$) are the coordinate eigenvectors.
Eq.(47) is called the Schmidt decomposition of $\left|\tau\right\rangle$ and
indicates $\left|\tau\right\rangle$ is an entangled state [26].
Eq.(47) leads to
$\displaystyle\left\langle m,n\right.\left|\tau\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dxe^{ix\sqrt{2}\tau_{2}}\left\langle
m\left|x+\frac{\tau_{1}}{\sqrt{2}}\right.\right\rangle\left\langle
n\left|x-\frac{\tau_{1}}{\sqrt{2}}\right.\right\rangle$
$\displaystyle=\left\langle
m\right|\left[\int_{-\infty}^{\infty}dxe^{ix\sqrt{2}\tau_{2}}\left|x+\frac{\tau_{1}}{\sqrt{2}}\right\rangle\left\langle
x-\frac{\tau_{1}}{\sqrt{2}}\right|\right]\left|n\right\rangle.$ (48)
It is interesting to notice that the integration […] in Eq.(48) is similar to
the single-mode Wigner operator,
$\Delta\left(x,p\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}due^{-iup}\left|x-\frac{u}{2}\right\rangle\left\langle
x+\frac{u}{2}\right|,$ (49)
and the left hand side of Eq.(48) corresponds to TVHP mode (L-G mode). It
might be expected that the L-G can be expressed in terms of the generalized
Wigner transform (GWT). Actually, after making variable replacement, Eq.(48)
can be rewritten as
$\left\langle m,n\right.\left|\tau\right\rangle=\pi\left\langle
m\right|\Delta\left(\frac{\tau_{1}}{\sqrt{2}},\frac{\tau_{2}}{\sqrt{2}}\right)\left(-1\right)^{N}\left|n\right\rangle.$
(50)
If we introduce the following GWT,
$W_{g}\left[f,v\right]\left(x,p\right)=\left\langle
f\right|\Delta\left(x,p\right)\left|v\right\rangle,$ (51)
which reduces to the usual Wigner transform under the condition
$\left|v\right\rangle=\left|f\right\rangle$, while for
$\left|v\right\rangle=\left(-1\right)^{n}\left|n\right\rangle$ and
$\left|f\right\rangle=\left|m\right\rangle$ Eq.(51) becomes the right hand
side of Eq.(50), which corresponds to the GWT.
On the other hand, note that Eqs. (22) and (42), the left hand side of Eq.(50)
can be put into (without loss of the generality, letting $m<n$)
$\displaystyle\left\langle m,n\right.\left|\tau\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{n}}{\sqrt{m!n!}}H_{m,n}\left(\tau,\tau^{\ast}\right)e^{-\left|\tau\right|^{2}/2}$
$\displaystyle=\left(-1\right)^{n+m}\sqrt{\frac{m!}{n!}}\tau^{\ast}{}^{n-m}L_{m}^{n-m}\left(\tau\tau^{\ast}\right)e^{-\left|\tau\right|^{2}/2},$
(52)
which indicates that the left hand side of Eq.(50) is just corresponding to
the L-G mode, as well as
$\left\langle
m\right.\left|x\right\rangle=\frac{e^{-x^{2}/2}}{\sqrt{2^{m}m!\sqrt{\pi}}}H_{m}\left(x\right)\equiv
h_{m}\left(x\right),$ (53)
where $H_{m}\left(x\right)$ is single variable Hermite polynomial, and
$h_{m}\left(x\right)$ just corresponds to the H-G mode, we have
$\displaystyle\sqrt{\frac{m!}{n!}}\tau^{\ast}{}^{n-m}L_{m}^{n-m}\left(\tau\tau^{\ast}\right)e^{-\left|\tau\right|^{2}/2}$
$\displaystyle=\frac{\left(-1\right)^{m}}{2}\int_{-\infty}^{\infty}due^{-iu\frac{\tau_{2}}{\sqrt{2}}}h_{m}\left(\frac{\tau_{1}}{\sqrt{2}}-\frac{u}{2}\right)h_{n}\left(\frac{\tau_{1}}{\sqrt{2}}+\frac{u}{2}\right)$
$\displaystyle=\left(-1\right)^{m}\pi
W_{g}\left[h_{m},h_{n}\right]\left(\tau_{1}/\sqrt{2},\tau_{2}/\sqrt{2}\right).$
(54)
Thus, we can conclude that the L-G mode can be obtained by the GWT of two
single-variable H-G modes. In addition, we should point out that the L-G mode
can also be generated by windowed Fourier transform (which is often used in
signal process) of two single-variable H-G modes by noticing the second line
of Eq.(47).
In summary, we have endowed the Laguerre-Gaussian (LG) mode with new physical
meaning in quantum optics, i.e., we find that it is just the wave function of
the common eigenvector of the orbital angular momentum and the total photon
number operators of 2-d oscillator in the entangled state representation. The
common eigenvector can be obtained by using beam splitter with the phase
difference $\phi=\pi/2$ between the reflected and transmitted fields. With the
aid of the Weyl ordering invariance under similar transforms, the Wigner
representation of LG is directly obtained. It is shown that its marginal
distributions can be calculated by the FrFT. In addition, L-G mode can also be
considered as the generalized Wigner transform of Hermite Gaussian modes by
using the Schmidt decomposition of the entangled state representation.
ACKNOWLEDGEMENT: Work supported by a grant from the Key Programs Foundation of
Ministry of Education of China (No. 210115) and the Research Foundation of the
Education Department of Jiangxi Province of China (No. GJJ10097). L.-Y. Hu’s
email address is hlyun2008@126.com.
## References
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* [18] L. Mandel, E. Wolf, _Optical Coherence and Quantum Optics_ (Cambridge University Press, 1995) and references therein; R. A. Campos, B. E. A. Saleh, M. C. Teich, _Phys. Rev. A_ 40 (1989) 1371.
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* [20] E. P .Wigner, _Phys. Rev_. 40 (1932) 749.
* [21] Li-yun Hu and Hong-yi Fan, _Europhys. Lett._ , 85 (2009) 60001; H.-Y. Fan, Ann. Phys. 323 (2008) 500.
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* [23] Wolfgang P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Birlin, 2001.
* [24] Li-yun Hu and Hong-yi Fan, _J. Mod. Opt._ 55 (2008) 2429.
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|
arxiv-papers
| 2010-08-28T08:29:29 |
2024-09-04T02:49:12.525471
|
{
"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/1008.4846"
}
|
1008.4898
|
# WiNV: A Framework for Web-based Interactive Scalable Network Visualization
Hassan Gobjuka Verizon
919 Hidden Ridge
Irving, TX 75083
Email: hasan.gobjuka@verizon.com Hassan Gobjuka Kamal A. Ahmat Verizon
Department of Information Technology 919 Hidden Ridge City University of New
York Irving, TX 75038 New York, NY 11101 hasan.gobjuka@verizon.com
kamal.ahmat@live.lagcc.cuny.edu
###### Abstract
In this paper we introduce WiNV - A framework for web-based interactive
scalable network visualization. WiNV enables a new class of rich and scalable
interactive cross-platform capabilities for visualizing large-scale networks
natively in a user’ s browser. Extensive experiments show that our system can
visualize networks that consist of tens of thousands of nodes while
maintaining fast, high-quality interaction.
## I Introduction
This paper presents a new state-of-the-art network visualization framework
which supports user interactions with large-scale networks in a web browser
without the need for plug-ins or special-purpose runtime systems. Our
framework supports the standard visual information browsing functionalities
that include overviewing, zooming, and editing. The WiNV framework supports
information discovery in two modes; the Standalone Mode and Google Earth Mode.
In the Standalone mode, the application runs as a native application. In the
Google Earth mode, WiNV visualizes the network on Google Earth enabling the
user to get better understanding of wide-spread networks. Interaction is used
to mold the network layout into the user’ s own mental model and for editing
the information being visualized such as device configurations or network
layout, if the underlying topology discovery algorithm is unable to discover
some connections.
## II Presentation
### II-A Background
Visualization of large scale graphs have been addressed in the research
community [1] and [6]; open source community [2, 7, 8]; and industry [9].
However, to our knowledge, most of the tools developed so far are general
purpose in such that they are not capable to visualize networks at the
detailed level may be required by users. Furthermore, none of these tools is
web-based, and thus, they don’t benefit from the advantages the web offers. We
have developed a web-based extensible framework which enables interactive
visualization of networks that may consist of hundreds of thousands of devices
natively in a web browser. Our framework was developed initially as a
presentation-tier for the Network Management System has been developed in [4]
and [5].
In multi-tiered web-based frameworks, presentation tiers (i.e. clients) are
classified into either thick or thin client. A thick client typically provides
most of the functionality needed for rendering the data in the browser. Thus,
most of the processing time is done in the client side and the interaction
with the back-end is relatively low. In thin-clients, on the other hand, most
of the processing is done on the back-end and consequently most of the front-
end resources can be used for presentation purposes, which in turn achieves
higher scalability. However, this approach requires constant communication
with the back-end. Web-based presentation-tier applications can also be
classified into native or plug-in applications. Examples of plug-ins that are
used for browser applications are Adobe Flash and Microsoft Silverlight. The
functionality of plug-in-based applications is restricted by the capabilities
of the base plug-in. Native applications don’t require any application to be
pre-installed and, consequently, they can utilize all the features provided by
technologies used such as DHTML, JSP, AJAX, and DWR [3]. We chose to develop
WiNV as thin-client native application but with the capability of integrating
it with other native frameworks such as Google Earth.
### II-B Architecture
As we stated earlier, the main contribution of this work is a scalable web-
based framework for visualizing very large networks while providing smooth
interaction and editing capability in a user’ s web-browser. The framework
architecture is shown in Figure 1. To make WiNV flexible and lightweight, the
architecture has been designed so that the minimal amount of client’s
resources are consumed. Furthermore, WiNV can run in any basic browser with no
external plug-ins. WiNV supports two modes: Standalone mode and Google Earth
mode. In both modes, most of computation including layout, interaction and
editing algorithms run on the back-end while the client only renders the
network. Network elements such as devices, hosts and links are generated in-
the-fly through the AJAX-based DWR technology and don’t require any web-page
refreshing if any changes are made, giving a very smooth interaction
experience. WiNV framework consists of the following tiers:
* •
Presentation Tier represents a web-browser running on a client machine.
Besides visualizing the network, the browser reacts to events triggered by the
user (e.g through the mouse) such as clicking on a network segment or creating
a network connection between two devices.
* •
Network Tier represents the communication between the presentation and server
tiers.
* •
Server Tier represents the “engine” of the framework. All algorithms used for
layout and other computations are loaded within the server tier.
* •
Persistent Tier is used to interact with the database. All changes made to the
network layout by the user are reflected to the persistent tier.
The WiNV architecture is modular and provides a set of extendable APIs that
can be used by any web-based application to visualize the network. The set of
APIs can be also used for plugging in multiple different layout algorithms.
These algorithms then can be selected through the user interface.
## III Demonstration
Figure 1: The scalable modular architecture used in WiNV.
The framework can receive the data from an XML file that contains network
representation, or it can be integrated to exchange the date with the middle
tier of a web-based application that implements the framework’s APIs. The demo
phases consist of (1) Rendering - Displaying the network layout based on the
initial data coming from the backend or after a change has been made; and (2)
Interaction - Capturing the user command and computing modifications to the
network. This process includes, for instance, zooming in, or out, changing
device and link configurations or editing the topology. If the framework is
set up in Google Earth mode, the user can view the network initially in very
high level and a detailed layout will be rendered when zooming in the area
that has the target network segment. At the very detailed level, the user
could view network device interfaces and direction interconnections among them
(i.e. physical network topology.) The user can also view VLAN and ISP level
configurations and device/link statistics. Figure 2 demonstrates high-level
network visualization in Google Earth mode. In Standalone mode, the networks
are usually grouped based on their IP prefixes or physical location (i.e IP
address longitude and latitude) and each group is displayed as a cloud. When
the user clicks on a cloud, it expands and that network is rendered at more
detailed level. At the most detailed level, all devices, hosts, and
peripherals in that segment are rendered. The number of levels, data
refreshing rate, and grouping methods are configurable and can be specified by
the user.
Figure 2: A screenshot of WiNV displaying high-level network layout in Google
Earth mode.
## References
* [1] D. Auber. Tulip. In P. Mutzel, M. J unger, and S. Leipert, editors, 9th Symp. Graph. Drawing, volume 2265 of Lecture Notes in Computer Science, pages 335 337. Springer-Verlag, 2001.
* [2] V. Batagelj, and A. Mrvar, Pajek - program for large network analysis, Connections, 21:47 57, 1998.
* [3] Direct Web Remoting, Open source online application, available at http://directwebremoting.org/dwr/index.html.
* [4] H. Gobjuka, Topology Discovery for Virtual Local Area Networks, in Proc. IEEE INFOCOM Mini-Conference 2010.
* [5] H. Gobjuka, and Y. Breitbart, Ethernet Topology Discovery for Networks with Incomplete Information, IEEE/ACM Transactions on Networking, 2010, In Press.
* [6] I. Herman, G. Melan on, and M. S. Marshall, Graph visualization and navigation in information visualization: A survey , IEEE Transactions on Visualization and Computer Graphics, 6(1):24-43, 2000.
* [7] J. O’Madadhain, D. Fisher, S. White, and Y. Boey, The JUNG (Java Universal Network/Graph) Framework, Technical Report UCI-ICS 03-17, School of Information and Computer Science University of California, Irvine.
* [8] P. Shannon, A. Markiel, O. Ozier, N. S. Baliga, J. T.Wang, D. Ramage, N. Amin, B. Schwikowski, and T. Ideker, Cytoscape: a software environment for integrated models of biomolecular interaction networks, Genome Res, 13(11):2498 2504, November 2003.
* [9] Tom Sawyer Software, Tom sawyer visualization, 2009.
|
arxiv-papers
| 2010-08-29T03:23:55 |
2024-09-04T02:49:12.532636
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hassan Gobjuka and Kamal Ahmat",
"submitter": "Kamal Ahmat",
"url": "https://arxiv.org/abs/1008.4898"
}
|
1008.4900
|
# Managing Clouds in Cloud Platforms
Hassan Gobjuka Verizon
919 Hidden Ridge
Irving, TX 75083
Email: hasan.gobjuka@verizon.com Kamal A. Ahmat Hassan Gobjuka Department of
Information Technology Verizon City University of New York 919 Hidden Ridge
New York, NY 11101 Irving, TX 75038 kamal.ahmat@live.lagcc.cuny.edu
hasan.gobjuka@verizon.com
## I Motivation
Managing cloud services is a fundamental challenge in today s virtualized
environments. These challenges equally face both providers and consumers of
cloud services. The issue becomes even more challenging in virtualized
environments that support mobile clouds. Cloud computing platforms such as
Amazon EC2 provide customers with flexible, on demand resources at low cost.
However, they fail to provide seamless infrastructure management and
monitoring capabilities that many customers may need. For instance, Amazon EC2
doesn’t fully support cloud services automated discovery and it requires a
private set of authentication credentials. Salesforce.com, on the other hand,
do not provide monitoring access to their underlying systems. Moreover, these
systems fail to provide infrastructure monitoring of heterogenous and legacy
systems that don’t support agents. In this work, we explore how to build a
cloud management system that combines heterogeneous management of virtual
resources with comprehensive management of physical devices. We propose an
initial prototype for automated cloud management and monitoring framework. Our
ultimate goal is to develop a framework that have the capability of
automatically tracking configuration and relationships while providing full
event management, measuring performance and testing thresholds, and measuring
availability consistently. Armed with such a framework, operators can make
better decisions quickly and more efficiently.
## II Challenges
These tasks are achieved through an agentless monitoring of the cloud’s
infrastructure. While traditional network management methods suffer from
inherited difficulties [1, 2], implementing seamless network management and
monitoring framework entails several new challenges:
* •
Discovering the relationship of virtualized resources to underlying physical
infrastructure.
* •
Minimizing the overhead of monitoring and problem determination across a
physical and virtualized infrastructure.
* •
Handling security-related constraints that may affect data collection is
probably one of the most serious issues.
* •
Response action should be taken regarding a particular virtual or physical
device within the hard response deadline time frame. In agentless-based
monitoring systems, this can be insured only by implementing high number of
threads, which in turn increases complexity.
* •
Dealing with infrastructure management issues such as root-cause analysis
becomes more complex.
## III Design and Implementation
We propose an event-based model where events are placed on an in-memory
publish/subscribe bus on the Management Server, enabling a high throughput of
events.
The event bus architecture, depicted in Figure 1 enables any authorized
mediator to create events on the bus, and any authorized consumer to access
events from the bus. Events on the bus show current status of infrastructure
components.
Figure 1: An initial prototype of our cloud management and monitoring system.
The framework will provide a set of APIs to simplify creation of consumer and
mediator applications. A set of language extensions and Web services will be
used to enable Perl, Ruby, or Java scripts to create events on the bus. To
support high level of reliability and scalability, the Distributed Collector
subsystem will be multi-threaded. Furthermore, events will are normalized from
any source into a common format, which will enable consistent processing.
## References
* [1] T. Benson, A. Akella, and D. A. Maltz, Unraveling the complexity of network management, In NSDI, 2009.
* [2] H. Gobjuka, and Y. Breitbart, Ethernet Topology Discovery for Networks with Incomplete Information, IEEE/ACM Transactions on Networking, 2010, In Press.
|
arxiv-papers
| 2010-08-29T03:35:24 |
2024-09-04T02:49:12.536111
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kamal A. Ahmat and Hassan Gobjuka",
"submitter": "Kamal Ahmat",
"url": "https://arxiv.org/abs/1008.4900"
}
|
1008.4946
|
# Scaling entropy and automorphisms with purely point spectrum111Partially
supported by the grants RFBR-08-01-00379-a and RFBR-09-01-12175-ofi-m.
A. M. Vershik
###### Abstract
We study the dynamics of metrics generated by measure-preserving
transformations. We consider sequences of average metrics and
$\epsilon$-entropies of the measure with respect to these metrics. The main
result, which gives a criterion for checking that the spectrum of a
transformation if purely point, is that the scaling sequence for the
$\epsilon$-entropies with respect to the averages of an admissible metric is
bounded if and only if the automorphism has a purely point spectrum. This
paper is one of a series of papers by the author devoted to the asymptotic
theory of sequences of metric measure spaces and its applications to ergodic
theory.
To the memory of my friend Misha Birman
###### Contents
1. 1 Introduction
2. 2 Admissible metrics
3. 3 Average and maximal metrics
4. 4 Entropy and scaling entropy
1. 4.1 The $\varepsilon$-entropy of a measure in a metric space
2. 4.2 Scaling sequence and scaling entropy
5. 5 Invariant metrics on groups and averages of admissible metrics
1. 5.1 Invariant metrics and discrete spectrum
2. 5.2 Admissible invariant metrics
3. 5.3 Admissibility of the average metric
6. 6 Criterion for the discreteness and continuity of the spectrum in terms of the scaling entropy
7. 7 Comparison with the traditional approach, the Pascal automorphism, and concluding remarks
1. 7.1 Supremum metrics
2. 7.2 Application of the discreteness criterion
3. 7.3 The dynamics of metrics
## 1 Introduction
Among the many mathematical and nonmathematical problems we have been
discussing with Misha Birman for many years after our acquaintance began in
the early 1960s, the most intriguing one was the parallel between scattering
theory and ergodic theory. Recently, I have returned to the (yet nonexistent)
“ergodic scattering theory” and some forgotten questions related to it.
However, this paper deals with another subject, which also correlates with M.
Sh. Birman’s research.
It is well known that the problem of deciding whether or not the spectrum of a
given (say, differential) operator is discrete is quite difficult. Several
remarkable early papers by M. Sh. Birman (in the first place, [1]) dealt with
exactly this problem. In ergodic theory and the theory of dynamical systems,
this problem (whether or not the system of eigenfunctions is complete) is also
very difficult. In what follows, we use the terminology common in ergodic
theory that slightly differs from the one adopted in operator theory: we use
the term “discrete spectrum” as a synonym for “pure point spectrum.” This is
justified by the fact that a discrete spectrum in the sense of operator theory
almost never appears in the theory of dynamical systems, since almost always
one deals with unitary operators.
As an example demonstrating the difficulty of this problem, we can mention the
theory of substitutions, or stationary adic transformations [2], on which an
extensive literature exists. The most intriguing problems concern
nonstationary adic transformations with subexponential growth of the number of
vertices in the corresponding Bratteli–Vershik diagram (see [2]). The simplest
and most popular example of such a transformation is the Pascal automorphism
defined in [3]; in this case, the measure space (i. e., the phase space of the
dynamical system) is the space of infinite paths in the Pascal graph endowed
with a Bernoulli measure, and the transformation sends a path to its successor
in the natural lexicographic order. In spite of many efforts, we still do not
know whether the corresponding unitary operator has a discrete (i. e., pure
point) or mixed spectrum, or, which seems most likely, its spectrum is purely
continuous. Attempts to directly construct its eigenfunctions failed; another
approach, based on the characterization of systems with discrete spectrum in
terms of Kushnirenko’s sequential entropy, [10] has not been carried out.
In the paper [14] (for more details, see [4, 8, 27]), we suggested a new
notion, the so-called _scaling entropy_ , which generalizes the notion of
Kolmogorov’s entropy. The main point is that we suggest to average the shifts
of the metric with respect to a given transformation and then compute the
$\varepsilon$-entropy of the average metric. The class of increasing sequences
of positive integers that normalize the growth of this $\varepsilon$-entropy
over all admissible metrics does not depend on the choice of an admissible
metric, so that the asymptotics of the growth of these sequences is a new
metric invariant of automorphisms. Admissible metrics are measurable metrics
satisfying some conditions that do or do not depend on the automorphism (see
Section 2). It is important that we consider not merely the
$\varepsilon$-entropy of a metric, but the $\varepsilon$-entropy of a metric
in a measure space. Admissible metrics play the same role as measurable
generating partitions in the classical theory of Kolmogorov’s entropy
according to Sinai’s definition. At first sight, the difference between using
partitions and metrics looks rather technical: a partition determines a
semimetric of a special form (a so-called “cut semimetric,” see [18]).
However, our approach has two important differences from the classical theory.
First, we use the $\varepsilon$-entropy of the iterated metric on a measure
space rather than the entropy of a partition; this is a generalization of
Kolmogorov’s entropy, which allows one to distinguish automorphisms with zero
entropy. The second, more important, difference is that we use the average
metric (rather than the supremum of metrics, which corresponds to the supremum
of partitions), which has no interpretation in terms of partitions and which
contains more information about the automorphism than the supremum of metrics.
In this paper, we give a necessary and sufficient condition for an
automorphism to have a discrete spectrum in terms of the scaling sequence. The
condition is that this sequence is bounded. This result generalizes a theorem
of S. Ferenczi [15, 16], who considered the measure-theoretic complexity of
symbolic systems by analogy with the ordinary complexity in symbolic dynamics.
Our approach substantially differs from that of [15]: we consider an arbitrary
admissible metric rather than the Hamming metric only, and, which is most
important, introduce the average metric and show that it is admissible in many
cases and, in particular, for the Hamming metric.
Thus criterion, i.e., the boundedness of the growth of the scaling entropy,
should first be applied to adic transformations, e.g., to the Pascal
automorphism (see [4]). Although it is not yet applied to checking that the
spectrum of the Pascal automorphism is not discrete, 222See the footnote in
the subsection 7.2, the corresponding combinatorics is already developed and
described in the recent paper [19], where a lower bound is obtained on the
scaling sequence of the sup metric for the same Pascal automorphism.
Supposedly, one can extend this bound ($\ln n$) to the average metric using
the same techniques.
A wider context is presented in our papers [4, 8], where we suggest a plan for
the study of the dynamics of metrics in a measure space as a source of new
invariants of automorphisms. It is important that the notion of scaling
entropy provides an answer to the question of whether or not the spectrum of a
given transformation is discrete.
Sections 2–4 are of general nature and are intended not only for the purposes
of this paper, which is devoted mostly to automorphisms with discrete
spectrum. Here we introduce our main objects: admissible metrics, averages,
scaling sequences, and the scaling entropy of an automorphism. In Section 5,
we study the dynamics of metrics on a group and find conditions under which
the average metric is admissible. The main result is given in Section 6, where
we present a criterion for checking whether the spectrum of an automorphism is
completely or partially discrete. In the last section, we sketch possible
applications, links to the ordinary construction of entropy, and general
remarks about the dynamics of metrics.
## 2 Admissible metrics
We consider various metrics and semimetrics on a measure space
$(X,{\mathfrak{A}},\mu)$. In what follows, it is assumed to be a standard
space with continuous measure $\mu$ and $\sigma$-algebra $\mathfrak{A}$ of
$\bmod 0$ classes of measurable sets, i.e., a Lebesgue space with continuous
measure in the sense of Rokhlin (see [24]). The space $X\times X$ is endowed
with the $\sigma$-algebra $\mathfrak{A}\times\mathfrak{A}$ and the measure
$\mu\times\mu$.
We define a class of (semi)metrics on a measure space, which plays an
important role in what follows.
###### Definition 1.
A measurable function $\rho:X\times X\rightarrow{\mathbb{R}}_{+}$ is called an
admissible (semi)metric if
1) $\rho$ is a (semi)metric in the ordinary sense on a subset
$X^{\prime}\subset X$ of full measure ($\mu X^{\prime}=1$), i.e.,
$\rho(x,y)\geq 0$, $\rho(x,y)=\rho(y,x)$, $\rho(x,y)+\rho(y,z)\geq\rho(x,z)$
for all triples $(x,y,z)\in X^{\prime}\times X^{\prime}\times X^{\prime}$,
$(\mu\times\mu)\\{(x,x),x\in X^{\prime}\\}=0$, and
$\int_{X}\int_{X}\rho(x,y)d\mu(x)d\mu(y)<\infty.$
In order to formulate the next condition, observe that if $\rho$ satisfies
Condition 1), then the partition $\psi_{\rho}$ of the space $X$ into the
classes of points $C_{x}=\\{y:\rho(x,y)=0\\}$, $x\in X$, is measurable. Hence
we have a well-defined quotient space $X_{\rho}\equiv X/{\psi_{\rho}}$ endowed
with the quotient metric denoted by the same letter $\rho$ and the quotient
measure $\mu_{\psi_{\rho}}\equiv\mu_{\rho}$. For the quotient space
$(X_{\rho},\rho,\mu_{\rho})$, Condition 1) is still satisfied.
2) The (completion of the) metric space $(X,\rho)$ (if $\rho$ is a metric) or
$(X_{\rho},\rho)$ (if $\rho$ on $X$ is a semimetric) is a Polish (= metric,
separable, complete) space with a Borel probability measure $\mu$
(respectively, $\mu_{\rho}$).
Following the measure-theoretic tradition, we must identify (semi)metrics (and
hence the corresponding spaces) if they coincide almost everywhere as
measurable functions on the space $(X\times X,\mu\times\mu)$. Of course, a
(semi)metric that coincides almost everywhere with an admissible (semi)metric
is admissible.333We can define the notion of almost metric as a measurable
function for which all axioms on metric satisfy for almost all pairs or
triples points. As F.Petrov noticed for each almost metric there exists the
admissible metric in our sense which almost everywhere coincided with it.
Condition 2) means that the $\sigma$-algebra of Borel sets in the metric space
$(X,\rho)$ (or $(X_{\rho},\rho)$ in the case of a semimetric) is dense in the
$\sigma$-algebra $\mathfrak{A}$ of all measurable sets and, therefore, the
measure $\mu$ (respectively, $\mu_{\rho}$) is a Borel probability measure. It
is obvious from the definition that a semimetric $\rho$ is admissible if and
only if the corresponding metric in the quotient space $X_{\rho}$ is
admissible.
An equivalent definition of an admissible metric is as follows: almost every
pair of points can be separated by balls of positive measure, or, in other
words, the $\sigma$-subalgebra generated by the open balls $\bmod 0$ separates
points of the space.
One can also formulate the admissibility condition in terms of the notion of a
pure function from our paper [6]:
###### Lemma 1.
A metric $\rho$ is admissible if and only if it satisfies Condition 1) and,
regarded as a function of two variables, is pure in the sense of [6]; the
latter means that the partition of the space on the classes of equivalence
$x\sim y\Leftrightarrow\mu\\{z:\rho(x,z)=\rho(y,z)\\}=1$ is the partition on
the separate points $mod0$. In other words: the map $x\mapsto\rho(x,.)$ is
injective from measure space $(X,\mu)$ to the classes of $mod0$ equal
functions of one variable. 444In other words, almost every point is uniquely
determined by the collection of distances from this point to all points of
some set of full measure (which may depend on the point).
If $\rho$ is a semimetric, then this condition must hold for the metric $\rho$
on the quotient space $(X_{\rho},\rho,\mu_{\rho})$.
Indeed, the purity condition implies that the $\sigma$-algebra of sets
generated by the balls separates points and hence is dense in the
$\sigma$-algebra $(X,{\mathfrak{A}},\mu)$; this also implies the separability.
The converse immediately follows from the definition of an admissible metric.
It is well known (see [24]) that if $(X,\rho)$ is a Polish space, then every
nondegenerate Borel probability measure $\mu$ on $X$ turns $(X,\rho)$ into a
Lebesgue space. In other words, the metric $\rho$ on a Polish space $(X,\rho)$
endowed with a Borel probability measure $\mu$ is an admissible metric on the
space $(X,\mu)$.
As in other our papers, in the definition of an admissible metric we reverse
the tradition and consider various metrics and semimetrics on a fixed measure
space rather than various measures on a given metric space. Recall that
triples $(X,\rho,\mu)$, consisting of a metric space endowed with a measure,
in M. Gromov’s book [5] were called $mm$-spaces, and in the paper [7], metric
triples or Gromov triples.
It is useful to regard admissible metrics as densities of some finite measures
equivalent to the measure $\mu\times\mu$ on $X\times X$:
$dM_{\mu,\rho}=\rho(x,y)d\mu(x)d\mu(y).$
If we set $\int_{X\times X}\rho(x,y)d\mu(x)d\mu(y)=1$, which can be done by
normalizing the metric, then the new measure is also a probability measure.
Observe the following important property of admissible metrics implied by this
interpretation.
###### Theorem 1.
For almost every point $x\in X$, there is a uniquely defined $\bmod 0$
conditional measure $\mu^{x}$ on $X$, which is given by the formula
$d\mu^{x}(A)=\int_{A}\rho(x,y)d\mu(y)$ for every measurable set $A\subset X$.
The family of measures $\\{\mu^{x};x\in X\\}$ satisfies the condition
$\mu(A)=\int_{X}\mu^{x}(A)d\mu(x).$
The metric $\rho$, regarded as a metric on the measure space $(X,\mu^{x})$, is
admissible.
###### Proof.
Consider the new measure $M_{\mu,\rho}$ on the space $X\times X$ and the
measurable partition into the classes of points $C^{x}=\\{(x,*)\\}\subset
X\times X$, and use the classical theorem on the existence of conditional
measures (see, e.g., [24]), which implies the desired formula and the
uniqueness $\bmod 0$ of the family of conditional measures. Now consider the
space $(X,\rho,\mu^{x})$ for a fixed $x$; the metric $\rho$ on this space is
admissible since the admissibility of a metric is obviously preserved under
replacing a measure with an equivalent one. ∎
The conditional measure $\mu^{x}(A)$ can be interpreted as the “average
distance,” or the conditional expectation of the distance from the set $A$ to
the point $x$.
In these terms, the condition that the metric, regarded as a function of two
variables, is pure means that the conditional expectations do not coincide
$\mod 0$ for almost all pairs of points.
Now we can give a convenient criterion of admissibility.
###### Proposition 1.
A measurable function $\rho(\cdot,\cdot)$ satisfying Condition 1) of
Definition 1 is admissible if and only if the following non-degeneracy
condition holds:
for $\mu$\- almost all $x$, the measure of arbitrary balls of positive radius
is positive: $\forall\varepsilon>0,\quad\mu\\{y:\rho(x,y)<\epsilon\\}>0$,
which is equivalent to:
$\mu^{x}([0,\epsilon])>0$
for almost all $x$ and all positive $\epsilon$.
###### Proof.
The fact that an admissible metric satisfies the condition in question was
observed above. Now assume that this condition is satisfied. We must prove
that the space $(X,\rho)$ (if $\rho$ is a metric) or the quotient space
$X_{\rho}$ (if $\rho$ is a semimetric) is separable. It suffices to consider
the case of a metric.
The condition stated in the proposition implies that for every $\epsilon$
there exists $\delta=\delta(\epsilon)$ with $\lim_{\epsilon\to
0}\delta(\epsilon)=0$ such that some set of measure $>1-\delta$ contains a
finite $\epsilon$-net. But this means exactly that the space is separable and
the measure is concentrated on a $\sigma$-compact set. ∎
Several examples. 1\. An important example is the following class of
semimetrics, which was in fact intensively used in entropy theory. Every
partition $\xi$ of a space $(X,\mu)$ into finitely or countably many
measurable sets gives rise to a semimetric:
$\rho_{\xi}(x,y)=\delta_{(\xi(x),\xi(y))},$
where $\xi(z)$ is the element of $\xi$ that contains $z$. In this case,
$X_{\rho}$ is a finite or countable metric space. Such (finite) semimetrics
are called cuts, and their linear combinations are called cut semimetrics (in
the terminology of [18]). It is easy to see that cut (semi)metrics are
admissible.
2\. The very important metric defined by the formula $\rho(x,y)={\rm const}$
for $x\neq y$ determines a discrete uncountable space. We will call it the
constant metric. The constant metric on a space with continuous measure is not
admissible. In this case, the $\sigma$-algebra generated by the open sets is
trivial and does not separate points of the space, i.e., a point is not
determined by the collection of distances to the other points.
3\. The condition defining an admissible metric can be strengthened by
requiring, in condition 1), that
$\int_{X}\int_{X}\rho(x,y)^{p}d\mu(x)d\mu(y)<\infty$
for $p>1$; in this case, we say that the metric is $p$-admissible. Let us say
that $\infty$-admissible metrics (semimetrics) are bounded; this class of
admissible metrics will be most useful in what follows.
4\. In combinatorial examples, it often suffices to consider metrics with
which the space is compact or precompact (or, in the case of a semimetric,
quasi-compact). For example, adic transformations [2, 3] act in the space of
infinite paths of an $\mathbb{N}$-graded graph, which is a totally
disconnected compact space.
The set of admissible metrics on a Lebesgue space $(X,\mu)$ with continuous
measure is a convex cone ${\cal R}(X,\mu)={\cal R}$ with respect to the
operation of taking a linear combination of metrics with nonnegative
coefficients. This cone is a canonical object (by the uniqueness of a Lebesgue
space up to isomorphism) and plays a role similar to the role of the simplex
of Borel probability measures in topological dynamics. In many cases, it
suffices to consider admissible (semi)metrics that produce compact spaces
(after completion), but we do not exclude the case of a noncompact space. One
may consider different topologies on the cone ${\cal R}$; the most natural of
them is the weak topology, in which an $\varepsilon$-neighborhood of a metric
$\rho$ is the collection of metrics
$\Bigl{\\{}\theta:(\mu\times\mu)\\{(x,y):|\rho(x,y)-\theta(x,y)|<\varepsilon\\}>1-\varepsilon\Bigr{\\}}.$
The property of being an admissible metric is invariant under measure-
preserving transformations: if a metric $\rho$ is admissible and a
transformation $T$ of the space $(X,\mu)$ preserves the measure $\mu$, then
the image $\rho^{T}$ of $\rho$, defined by the formula
$\rho^{T}(\cdot,\cdot)=\rho(T\cdot,T\cdot)$, is also admissible. Thus the
group of (classes of) measure-preserving transformations acts on the cone
${\cal R}$ in a natural way.
## 3 Average and maximal metrics
Let $T$ be a measure-preserving transformation (in what follows, it will be an
automorphism). When considering automorphisms or groups of automorphisms in
spaces with admissible semimetrics, it is natural to assume that there exist
an invariant set of full measure on which the (semi)metric is admissible in
the sense of our definition.
In addition to the notion of admissible (semi)metrics, we define the class of
$T$-admissible metrics. The $T$-admissibility condition must be invariant, in
the sense that if $\cal M$ is the class of $T$-admissible metrics, then
$V{\cal M}\equiv\\{\rho:\rho(x,y)=\rho_{1}(Vx,Vy),\;\rho_{1}\in\cal M\\}$
is the class of $VTV^{-1}$-admissible metrics in the same sense. Among many
possible versions, we choose the class of admissible (semi)metrics for which
$T$ is a Lipschitz transformation almost everywhere: there exists a positive
constant $C$ such that for $(\mu\times\mu)$-almost all pairs $(x,y)$, the
condition
$\rho(Tx,Ty)\leq C\rho(x,y)$
holds; let us say that metrics from this class are Lipschitz $T$-admissible
(semi)metrics.
The choice of an appropriate class depends on the problem under consideration
and the properties of the automorphism. For example, in the case of adic
automorphisms, it is most convenient to consider the class of Lipschitz
metrics. This class can also be defined for countable groups of automorphisms.
For an arbitrary admissible semimetric $\rho$, we have defined the partition
$\psi_{\rho}$ of the space $(X,\mu)$. In the same way, given an arbitrary
automorphism $T$ of the space $(X,\mu)$, we consider the $T$-invariant
partition $\psi_{\rho}^{T}=\bigvee_{k=0}^{\infty}T^{k}\psi_{\rho}$. We say
that the semimetric $\rho$ is generating for $T$ if $\psi_{\rho}^{T}$ is the
partition into separate points $\bmod 0$ (which we denote by $\varepsilon$).
If $\rho$ is the metric generated by a finite partition, then this is the
ordinary definition of a generator (see [25]).
Let us define the average metric and the $\sup$-metric for a given
automorphism.
###### Definition 2.
Let $T$ be an automorphism of a space $(X,\mu)$, and let $\rho$ be an
admissible metric.
The average metric $\rho_{n}^{T}$ is defined by the formula
$\hat{\rho}_{n}^{T}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y).$
The $\sup$-metric is defined by the formula
${\bar{\rho}}_{n}^{T}(x,y)=\sup_{0\leq k<n}\rho(T^{k}x,T^{k}y).$
The following important result is a direct corollary of the pointwise ergodic
theorem.
###### Theorem 2.
For any automorphism $T$ and any admissible (semi)metric $\rho$, the limit of
the sequence of average (semi)metrics $\rho^{T}_{n}$, which we denote by
$\hat{\rho}$, exists almost everywhere in the space $(X\times
X,\mu\times\mu)$:
$\hat{\rho}^{T}(x,y)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y);$
$\hat{\rho}$ is a metric if and only if $\rho$ is a metric or a generating
semimetric.
The existence of the limit follows from the fact that the integral
$\int_{X}\int_{X}\rho(x,y)d\mu(x)d\mu(y)$ is finite and the ergodic theorem.
###### Definition 3.
The metric
$\hat{\rho}^{T}(x,y)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y)$
(the limit exists $\mu$-a.e.) is called the average, or the $l^{1}$-average of
$\rho$ with respect to the automorphism $T$.
The metric
$\bar{\rho}^{T}(x,y)=\sup_{k\geq 0}\rho(T^{k}x,T^{k}x)$
is called the limiting $\sup$-metric of $\rho$ with respect to $T$.
It is clear that $\hat{\rho}^{T}$ and $\bar{\rho}^{T}$ satisfy all conditions
of the definition of a (semi)metric; in what follows, we will consider only
admissible metrics or generating semimetrics $\rho$, so that $\hat{\rho}^{T}$,
and $\bar{\rho}^{T}$ for an ergodic automorphism $T$, are metrics. The
superscript $T$ in the notation for the average and $\sup$-metrics will be
omitted if the automorphism is clear from the context. It is also obvious that
$\hat{\rho}\leq\bar{\rho}$. However, the metrics $\hat{\rho},\bar{\rho}$ may
not be admissible even if the (semi)metric $\rho$ is admissible.
In what follows, we will mainly consider the average metric $\hat{\rho}$. In
most interesting cases, namely if $T$ is weakly mixing, i.e., its spectrum in
the orthogonal complement to the space of constants is continuous, this metric
is constant and hence not admissible; the $\sup$-metric may be constant even
for automorphisms with discrete spectrum. However, in all cases we will be
interested not in the limiting metrics themselves, but in the asymptotic
behavior of the averages $\rho_{n}^{T}$ as $n\to\infty$. As we will see,
automorphisms with discrete spectrum stand apart, since for them the average
metric is often admissible.
It is clear that $\hat{\rho}$ is nothing else but the projection of the
function $\rho$, regarded as an element of the space $L^{1}(X\times
X,\mu\times\mu)$, to the subspace of $(T\times T)$-invariant functions, i.e.,
the expectation of the metric $\rho$ with respect to the subspace of invariant
functions on $X\times X$. This space consists of constants if and only if $T$
has no nontrivial eigenfunctions (in other words, $T$ is weakly mixing). In
this case, $\hat{\rho}$ is almost everywhere a constant, which is equal to the
average $\rho$-distance between the points of the space $X$. At the same time,
if $T$ is not weakly mixing, then the spectrum of $T$ contains a discrete
component and $\hat{\rho}$ may be a nonconstant $T$-invariant (semi)metric. In
this case, one may obtain bounds on $\hat{\rho}$ using Fourier analysis.
Definitions and Lemma 1 imply the following lemma.
###### Lemma 2.
Let $\rho$ be an admissible Lipschitz metric for an automorphism $T$; then the
average metric $\hat{\rho}$ is also admissible and Lipschitz for $T$.
We do not prove this assertion, because we do not need it in this paper.
Here is an example of computing the average metric generated by the cut
semimetric in the case of a rotation of the circle.
Example. Let $X={\mathbb{T}}^{1}={\mathbb{R}}/\mathbb{Z}$, and let $\lambda\in
X$ be an irrational number. Consider the semimetric
$\rho(x,y)=|\chi_{A}(x)-\chi_{A}(y)|$, where $\chi_{A}$ is the indicator of a
measurable set $A\subset X$; the metric $\rho(x,y)$ is $T$-admissible in the
sense of our definition for the shift $T_{\lambda}$ by any irrational number
$\lambda$. The corresponding average metric is shift-invariant and looks as
follows: $\hat{\rho}(x,y)=m[(A+x)\Delta\mu(A+y)]=m[A\Delta(A+x-y)]$; it is
obviously admissible.
## 4 Entropy and scaling entropy
### 4.1 The $\varepsilon$-entropy of a measure in a metric space
Recall that the $\varepsilon$-entropy of a compact metric space $(X,\rho)$ is
the function $\varepsilon\mapsto H_{\rho}(\varepsilon)$ whose value is equal
to the minimum number of points in an $\varepsilon$-net of $(X,\rho)$.
###### Definition 4.
The $\varepsilon$-entropy of a measure space $(X,\mu)$ with an admissible
metric $\rho$ is the function
$\varepsilon\mapsto
H_{\rho,\varepsilon}(\mu)=\inf\\{H(\nu):k_{\rho}(\nu,\mu)<\varepsilon\\},$
where $\nu$ ranges over the set of all discrete measures with finite entropy
and $k_{\rho}(\cdot,\cdot)$ is the Kantorovich metric on the space of Borel
probability measures on $(X,\rho)$. The entropy of a discrete measure
$\nu=\sum_{i}c_{i}\delta_{x_{i}}$ is defined in the usual way:
$H(\nu)=-\sum_{i}c_{i}\ln c_{i}$.
For our purposes, it is more convenient to use in the above definition another
characteristic instead of $H(\mu)$, namely,
$H^{\prime}_{\rho,\varepsilon}(\mu)=\min\Bigl{\\{}\ln k:\exists
X^{\prime},\mu(X^{\prime})>1-\varepsilon,\;\exists\\{x_{i}\\}_{1}^{k}:X^{\prime}\subset\bigcup_{i=1}^{k}V_{\varepsilon}(x_{i})\Bigr{\\}},$
where $V_{\varepsilon}(x)$ is the $\varepsilon$-ball centered at $x$; thus
$\\{x_{1},x_{2},\dots,x_{k}\\}$ is an $\varepsilon$-net in $X^{\prime}$. The
finiteness of $H^{\prime}$ follows from the fact that a Borel probability
measure in a Polish space is concentrated, up to $\varepsilon$, on a compact
set; $H^{\prime}$ is more convenient for computations than $H$.
In this paper, we use the following simple inequality.
###### Lemma 3.
For every compact metric space $(X,\rho)$ and every nondegenerate Borel
measure $\mu$, the following inequality holds:
$H_{\rho,(d+1)\varepsilon}(\mu)\leq H^{\prime}_{\rho,\varepsilon}(\mu),$
where $d$ is the diameter of the space.
###### Proof.
Assume that the diameter of the compact space does not exceed $1$. Assume that
the measure of a set $X^{\prime}$ is greater than $1-\varepsilon$ and
$X^{\prime}\subset\bigcup_{i=1}^{k}V_{\varepsilon}(x_{i})$. Thus the points
$x_{1},\dots,x_{k}$ form an $\varepsilon$-net in $X^{\prime}$. Consider the
discrete measure $\nu$ supported by the points $x_{1},\dots,x_{k}$ with
charges $\nu(x_{i})$ equal to the measures $\mu(V(x_{i}))$ of the
corresponding balls (if two balls have a nonempty intersection, then we
distribute the measure of the intersection proportionally between their
centers). Choose an arbitrary point $x_{\infty}$ and set its charge equal to
$1-\mu(X^{\prime})$. Now consider the Monge–Kantorovich transportation problem
with input measure $\mu$ and output measure $\nu$. It is easy to see that we
have in fact determined an admissible plan $\Psi$ for this problem: the
transportation from a point $x\in X^{\prime}$ goes to the points $x_{i}$ for
which $x\in V_{\varepsilon}(x_{i})$, and the remaining part of the measure
$\mu$ on the set $X\setminus X^{\prime}$ goes to the point $x_{\infty}$. It is
easy to compute the cost of this plan; this gives a bound on the Kantorovich
distance between the measures $\nu$ and $\mu$:
$k_{\rho}(\nu,\mu)\leq\varepsilon(1-\varepsilon)+\varepsilon<2\varepsilon.$
On the other hand, we have $H(\nu)\leq\ln
k=H^{\prime}_{\rho,\varepsilon}(\mu)$. ∎
### 4.2 Scaling sequence and scaling entropy
Let us define the notion of a scaling sequence for the entropy of an
automorphism. If an automorphism $T$ is fixed, we omit the superscript in the
notation for the average entropy and white simply $\hat{\rho}_{n}$.
###### Definition 5.
Let $T$ be an automorphism of a Lebesgue space $(X,\mu)$ with a $T$-invariant
measure $\mu$. By definition, the class of scaling sequences for the
automorphism $T$ and a given (semi)metric $\rho$ on $X$ is the class, denoted
by ${\cal H}_{\rho,\varepsilon}(T)$, of increasing sequences of positive
numbers $\\{c_{n},n\in{\mathbb{N}}\\}$ such that
${\cal
H}_{\rho,\varepsilon}(T)=\Bigl{\\{}\\{c_{n}\\}:0<\liminf_{n\to\infty}\frac{H_{\hat{\rho}_{n},\varepsilon}(\mu)}{c_{n}}\leq\limsup_{n\to\infty}\frac{H_{\hat{\rho}_{n},\varepsilon}(\mu)}{c_{n}}<\infty\Bigr{\\}}.$
In many cases, the class of scaling sequences for a given metric $\rho$ does
not depend on sufficiently small $\varepsilon$. In this case, it is obvious
that all sequences $\\{c_{n}\\}$ from ${\cal H}_{\rho}(T)$ are equivalent.
###### Definition 6.
Assume that for a given ergodic automorphism $T$ of a space $(X,\mu)$ there
exists a (semi)metric $\rho_{0}$ such that the class of scaling sequences for
$\rho_{0}$ is the maximal one (i.e., for any other (semi)metric, sequences
$\\{c_{n}\\}$ from the corresponding class grow not faster than for $\rho$).
In symbols, we write this fact as
${\cal H}_{\rho_{0}}(T)=\sup_{\rho}{\cal H}_{\rho}(T).$
Then we say that ${\cal H}_{\rho_{0}}(T)$ is the class of scaling sequences
for the automorphism $T$ and the metric $\rho_{0}$ is $T$-maximal.
It seems that such a metric exists for every automorphism. If we have chosen
some $T$-maximal scaling sequence and the corresponding limit of entropies
does exist, then it is called the scaling entropy.
###### Conjecture 1.
For every ergodic automorphism $T$, a generic $T$-admissible Lipschitz metric
is $T$-maximal. In particular, for a $K$-automorphism (i.e., an automorphism
with completely positive entropy), the scaling sequence is equivalent to the
sequence $c_{n}=h(T)n$, where $h(T)$ is the entropy of $T$, for every
Lipschitz metric.
A preparatory result in this direction was obtained in [17].
In this paper, we will prove that for an automorphism with purely discrete
spectrum and a $T$-admissible metric, the class ${\cal H}_{\rho}(T)$ of
scaling sequences is the class of bounded sequences.
## 5 Invariant metrics on groups and averages of admissible metrics
### 5.1 Invariant metrics and discrete spectrum
Let us recall some known facts about ergodic automorphisms with discrete
spectrum. It obviously follows from the character theory of commutative groups
that the spectrum of a translation on a compact Abelian group is discrete. By
the classical von Neumann theorem, the converse is also true: an ergodic
automorphism with discrete spectrum is metrically isomorphic to the
translation $T$ on a compact Abelian group $G$ endowed with the Haar measure
$m$ by an element whose powers form a dense subgroup:
$x\mapsto Tx=x+g,\quad\operatorname{Cl}\\{ng,\,n\in\mathbb{Z}\\}=G$
(we use the additive notation). Note that on a compact group $G$ there are
many metrics that are invariant under the whole group of translations and
determine the standard group topology. We will need the following assertion
(which is, possibly, partially known).
###### Proposition 2.
The spectrum of an ergodic automorphism $T$ of a measure space $(X,\mu)$ for
which there exists a $T$-invariant admissible semimetric $\rho$ contains a
discrete component. Moreover, if $\rho$ is a metric, then the spectrum of $T$
is discrete and, consequently, $T$ is isomorphic to a translation on a compact
Abelian group.
###### Proof.
Since an admissible (semi)metric lies in the space $L^{1}(X\times
X,\mu\times\mu)$, the tensor square of the operator $U_{T}$, which corresponds
to the automorphism $T\times T$, has nonconstant eigenfunctions. This can
happen only if the spectrum of the unitary operator $U_{T}$ contains a
discrete component, and the first claim is proved. In other words, $T$ is an
extension of some quotient automorphism with discrete spectrum, which may
coincide with $T$ itself. This means that $T$ is a skew product over a base
with discrete spectrum. Denote by $H\subset L^{2}(X,\mu)$ the subspace spanned
by all eigenfunctions of $U_{T}$. All invariant functions of the operator
$U_{T}\otimes U_{T}$ belong to the tensor square $H\otimes H$; hence these
functions, regarded as functions of two variables, do not change when the
argument ranges over a fiber of the skew product. But if these fibers are not
single point sets, it follows that the metric does not distinguish points in
fibers and hence is a semimetric. Thus if $\rho$ is a metric, then each fiber
necessarily consists of a single point, the spectrum of $T$ is purely
discrete, and, by the ergodicity, $T$ is isomorphic to a translation on a
group. ∎
Let us supplement this proof with an important refinement. Assume that the
automorphism $T$ has an orbit that is everywhere dense with respect to the
metric $\rho$, i.e., $T$ is topologically transitive (though we do not assume
that it is a priori continuous). It is clear from above that this condition
follows in fact from the existence of an invariant metric. Since the metric is
admissible, we may assume without loss of generality that $(X,\rho)$ is a
Polish space. Consider the dense orbit $O=\\{T^{n}x,\,n\in\mathbb{Z}\\}$ of
some point $x$. The restriction of $\rho$ to $O$ is a translation-invariant
metric on the group $\mathbb{Z}$, and all translations are isometries. Hence
the completion of $O$ is an Abelian group to which we can extend the
translations and their limits. Therefore $X$ is a Polish monothetic group.555A
topological group that contains a dense infinite cyclic subgroup is called
monothetic. Note that there are many non-locally compact monothetic groups on
which there is an invariant metric, but there is no invariant measure. A
recent example is the Urysohn universal space regarded as a commutative group,
see [9]. Obviously, the measure $\mu$ is invariant under the action of the
closures of powers of $T$, i.e., it is an invariant probability measure on the
whole group. Hence, by Weil’s theorem, $X$ is a compact commutative group and
$T$ is the translation by an element whose powers are everywhere dense.
### 5.2 Admissible invariant metrics
As we have already observed, for weakly mixing automorphisms, the average of
every metric is constant, since there are no other invariant metrics. However,
for automorphims whose spectra contain a discrete component or are purely
discrete, there are many invariant (semi)metrics. Hence, in order to study
such automorphisms, we should investigate the question when the average metric
for an automorphism $T$ with a discrete spectrum is admissible.
Given a translation-invariant metric $\rho$ on a compact commutative group
with the Haar measure, consider the function
$\phi_{\rho}(r)=\rho(x,x+r)=\rho(0,r)$. In the example from Section 3, it
looked as $\phi(r)\equiv\hat{\rho}(x,x+r)=m[A\Delta(A+r)]$.
One can easily write down necessary and sufficient algebraic conditions on a
measurable function $\phi$ that guarantee that it can be written as
$\phi_{\rho}$ for a measurable invariant metrics:
$\phi(\textbf{0})=0,\;\phi(x)\geq
0,\;\phi(-x)=\phi(x),\;\phi(x)+\phi(y)\geq\phi(x+y).$
We will not need these conditions; the only important fact is that the
admissibility condition from Lemma 1 can easily be reformulated in terms of
this function.
###### Theorem 3.
An invariant measurable (semi)metric on a commutative compact group
(satisfying Condition 1 from Definition 1) is admissible if and only if any of
the following conditions holds.
1. The corresponding function $\phi_{\rho}$ is measurable, and
$\mu\\{z:\phi_{\rho}(z)\neq\phi_{\rho}(g+z)\\}>0$
for almost all $g$.
2.
$\operatorname{ess\,inf}_{g\in V\setminus 0}\phi_{\rho}(g)=0,$
where $V$ is an arbitrary neighborhood of the zero of the group in the
standard topology.
###### Proof.
Condition 1 is exactly equivalent to the condition of Lemma 1. It is useful to
give a direct proof that this condition is necessary. Assume that it is not
satisfied, i.e., for all elements $g$ from some set of positive Haar measure,
$\phi_{\rho}(z)=\phi_{\rho}(g+z)$ for almost all $z$. Therefore, the
measurable function $\phi$ is constant on cosets of the subgroup generated by
$g$. However, every set of positive Haar measure in a nondiscrete Abelian
compact group on which there is an ergodic translation contains an element $g$
that generates a dense cyclic subgroup. Then, since the function $\phi_{\rho}$
is measurable, it follows that it is constant almost everywhere. But if the
function $\phi_{\rho}$ is constant, then the metric $\rho$ is also constant
and hence not admissible.
Let us prove that Condition 2 of the theorem is necessary. Consider the
function $\phi_{\rho}$ for an invariant admissible metric $\rho$. Assume that
$\theta=\operatorname{ess\,inf}$ from Condition 2 is positive; then, using the
invariance of $\rho$, we can construct a continuum of points lying at a fixed
positive $\rho$-distance greater than $\theta$, which contradicts the
admissibility. For the same reason, the set of values of $\phi_{\rho}$ is
dense in some neighborhood of the zero. The fact that Condition 2 is
sufficient follows from Proposition 1. ∎
The last assertion implies the following corollary.
###### Corollary 1.
For every admissible invariant metric there exists a sequence of group
elements that converges to zero both in the standard topology and with respect
to the (semi)metric.
### 5.3 Admissibility of the average metric
Now we can explicitly write down the condition that guarantees the
admissibility of the average, i.e., invariant, metric in terms of the original
metric. Consider an arbitrary measurable (non necessarily admissible) metric
$\rho$ on a compact commutative group $G$ and an ergodic translation $T$ by an
element $g$. Let us write down an expression for the average metric:
$\displaystyle{\hat{\rho}}^{T}(x,y)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n}\rho(x+kg,y+kg)=\int_{G}\rho(g+x,g+y)dm(g)$
$\displaystyle=$ $\displaystyle\int_{G}\rho(z,z+y-x)dm(z)$
(we have used the fact that the measure $m$ is invariant). Hence the function
$\phi(r)$, regarded as a function of $r$, is measurable and has the form
$\phi(r)=\int_{G}\rho(z,z+r)dm(z)=\hat{\rho}(x,y);\quad y-x=r.$
Obviously, $\hat{\rho}$ is a measurable function on the group $G\times G$. Now
we can check the admissibility condition for the average metric $\hat{\rho}$.
###### Definition 7.
We say that an admissible metric $\rho$ on a compact commutative group $G$
with Haar measure $m$ is semicontinuous at zero in mean if
$\liminf_{r\to 0}\int_{G}\rho(x,x+r)dm(x)=0;$
we say that it is semicontinuous at zero in measure if the following condition
holds (in which “meas” means convergence in measure):
$\liminf_{r\to 0}(meas)\rho(x,x+r)=0.$
Note that the second condition follows from the first one, and that both
conditions are stated in purely group terms, i.e., do not depend on the
particular translation $T$.
Thus we have the following admissibility criterion for the average metric.
###### Proposition 3.
The average metric $\hat{\rho}^{T}$ for an ergodic translation $T$ on a
compact commutative group $G$ is admissible if and only if the original
(semi)metric $\rho$ is admissible and semicontinuous at zero in mean.
###### Proof.
The “if” part is proved above; the “only if” part follows from the equality
$\phi_{\hat{\rho}}(r)=\int_{G}\rho(x,x+r)dm(x)$ and the previous proposition.
∎
Now we are ready to formulate and prove the following important fact.
###### Theorem 4.
For every bounded admissible metric on a compact commutative group $G$, the
average metric is admissible.
###### Proof.
Assume that the metric $\rho$ is not semicontinuous at zero and there exists a
positive number $c>0$ such that
$\liminf_{r\to 0}\int_{G}\rho(x,x+r)dm(x)>c.$
Assume that the metric is normalized so that the diameter of the space $X$ is
equal to 1. Then it follows from our assumption that for sufficiently small
(i.e., belonging to a small neighborhood of the zero in the group $G$) $r$
there exists a (depending on $r$) subset in $X$ of measure $\alpha$, which
does not depend on $r$ and is greater than $\frac{c}{2}$, on which
$\rho(x,x+r)>\frac{c}{2}$. Indeed,
$1\cdot\alpha+(1-\alpha)\frac{c}{2}>\int_{G}\rho(x,x+r)dm(x)>c$. But since the
group is compact, there is a set of positive measure for all points $x$ of
which the inequality $\rho(x,x+r)>\frac{c}{2}$ holds for all sufficiently
small $r$ from some set of positive measure. This in turn contradicts the
admissibility of the metric $\rho$ in the formulation of Proposition 1:
arbitrarily small values $\phi(r)$ for small $r$ cannot interlace with values
greater than $\frac{c}{2}$ by the triangle inequality. Thus we have proved
that $\rho$ is semicontinuous. ∎
Question. Does there exist an unbounded admissible metric $\rho$ on the circle
$S^{1}={\mathbb{R}}/{\mathbb{Z}}$ for which the average metric $\hat{\rho}$ is
not admissible? Does there exist an unbounded $p$-admissible metric, with
$1<p<\infty$, on a compact Abelian group for which the average metric is not
admissible?666While the paper was in press, F. Petrov and P. Zatitskiy proved
that the average of any (in particular, unbounded) admissible metric on the
circle is admissible. Thus the additional assumptions that the average metric
is admissible in Proposition 2 and Theorem 5 are superfluous.
## 6 Criterion for the discreteness and continuity of the spectrum in terms
of the scaling entropy
Now we formulate our main result.
###### Theorem 5.
For an ergodic automorphism with discrete spectrum realized as a translation
on a compact commutative group with an arbitrary bounded admissible metric,
or, more generally, with a metric for which the average metric is admissible,
the scaling sequence is bounded.
###### Proof.
Since the set $B_{\varepsilon}=\phi^{-1}([0,\varepsilon])$ is, by definition,
the ball of radius $\varepsilon$ centered at $\textbf{0}\in G$ in the metric
$\hat{\rho}$, which is admissible by assumption, it follows that
$mB_{\varepsilon}>0$, since the metric is nondegenerate (see the definition of
an admissible metric). But the sum $B_{\varepsilon}+B_{\varepsilon}$, like the
sum $A+A$ for every set $A$ of positive Haar measure in a locally compact
group, contains a neighborhood $V$ of the zero in the standard topology (see,
e.g., [28]). It follows from the triangle inequality that
$B_{\varepsilon}+B_{\varepsilon}\subset B_{2\varepsilon}$, so that $V\subset
B_{2\varepsilon}$. Since $\varepsilon>0$ is arbitrary, we see that the
topology on $G$ determined by the average metric $\hat{\rho}$ coincides with
the standard topology, i.e., in the topology determined by $\hat{\rho}$, the
group $G$ is compact and contains a finite $\varepsilon$-net for every
$\varepsilon$.
The pointwise a.e. convergence
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\rho(x+kg,y+kg)={\hat{\rho}}_{T}(x,y),$
which follows from the pointwise ergodic theorem, implies that the number of
points in an $\varepsilon$-net for $(G,\rho_{n})$ tends to the number of
points in an $\varepsilon$-net for $(G,\hat{\rho})$. This means that the
sequence $H_{\rho_{n},\varepsilon}(X)$ converges to
$H_{\hat{\rho},\varepsilon}(G)$. From the inequalities of Lemma 3 we see that
the sequence $H_{\rho_{n},\varepsilon}(\mu)$ is bounded as $n\to\infty$, and
thus the scaling sequence for the automorphism $T$, which acts on the metric
triple $(G,\rho,m)$, is bounded. ∎
Combining this theorem with the previous one, we obtain the following result.
###### Theorem 6.
An ergodic automorphism $T$ has a discrete spectrum if and only if the scaling
sequence for $T$ is bounded for some, and hence for every, bounded admissible
metric.
###### Proof.
Above we have proved that if an automorphism has a discrete spectrum and the
average metric is admissible, then the scaling sequence is bounded. But the
average metric is always admissible provided that the original metric is
bounded and admissible.
Assume that the scaling sequence is bounded for an automorphism $T$ and an
admissible metric $\rho$. Recall that the average metric is indeed a metric
(and not a semimetric). Consequently, the space $(X,\hat{\rho})$ is
precompact, and hence the metric $\hat{\rho}$ is admissible. Since it is
$T$-invariant, it follows from Theorem 1 that $T$ has a purely discrete
spectrum. ∎
Combining the last theorems with the previous results yields a criterion for
the discreteness and continuity of the spectrum in terms of the automorphism
$T$ and an arbitrary admissible metric.
###### Theorem 7.
Let $T$ be an ergodic automorphism, and let $\rho$ be a bounded admissible
(semi)metric. If the corresponding scaling sequence is not bounded, then the
spectrum of $T$ contains a continuous component. If the scaling sequence is
not bounded for every admissible (semi)metric, then the spectrum of $T$ is
purely continuous.
In the next section, we will show how one could apply this criterion.
## 7 Comparison with the traditional approach, the Pascal automorphism, and
concluding remarks
### 7.1 Supremum metrics
The entropy theory of dynamical systems, developed mainly by Kolmogorov,
Sinai, and Rokhlin, essentially uses the tools of the theory of measurable
partitions. In Sinai’s definition, the entropy appears as an asymptotic
invariant of the dynamics of finite partitions under the automorphism:
$\lim_{n}\frac{H(\bigvee_{k=0}^{n-1}T^{k}\xi)}{n}=h(T,\xi).$
As a result of this theory, the study of the class of automorphisms with
completely positive entropy was differentiated into a separate field, whose
methods do not apply to automorphisms with zero entropy. For example, one
cannot obtain a new invariant for such automorphisms following the same
scheme. This can be seen from the following simple fact.
###### Proposition 4.
For every transformation $T$ and every increasing sequence of positive numbers
$\\{c_{n},\,n\in\mathbb{N}\\}$ satisfying the condition
$lim_{n}\frac{c_{n}}{n}=0$, there exists a generating partition $\xi$ such
that
$\lim_{n}\frac{H(\bigvee_{k=1}^{n}T^{k}\xi)}{c_{n}}=\infty.$
This means that the maximum growth of the entropy
$H(\bigvee_{k=1}^{n}T^{k}\xi)$ either is linear (for automorphisms with
positive entropy), or, in the case it is sublinear, it is arbitrarily close to
linear for every automorphism. Thus we obtain no new information.
The metric corresponding to the supremum (product)
$\xi_{n}=\bigvee_{k=1}^{n}T^{k}\xi$ of partitions is the supremum of the
shifted metrics: ${\bar{\rho}}_{n}^{T}(x,y)=\sup_{0\leq
k<n}\rho(T^{k}x,T^{k}y)$. Hence, following our plan, we can use the
$\varepsilon$-entropy of the metric ${\bar{\rho}}_{n}^{T}(x,y)$ instead of the
entropy of the partition $\xi_{n}$ itself. Then, using the definitions from
Section 4, for a given metric $\rho$ we can introduce an analog of the
function ${\cal H}_{\rho,\varepsilon}(T)$ with the metric $\hat{\rho}_{n}$
replaced by $\bar{\rho}_{n}$:
${\cal\bar{H}}_{\rho,\varepsilon}(T)=\Bigl{\\{}\\{c_{n}\\}:0<\liminf_{n\to\infty}\frac{H_{\bar{\rho}_{n},\varepsilon}(\mu)}{c_{n}}\leq\limsup_{n\to\infty}\frac{H_{\bar{\rho}_{n},\varepsilon}(\mu)}{c_{n}}<\infty\Bigr{\\}}.$
In this way we define the class of $\sup$-scaling sequences $\bar{c}_{n}$ for
a given metric $\rho$. This also allows us to extend the classical entropy
theory following the above scheme. Though it is somewhat easier to deal with
the sup-metric than with the average metric, the former is much less useful
than the latter. The metric $\bar{\rho}$ more often happens to be constant for
an automorphism with discrete spectrum, while, as we have seen, $\hat{\rho}$
is always admissible if the original metric is bounded. Let us illustrate the
important difference between the operations of taking the average and supremum
metrics by the following example.
Example. Let $T$ be an irrational rotation of the unit circle, and let $\rho$
be the semi-metric corresponding to a generating two-block partition (i.e., a
partition into two sets of positive measure); the semi-metric $\rho$ is
$T$-admissible, hence, as we have seen, $\hat{\rho}$ is an invariant
admissible metric. At the same time, $\bar{\rho}$ is the constant metric. This
means that the scaling sequence $c_{n}$ is bounded, but $\bar{c}_{n}$ (the
scaling sequence for the sup-metric) is not; namely, we have
$\bar{c}_{n}\sim\ln n$. Thus the difference manifests itself even in the case
of a discrete spectrum.
Does in make sense to use intermediate averages, e.g., the $l^{p}$-averages
$\lim\Bigl{[}\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y)^{p}\Bigr{]}^{\frac{1}{p}}=\hat{\rho}^{p}(x,y)$
for $p\in(1,\infty)$, instead of $l^{1}$? Apparently, they do not lead to any
new effects: these metrics behave in the same way as the $l^{1}$-average
metric. For instance, in the above example, the $p$-average of the metric
determined by a two-block partition of the unit circle is
$\hat{\rho}^{p}(x,y)=\\{m[A\Delta(A+x-y)]\\}^{\frac{1}{p}}$. For $p=\infty$
(i.e., the sup-metric), the picture is completely different, as in other
interpolation theories. Thus in entropy theory, the use of average metrics
substantially supplements the classical considerations.
### 7.2 Application of the discreteness criterion
As noted above, the problem of determining whether or not the spectrum of an
automorphism is discrete, is not at all simple. Theorems 5–7 provide
convenient non-spectral criteria for checking that the spectrum is not purely
discrete; for this, one should bound the entropy from below for one admissible
metric satisfying the conditions of Section 2 by a sequence that grows
arbitrarily slow with $n$.
One of the intriguing examples of automorphisms for which the discreteness of
the spectrum has not been neither proved nor disproved since the 1980s is the
Pascal automorphism. It was introduced by the author in 1980 (see [2, 3]) as
an example of an adic transformation, and is defined as a natural
transformation in the space of paths in the Pascal graph regarded as a
Bratteli–Vershik diagram with lexicographic ordering of paths. One can give a
short combinatorial description of this transformation by encoding these paths
with sequences of zeros and ones and identifying the space of paths with the
compact space $X=\\{0;1\\}^{\infty}=\textbf{Z}_{2}$. Then the Pascal
automorphism is defined by the formula
$T(\\{1^{i}0^{j}1**\\}=\\{0^{j-1}1^{i+1}0**\\});$
here $i\geq 0$, $j>0$, and the domain of $T$ and $T^{-1}$ is the whole $X$
except for the countable set of sequences having finitely many zeros or ones.
The most natural metric on $X$ is the 2-adic metric
$\rho(\\{x_{k}\\},\\{y_{k}\\})=2^{-n}$, where $n$ is the first digit with
$x_{k}\neq y_{k}$. This metric is admissible, and the Pascal automorphism
satisfies the Lipschitz condition almost everywhere. The orbits of this
automorphism coincide with the orbits of the action of the infinite symmetric
group. The Bernoulli measures are $T$-invariant. The spectrum of the Pascal
transformation was studied in the papers [21, 22, 23, 26], where some
interesting properties were established (e.g., it was proved that $T$ is
loosely Bernoulli, the complexity of $T$ was computed, etc.), but the question
about the type of the spectrum remains open. 777Note on the translation: The
answer is known now — the spectrum of Pascal automorphism is continuous. In
[4, 8] it was conjectured that the study of the behavior of scaling sequences
may turn to be useful. The corresponding plan was carried out in [19], but in
that paper a logarithmic lower bound was obtained on the scaling sequence for
the sup-metric, and not for the average metric; this is not sufficient for the
conclusion that the spectrum is not discrete. Nevertheless, one may hope that
the combinatorics developed in [19] will help to prove that the scaling
sequence is unbounded also for the $\varepsilon$-entropy of the average
metric, which, by our theorem, would imply that the spectrum is not discrete.
There are many adic transformations similar to the Pascal automorphism for
which the same question is also of great interest. For example, if we replace
the Pascal graph with its multidimensional analog or the Young graph, we will
obtain automorphisms that supposedly have continuous spectra. As observed
above, in order to prove that there are no nontrivial eigenfunctions, one
should obtain a growing lower bound on the scaling sequence not for one, but
for all (or for some representative set of) bounded admissible (semi)metrics.
### 7.3 The dynamics of metrics
Recall that the general approach that consists in studying the asymptotic
behavior of metrics is not exhausted by considering the asymptotics of the
$\varepsilon$-entropy of the average or supremum metric, i.e., does not reduce
to studying the growth of scaling sequences; this is only its simplest
version. In fact, we consider the original measure space
$(X,{\mathfrak{A}},\mu)$ with an action of an automorphism $T$ (or a group of
automorphisms $G$), fix an appropriate metric $\rho$, and study the sequence
of metric triples
$(X,\rho^{T}_{n},\mu),\quad\mbox{where
}\rho_{n}^{T}(x,y)=\frac{1}{n}\sum_{k=0}^{n-1}\rho(T^{k}x,T^{k}y).$
The conjecture is that, for a fixed measure and a fixed automorphism (or group
of automorphisms), the asymptotic properties of this sequence of metric
triples do not depend (or weakly depend) on the choice of an individual
admissible metric from a wide class. These properties include not only the
scaling entropy, but also more complicated characteristics of the sequence,
say the mutual properties of several consecutive metric triples. Since the
classification of metric triples up to measure-preserving isometry is known
(see [5, 7]), one may hope to apply it to this problem. In this field there
are many traditional and nontraditional questions. For example, what is the
distribution of the fluctuations of the sequence of average metrics, regarded
as functions of two variables on $(X\times X,\mu\times\mu)$, as they converge
to the constant metric (for weakly mixing transformations, e.g.,
$K$-automorphisms)? What can be said about the asymptotic properties of
neighboring pairs of metric triples (with indices $n$ and $n+1$)? Etc.
In conclusion, it is worth mentioning that the concept of scaling entropy
appeared in connection with the classification of filtrations in [14] and was
used in [27]. In terms of the present paper, the scaling entropy for
filtrations, i.e., decreasing sequences of measurable partitions or
$\sigma$-algebras, is the scaling entropy for an action of a locally finite
group such as $\sum{\mathbb{Z}}/2$ instead of an action of $\mathbb{Z}$
considered here. The definitions we have given for an action of $\mathbb{Z}$
essentially coincide with those given in [14] for locally finite groups.
## References
* [1] M. S. Birman, The spectrum of singular boundary problems, Mat. Sb., N. Ser. 55(97), No. 2, (1961), 125–174. English Translation in: Eleven Papers on Analysis, Amer. Math. Soc. Transl. 53, 23–80, AMS, Providence, RI, 1966.
* [2] A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators. Dokl. Akad. Nauk SSSR 259, No. 3, 526–529 (1981). English translation: Sov. Math. Dokl. 24, 97–100 (1981).
* [3] A. M. Vershik, A theorem on periodic Markov approximation in ergodic theory. Zapiski Nauchn. Semin. LOMI 115, 72–82 (1982). English translation: J. Sov. Math. 28, 667–674 (1985).
* [4] A. M. Vershik, Dynamics of metrics in measure spaces and their asymptotic invariants. Markov Proc. Related Fields, 16, No. 1, 169–185 (2010).
* [5] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhauser, 1999.
* [6] A. M. Vershik, Classification of measurable functions of several variables and invariantly distributed random matrices, Funkts. Anal. Prilozh. 36, No. 2, 12–28 (2002). English translation: Funct. Anal. Appl. 36, No. 2, 93–105 (2002).
* [7] A. M. Vershik, Random metric spaces and universality, Uspekhi Mat. Nauk 59, No. 2(356), 65–104 (2004). English translation: Russian Math. Surveys 59, No. 2, 259–295 (2004).
* [8] A. M. Vershik, Information, entropy, dynamics, in: Mathematics of the 20th Century: A View from Petersburg [in Russian], MCCME, 2010, pp. 47–76.
* [9] P. Cameron and A. M. Vershik, Some isometry groups of Urysohn spaces. Ann. Pure Appl. Logic 143, No. 1–3, 70–78 (2006).
* [10] A. G. Kushnirenko, Metric invariants of entropy type, Uspekhi Mat. Nauk 22, No. 5, 57–65 (1967). English translation: Russian Math. Surv. 22, No. 5, 53–61 (1967).
* [11] A. M. Vershik, Orbit theory, locally finite permutations, and Morse arithmetics, Contemp. Math. 532, 115–136 (2010).
* [12] J. Feldman, r-entropy, equipartition and Ornstein’s isomorphism theorem in $R^{n}$. Israel J. Math. 36, 321–345 (1980).
* [13] M. Ratner, Some invariants of Kakutani equivalence. Israel J. Math. 38, 232–240 (1981).
* [14] A. M. Vershik, Dynamic theory of growth in groups: Entropy, boundaries, examples Uspekhi Mat. Nauk 55, No. 4(334), 59–128 (2000). English translation: Russian Math. Surv. 55, No. 4, 667–733 (2000).
* [15] S. Ferenczi, Measure theoretic complexity of ergodic systems, Israel J. Math. 100, 189–207 (1997).
* [16] S. Ferenczi and K. Park, Entropy dimensions and a class of constructive examples, Discr. Cont. Dyn. Syst. 17, 133–141 (2007).
* [17] M. Keith, a private communication.
* [18] M. M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer, 1997.
* [19] A. A. Lodkin, I. E. Manaev, and A. R. Minabutdinov, Asymptotic behavior of the scaling sequence of the Pascal adic transformation, Zapiski Nauchn. Semin. POMI 378, 58–72 (2010). English translation to appear in J. Math. Sci.
* [20] B. Glasner, B. Tsirelson, and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise, Israel J. Math. 148, 305–329 (2005).
* [21] S. Bailey, M. Keane, K. Petersen, and I. Salama, Ergodicity of the adic transformation on the Euler graph, Math. Proc. Camb. Phil. Soc. 141, 231–238 (2006).
* [22] K. Peterson and A. Varchenko, The Euler adic dynamical system and path counts in the Euler graph, to appear in Tokyo J. Math., 2010.
* [23] K. Peterson and X. Mela, Dynamical properties of the Pascal adic transformation, Ergodic Theory Dynam. Systems 25, 227–256 (2005).
* [24] V. A. Rokhlin, On the fundamental ideas of measure theory, Mat. Sb., N. Ser. 25(67), No. 2, 107–150 (1949).
* [25] V. A. Rokhlin, Lectures on the entropy theory of measure-preserving transformations, Uspekhi Mat. Nauk 22, No. 5(137), 3–56 (1967). English translation: Russian Math. Surveys 22, No. 5, 1–52 (1967).
* [26] E. Janvresse and T. de la Rue, The Pascal adic transformation is loosely Bernoulli, Ann. Inst. H. Poincaré Prob. Stat. 40, No. 2, 133–139 (2004).
* [27] A. M. Vershik and A. D. Gorbulsky, Scaled entropy of filtrations of sigma-fields, Teor. Veroyatnost. i Primenen 52, No. 3, 446–467 (2007). English translation: Probab. Theory Appl. 52, No. 3, 493–508 (2008).
* [28] A. Weil, L’integration dans les groupes topologiqes et ses applications. Paris, 1940.
|
arxiv-papers
| 2010-08-29T18:05:19 |
2024-09-04T02:49:12.542284
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.Vershik",
"submitter": "Anatoly Vershik M",
"url": "https://arxiv.org/abs/1008.4946"
}
|
1008.4987
|
# The double charm decays of B Mesons in the mSUGRA model
Lin-Xia Lü1, Zhen-Jun Xiao2, Shuai-Wei Wang1 and Wen-Jun Li3
1 Physics and electronic engineering college, Nanyang Normal University,
Nanyang, Henan 473061, P.R. China
2 Department of Physics and Institute of Theoretical Physics, Nanjing Normal
University,
Nanjing, Jiangsu 210046, P.R. China
3 Department of Physics, Henan Normal University,
Xinxiang, Henan 453007, P.R. China E-mail: lvlinxia@sina.comE-mail:
xiaozhenjun@njnu.edu.cn
###### Abstract
Based on the low energy effective Hamiltonian with naive factorization, we
calculate the branching ratios(BRs) and CP asymmetries (CPAs) for the twenty
three double charm decays $B/B_{s}\to D^{(*)}_{(s)}D^{(*)}_{(s)}$ in both the
standard model (SM) and the minimal supergravity (mSUGRA) model. Within the
considered parameter space, we find that (a) the theoretical predictions for
the BRs, CPAs and the polarization fractions in the SM and the mSUGRA model
are all consistent with the currently available data within $\pm 2\sigma$
errors; (b) For all the considered decays, the supersymmetric contributions in
the mSUGRA model are very small, less than $7\%$ numerically. It may be
difficult to observe so small SUSY contributions even at LHC.
###### pacs:
13.25.Hw, 14.40.Lb, 12.60.Jv, 12.15.Ji
## I Introduction
Within the standard model (SM), the double charm decays of $B_{u,d}$ and
$B_{s}$ Mesons considered here are dominated by the color-favored “Tree”
transition $b\to c\bar{c}d(s)$, while the color-suppressed “Penguin”
transition is generally small. If the penguin contribution was absent, the
mixing induced CP asymmetry (CPA), denoted as $S_{f}$, would be proportional
to $\sin(2\beta)$, while the direct CPA, denoted as $C_{f}$, would be zero. In
some new physics models beyond the SM, the penguin contributions can be large
and may change the SM predictions for the branching ratios and the CP
asymmetries (CPA) significantly. The study of these double charm $B/B_{s}$
meson decays therefore plays an important role in testing the SM as well as
searching for the signals of the new physics (NP).
Experimentally, the BaBar and Belle Collaboration have reported the
measurement of the direct CPA in $B^{0}\to D^{+}D^{-}$ decay
$\displaystyle\mathcal{C}(B^{0}\to D^{+}D^{-})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-0.91\pm 0.23\pm 0.06&\mbox{(Belle
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Fratina:2007zk}{\@@citephrase{(}}{\@@citephrase{)}}})},\\\
-0.07\pm 0.23\pm 0.03&\mbox{(BaBar \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{:2008aw}{\@@citephrase{(}}{\@@citephrase{)}}})}.\end{array}\right.$
(3)
It is easy to see that Belle found an evidence of CP violation in $B^{0}\to
D^{+}D^{-}$ at the $4.1\sigma$ level Fratina:2007zk , but BaBar did not
:2008aw . On the other hand, such a large direct CPA in $B^{0}\to D^{+}D^{-}$
decay has not been observed in the measurements for other similar decay modes:
such as $\bar{B}^{0}\to D^{{}^{(*)+}}D^{{}^{(*)-}}$, $B^{-}\to
D^{{}^{(*)0}}D^{{}^{(*)-}}$ and $\bar{B}^{0}_{s}\to
D^{{}^{(*)+}}_{s}D^{{}^{(*)-}}$ :2008aw ; Miyake:2005qb ; Aushev:2004uc ;
Aubert:2007rr ; Vervink:2008dv ; Aubert:2006ia ; Abe:2007sk ; Majumder:2005gy
, although they have the same flavor structures as $B^{0}\to D^{+}D^{-}$ at
the quark level. In the SM, the direct CPA’s should be naturally very small in
size because the penguin contributions are small. If the large CP violation in
$B^{0}\to D^{+}D^{-}$ from Belle is true, it would establish the presence of
new physics.
Up to now, by using the low-energy effective hamiltonian and various
factorization hypothesis, many investigations on the decays of B to double-
charm states have been carried out in the framework of the SM LuCaidian10 ;
Liying or some popular new physics models Zwicky:2007vv ; Fleischer:2007zn ;
Gronau:2008ed ; ruminwang:SUSY2009 .
In this paper, we will present our systematic calculation of the branching
ratios and CP violations for double charm decays $B/B_{s}\to
D^{(*)}_{(s)}D^{(*)}_{(s)}$ in the minimal supergravity (mSUGRA) model
nills1984 . In the framework of the mSUGRA model, the new physics
contributions to the semileptonic, leptonic and radiative rare B decays and
the charmless two-body B-meson decays have been investigated in previous works
hmdx98 ; tyy96 ; tyy97 ; huang03 ; zw04 . For the two-body $B\to M_{1}M_{2}$
decays, the new physics part of the Wilson coefficients
$C_{k}(k=3,\cdots,6)$,$C_{7\gamma}$ and $C_{8g}$ in the mSUGRA model can be
found in Ref. zw04 .
The usual route to calculate the decay amplitude for non-leptonic two-body B
decays is to start from the low energy effective Hamiltonian for $\Delta B=1$
decays. With the operator product expansion method, the relevant $\Delta B=1$
effective Hamiltonian can be factorized into the Wilson coefficients
$C_{i}(\mu)$ times the four-quark operators $Q_{i}(\mu)$. As to $C_{i}(\mu)$,
they have been evaluated to next-to-leading order with the perturbation theory
and renormalization group method. The remanent and also intractable problem is
to calculate the hadronic matrix elements of these four-quark operators. Up to
now, many methods have been put forward to settle this problem, such as the
naive or generalized factorization approach NF ; gfa , QCD factorization
approach (QCDF) qcdf ; bn03a and the perturbative QCD (PQCD) approachpqcd .
For the strong phase, which is important for the CP violation prediction, is
quite sensitive to these various approaches, and different approaches may lead
to quite different results. In this paper, we will use the naive factorization
method, which is expected to be reliable for the color-allowed amplitudes,
which are dominant contributions in these double charm decays.
This paper is organized as follows. In the next section we will give a brief
review for the mSUGRA model. In Sec. III, we introduce the basic formulas for
calculating the branching ratios, the polarization fractions and the CP
violation in the considered $B/B_{s}\to D^{(*)}_{(s)}D^{(*)}_{(s)}$ decays. In
Sec. IV, we present the numerical results for the double charm decays of
B-meson in both the SM and the mSUGRA model. The conclusions are included in
the final section.
## II outline of the mSUGRA model
In the minimal supersymmetry model (MSSM), the most general superpotential
takes the form nills1984 ; msugra
$\displaystyle{\cal
W}=\varepsilon_{\alpha\beta}\left[f_{Uij}Q_{i}^{\alpha}H_{2}^{\beta}U_{j}+f_{Dij}H_{1}^{\alpha}Q_{i}^{\beta}D_{j}+f_{Eij}H_{1}^{\alpha}L_{i}^{\beta}E_{j}-\mu
H_{1}^{\alpha}H_{2}^{\beta}\right],$ (4)
a set of terms which explicitly but softly break SUSY should be added to the
supersymmetric Lagrangian. A general form of the soft SUSY-breaking terms is
given as
$\displaystyle-{\cal L}_{soft}$ $\displaystyle=$
$\displaystyle\left(m^{2}_{Q}\right)_{ij}\tilde{q}^{+}_{Li}\tilde{q}_{Lj}+\left(m^{2}_{U}\right)_{ij}\tilde{u}^{*}_{Ri}\tilde{u}_{Rj}+\left(m^{2}_{D}\right)_{ij}\tilde{d}^{*}_{Ri}\tilde{d}_{Rj}+\left(m^{2}_{L}\right)_{ij}\tilde{l}^{+}_{Li}\tilde{l}_{Lj}$
(5)
$\displaystyle+\left(m^{2}_{E}\right)_{ij}\tilde{e}^{*}_{Ri}\tilde{e}_{Rj}+\Delta^{2}_{1}h_{1}^{+}h_{1}+\Delta^{2}_{2}h_{2}^{+}h_{2}$
$\displaystyle+\varepsilon_{\alpha\beta}\left[A_{Uij}\tilde{q}^{\alpha}_{Li}h^{\beta}_{2}\tilde{u}^{*}_{Rj}+A_{Dij}h^{\alpha}_{1}\tilde{q}^{\beta}_{Li}\tilde{d}^{*}_{Rj}+A_{Eij}h^{\alpha}_{1}\tilde{l}^{\beta}_{Li}\tilde{e}^{*}_{Rj}+B\mu
h^{\alpha}_{1}h^{\beta}_{2}\right]$
$\displaystyle+\frac{1}{2}m_{\tilde{B}}\tilde{B}\tilde{B}+\frac{1}{2}m_{\tilde{W}}\tilde{W}\tilde{W}+\frac{1}{2}m_{\tilde{G}}\tilde{G}\tilde{G}+H.C.$
where $\tilde{q}_{Li}$, $\tilde{u}^{*}_{Ri}$, $\tilde{d}^{*}_{Ri}$,
$\tilde{l}_{Li}$, $\tilde{e}^{*}_{Ri}$, $h_{1}$ and $h_{2}$ are scalar
components of chiral superfields $Q_{i}$, $U_{i}$, $D_{i}$, $L_{i}$, $E_{i}$,
$H_{1}$ and $H_{2}$ respectively, and $\tilde{B}$, $\tilde{W}$ and $\tilde{G}$
are $U(1)_{Y}$, $SU(2)_{L}$, and $SU(3)_{C}$ gauge fermions.
In order to avoid severe phenomenological problems, such as large flavor
changing neutral currents (FCNC), unacceptable amount of additional CP
violation and so on, a set of assumptions are added to the unconstrained MSSM
in the mSUGRA model. One underlying assumption is that SUSY-breaking occurs in
a hidden sector which communicates with the visible sector only through
gravitational interactions. The free parameters in the MSSM are assumed to
obey a set of boundary conditions at the Grand Unification scale
$M_{X}$nills1984 ; msugra
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\alpha_{2}=\alpha_{3}=\alpha_{X},$
$\displaystyle(m^{2}_{Q})_{ij}$ $\displaystyle=$
$\displaystyle(m^{2}_{U})_{ij}=(m^{2}_{D})_{ij}=(m^{2}_{L})_{ij}=(m^{2}_{E})_{ij}=(m^{2}_{0})\delta_{ij},$
$\displaystyle\Delta^{2}_{1}$ $\displaystyle=$
$\displaystyle\Delta^{2}_{2}=m^{2}_{0},$ $\displaystyle A_{Uij}$
$\displaystyle=$ $\displaystyle f_{Uij}A_{0},\ \ A_{Dij}=f_{Dij}A_{0},\ \
A_{Eij}=f_{Eij}A_{0},$ $\displaystyle m_{\tilde{B}}$ $\displaystyle=$
$\displaystyle m_{\tilde{W}}=m_{\tilde{G}}=m_{\frac{1}{2}}$ (6)
where $\alpha_{i}=g^{2}_{i}/(4\pi)$, while $g_{i}$ (i=1,2,3) denotes the
coupling constant of the $U(1)_{Y}$, $SU(2)_{L}$, $SU(3)_{C}$ gauge group,
respectively. Besides the three parameters $m_{\frac{1}{2}}$, $m_{0}$ and
$A_{0}$, the bilinear coupling B and the supersymmetric Higgs(ino) mass
parameter $\mu$ in the supersymmetric sector should also be determined. By
requiring the radiative electroweak symmetry-breaking (EWSB) takes place at
the low energy scale, both of them are obtained except for the sign of $\mu$.
At this stage, only four continuous free parameters and an unknown sign are
left in the mSUGRA model
$\displaystyle\tan\beta,m_{\frac{1}{2}},m_{0},A_{0},sign(\mu).$ (7)
According to the previous studies about the constraints on the parameter space
of the mSUGRA model lepa ; lepb ; spa ; sps1 ; zw04 ; ali02 , we choose two
sets of typical mSUGRA points as listed in Table 1.
Table 1: Two typical sets of SUSY parameters to be used in the numerical calculation. CASE | $m_{0}$ | $m_{\frac{1}{2}}$ | $A_{0}$ | $\tan\beta$ | $Sign[\mu]$ | $R_{7}$
---|---|---|---|---|---|---
A | $300$ | $300$ | $0$ | $2$ | $-$ | $1.10$
B | $369$ | $150$ | $-400$ | $40$ | $+$ | $-0.93$
## III Effective Hamiltonian and observables
In this section, we will give a brief review of the theoretical framework of
the low energy effective Hamiltonian and the factorized matrix elements as
well as the decay amplitudes for $\Delta B=1$ decays.
### III.1 Effective Hamiltonian in the SM and mSUGRA model
In the SM, the low energy effective Hamiltonian for $\Delta B=1$ transition at
a scale $\mu$ is given by Buchalla:1995vs
$\displaystyle\mathcal{H}^{\rm SM}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}\lambda_{p}\Biggl{\\{}C_{1}Q_{1}^{p}+C_{2}Q_{2}^{p}+\sum_{i=3}^{10}C_{i}Q_{i}+C_{7\gamma}Q_{7\gamma}+C_{8g}Q_{8g}\Biggl{\\}}+h.c.,$
(8)
here $\lambda_{p}=V_{pb}V_{pq}^{*}$ for $b\to q$ transition
$(p\in\\{u,c\\},q\in\\{d,s\\})$. The detailed definition of the operators can
be found in Ref. Buchalla:1995vs . Within the SM and at the scale $M_{W}$, the
Wilson coefficients $C_{1}(M_{W}),\cdot\cdot\cdot,C_{10}(M_{W})$,
$C_{7\gamma}(M_{W})$ and $C_{8g}(M_{W})$ have been given, for example, in Ref.
Buchalla:1995vs . By using QCD renormalization group equations, it is
straightforward to run Wilson coefficients $C_{i}(M_{W})$ from the scale
$\mu\sim O(M_{W})$ down to the lower scale $\mu\sim O(m_{b})$.
In the mSUGRA model, there are four kinds of SUSY contributions to the $b\to
d(s)$ transition at the one-loop level, depending on the virtual particles
running in the penguin diagrams:
* (i) the charged Higgs boson $H^{\pm}$ and up-type quarks $u,c,t$;
* (ii) the charginos $\tilde{\chi}^{\pm}_{1,2}$ and the up-type squarks $\tilde{u},\tilde{c},\tilde{t}$;
* (iii) the neutralinos $\tilde{\chi}^{0}_{1,2,3,4}$ and the down-type quarks $\tilde{d},\tilde{s},\tilde{b}$;
* (iv) the gluinos $\tilde{g}$ and the down-type quarks $\tilde{d},\tilde{s},\tilde{b}$.
In general, the Wilson coefficients after the inclusion of various
contributions can be expressed as
$\displaystyle
C_{i}(\mu_{W})=C_{i}^{SM}+C_{i}^{H^{-}}+C_{i}^{\tilde{\chi}^{-}}+C_{i}^{\tilde{\chi}^{0}}+C_{i}^{\tilde{g}},$
(9)
where $C_{i}^{H^{-}},C_{i}^{\tilde{\chi}^{-}},C_{i}^{\tilde{\chi}^{0}}$ and
$C_{i}^{\tilde{g}}$ denote the Wilson coefficients induced by the penguin
diagrams with the exchanges of the charged Higgs $H^{\pm}$, the chargino
$\tilde{\chi}^{\pm}_{1,2}$, the neutralino $\tilde{\chi}^{0}_{1,2,3,4}$ and
the gluino $\tilde{g}$, respectively. The detailed expressions of these Wilson
coefficients can be found in Ref. zw04 .
### III.2 Decay amplitudes in naive factorization
The decay amplitudes of $B\to D^{{}^{(*)}}D^{{}^{(*)}}_{q}$ in the SM within
the naive factorization can be written as NF
$\displaystyle\mathcal{M}^{\rm SM}(B\to
D^{{}^{(*)}}D^{{}^{(*)}}_{q})=\frac{G_{F}}{\sqrt{2}}\left(\lambda_{c}a_{1}^{c}+\sum_{p=u,c}\lambda_{p}\left[a_{4}^{p}+a_{10}^{p}+\xi(a_{6}^{p}+a_{8}^{p})\right]\right)A_{[BD^{{}^{(*)}},D^{{}^{(*)}}_{q}]},$
(10)
where the coefficients $a^{p}_{i}=\left(C_{i}+\frac{C_{i\pm
1}}{N_{c}}\right)+P^{p}_{i}$ with the upper (lower) sign applied when $i$ is
odd (even), and $P^{p}_{i}$ account for penguin contributions. The
factorization parameter $\xi$ in Eq. (10) arises from the transformation of
$(V-A)(V+A)$ currents into $(V-A)(V-A)$ ones for the penguin operators. It
depends on properties of the final-state mesons involved and is defined as
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}+\frac{2m^{2}_{D_{q}}}{(\bar{m}_{c}+\bar{m}_{q})(\bar{m}_{b}-\bar{m}_{c})}&~{}~{}\mbox{($DD_{q}$)},\\\
0&~{}~{}\mbox{($DD^{*}_{q}$)},\\\
-\frac{2m^{2}_{D_{q}}}{(\bar{m}_{c}+\bar{m}_{q})(\bar{m}_{b}+\bar{m}_{c})}&~{}~{}\mbox{($D^{*}D_{q}$)},\\\
0&~{}~{}\mbox{($D^{*}D^{*}_{q}$)}.\\\ \end{array}\right.$ (15)
The term $A_{[BD^{{}^{(*)}},D^{{}^{(*)}}_{q}]}$ in Eq. (10) is the factorized
matrix element. For $B\to D^{{}^{(*)}}D^{{}^{(*)}}_{q}$ decay mode, it can be
written as
$\displaystyle
A_{[BD^{{}^{(*)}},D^{{}^{(*)}}_{q}]}\equiv\left<D^{{}^{(*)}}_{q}|\bar{q}\gamma^{\mu}(1-\gamma_{5})c|0\right>\left<D^{{}^{(*)}}|\bar{c}\gamma_{\mu}(1-\gamma_{5})b|B\right>.$
(16)
The decay constants and form factors NF ; Neubert:1991xw are usually defined
as
$\displaystyle\langle
D_{q}(p_{{}_{D_{q}}})|\bar{q}\gamma^{\mu}\gamma_{5}c|0\rangle$
$\displaystyle=$ $\displaystyle-if_{{}_{D_{q}}}p^{\mu}_{{}_{D_{q}}},$ (17)
$\displaystyle\langle
D^{*}_{q}(p_{{}_{D^{*}_{q}}})|\bar{q}\gamma^{\mu}c|0\rangle$ $\displaystyle=$
$\displaystyle f_{{}_{D^{*}_{q}}}p^{\mu}_{{}_{D^{*}_{q}}},$ (18)
$\displaystyle\langle D(p_{{}_{D}})|\bar{c}\gamma_{\mu}b|B(p_{{}_{B}})\rangle$
$\displaystyle=$
$\displaystyle\frac{m^{2}_{B}-m^{2}_{{}_{D}}}{q^{2}}q_{\mu}F_{0}(q^{2})+\left[(p_{{}_{B}}+p_{{}_{D}})_{\mu}-\frac{m^{2}_{B}-m^{2}_{{}_{D}}}{q^{2}}q_{\mu}\right]F_{1}(q^{2}),\
\ \ $ (19) $\displaystyle\langle
D^{*}(p_{{}_{D^{*}}},\varepsilon^{\ast})|\bar{c}\gamma_{\mu}b|B(p_{{}_{B}})\rangle$
$\displaystyle=$
$\displaystyle\frac{2V(q^{2})}{m_{B}+m_{{}_{D^{*}}}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p_{{}_{B}}^{\alpha}p_{{}_{D^{*}}}^{\beta},$
(20) $\displaystyle\langle
D^{*}(p_{{}_{D^{*}}},\varepsilon^{\ast})|\bar{c}\gamma_{\mu}\gamma_{5}b|B(p_{{}_{B}})\rangle$
$\displaystyle=$ $\displaystyle
i\left[\varepsilon_{\mu}^{\ast}(m_{B}+m_{{}_{D^{*}}})A_{1}(q^{2})-(p_{{}_{B}}+p_{{}_{D^{*}}})_{\mu}({\varepsilon^{\ast}}\cdot{p_{{}_{B}}})\frac{A_{2}(q^{2})}{m_{B}+m_{{}_{D^{*}}}}\right]$
(21) $\displaystyle-
iq_{\mu}({\varepsilon^{\ast}}\cdot{p_{{}_{B}}})\frac{2m_{{}_{D^{*}}}}{q^{2}}[A_{3}(q^{2})-A_{0}(q^{2})],$
where $q=p_{B}-p_{D^{{}^{(*)}}}$. In terms of decay constants and form
factors, the matrix element $A_{[BD^{{}^{(*)}},D^{{}^{(*)}}_{q}]}$ can be
written as follows
$\displaystyle
A_{[BD^{{}^{(*)}},D^{{}^{(*)}}_{q}]}=\left\\{\begin{array}[]{ll}if_{D_{q}}(m_{B}^{2}-m^{2}_{D})F_{0}(m^{2}_{D_{q}}),&(DD_{q}),\\\
2f_{D^{{}^{*}}_{{}_{q}}}m_{B}|p_{c}|F_{1}(m^{2}_{D^{{}^{*}}_{q}}),&(DD^{*}_{q}),\\\
-2f_{D_{q}}m_{B}|p_{c}|A_{0}(m^{2}_{D_{q}}),&(D^{*}D_{q}),\\\
-if_{D^{{}^{*}}_{{}_{q}}}m_{D^{{}^{*}}_{q}}\biggl{[}(\varepsilon_{D^{{}^{*}}}^{\ast}\cdot\varepsilon_{D^{{}^{*}}_{q}}^{\ast})(m_{B}+m_{D^{{}^{*}}})A_{1}(m_{D^{{}^{*}}_{q}}^{2})\\\
\hskip 56.9055pt-(\varepsilon_{D^{{}^{*}}}^{\ast}\cdot
p_{D^{{}^{*}}_{q}})(\varepsilon_{D^{{}^{*}}_{q}}^{\ast}\cdot
p_{D^{{}^{*}}})\frac{2A_{2}(m^{2}_{D^{{}^{*}}_{q}})}{m_{B}+m_{D^{{}^{*}}}}\biggr{.}\\\
\hskip
56.9055pt\left.+i\epsilon_{\mu\nu\alpha\beta}\varepsilon_{D^{{}^{*}}_{q}}^{\ast\mu}\varepsilon_{D^{{}^{*}}}^{\ast\nu}p_{D^{{}^{*}}_{q}}^{\alpha}p_{D^{{}^{*}}}^{\beta}\frac{2V(m^{2}_{D^{{}^{*}}_{q}})}{m_{B}+m_{D^{{}^{*}}}}\right],&(D^{*}D^{*}_{q}).\end{array}\right.$
(28)
For the penguin contributions, we will consider not only QCD and electroweak
penguin operator contributions but also the contributions from the
electromagnetic and chromomagnetic dipole operators $Q_{7\gamma}$ and
$Q_{8g}$, as defined by the factor $P^{p}_{i}$NF :
$\displaystyle P_{1}^{c}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
P_{4}^{p}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{9\pi}\left\\{C_{1}\left[\frac{10}{9}-G_{D^{(*)}_{q}}(m_{p})\right]-2F_{1}C^{eff}_{8g}\right\\},$
$\displaystyle P_{6}^{p}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{9\pi}\left\\{C_{1}\left[\frac{10}{9}-G_{D^{(*)}_{q}}(m_{p})\right]-2F_{2}C^{eff}_{8g}\right\\},$
$\displaystyle P_{8}^{p}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{e}}{9\pi}\frac{1}{N_{c}}\left\\{(C_{1}+N_{c}C_{2})\left[\frac{10}{9}-G_{D^{(*)}_{q}}(m_{p})\right]-3F_{2}C^{eff}_{7\gamma}\right\\},$
$\displaystyle P_{10}^{p}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{e}}{9\pi}\frac{1}{N_{c}}\left\\{(C_{1}+N_{c}C_{2})\left[\frac{10}{9}-G_{D^{(*)}_{q}}(m_{p})\right]-3F_{1}C^{eff}_{7\gamma}\right\\},$
(29)
with the penguin loop-integral function $G_{D^{(*)}_{q}}(m_{p})$ defined as
$\displaystyle G_{D^{(*)}_{q}}(m_{p})$ $\displaystyle=$
$\displaystyle\int^{1}_{0}duG(m_{p},k)\Phi_{D^{(*)}_{q}}(u),$ (30)
$\displaystyle G(m_{p},k)$ $\displaystyle=$
$\displaystyle-4\int^{1}_{0}dxx(1-x)\mbox{ln}\left[\frac{m_{p}^{2}-k^{2}x(1-x)}{m^{2}_{b}}-i\epsilon\right],$
(31)
where
$k^{2}=m^{2}_{c}+\bar{u}(m^{2}_{b}-m^{2}_{c}-m^{2}_{M_{2}})+\bar{u}^{2}m^{2}_{M_{2}}$
is the penguin momentum transfer with $\bar{u}\equiv 1-u$. In the function
$G_{D^{(*)}_{q}}(m_{p})$, we have used a $D^{(*)}_{q}$ meson-emitting
distribution amplitude
$\Phi_{D^{(*)}_{q}}(u)=6u(1-u)[1+a_{D^{(*)}_{q}}(1-2u)]$, in stead of keeping
$k^{2}$ as a free parameter as usual. The constants $F_{1}$ and $F_{2}$ in Eq.
(29) are defined by NF
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\int^{1}_{0}du\Phi_{D_{q}}(u)\frac{m_{b}}{m_{b}-m_{c}}\frac{m^{2}_{b}-um^{2}_{D_{q}}-2m^{2}_{c}+m_{b}m_{c}}{k^{2}}&~{}~{}\mbox{($DD_{q}$)},\\\
\int^{1}_{0}du\Phi_{D^{*}_{q}}(u)\frac{m_{b}}{k^{2}}\left(\bar{u}m_{b}+\frac{2um_{D^{*}_{q}}}{m_{b}-m_{c}}\epsilon_{2}^{*}\cdot
p_{1}-um_{c}\right)&~{}~{}\mbox{($DD^{*}_{q}$)},\\\
\int^{1}_{0}du\Phi_{D_{q}}(u)\frac{m_{b}}{m_{b}+m_{c}}\frac{m^{2}_{b}-um^{2}_{D_{q}}-2m^{2}_{c}-m_{b}m_{c}}{k^{2}}&~{}~{}\mbox{($D^{*}D_{q}$)},\\\
\int^{1}_{0}du\Phi_{D^{*}_{q}}(u)\frac{m_{b}}{k^{2}}\left(\bar{u}m_{b}+\frac{2um_{D^{*}_{q}}}{m_{b}+m_{c}}\epsilon_{2}^{*}\cdot
p_{1}+um_{c}\right)&~{}~{}\mbox{($D^{*}D^{*}_{q}$)},\\\ \end{array}\right.$
(36) $\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\int^{1}_{0}du\Phi_{D_{q}}(u)\frac{m_{b}}{k^{2}}[\bar{u}(m_{b}-m_{c})+m_{c}]&~{}~{}\mbox{($DD_{q}$)},\\\
0&~{}~{}\mbox{($DD^{*}_{q}$)},\\\
\int^{1}_{0}du\Phi_{D_{q}}(u)\frac{m_{b}}{k^{2}}[\bar{u}(m_{b}+m_{c})-m_{c}]&~{}~{}\mbox{($D^{*}D_{q}$)},\\\
0&~{}~{}\mbox{($D^{*}D^{*}_{q}$)},\\\ \end{array}\right.$ (41)
where $\epsilon_{2L}^{*}\cdot
p_{1}\approx(m_{b}^{2}-m^{2}_{M^{*}_{q}}-m_{c}^{2})/(2m_{M^{*}_{q}})$ and
$\epsilon_{2T}^{*}\cdot p_{1}=0$ for $B\to D^{*}D^{*}_{q}$ decays.
### III.3 Observables of $B\to M_{1}M_{2}$ decays
In the $B$ meson rest frame, the branching ratios of two-body $B$ meson decays
can be written as
$\displaystyle\mathcal{B}(B\to
D^{{}^{(*)}}D^{{}^{(*)}}_{{}_{q}})=\frac{\tau_{B}}{8\pi}\frac{|p_{c}|}{m_{B}^{2}}\left|\mathcal{M}(B\to
D^{{}^{(*)}}D^{{}^{(*)}}_{{}_{q}})\right|^{2},$ (42)
where $\tau_{B}$ is the $B$ meson lifetime, and $|p_{c}|$ is the magnitude of
momentum of particle $M_{1}$ and $M_{2}$ in the B rest frame and written as
$\displaystyle|p_{c}|=\frac{\sqrt{[m_{B}^{2}-(m_{D^{{}^{(*)}}}+m_{D^{(*)}_{q}})^{2}][m_{B}^{2}-(m_{D^{{}^{(*)}}}-m_{D^{(*)}_{q}})^{2}]}}{2m_{B}}.$
(43)
In $B\to D^{*}D^{*}_{q}$ decays, one generally should evaluate three
amplitudes as $\mathcal{M}_{0,\pm}$ in the helicity basis or as
$\mathcal{M}_{L,\parallel,\perp}$ in the transversity basis, which are related
by $\mathcal{M}_{L}=\mathcal{M}_{0}$ and
$\mathcal{M}_{\parallel,\perp}=\frac{\mathcal{M}_{+}\pm\mathcal{M}_{-}}{\sqrt{2}}$.
Then we have
$\displaystyle\left|\mathcal{M}(B\to
D^{*}D^{*}_{q})\right|^{2}=|\mathcal{M}_{0}|^{2}+|\mathcal{M}_{+}|^{2}+|\mathcal{M}_{-}|^{2}=|\mathcal{M}_{L}|^{2}+|\mathcal{M}_{\parallel}|^{2}+|\mathcal{M}_{\perp}|^{2}.$
(44)
The longitudinal polarization fraction $f_{L}$ and transverse polarization
fraction $f_{\perp}$ are defined by
$\displaystyle f_{L,\perp}(B\to D^{*}D^{*}_{q})$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{L,\perp}}{\Gamma}=\frac{|\mathcal{M}_{L,\perp}|^{2}}{|\mathcal{M}_{L}|^{2}+|\mathcal{M}_{\parallel}|^{2}+|\mathcal{M}_{\perp}|^{2}}.$
(45)
In charged $B$ meson decays, where mixing effects are absent, the only
possible source of CPAs is
$\displaystyle\mathcal{A}_{\rm CP}^{k,{\rm
dir}}=\frac{\left|\mathcal{M}_{k}(B^{-}\rightarrow\overline{f})/\mathcal{M}_{k}(B^{+}\rightarrow
f)\right|^{2}-1}{\left|\mathcal{M}_{k}(B^{-}\rightarrow\overline{f})/\mathcal{M}_{k}(B^{+}\rightarrow
f)\right|^{2}+1},$ (46)
and $k=L,\parallel,\perp$ for $B^{-}\to D^{*}D^{*}_{q}$ decays and $k=L$ for
$B^{-}_{u}\to DD_{q},DD^{*}_{q},D^{*}D_{q}$ decays. Then for $B^{-}_{u}\to
D^{*}D^{*}_{q}$ decays, we have
$\displaystyle\mathcal{A}_{\rm CP}^{+,{\rm dir}}(B\to D^{*}D^{*}_{q})$
$\displaystyle=$ $\displaystyle\frac{\mathcal{A}_{\rm CP}^{\parallel,{\rm
dir}}|\mathcal{M}_{\parallel}|^{2}+\mathcal{A}_{\rm CP}^{L,{\rm
dir}}|\mathcal{M}_{L}|^{2}}{|\mathcal{M}_{\parallel}|^{2}+|\mathcal{M}_{L}|^{2}}.$
(47)
For neutral $B_{q}$ meson decays, the situation becomes complicated because of
$B^{0}_{q}-\bar{B}^{0}_{q}$ mixing, and have been studied by many authors. We
do not repeat the lengthy discussions here, one can see Refs. Gronau:1989zb ;
Soto:1988hf ; Palmer:1994ec ; Ali:1998gb for details.
## IV Numerical calculations
### IV.1 Input parameters
* •
CKM matrix elements: In numerical calculation, we will use the following
values which given as CKMfit
$\displaystyle|V_{ud}|$ $\displaystyle=$ $\displaystyle
0.9743,\quad|V_{us}|=0.2252,\quad|V_{ub}|=0.0035,$ $\displaystyle|V_{cd}|$
$\displaystyle=$ $\displaystyle
0.2251,\quad|V_{cs}|=0.9735,\quad|V_{cb}|=0.0412,$ $\displaystyle|V_{td}|$
$\displaystyle=$ $\displaystyle
0.0086,\quad|V_{ts}|=0.0404,\quad|V_{tb}|=0.9991,$ $\displaystyle\beta$
$\displaystyle=$
$\displaystyle(21.58^{+0.91}_{-0.81})^{\circ},\quad\gamma=(67.8_{-3.9}^{+4.2})^{\circ}.$
(48)
* •
Quark masses. When calculating the decay amplitudes, the pole and current
quark masses will be used. For the former, we will use
$m_{u}=4.2{\rm MeV},\ \ m_{c}=1.5{\rm GeV},\ \ m_{t}=175{\rm GeV},$
$m_{d}=7.6{\rm MeV},\ \ m_{s}=0.122{\rm GeV},\ \ m_{b}=4.62{\rm GeV}.$
The current quark mass depends on the renormalization scale. In the
$\overline{MS}$ scheme and at a scale of 2GeV, we fix
$\overline{m}_{u}(2{\rm GeV})=2.4{\rm MeV},\ \ \overline{m}_{d}(\rm
2GeV)=6{\rm MeV},$ $\overline{m}_{s}(2{\rm GeV})=105{\rm MeV},\ \
\overline{m}_{b}(\overline{m}_{b})=4.26{\rm GeV},$
and then employ the formulae in Ref.Buchalla:1995vs
$\displaystyle\overline{m}(\mu)=\overline{m}(\mu_{0})\left[\frac{{\alpha_{s}(\mu)}}{{\alpha_{s}(\mu_{0})}}\right]^{\frac{{\gamma_{m}^{(0)}}}{{2\beta_{0}}}}\left[1+\left(\frac{{\gamma_{m}^{(1)}}}{{2\beta_{0}}}-\frac{{\beta_{1}\gamma_{m}^{(0)}}}{{2\beta_{0}^{2}}}\right)\frac{{\alpha_{s}(\mu)-\alpha_{s}(\mu_{0})}}{{4\pi}}\right]$
(49)
to obtain the current quark masses at any scale. The definitions of
$\alpha_{s}$, $\gamma_{m}^{(0)}$, $\gamma_{m}^{(1)}$, $\beta_{0}$, and
$\beta_{1}$ can be found in Ref.Buchalla:1995vs .
* •
Decay constants: The decay constants of $D^{*}_{q}$ mesons have not been
directly measured in experiments so far. In the heavy-quark limit
$(m_{c}\to\infty)$, spin symmetry predicts that $f_{D^{*}_{q}}=f_{D_{q}}$, and
most theoretical predictions indicate that symmetry-breaking corrections
enhance the ratio $f_{D^{*}_{q}}/f_{D_{q}}$ by $10\%-20\%$ Neubert:1993mb ;
Neubert:1996qg . In this paper, we will take $f_{D}=0.201\pm 0.017{\rm GeV}$,
$f_{D_{s}}=0.249\pm 0.016{\rm GeV}$ and $f_{D^{*}_{q}}=f_{D_{q}}$ as our input
values.
* •
Distribution amplitudes: The distribution amplitudes of $D^{(*)}_{q}$ mesons
are less constrained, and we use the shape parameter $a_{D^{(*)}}=0.7\pm 0.2$
and $a_{D^{(*)}_{s}}=0.3\pm 0.2$.
* •
Form factors: For the form factors involving $B\to D^{(*)}$ transitions, we
take expressions which include perturbative QCD corrections induced by hard
gluon vertex corrections of $b\to c$ transitions and power corrections in
orders of $1/m_{b,c}$ Neubert:1991xw ; Neubert:1992tg . As for Isgur-Wise
function $\xi(\omega)$, we use the fit result
$\xi(\omega)=1-1.22(\omega-1)+0.85(\omega-1)^{2}$ from Ref. Cheng:2003sm .
* •
Mass and lifetimes: For B and D meson masses, the lifetimes, we use the
following as input parameters pdg2008 .
$\displaystyle m_{{}_{B_{u}}}$ $\displaystyle=$ $\displaystyle 5.279{\rm
GeV},\;\;\;m_{{}_{B_{d}}}=5.280{\rm GeV},\;\;\;m_{{}_{B_{s}}}=5.366{\rm GeV},$
$\displaystyle M_{D^{0}}$ $\displaystyle=$ $\displaystyle 1.865{\rm
GeV},\;\;\;M_{D^{+}}=1.870{\rm GeV},\;\;\;M_{D^{+}_{s}}=1.969{\rm GeV},$
$\displaystyle M_{D^{*0}}$ $\displaystyle=$ $\displaystyle 2.007{\rm
GeV},\;\;\;M_{D^{*+}}=2.010{\rm GeV},\;\;\;M_{D^{*+}_{s}}=2.107{\rm GeV}$
$\displaystyle\tau_{{}_{B_{u}}}$ $\displaystyle=$ $\displaystyle(1.638){\rm
ps},\;\;\;\tau_{{}_{B_{d}}}=(1.530){\rm ps},$ $\displaystyle\tau_{{}_{B_{s}}}$
$\displaystyle=$ $\displaystyle(1.425^{+0.041}_{-0.041}){\rm ps}.$ (50)
Using the input parameters given above, we then present the numerical results
and make some theoretical analysis for double charm $B_{u,d}$ and $B_{s}$
decay processes.
### IV.2 data and theoretical prediction
#### IV.2.1 $b\to c\bar{c}d$ decays
In the SM, $\bar{B}^{0}_{d}\to D^{{}^{(*)+}}D^{{}^{(*)-}}$, $B^{-}_{u}\to
D^{{}^{(*)0}}D^{{}^{(*)-}}$ and $\bar{B}^{0}_{s}\to
D^{{}^{(*)+}}_{s}D^{{}^{(*)-}}$ decays are dominated by the tree $b\to
c\bar{c}d$ transition, and receive additional $b\to c\bar{c}d$ penguin diagram
contributions.
In Table 2, we show the theoretical predictions for the $CP$-averaged
branching ratios and the polarization fractions in SM and mSUGRA model. The
weighted averages of the relevant experimental data pdg2008 are given in the
last column in both the Table 2 and Table 3. The data with a star in the top
right corner denote the BaBar measurement only, while that with two stars are
the Belle measurements only. The central values of the theoretical predictions
are obtained at the scale $\mu=m_{b}$, while the two errors are induced by the
uncertainties of $f_{D}=0.201\pm 0.017{\rm GeV}$ and $\gamma=67.8^{\circ}\pm
20^{\circ}$.
From the numerical results and the data as given in Table 2, we have the
following remarks on the branching ratios and the polarization fractions of
$b\to c\bar{c}d$ double charm decays:
1. (i) The SUSY contributions to the branching ratios of the considered decays are indeed very small, less than $5\%$, which is consistent with the general expectation since these decays are all ”tree” dominated decay processes.
2. (ii) Thhe theoretical predictions of the Br’s in both the SM and the mSUGRA model are consistent with the experimental measurements within $\pm 2\sigma$ errors. The central value of the theoretical prediction for $Br(\bar{B}^{0}_{d}\to D^{+}D^{-})$ ($Br(B^{-}_{u}\to D^{*0}D^{-})$) is, however, much larger (smaller ) than that of the corresponding measurement. This point will be clarified by the forthcoming LHC experiments.
3. (iii) The SUSY contributions to the polarization fractions of these decays in mSUGRA model are very small, less than $2\%$, and can be neglected safely. Only the central values are presented here since they are not sensitive to the variations of the form factors and the weak phase $\gamma$, which can be seen from the definition of the polarization fraction.
Table 2: Theoretical predictions for CP-averaged branching ratios (in units of $10^{-4}$), polarization (in percent) for $b\to c\bar{c}d$ decays in the SM and mSUGRA model. The last column shows currently available data pdg2008 . Observables | SM | mSUGRA | Data
---|---|---|---
| | (A) | (B) |
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{+}D^{-})$ | $3.26^{+0.57+0.10}_{-0.53-0.12}$ | $3.27^{+0.58+0.10}_{-0.53-0.11}$ | $3.15^{+0.55+0.08}_{-0.51-0.13}$ | $2.1\pm 0.3$
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{*\pm}D^{\mp})$ | $5.92^{+1.05+0.01}_{-0.95-0.01}$ | $5.93^{+1.04+0.01}_{-0.96-0.01}$ | $5.91^{+1.04+0.01}_{-0.96-0.01}$ | $6.1\pm 1.5$
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{*+}D^{*-})$ | $7.24^{+1.28+0.06}_{-1.17-0.06}$ | $7.25^{+1.28+0.06}_{-1.17-0.06}$ | $7.19^{+1.26+0.06}_{-1.17-0.06}$ | $8.2\pm 0.9$
$\mathcal{B}(B^{-}_{u}\to D^{0}D^{-})$ | $3.48^{+0.61+0.11}_{-1.20-0.78}$ | $3.50^{+0.62+0.10}_{-0.57-0.12}$ | $3.37^{+0.59+0.11}_{-0.55-0.14}$ | $3.8\pm 0.4$
$\mathcal{B}(B^{-}_{u}\to D^{*0}D^{-})$ | $3.43^{+0.60+0.03}_{-0.51-0.07}$ | $3.43^{+0.60+0.02}_{-0.56-0.02}$ | $3.44^{+0.61+0.03}_{-0.55-0.02}$ | $6.3\pm 1.4\pm 1.0^{*}$
$\mathcal{B}(B^{-}_{u}\to D^{0}D^{*-})$ | $2.92^{+0.51+0.02}_{-0.15-0.03}$ | $2.92^{+0.52+0.02}_{-0.47-0.02}$ | $2.89^{+0.51+0.03}_{-0.47-0.03}$ | $3.9\pm 0.5$
$\mathcal{B}(B^{-}_{u}\to D^{*0}D^{*-})$ | $7.75^{+1.36+0.05}_{-1.16-0.07}$ | $7.76^{+1.36+0.06}_{-1.26-0.07}$ | $7.68^{+1.36+0.07}_{-1.22-0.07}$ | $8.1\pm 1.2\pm 1.2^{*}$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{-})$ | $3.22^{+0.51+0.10}_{-0.52-0.11}$ | $3.24^{+0.57+0.09}_{-0.53-0.12}$ | $3.11^{+0.55+0.11}_{-0.50-0.13}$ | $-$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{-})$ | $3.13^{+0.55+0.02}_{-0.51-0.02}$ | $3.13^{+0.55+0.02}_{-0.51-0.02}$ | $3.14^{+0.55+0.02}_{-0.51-0.02}$ | $-$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{*-})$ | $2.67^{+0.48+0.03}_{-0.43-0.02}$ | $2.68^{+0.47+0.02}_{-0.44-0.03}$ | $2.65^{+0.47+0.02}_{-0.45-0.03}$ | $-$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-})$ | $7.12^{+1.26+0.07}_{-1.15-0.06}$ | $7.13^{+1.26+0.06}_{-1.15-0.06}$ | $7.07^{+1.24+0.06}_{-1.15-0.08}$ | $-$
$f_{L}(\bar{B}^{0}_{d}\to D^{*+}D^{*-})$ | $53.86$ | $53.87$ | $53.79$ | $57.0\pm 8.0\pm 2.0^{**}$
$f_{L}(B^{-}_{u}\to D^{*0}D^{*-})$ | $53.88$ | $53.89$ | $53.81$ | $-$
$f_{L}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-})$ | $53.88$ | $53.89$ | $53.81$ | $-$
$f_{\perp}(\bar{B}^{0}_{d}\to D^{*+}D^{*-})$ | $5.51$ | $5.50$ | $5.51$ | $15.0\pm 2.5$
$f_{\perp}(B^{-}_{u}\to D^{*0}D^{*-})$ | $5.52$ | $5.52$ | $5.53$ | $-$
$f_{\perp}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-})$ | $5.20$ | $5.20$ | $5.21$ | $-$
Table 3: Theoretical predictions of CPAs (in percent) for the exclusive color-allowed $b\to c\bar{c}d$ decays. The last column shows the word averages pdg2008 . Observables | SM | mSUGRA | Data
---|---|---|---
| | (A) | (B) |
$\mathcal{S}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{+}D^{-})$ | $-75.3^{+1.4+1.4}_{-1.5-0.6}$ | $-75.1^{+1.3+1.3}_{-1.3-0.6}$ | $-76.3^{+1.3+1.6}_{-1.2-0.7}$ | $-87\pm 26$
$\mathcal{S}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{*+}D^{-})$ | $-68.4^{+0.2+0.3}_{-0.3-0.2}$ | $-68.4^{+0.2+0.3}_{-0.3-0.2}$ | $-68.5^{+0.2+0.3}_{-0.3-0.2}$ | $-61\pm 19$
$\mathcal{S}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{+}D^{*-})$ | $-68.4^{+0.1+0.2}_{-0.4-0.2}$ | $-68.4^{+0.1+0.2}_{-0.4-0.2}$ | $-68.5^{+0.2+0.2}_{-0.4-0.2}$ | $-78\pm 21$
$\mathcal{S}^{+}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{*+}D^{*-})$ | $-70.2^{+0.4+0.4}_{-0.6-0.1}$ | $-70.1^{+0.5+0.3}_{-0.6-0.2}$ | $-70.4^{+0.4+0.4}_{-0.7-0.2}$ | $-81\pm 14$
$\mathcal{C}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{+}D^{-})$ | $-4.4^{+0.3+1.0}_{-0.4-0.5}$ | $-4.4^{+0.3+1.0}_{-0.4-0.5}$ | $-4.5^{+0.3+1.0}_{-0.4-0.6}$ | $-48\pm 42$
$\mathcal{C}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{*+}D^{-})$ | $7.8^{+0.3+0.7}_{-0.6-0.6}$ | $7.7^{+0.3+0.7}_{-0.6-0.6}$ | $8.3^{+0.3+0.8}_{-0.6-0.7}$ | $-9\pm 22$
$\mathcal{C}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{+}D^{*-})$ | $-8.4^{+1.1+0.7}_{-1.1-0.8}$ | $-8.3^{+1.1+0.7}_{-1.1-0.8}$ | $-8.9^{+1.0+0.8}_{-1.0-0.9}$ | $7\pm 14$
$\mathcal{C}^{+}(B^{0}_{d},\bar{B}^{0}_{d}\to D^{*+}D^{*-})$ | $-1.2^{+0.2+0.2}_{-0.4-0.1}$ | $-1.2^{+0.2+0.2}_{-0.4-0.1}$ | $-1.2^{+0.2+0.2}_{-0.4-0.1}$ | $-7\pm 9$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{0}D^{-})$ | $4.4^{+0.4+1.0}_{-0.3-0.2}$ | $4.4^{+0.4+0.5}_{-0.3-1.0}$ | $4.5^{+0.4+0.6}_{-0.3-1.0}$ | $-3\pm 7$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{*0}D^{-})$ | $-0.6^{+0.4+0.1}_{-0.2-0.1}$ | $-0.6^{+0.4+0.1}_{-0.2-0.1}$ | $-0.6^{+0.4+0.1}_{-0.2-0.1}$ | $13\pm 18\pm 4^{*}$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{0}D^{*-})$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $3\pm 10$
$\mathcal{A}^{+,dir}_{CP}(B^{-}_{u}\to D^{*0}D^{*-})$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $-15\pm 11\pm 2^{*}$
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{-})$ | $4.4^{+0.4+0.5}_{-0.3-1.0}$ | $4.4^{+0.4+0.5}_{-0.3-0.6}$ | $4.5^{+0.4+0.6}_{-0.3-1.0}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{-})$ | $-0.6^{+0.4+0.1}_{-0.2-0.2}$ | $-0.6^{+0.4+0.1}_{-0.2-0.1}$ | $-0.6^{+0.4+0.1}_{-0.2-0.1}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{*-})$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $-$
$\mathcal{A}^{+,{\rm dir}}_{\rm CP}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-})$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $1.2^{+0.4+0.1}_{-0.2-0.2}$ | $-$
In Table 3, we present the theoretical predictions for the CPAs in the
framework of the SM and the mSUGRA model. The currently available data are
also listed in the last column. The uncertainties come from the scale
$m_{b}/2\leq\mu\leq 2m_{b}$ and the weak angle $\gamma=67.8^{\circ}\pm
20^{\circ}$. From the numerical results and the data, we find that
1. (i) Just as generally expected based on the SM, the direct CPAs $C_{f}$ are indeed quite small, while the mixing-induced CPAs of all considered decays are close to $-0.7$: i.e. $S_{f}\approx\sin(2\beta)\approx-0.7$.
2. (ii) The SUSY contributions to all considered decays are less than $7\%$. The new physics contributions is not sensitive to the variation of the scale $\mu$ and the weak angle $\gamma$.
3. (iii) The theoretical predictions in the SM and mSUGRA model are all consistent with the experimental measurements within $\pm 1\sigma$ error. Of course, the errors of currently available data are very large now.
#### IV.2.2 $b\to c\bar{c}s$ decays
The twelves decay modes $\bar{B}^{0}_{d}\to D^{{}^{(*)+}}D^{{}^{(*)-}}_{s}$,
$B^{-}_{u}\to D^{{}^{(*)0}}D^{{}^{(*)-}}_{s}$ and $\bar{B}^{0}_{s}\to
D^{{}^{(*)+}}_{s}D^{{}^{(*)-}}_{s}$ are the tree-dominated processes, and also
receive the additional $b\to c\bar{c}s$ penguin contributions.
Table 4: Theoretical predictions for CP-averaged $\mathcal{Br}$ (in units of $10^{-3}$) and polarization fractions (in units of $10^{-2}$) of exclusive color-allowed $b\to c\bar{c}s$ decays in the SM and the mSUGRA model. The last column corresponds to the world averages pdg2008 . Observables | SM | mSUGRA | Data
---|---|---|---
| | (A) | (B) |
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{+}D^{-}_{s})$ | $8.77^{+1.16+0.02}_{-1.09-0.02}$ | $8.83^{+1.17+0.02}_{-1.10-0.02}$ | $8.39^{+1.11+0.02}_{-1.36-0.02}$ | $7.4\pm 0.7$
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{*+}D^{-}_{s})$ | $8.78^{+1.16+0.01}_{-1.10-0.01}$ | $8.77^{+1.17+0.01}_{-1.09-0.01}$ | $8.78^{+1.17+0.01}_{-1.09-0.01}$ | $8.2\pm 1.1$
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{+}D^{*-}_{s})$ | $7.30^{+0.97+0.01}_{-0.91-0.01}$ | $7.31^{+0.97+0.01}_{-0.91-0.01}$ | $7.22^{+0.96+0.01}_{-0.90-0.01}$ | $7.5\pm 1.6$
$\mathcal{B}(\bar{B}^{0}_{d}\to D^{*+}D^{*-}_{s})$ | $21.2^{+2.8}_{-2.6}\pm 0.0$ | $21.2^{+2.8}_{-2.6}\pm 0.0$ | $20.9^{+2.8}_{-2.6}\pm 0.0$ | $17.8\pm 1.4$
$\mathcal{B}(B^{-}_{u}\to D^{0}D^{-}_{s})$ | $9.38^{+1.24+0.01}_{-1.17-0.02}$ | $9.44^{+1.25+0.01}_{-1.18-0.02}$ | $8.97^{+1.19+0.02}_{-1.12-0.02}$ | $10.2\pm 1.7$
$\mathcal{B}(B^{-}_{u}\to D^{*0}D^{-}_{s})$ | $9.40^{+1.24+0.01}_{-1.17-0.01}$ | $9.39^{+1.25+0.01}_{-1.17-0.01}$ | $9.40^{+1.25+0.01}_{-1.17-0.01}$ | $8.4\pm 1.7$
$\mathcal{B}(B^{-}_{u}\to D^{0}D^{*-}_{s})$ | $7.82^{+1.04+0.01}_{-0.97-0.01}$ | $7.83^{+1.04+0.01}_{-0.97-0.01}$ | $7.73^{+1.03+0.01}_{-0.96-0.01}$ | $7.8\pm 1.6$
$\mathcal{B}(B^{-}_{u}\to D^{*0}D^{*-}_{s})$ | $22.6^{+3.0}_{-2.8}\pm 0.0$ | $22.7^{+3.0}_{-2.8}\pm 0.0$ | $22.4^{+3.0}_{-2.8}\pm 0.0$ | $17.4\pm 2.3$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s})$ | $8.68^{+1.15+0.02}_{-1.08-0.02}$ | $8.73^{+1.16+0.02}_{-1.08-0.02}$ | $8.30^{+1.10+0.02}_{-1.03-0.02}$ | $11\pm 4$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{-}_{s})$ | $8.74^{+1.16+0.01}_{-1.09-0.01}$ | $8.73^{+1.16+0.01}_{-1.08-0.01}$ | $8.75^{+1.16+0.01}_{-1.09-0.01}$ | $-$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{+}_{s}D^{*-}_{s})$ | $7.16^{+0.95+0.01}_{-0.89-0.01}$ | $7.17^{+0.98+0.01}_{-0.88-0.01}$ | $7.08^{+0.94+0.01}_{-0.88-0.01}$ | $<121$
$\mathcal{B}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-}_{s})$ | $20.8^{+2.8}_{-2.6}\pm 0.0$ | $20.8^{+2.8}_{-2.6}\pm 0.0$ | $20.6^{+2.7}_{-2.6}\pm 0.0$ | $<257$
$f_{L}(\bar{B}^{0}_{d}\to D^{*+}D^{*-}_{s})$ | $51.68$ | $51.70$ | $51.58$ | $52\pm 5$
$f_{L}(B^{-}_{u}\to D^{*0}D^{*-}_{s})$ | $51.70$ | $51.72$ | $51.61$ | $-$
$f_{L}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-}_{s})$ | $51.70$ | $51.71$ | $51.60$ | $-$
$f_{\perp}(\bar{B}^{0}_{d}\to D^{*+}D^{*-}_{s})$ | $5.50$ | $5.50$ | $5.51$ | $-$
$f_{\perp}(B^{-}_{u}\to D^{*0}D^{*-}_{s})$ | $5.51$ | $5.51$ | $5.52$ | $-$
$f_{\perp}(\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-}_{s})$ | $5.19$ | $5.18$ | $5.20$ | $-$
Table 5: Theoretical predictions for CPAs (in percent) of exclusive color-allowed $b\to c\bar{c}s$ decays in the SM and the mSUGRA model. Observables | SM | mSUGRA | Data
---|---|---|---
| | (A) | (B) |
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{d}\to D^{+}D^{-}_{s})$ | $-0.26^{+0.02+0.05}_{-0.03-0.02}$ | $-0.26^{+0.02+0.05}_{-0.01-0.02}$ | $-0.27^{+0.02+0.06}_{-0.02-0.02}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{d}\to D^{*+}D^{-}_{s})$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(\bar{B}^{0}_{d}\to D^{+}D^{*-}_{s})$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-0.07^{+0.02+0.02}_{-0.01-0.01}$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-$
$\mathcal{A}^{+,{\rm dir}}_{\rm CP}(\bar{B}^{0}_{d}\to D^{*+}D^{*-}_{s})$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-0.07^{+0.02+0.02}_{-0.01-0.01}$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{0}D^{-}_{s})$ | $-0.26^{+0.02+0.05}_{-0.03-0.02}$ | $-0.26^{+0.02+0.05}_{-0.01-0.02}$ | $-0.27^{+0.02+0.06}_{-0.02-0.02}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{*0}D^{-}_{s})$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $0.03^{+0.02+0.01}_{-0.02-0.01}$ | $-$
$\mathcal{A}^{\rm dir}_{\rm CP}(B^{-}_{u}\to D^{0}D^{*-}_{s})$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-0.07^{+0.02+0.02}_{-0.01-0.01}$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-$
$\mathcal{A}^{+,{\rm dir}}_{\rm CP}(B^{-}_{u}\to D^{*0}D^{*-}_{s})$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-0.07^{+0.02+0.02}_{-0.01-0.01}$ | $-0.07^{+0.02+0.02}_{-0.02-0.01}$ | $-$
$\mathcal{S}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s})$ | $0.53^{+0.11+0.11}_{-0.12-0.12}$ | $0.51^{+0.06+0.04}_{-0.11-0.10}$ | $0.62^{+0.11+0.06}_{-0.11-0.12}$ | $-$
$\mathcal{S}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{*+}_{s}D^{-}_{s})$ | $0.93^{+0.02+0.02}_{-0.01-0.02}$ | $0.93^{+0.01+0.02}_{-0.06-0.02}$ | $0.94^{+0.02+0.02}_{-0.01-0.02}$ | $-$
$\mathcal{S}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{+}_{s}D^{*-}_{s})$ | $-0.94^{+0.03+0.02}_{-0.01-0.01}$ | $-0.94^{+0.10+0.02}_{-0.01-0.02}$ | $-0.93^{+0.02+0.02}_{-0.03-0.02}$ | $-$
$\mathcal{S}^{+}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-}_{s})$ | $0.13^{+0.04+0.01}_{-0.04-0.03}$ | $0.12^{+0.03+0.01}_{-0.03-0.02}$ | $0.14^{+0.05+0.02}_{-0.03-0.02}$ | $-$
$\mathcal{C}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{+}_{s}D^{-}_{s})$ | $0.26^{+0.03+0.05}_{-0.02-0.02}$ | $0.26^{+0.01+0.02}_{-0.02-0.05}$ | $0.27^{+0.02+0.02}_{-0.02-0.06}$ | $-$
$\mathcal{C}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{*+}_{s}D^{-}_{s})$ | $9.91^{+0.91+0.05}_{-1.14-0.04}$ | $9.82^{+0.21+0.04}_{-1.15-0.05}$ | $10.52^{+0.89+0.05}_{-1.12-0.05}$ | $-$
$\mathcal{C}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{+}_{s}D^{*-}_{s})$ | $-9.93^{+1.16+0.01}_{-0.95-0.04}$ | $-9.84^{+1.18+0.05}_{-0.25-0.03}$ | $-10.54^{+1.14+0.05}_{-0.93-0.05}$ | $-$
$\mathcal{C}^{+}(B^{0}_{s},\bar{B}^{0}_{s}\to D^{*+}_{s}D^{*-}_{s})$ | $0.07^{+0.01+0.01}_{-0.02-0.02}$ | $0.07^{+0.02+0.01}_{-0.01-0.02}$ | $0.07^{+0.01+0.01}_{-0.02-0.02}$ | $-$
In Table 4, we present the theoretical predictions for the CP-averaged
branching ratios and the polarization fractions in the framework of the SM and
the mSUGRA model. The last column in table 4 correspond to the world averages
pdg2008 . The theoretical predictions for CP asymmetries of considered decays
are given in Table 5, although they have not been measured yet. The central
values of the theoretical predictions are obtained at the scale $\mu=m_{b}$,
while the two errors are induced by the uncertainties of $f_{D}=0.201\pm
0.017{\rm GeV}$ and $\gamma=67.8^{\circ}\pm 20^{\circ}$.
From the numerical results and currently available data, one can see that
* (i) For the Br’s and CPAs, the SUSY contributions again are very small for all considered decays, less than $3\%$ numerically. The theoretical predictions in both the SM and the mSUGRA model are all consistent with currently available data within one or two standard deviations.
* (ii) The direct CP violations $\mathcal{C}(B^{0}_{s}\to D^{*+}_{s}D^{-}_{s})$ and $\mathcal{C}(B^{0}_{s}\to D^{+}_{s}D^{*-}_{s})$ are at the $\pm 10\%$ level and to be tested by the LHC experiments. And the CP asymmetries for the remaining ten decays are very small, about $10^{-3}$ or $10^{-4}$ numerically, since the penguin effects are doubly Cabibbo-suppressed for the color-allowed $b\to c\bar{c}s$ decays.
## V Summary
In this paper, we have investigated the new contributions to the branching
rations, polarization fractions and CP asymmetries of the twenty three double
charm decays $B/B_{s}\to D^{(*)}_{(s)}D^{(*)}_{(s)}$ in the SM and the mSUGRA
model by employing the effective hamiltonian for $\Delta B=1$ transition and
the naive factorization approach.
From the numerical results and the phenomenological analysis, the following
conclusions can be reached:
1. (i) For the exclusive double charm decays $B/B_{s}\to D^{(*)}_{(s)}D^{(*)}_{(s)}$ studied in this paper, the SUSY contributions in the mSUGRA model are very small, less than $7\%$ numerically. It may be difficult to observe so small SUSY contributions even at LHC.
2. (ii) All the theoretical predictions in the SM and mSUGRA model are consistent with the experimental measurements within $\pm 2\sigma$ errors.
3. (iii) The theoretical predictions in both the SM and mSUGRA model still have large theoretical uncertainties. The dominant errors are induced by the uncertainties of the form factors $f_{D}$ or $f_{D_{s}}$.
###### Acknowledgements.
We are grateful to Wen-juan Zou for valuable help. This work is partially
supported by the National Natural Science Foundation of China under Grant No.
10947020, and by Foundation of Henan Educational Committee for Youth Backbone
Scholars in Colleges and Universities, and by the Natural Science Foundation
of the Eduction Department of Henan Province under Grant No. 2010A140012.
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|
arxiv-papers
| 2010-08-30T02:40:28 |
2024-09-04T02:49:12.551858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lin-Xia L\\\"u, Zhen-Jun Xiao, Shuai-Wei Wang and Wen-Jun Li",
"submitter": "Linxia Lu",
"url": "https://arxiv.org/abs/1008.4987"
}
|
1008.5238
|
# Invariant and hyperinvariant subspaces for amenable operators
Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn , YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and You Qing Ji Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA jiyq@jlu.edu.cn
(Date: 7.04. 2010)
# Invariant and hyperinvariant subspaces for amenable operators
Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn , YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and You Qing Ji Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA jiyq@jlu.edu.cn
(Date: 7.04. 2010)
###### Abstract.
There has been a long-standing conjecture in Banach algebra that every
amenable operator is similar to a normal operator. In this paper, we study the
structure of amenable operators on Hilbert spaces. At first, we show that the
conjecture is equivalent to every non-scalar amenable operator has a non-
trivial hyperinvariant subspace and equivalent to every amenable operator is
similar to a reducible operator and has a non-trivial invariant subspace; and
then, we give two decompositions for amenable operators, which supporting the
conjecture.
###### Key words and phrases:
Amenable; Invariant subspaces; Hyperinvariant subspaces; Reduction property
###### 2000 Mathematics Subject Classification:
47C05 (46H35 47A65 47A66 47B15)
Supported by NCET(040296), NNSF of China(10971079) and the Specialized
Research Fund for the Doctoral Program of Higher Education(20050183002)
## 1\. Introduction
Throughout this paper, $\mathfrak{H}$ denotes a complex separable infinite-
dimension Hilbert space and $\mathfrak{B}(\mathfrak{H})$ denotes the bounded
linear operators on $\mathfrak{H}$. For an algebra $\mathfrak{A}$ in
$\mathfrak{B}(\mathfrak{H})$, we write $\mathfrak{A}^{\prime}$ for the
commutant of $\mathfrak{A}$ (i.e.,
$\mathfrak{A}^{\prime}=\\{B\in\mathfrak{B}(\mathfrak{H}),BA=AB~{}~{}\textup{for
all}A\in\mathfrak{A}\\})$ and $\mathfrak{A}^{\prime\prime}$ for the double
commutant of $\mathfrak{A}$ (i.e.,
$\mathfrak{A}^{\prime\prime}=(\mathfrak{A}^{\prime})^{\prime}$). We also write
Lat$\mathfrak{A}$ for the collection of those subspaces which are invariant
for every operator in $\mathfrak{A}$. We say $\mathfrak{A}$ is completely
reducible if for every subspace $M$ in Lat$\mathfrak{A}$ there exists $N$ in
Lat$\mathfrak{A}$ such that $\mathfrak{H}=M\dot{+}N$ (i.e., $M\cap N=\\{0\\}$
and $\mathfrak{H}$ is the algebraic direct sum of $M$ and $N$); $\mathfrak{A}$
is reducible if for every subspace $M$ in Lat$\mathfrak{A}$ we have $M^{\bot}$
(the orthogonal complement of $M$) in Lat$\mathfrak{A}$; $\mathfrak{A}$ is
transitive if Lat$\mathfrak{A}=\\{\\{0\\},\mathfrak{H}\\}$. If
$T\in\mathfrak{B}(\mathfrak{H})$, we say that a subspace $M$ of $\mathfrak{H}$
is a hyperinvariant subspace for $T$ if $M$ is invariant under each operator
in $\mathfrak{A}_{T}^{\prime}$; $M$ is a reducible subspace for $T$ if
$M,M^{\bot}\in$ Lat$T$.
The concept of amenable Banach algebras was first introduced by B. E. Johnson
in [10]. Suppose that $\mathfrak{A}$ is a Banach algebra. A Banach
$\mathfrak{A}$-bimodule is a Banach space $X$ that is also an algebra
$\mathfrak{A}$-bimodule for which there exists a constant $K>0$ such that
$||a\cdot x||\leq K||a||||x||$ and $||x\cdot a||\leq K||a||||x||$ for all
$a\in\mathfrak{A}$ and $x\in X$. We note that $X^{*}$, the dual of $X$, is a
Banach $\mathfrak{A}$-bimodule with respect to the dual actions
$[a\cdot f](x)=f(x\cdot a),[f\cdot a](x)=f(a\cdot x),a\in\mathfrak{A},x\in
X,f\in X^{*}.$
Such a Banach $\mathfrak{A}$-bimodule is called a dual
$\mathfrak{A}$-bimodule.
A derivation $D:\mathfrak{A}\rightarrow X$ is a continuous linear map such
that $D(ab)=a\cdot D(b)+D(a)\cdot b$, for all $a,b\in\mathfrak{A}$. Given
$x\in X$, the inner derivation $\delta_{x}:\mathfrak{A}\rightarrow X$, is
defined by $\delta_{x}(a)=a\cdot x-x\cdot a$.
According to Johnson s original definition, a Banach algebra $\mathfrak{A}$ is
amenable if every derivation from $\mathfrak{A}$ into the dual
$\mathfrak{A}$-bimodule $X^{*}$ is inner for all Banach
$\mathfrak{A}$-bimodules $X$. If $T\in\mathfrak{B}(\mathfrak{H})$, denote the
norm-closure of span$\\{T^{k}:k\in\\{0\\}\cup\mathbb{N}\\}$ by
$\mathfrak{A}_{T}$, where $\mathbb{N}$ is the set of natural numbers, $T$ is
said to be an amenable operator, if $\mathfrak{A}_{T}$ is an amenable Banach
algebra. Ever since its introduction, the concept of amenability has played an
important role in research in Banach algebras, operator algebras and harmonic
analysis. There has been a long-standing conjecture in the Banach algebra
community, stated as follows:
###### Conjecture 1.1.
A commutative Banach subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable if
and only if it is similar to a $C^{*}$-algebra.
One of the first result in this direction is due to Willis [16]. Willis showed
that if $T$ is an amenable compact operator, then $T$ is similar to a normal
operator. In [7] Gifford studied the reduction property for operator algebras
consisting of compact operators and showed that if such an algebra is amenable
then it is similar to a $C^{*}$-algebras. In the recent papers [5], [6]
Farenick, Forrest and Marcoux showed that if $T$ is similar to a normal
operator, then $\mathfrak{A}_{T}$ is amenable if and only if
$\mathfrak{A}_{T}$ is similar to a $C^{*}$-algebra and the spectrum of $T$ has
connected complement and empty interior; If $T$ is a triangular operator with
respect to an orthonormal basis of $\mathfrak{H}$, then $\mathfrak{A}_{T}$ is
amenable if and only if $T$ is similar to a normal operator whose spectrum has
connected complement and empty interior. For further details see [5] and [6].
In this paper, we give the characterization of the structure of amenable
operators. At first, we use the reduction theory of von Neumann to give two
equivalent descriptions for Conjecture 1.1; and then, we give two
decompositions for amenable operators, which supporting the Conjecture 1.1.
## 2\. An equivalent formulation of the conjecture 1.1
In this section we use the reduction theory of von Neumann to give two
equivalent descriptions for Conjecture 1.1. We obtain that every amenable
operator is similar to a normal operator if and only if every non-scalar
amenable operator has a non-trivial hyperinvariant subspace if and only if
every amenable operator is similar to a reducible operator and has a non-
trivial invariant subspace.
In order to proof the main theorem, we need to introduce von Neumann’s
reduction theory [15] and some lemmas.
Let
$\mathfrak{H_{1}}\subseteq\mathfrak{H_{2}}\subseteq\cdots\subseteq\mathfrak{H_{\infty}}$
be a sequence of Hilbert spaces chosen once and for all, $\mathfrak{H_{n}}$
having the dimension $n$. Let $\mu$ be a finite positive regular measure
defined on the Borel sets of a separable metric space $\wedge$, and let
$\\{E_{n}\\}_{n=1}^{\infty}$ be a collection of disjoint Borel sets of
$\wedge$ with union $\wedge$. Then the symbol
$\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$
denotes the set of all functions $f$ defined on $\wedge$ such that
(1)$f(\lambda)\in\mathfrak{H}_{n}\subseteq\mathfrak{H}_{\infty}$ if
$\lambda\in E_{n}$;
(2)$f(\lambda)$ is a $\mu$-measurable function with values in
$\mathfrak{H}_{\infty}$;
(3) $\int_{\wedge}^{\oplus}|f(\lambda)|^{2}\mu(d\lambda)<\infty$.
We put
(4)$(f,g)=\int_{\wedge}^{\oplus}(f(\lambda),g(\lambda))\mu(d\lambda).$
The set of functions thus defined is called the direct integral Hilbert space
with measure $\mu$ and dimension sets $\\{E_{n}\\}$ and denoted by
$\mathfrak{H}=\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$.
An operator on $\mathfrak{H}$ is said to be decomposable if there exists a
strongly $\mu$-measurable operator-value function $A(\cdot)$ defined on
$\wedge$ such that $A(\lambda)$ is a bounded operator on the space
$\mathfrak{H}(\lambda)=\mathfrak{H}_{n}$ when $\lambda\in E_{n}$, and for all
$f\in\mathfrak{H}$, $(Af)(\lambda)=A(\lambda)f(\lambda)$. We write
$A=\int_{\wedge}^{\oplus}A(\lambda)\mu(d\lambda)$ for the equivalence class
corresponding to $A(\cdot)$. If $A(\lambda)$ is a scalar multiple of the
identity on $\mathfrak{H}(\lambda)$ for almost all $\lambda$, then $A$ is
called diagonal. The collection of all diagonal operator is called the
diagonal algebra of $\wedge$. In [15]I.3, Schwartz showed that an operator $A$
on Hilbert space
$\mathfrak{H}=\int_{\wedge}^{\oplus}\mathfrak{H}(\lambda)\mu(d\lambda)$ is
decomposable if and only if $A$ belong to the commutant of the diagonal
algebra of $\wedge$. And $||A||=\mu-ess.sup_{\lambda\in\wedge}||A(\lambda)||$.
In [1], Azoff, Fong and Gilfeather used von Neumann’s reduction theory to
define the reduction theory for non-selfadjoint operator algebras: Fix a
partitioned measure space $\wedge$ and let $\mathfrak{D}$ be the corresponding
diagonal algebra. Given an algebra $\mathfrak{A}$ of decomposable operators.
Each operator $A\in\mathfrak{A}$ has a decomposition
$A=\int_{\wedge}^{\oplus}A(\lambda)\mu(d\lambda)$. Chosse a countable
generating set $\\{A_{n}\\}$ for $\mathfrak{A}$. let $\mathfrak{A}(\lambda)$
be the strongly closed algebra generated by the $\\{A_{n}(\lambda)\\}$.
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ is
called the decomposition of $\mathfrak{A}$ respect to $\mathfrak{D}$. A
decomposition
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ of
an algebra is said to be maximal if the corresponding diagonal algebra is
maximal among the abelian von Neumman subalgebras of $\mathfrak{A}^{\prime}$.
The following lemma is a basic result in [1] which will be used in this paper.
###### Lemma 2.1.
([1], Theorem 4.1) Let
$\mathfrak{A}\sim\int_{\wedge}^{\oplus}\mathfrak{A}(\lambda)\mu(d\lambda)$ be
a decomposition of a reductive algebra. Then almost all of
$\\{\mathfrak{A}(\lambda)\\}$ are reducible. In particular, if the
decomposition is maximal, then almost all of the algebras
$\\{\mathfrak{A}(\lambda)\\}$ are transitive.
In [7] Gifford studied the reduction property for operator algebras and
obtained the following result:
###### Lemma 2.2.
([7] Lemma 4.4, Lemma 4.12) If $\mathfrak{A}$ is a commutative amenable
operator algebra, then $\mathfrak{A}^{\prime},\mathfrak{A}^{\prime\prime}$ are
complete reducible and there exists $M\geq 1$ so that for any idempotent
$p\in\mathfrak{A}^{\prime\prime}$ $||p||\leq M$.
Assume $\mathfrak{A}$ is a operator algebra, let $P(\mathfrak{A})$ denote the
idempotents in $\mathfrak{A}$ and $\mathcal{P}(\mathfrak{A})$ denote the
strongly closed algebra generated by $P(\mathfrak{A})$. We get the following
lemma:
###### Lemma 2.3.
If $\mathfrak{A}$ is a commutative amenable operator algebra, then
$\mathcal{P}(\mathfrak{A^{\prime\prime}})$ is similar to an abelian von
Neumann algebra.
###### Proof.
By Lemma 2.2 and [2, Corollary 17.3], it follows that there exists
$X\in\mathfrak{B}(\mathfrak{H})$ such that $XpX^{-1}$ is selfadjoint for each
$p\in P(\mathfrak{A^{\prime\prime}})$. Hence
$\mathcal{P}(\mathfrak{A^{\prime\prime}})$ is similar to a abelian von Neumann
algebra. ∎
###### Lemma 2.4.
([5]) Let $\mathfrak{A}$ and $\mathfrak{B}$ be Banach algebras and suppose
that $\varphi:\mathfrak{A}\longrightarrow\mathfrak{B}$ is a continuous
homomorphism with $\varphi(\mathfrak{A})$ dense in $\mathfrak{B}$. If
$\mathfrak{A}$ is amenable, then $\mathfrak{B}$ is amenable.
###### Notation 2.5.
From Lemma 2.3,2.4 we always assume that
$\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$ is a abelian von Neumann
algebra, and $\mathfrak{A}_{T}^{\prime}$ is a reducible operator algebra in
this section.
Now we will proof the main result of this section:
###### Theorem 2.6.
The following are equivalent:
(1) Every amenable operator is similar to a normal operator;
(2) Every non-scalar amenable operator has a non-trivial hyperinvariant
subspace;
(3) Every amenable Banach algebra which is generated by an operator is similar
to a $C^{*}$-algebra.
###### Proof.
$(1)\Leftrightarrow(3)$ and $(1)\Rightarrow(2)$ is clear by [5]. Therefore, in
order to establish the theorem it suffices to show the implications
$(2)\Rightarrow(1)$.
Assume (2), by Lemma 2.3 choose a maximal decomposition for
$\mathfrak{A}_{T}^{\prime}\sim\int_{\wedge}^{\oplus}\mathfrak{A}_{T}^{\prime}(\lambda)\mu(d\lambda)$
respect to the diagonal algebra
$\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$.
Assume $T\sim\int_{\wedge}^{\oplus}T(\lambda)\mu(d\lambda)$ is the
decomposition for $T$. Let $\\{p_{n}\\}_{n=1}^{\infty}$ denote the all
rational polynomials. Then
$p_{n}(T)\sim\int_{\wedge}^{\oplus}p_{n}(T)(\lambda)\mu(d\lambda)$ is
decomposable for all $n$ and there exists a measurable $E\subseteq\wedge$ such
that $\mu(\wedge-E)=0$ and for any $\lambda\in E$ we have
$p_{n}(T)(\lambda)=p_{n}(T(\lambda))$ and
$||p_{n}(T)(\lambda)||\leq||p_{n}(T)||$ by [15, Lemma I.3.1, I.3.2]. Define a
mapping
$\varphi_{\lambda}:\mathfrak{A}_{T}\rightarrow\mathfrak{A}_{T(\lambda)}$ by
$\varphi_{\lambda}(p_{n}(T))=p_{n}(T(\lambda))$ for each rational polynomial
$p_{n}$ and $\lambda\in E$. Note that $||p_{n}(T(\lambda))||\leq||p_{n}(T)||$
for each rational polynomial $p_{n}$ and furthermore, $\\{p_{n}(T)\\}$ is
dense in $\mathfrak{A}_{T}$. Hence, $\varphi_{\lambda}$ is well-defined and
$\varphi_{\lambda}$ a continuous homomorphism with $\varphi(\mathfrak{A}_{T})$
dense in $\mathfrak{A}_{T(\lambda)}$. By Lemma 2.4, $T(\lambda)$ is amenable
for almost all $\lambda$.
Now for almost all $\lambda$, $T(\lambda)$ is amenable and
$\mathfrak{A}_{T}^{\prime}(\lambda)\subseteq\mathfrak{A}_{T(\lambda)}^{\prime}$
and $\mathfrak{A}_{T}^{\prime}(\lambda)$ is transitive by Lemma 2.1. Thus
almost all of $T(\lambda)$ are scalar operators, i.e. $T$ is a normal
operator. ∎
###### Corollary 2.7.
Every amenable operator is similar to a normal operator if and only if there
exists a non-trivial idempotent in the double-commutant of every non-scalar
amenable operator.
###### Remark 2.8.
In [5], Farenick, Forrest and Marcoux showed that if
$T\in\mathfrak{B}(\mathfrak{H})$ is amenable and similar to a normal operator
$N$, then the spectrum of $N$ has connected complement and empty interior.
According to [14, Theorem 1.23], $N$ is a reducible operator. Hence, there
exists an invertible operator $X\in\mathfrak{B}(\mathfrak{H})$ such that
$\mathfrak{A}_{XTX^{-1}}^{\prime\prime}$ is a reducible algebra. The following
theorem give the equivalent description for Conjecture 1.1 from the existence
of invariant subspace for amenable operators.
###### Theorem 2.9.
The following are equivalent:
(1) Every amenable operator is similar to a normal operator;
(2) For every amenable operator $T\in\mathfrak{B}(\mathfrak{H})$, there exists
an invertible operator $X\in\mathfrak{B}(\mathfrak{H})$ such that
$\mathfrak{A}_{XTX^{-1}}^{\prime\prime}$ is a reducible algebra and $T$ has a
non-trivial invariant subspace.
###### Proof.
$(1)\Rightarrow(2)$ is clear by Remark 2.8.
$(2)\Rightarrow(1)$ is trivial modifications adapt the proof of theorem 2.6. ∎
###### Remark 2.10.
According to theorem 2.6, 2.9, we obtain that the Conjecture 1.1 for operator
algebra which is generated by an operator is equivalent to the following
statements:
(1) Every amenable operator $T$ has a non-trivial invariant subspace and
renorm $\mathfrak{H}$ with an equivalent Hilbert space norm so that under this
norm Lat$\mathfrak{A}_{T}$ becomes orthogonally complemented;
(2) Every non-scalar amenable operator has a non-trivial hyperinvariant
subspace.
## 3\. Decomposition of Amenable operators
In this section, we get two decompositions for amenable operators and prove
that the two decompositions are the same which supporting Conjecture 1.1.
At first, we summarize some of the details of multiplicity theory for abelian
von Neumann algebras. For the most part, we will follow [3]. If $A$ is an
operator on a Hilbert space $\mathfrak{K}$ and $n$ is a cardinal number, Let
$\mathfrak{K}^{n}$ denote the orthogonal direct sum of $n$ copies of
$\mathfrak{K}$, and $A^{(n)}$ be the operator on $\mathfrak{K}^{n}$ which is
the direct sum of $n$ copies of $A$. Whenever $\mathfrak{A}$ is an operator
algebra on $\mathfrak{K}$, $\mathfrak{A}^{(n)}$ denotes the algebra
$\\{A^{(n)},A\in\mathfrak{A}\\}$. An abelian von Neumann algebra
$\mathfrak{B}$ is of uniform multiplicity $n$ if it is (unitary equivalent to)
$\mathfrak{A}^{(n)}$ for some maximal abelian von Neumann algebra
$\mathfrak{A}$. By [3], for any abelian von Neumann algebra $\mathfrak{A}$,
there exists a sequens of regular Borel measures $\\{\mu_{n}\\}$ on a sequens
of separable metric space $\\{X_{n}\\}$ such that $\mathfrak{A}$ is unitary
equivalent to
$\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}\oplus\mathfrak{B}_{\infty}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $1\leq n\leq\infty$. For further details see [3, II.3].
###### Proposition 3.1.
Suppose that $T$ is amenable operator and $\mathfrak{A}_{T}^{\prime}$ contains
a subalgebra which is similar to an abelian von Neumman algebra with no direct
summand of uniform multiplicity infinite, then $T$ is similar to a normal
operator.
###### Proof.
For the sake of simplicity, we assume $\mathfrak{A}_{T}^{\prime}$ contains a
subalgebra $\mathfrak{B}$ which is an abelian von Neumman subalgebra with no
direct summand of uniform multiplicity infinite. Trivial modifications adapt
the proof to the more general case.
By [3, II.3], there exists a sequens of regular Borel measures $\\{\mu_{n}\\}$
on a sequens of separable metric space $\\{X_{n}\\}$ such that $\mathfrak{B}$
is unitarily equivalent to $\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $n$. Hence, $T=\sum_{n=1}^{\infty}\oplus T_{n}$, where
$T_{n}\in\mathfrak{B}_{n}^{\prime}$. It suffices to show that $T_{n}$ is
similar to a normal operator for all $n$, then by [4, Corollary 26], it
follows that $T$ is similar to a normal operator.
Since $T\in\mathfrak{B}_{n}^{\prime}$, according to [14, Theorem 7.20], for
any $1\leq n<\infty$ there exists a unitary operator
$U_{n}\in\mathfrak{B}_{n}^{\prime}$ such that
$U_{n}T_{n}(U_{n})^{-1}=\left[\begin{array}[]{ccccc}N_{11}&N_{12}&\cdots&\cdots&N_{1n}\\\
0&N_{22}&\cdots&\cdots&N_{2n}\\\ 0&0&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&\vdots\\\ 0&0&\cdots&0&N_{nn}\\\
\end{array}\right]$
where $N_{ij}$ is a normal operator for all $1\leq i,j\leq n$. By [17,
Proposition 3.1], it follows that $T_{n}$ is similar to
$\oplus_{i=1}^{n}N_{ii}$, i.e. $T_{n}$ is similar to a normal operator for all
$n$. ∎
###### Corollary 3.2.
Assume $T$ is amenable operator, then there exists hyperinvariant subspaces
$M,N$ of $T$ such that $T$ has the form $T=T_{1}\dot{+}T_{2}$ respect to the
space decomposition $\mathfrak{H}=M\dot{+}N$, where $T_{1},T_{2}$ are amenable
operators, $T_{1}$ is similar to a normal operator and
$\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar to an abelian
von Neumman algebra with uniform multiplicity infinite.
The proof of the following lemma is straightforward and we omit it.
###### Lemma 3.3.
Suppose that $\mathfrak{A}$ is a completely reductive operator algebra and
$p\in P(\mathfrak{A}^{\prime})$. Then $p\mathfrak{A}$ is a completely
reductive operator algebra on $\textup{Ran}p$.
We are in need of the following propositions before we can address the main
theorem of this section.
###### Proposition 3.4.
Assume that $T$ is a amenable operator and there exists a space decomposition
$\mathfrak{H}=M\dot{+}N$ such that $T$ has the matrix form
$T=\left[\begin{array}[]{cc}T_{1}&\\\
&T_{2}\end{array}\right]\begin{matrix}\mbox{$M$}\\\ \mbox{$N$}\\\
\end{matrix}$. Then $T$ is similar to a normal operator if and only if $T_{1}$
and $T_{2}$ are similar to normal operators.
###### Proof.
Assume that $T$ has the matrix form
$T=\left[\begin{array}[]{cc}T_{1}&T_{12}\\\
&\widetilde{T_{2}}\end{array}\right]\begin{matrix}\mbox{$M$}\\\
\mbox{$M^{\bot}$}\\\ \end{matrix}$
respect to the space decomposition $\mathfrak{H}=M\oplus M^{\bot}$. By [17,
Lemma 2.8], there exists an invertible operator
$S=\left[\begin{array}[]{cc}I&S_{12}\\\
&I\end{array}\right]\begin{matrix}\mbox{$M$}\\\ \mbox{$M^{\bot}$}\\\
\end{matrix}$ such that $S^{-1}TS=\left[\begin{array}[]{cc}T_{1}&\\\
&\widetilde{T_{2}}\end{array}\right]\begin{matrix}\mbox{$M$}\\\
\mbox{$M^{\bot}$}\\\ \end{matrix}$. Assume that $S$ has the matrix form
$S=\begin{array}[]{cc}\begin{array}[]{cc}M&M^{\bot}\end{array}\\\
\left[\begin{array}[]{cc}I&\\\
&S_{1}\end{array}\right]\end{array}\begin{matrix}\\\ \mbox{$M$}\\\
\mbox{$N$}\\\ \end{matrix}$, we obtain that
$T_{2}=S_{1}\widetilde{T_{2}}S_{1}^{-1}$. By [9, propsition 6.5], we get that
$T$ is similar to a normal operator if and only if $T_{1}$ and $T_{2}$ are
similar to normal operators. ∎
###### Proposition 3.5.
Suppose that $T$ is an amenable operator, $M_{1}\in$
Lat$\mathfrak{A}_{T}^{\prime}$ and $M_{2}\in$
Lat$\mathfrak{A}_{T}^{\prime\prime}$. Then $M_{1}+M_{2}$ is closed.
Moreover, if $T|_{M_{1}}$ and $T|_{M_{2}}$ are similar to normal operators,
then $T|_{M_{1}+M_{2}}$ is similar to a normal operator.
###### Proof.
Let $N_{0}=M_{1}\cap M_{2}$, according to Lemma 3.3, there exists $N\in$
Lat$\mathfrak{A}_{T}^{\prime\prime}$ such that $M_{2}=N_{0}\dot{+}N$. Choose
$q\in P(\mathfrak{A}_{T}^{\prime})$, such that $\textup{Ran}q=N$. By the
assumption, $M_{1}\in$ Lat$\mathfrak{A}_{T}^{\prime}$. Hence $qM_{1}\subset
M_{1}\cap N=\\{0\\}$. Therefore $M_{1}\subset(I-q)\mathfrak{H}$. We see that
$M_{1}+M_{2}=M_{1}\dot{+}N$ is closed. This establishes the first statement of
the proposition.
Since $T|_{M_{2}}$ is similar to a normal operator, by proposition 3.4, we get
that $T|_{N}$ is similar to a normal operator. By the assumption $T|_{M_{1}}$
is similar to a normal operator, using proposition 3.4 again, we obtain that
$T|_{M_{1}+M_{2}}=T|_{M_{1}\dot{+}N}$ is similar to a normal operator. ∎
Now we will obtain the main theorem of this section.
###### Theorem 3.6.
Assume $T$ is an amenable operator, then there exists hyperinvariant subspaces
$M_{1},M_{2}$ of $T$ such that $T$ has the form $T=T_{1}\dot{+}T_{2}$ respect
to the space decomposition $\mathfrak{H}=M_{1}\dot{+}M_{2}$ and satisfies
that:
(1) $T_{1},T_{2}$ are amenable operators;
(2) If $M$ is a hyperinvariant subspace of $T$ and $T|_{M}$ is similar to a
normal operator, then $M\subseteq M_{1}$, i.e. $M_{1}$ is the largest
hyperinvariant subspace on which $T$ is similar to a normal operator;
(3) For any $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$,
$T_{2}|_{\textup{Ran}q}$ is not similar to a normal operator;
(4) $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite;
(5)
$\mathfrak{A}_{T}^{\prime}=\mathfrak{A}_{T_{1}}^{\prime}\dot{+}\mathfrak{A}_{T_{2}}^{\prime}$,
$\mathfrak{A}_{T}^{\prime\prime}=\mathfrak{A}_{T_{1}}^{\prime\prime}\dot{+}\mathfrak{A}_{T_{2}}^{\prime\prime}$;
(6) There exists no nonzero compact operator in
$\mathfrak{A}_{T_{2}}^{\prime}$.
###### Proof.
Case1. For any $p\in P(\mathfrak{A}_{T}^{\prime\prime})$, $T|_{\textup{Ran}p}$
is not similar to normal operator. According to the proof of Proposition 3.1,
we obtain that $\mathcal{P}(\mathfrak{A}_{T}^{\prime\prime})$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite. Let $M_{1}=0$.
Case2. There exists $p\in P(\mathfrak{A}_{T}^{\prime\prime})$ such that
$T|_{\textup{Ran}p}$ is similar to normal operator. Then, by Zorn’s Lemma and
the same method in the proof of [4, Corollary 26], we can show that there
exists an element $p_{0}\in P(\mathfrak{A}_{T}^{\prime\prime})$ which is
maximal with respect to the property that $T|_{\textup{Ran}p_{0}}$ is similar
to a normal operator. Using Proposition 3.5, $\textup{Ran}p_{0}$ is the
largest hyperinvariant subspace of $T$ on which $T$ is similar to a normal
operator. Hence, $T$ has the form $T=T_{1}\dot{+}T_{2}$ with respect to the
space decomposition $\mathfrak{H}=\textup{Ran}p_{0}\dot{+}\textup{Ker}p_{0}$
where $T_{1}$ is similar to a normal operator, $T_{1},T_{2}$ are amenable
operators. Let $M_{1}=\textup{Ran}p_{0},M_{2}=\textup{Ker}p_{0}$.
Next we will prove that for any $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$,
$T_{2}|_{\textup{Ran}q}$ is not similar to normal operator. Then according to
Proposition 3.1 $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime\prime})$ is similar
to an abelian von Neumman algebra with uniform multiplicity infinite.
Indeed, if there exists $q\in P(\mathfrak{A}_{T_{2}}^{\prime\prime})$ such
that $T_{2}|_{\textup{Ran}q}$ is similar to a normal operator and $q$ has the
form $q=\left[\begin{array}[]{cc}I&0\\\ 0&0\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}q$}\\\
\mbox{$\textup{Ker}q$}\\\ \end{matrix}$. Then for any
$A\in\mathfrak{A}_{T}^{\prime}$, $A$ has the form
$A=\left[\begin{array}[]{ccc}A_{11}&&\\\ &A_{22}&\\\ &&A_{33}\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}p_{0}$}\\\
\mbox{$\textup{Ran}q$}\\\ \mbox{$\textup{Ker}q$}\\\ \end{matrix}.$
Let
$R=\left[\begin{array}[]{ccc}I&&\\\ &I&\\\ &&0\\\
\end{array}\right]\begin{matrix}\mbox{$\textup{Ran}p_{0}$}\\\
\mbox{$\textup{Ran}q$}\\\ \mbox{$\textup{Ker}q$}\\\ \end{matrix}.$
Then $R\in P(\mathfrak{A}_{T}^{\prime\prime})$. By the assumption
$T|_{\textup{Ran}R}$ is similar to a normal operator which contradicts to the
maximal property of $p_{0}$.
At last we will prove that there exists no nonzero compact operator in
$\mathfrak{A}_{T_{2}}^{\prime}$.
Indeed, if there exists a nonzero compact operator
$k_{0}\in\mathfrak{A}_{T_{2}}^{\prime}$, let $L_{1}$ denote the subspace
spanned by the ranges of all compact operators in
$\mathfrak{A}_{T_{2}}^{\prime}$, and $L_{2}$ the intersection of their kernel,
by [13, Lemma 3.1], both $L_{1},L_{2}$ lie in
Lat$\mathfrak{A}_{T_{2}}^{\prime}$ and
$L_{1}\dot{+}L_{2}=\textup{{Ker}}p_{0}$. Considering the restricting
$T_{2}|_{L_{1}}$, assume $T_{21}=T_{2}|_{L_{1}}$, then $T_{21}$ is an amenable
operator and $\mathfrak{A}_{T_{21}}^{\prime}$ contain a sufficient set of
compact operators. By Lemma 2.2 and [12, Theorem 9], $T_{21}$ is similar to a
normal operator which contradicts to the above discussion. ∎
Trivial modifications adapt the proof of Theorem 3.6, we obtain the following
theorem which decomposes amenable operators by the invariant subspaces of
them. The proof is similar to Theorem 3.6 and we omit it.
###### Theorem 3.7.
Assume $T$ is an amenable operator, then there exists invariant subspaces
$N_{1},N_{2}$ of $T$ such that $T$ has the form $T=A_{1}\dot{+}A_{2}$ respect
to the space decomposition $\mathfrak{H}=N_{1}\dot{+}N_{2}$ and satisfies
that:
(1) $A_{1},A_{2}$ are amenable operators;
(2) If $N$ is an invariant subspace of $T$ such that $N_{1}\subseteq N$ and
$T|_{N}$ is similar to a normal operator, then $N=N_{1}$, i.e. $N_{1}$ is the
maximal invariant subspace on which $T$ is similar to a normal operator;
(3) For any $q\in P(\mathfrak{A}_{T_{2}}^{\prime})$, $T_{2}|_{\textup{Ran}q}$
is not similar to a normal operator;
(4) If $\mathcal{P}(\mathfrak{A}_{T_{2}}^{\prime})$ contains a subalgebra
which is similar to an abelian von Neumman algebra then the von Neumman
algebra has the uniform multiplicity infinite.
###### Remark 3.8.
If the answer to Conjecture 1.1 is positive, by Theorem 2.6, every amenable is
similar to a normal operator. Then, for the above theorem
$M_{1}=N_{1}=\mathfrak{H}$. That is to say, the two decompositions of theorem
3.6 and 3.7 are the same. The remainder of this section, we will prove that
the two decompositions are the same which supporting Conjecture 1.1.
###### Lemma 3.9.
[4] If $T\in\mathfrak{B}(\mathfrak{H})$ is an amenable operator and there
exist a one-to-one bounded linear map
$W:\mathfrak{H}\rightarrow\mathfrak{H}_{2}$, a bounded linear map
$V:\mathfrak{H}_{1}\rightarrow\mathfrak{H}$ with dense range and operators
$S_{1}\in\mathfrak{B}(\mathfrak{H}_{1})$,
$S_{2}\in\mathfrak{B}(\mathfrak{H}_{2})$ which are similar to normal operators
such that $TV=VS_{1}$ and $WT=S_{2}W$, then $T$ is similar to a normal
operator.
###### Corollary 3.10.
Assume $T=B_{1}B_{2}$ is an amenable operator, where $B_{1},B_{2}$ are
positive operators, then $T$ is similar to a normal operator.
###### Proof.
Assume $B_{1},B_{2}$ have the forms
$B_{2}=\left[\begin{array}[]{cc}0&\\\ &\widetilde{B_{2}}\\\
\end{array}\right],B_{1}=\left[\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{12}^{*}&B_{22}\\\ \end{array}\right],$
respect to the space decomposition
$\mathfrak{H}=\textup{Ker}B_{2}\oplus(\textup{Ker}B_{2})^{\bot}$ where
$\widetilde{B_{2}}$ is one-to-one and $B_{11},B_{22}$ are positive operators.
Thus $T$ has the form $T=\left[\begin{array}[]{cc}0&B_{12}\widetilde{B_{2}}\\\
0&B_{22}\widetilde{B_{2}}\\\ \end{array}\right]$ respect to the space
decomposition. Since $T$ is an amenable operator, by [17, Lemma 2.8], $T$ is
similar to $\left[\begin{array}[]{cc}0&0\\\ 0&B_{22}\widetilde{B_{2}}\\\
\end{array}\right]$. Thus without loss of generality, we may assume that
$B_{2}$ is one-to-one.
Assume that $B_{1},B_{2}$ has the form
$B_{1}=\left[\begin{array}[]{cc}\widetilde{B_{1}}&\\\ &0\\\
\end{array}\right],B_{2}=\left[\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{12}^{*}&B_{22}\\\ \end{array}\right],$
respect to the space decomposition
$\mathfrak{H}=(\textup{Ker}B_{1})^{\bot}\oplus\textup{Ker}B_{1}$ where
$\widetilde{B_{1}}$ is one-to-one and has dense range and $B_{11},B_{22}$ are
positive operators. Thus $T$ has the form
$T=\left[\begin{array}[]{cc}\widetilde{B_{1}}B_{11}&\widetilde{B_{1}}B_{12}\\\
0&0\\\ \end{array}\right]$ respect to the space decomposition. Since $T$ is an
amenable operator, by [17, Lemma 2.8], $T$ is similar to
$\left[\begin{array}[]{cc}\widetilde{B_{1}}B_{11}&0\\\ 0&0\\\
\end{array}\right]$ and there exists an operator $S$ such that
$\widetilde{B_{1}}B_{12}=\widetilde{B_{1}}B_{11}S$. Note that
$\widetilde{B_{1}},B_{2}$ are one-to-one, hence $B_{12}=B_{11}S$, and $B_{11}$
is one-to-one. Thus without loss of generality, we may assume that $B_{1}$ has
dense range and $B_{2}$ is one-to-one.
Note that
$B_{1}^{\frac{1}{2}}B_{2}B_{1}^{\frac{1}{2}},B_{2}^{\frac{1}{2}}B_{1}B_{2}^{\frac{1}{2}}$
are positive operators and
$TB_{1}^{\frac{1}{2}}=B_{1}^{\frac{1}{2}}B_{1}^{\frac{1}{2}}B_{2}B_{1}^{\frac{1}{2}}$
and
$B_{2}^{\frac{1}{2}}T=B_{2}^{\frac{1}{2}}B_{1}B_{2}^{\frac{1}{2}}B_{2}^{\frac{1}{2}}$,
by Lemma 3.9, $T$ is similar to a normal operator. ∎
###### Theorem 3.11.
The two decompositions for an amenable operator in Theorem 3.6, 3.7 are the
same.
###### Proof.
According to Theorem 3.6, 3.7, and Proposition 3.5, it is suffices to proof
that $N_{1}\in Lat\mathfrak{A}_{T}^{\prime}$.
In fact, if not. $T$ has the form $T=\left[\begin{array}[]{cc}T_{1}&\\\
&T_{2}\\\ \end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}$ and there exists $S=\left[\begin{array}[]{cc}0&0\\\ Y&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}\in\mathfrak{A}_{T}^{\prime}$ where $Y\neq 0$. Note that $S$ and
$T$ have the form
$S=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\ 0&0&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\
\mbox{$\overline{\textup{Ran}Y}$}\\\
\mbox{$N_{2}\ominus\overline{\textup{Ran}Y}$}\\\
\end{matrix},T=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&T_{22}\\\
&T_{23}&T_{24}\\\ \end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\
\mbox{$\overline{\textup{Ran}Y}$}\\\
\mbox{$N_{2}\ominus\overline{\textup{Ran}Y}$}\\\ \end{matrix},$
where $\tilde{Y}$ has dense range. Note that $TS=ST$, we get that $T_{23}=0$.
Since $T$ is amenable, by [17, Lemma 2.8] there exists an operator
$B:N_{2}\ominus\overline{\textup{Ran}Y}\rightarrow\overline{\textup{Ran}Y}$
such that
$\left[\begin{array}[]{ccc}I&0&0\\\ &I&B\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&T_{22}\\\
&&T_{24}\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-B\\\
&&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{21}&0\\\
&&T_{24}\\\ \end{array}\right].$
Moreover,
$\left[\begin{array}[]{ccc}I&0&0\\\ &I&B\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\ 0&0&0\\\
\end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-B\\\ &&I\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y}&0&0\\\
0&0&0\\\ \end{array}\right].$
Hence, we can assume that $Y$ has dense range. Using $T$ is amenable again,
there exists $L=\left[\begin{array}[]{cc}0&X\\\ 0&0\\\
\end{array}\right]\begin{matrix}\mbox{$N_{1}$}\\\ \mbox{$N_{2}$}\\\
\end{matrix}\in\mathfrak{A}_{T}^{\prime}$, where $X\neq 0$, by [7, lemma
4.11]. Similar to the decomposition to $S$ and $T$, we get that $S$, $L$ and
$T$ have the form
$S=\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y_{1}}&0&0\\\
\tilde{Y_{2}}&0&0\\\
\end{array}\right],L=\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right],T=\left[\begin{array}[]{ccc}T_{1}&0&0\\\
&T_{31}&T_{32}\\\ &T_{33}&T_{34}\\\ \end{array}\right],$
respect to the space decomposition
$\mathfrak{H}=N_{1}\oplus\textup{Ker}X\oplus(N_{2}\ominus\textup{Ker}X)$,
where $\tilde{X}$ is one-to-one, and $\tilde{Y_{1}},\tilde{Y_{2}}$ has dense
range. Note that $LT=TL$, we get that $T_{33}=0$. Using $T$ is amenable again,
there exists an operator $C:N_{2}\ominus\textup{Ker}X\rightarrow\textup{Ker}X$
such that
$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{31}&T_{32}\\\
&&T_{34}\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-C\\\
&&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}T_{1}&0&0\\\ &T_{31}&0\\\
&&T_{34}\\\ \end{array}\right]$
$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ &I&-C\\\ &&I\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0&0&\tilde{X}\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right]$
and
$\left[\begin{array}[]{ccc}I&0&0\\\ &I&C\\\ &&I\\\
\end{array}\right]\left[\begin{array}[]{ccc}0&0&0\\\ \tilde{Y_{1}}&0&0\\\
\tilde{Y_{2}}&0&0\\\ \end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\
&I&-C\\\ &&I\\\ \end{array}\right]=\left[\begin{array}[]{ccc}0&0&0\\\
\tilde{Y_{1}}+C\tilde{Y_{2}}&0&0\\\ \tilde{Y_{2}}&0&0\\\ \end{array}\right].$
Moreover,
$\tilde{Y_{2}}T_{1}=T_{34}\tilde{Y_{2}},T_{1}\tilde{X}=\tilde{X}T_{34}$, and
$T_{1}$ is similar to a normal operator, by Lemma 3.9, $T_{34}$ is similar to
a normal operator, which contracts to Theorem 3.7. ∎
###### Corollary 3.12.
Assume $T$ is an amenable operator, then $M$ is a maximal invariant subspace
such that $T|_{M}$ is similar to a normal operator if and only if $M$ is the
largest invariant subspace such that $T|_{M}$ is similar to a normal operator.
###### Corollary 3.13.
Assume $T$ is an amenable operator and which is quasisimilar to a compact
operator, then $T$ is similar to a normal operator.
###### Proof.
Suppose, $TV=VK,WT=KW$ with $V,W$ injective operators having dense ranges and
$K$ is a compact operator. Then $TVKW=VKWT$. Let $C=VKW$,
$C\in\mathfrak{A}_{T}^{\prime}$, and $C$ is a compact operator. According to
Theorem 3.6, $C$ has the form $\left[\begin{array}[]{cc}C_{1}&\\\ &0\\\
\end{array}\right]$ respect to the space decomposition in the Theorem. If
$Cx=0$, $VWTx=Cx=0$, thus $Tx=0$. It follows that there is no part of $T_{2}$,
i.e. $T$ is similar to a normal operator. ∎
## 4\. (Essential) operator valued roots of abelian analytic functions
In this section, we will study the structure of an operator which is an
(essential) operator valued roots of abelian analytic functions and then we
get that if such an operator is also amenable, then it is similar to a normal
operator. In [8] Gilfeather introduce the concept of operator valued roots of
abelian analytic functions as follows: Let $\mathfrak{A}$ is an abelian von
Neumann algebra and $\psi(z)$, an $\mathfrak{A}$ valued analytic function on a
domain $\mathcal{D}$ in the complex plane. We may decompose $\mathfrak{A}$
into a direct integral of factors such that for $A\in\mathfrak{A}$, there
exists a unique $g\in L_{\infty}(\wedge,\mu)$ such that
$\mathfrak{A}=\int_{\wedge}^{\oplus}g(\lambda)I(\lambda)\mu(d\lambda)$. If
$T\in\mathfrak{A}^{\prime}$ and $\sigma(T)\subseteq\mathcal{D}$, let
$\psi(T)=(2\pi i)^{-1}\int_{\wedge}(T-zI)^{-1}\psi(z)dz.$
An operator $T$ is called a (essential)roots of the abelian analytic function
$\psi$, if $\psi(T)=0$(compact, respectively). The structure of roots of a
locally nonzero abelian analytic function has been given in [8], in this
section we main study the structure of essential roots of a locally nonzero
abelian analytic function.
###### Lemma 4.1.
Assume $T\in\mathfrak{B}(\mathfrak{H})$, $f$ is a locally nonzero analytic
function on the neighborhood of $\sigma(T)$ and assume $f(T)$ is a compact
operator, then $T$ is a polynomial compact operator.
###### Proof.
Let $\widehat{T}$ denote the image of $T$ in the Calkin algebra, then
$\widehat{f(T)}=0$. Since $f$ is a locally nonzero analytic function on
$\sigma(T)$, there exists a polynomial $p$ such that $\widehat{p(T)}=0$. i.e.
$T$ is a polynomial compact operator. ∎
###### Theorem 4.2.
Let $\psi$ be a locally nonzero abelian analytic function on $\mathcal{D}$
taking values in the von Neumann algebra $\mathfrak{A}$. If $T$ is an
essential roots of $\psi$ and is amenable, then $T$ is similar to a normal
operator.
###### Proof.
Since $\mathfrak{A}$ is an abelian von Neumann algebra, $\mathfrak{A}$ is
unitarily equivalent to
$\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n}\oplus\mathfrak{B}_{\infty}$, where
$\mathfrak{B}_{n}$ is a von Neumman algebra which has uniform multiplicity $n$
for all $1\leq n\leq\infty$. Note $T\in\mathfrak{A}^{\prime}$ is an amenable
operator, thus $T=T_{1}\oplus T_{2}$, where $T_{1}$ is similar to a normal
operator, and
$T_{1}\in(\sum_{n=1}^{\infty}\oplus\mathfrak{B}_{n})^{\prime},T_{2}\in\mathfrak{B}_{\infty}^{\prime}$.
Let $\sigma_{1}(\sigma_{2})$ denote the continuous (atom, respectively) parts
of the spectrum of $\mathfrak{B}_{\infty}$, then
$\mathfrak{B}_{\infty}=\mathfrak{C}_{\infty}\oplus\mathfrak{D}_{\infty}$,
where $\mathfrak{C}_{\infty}$ and $\mathfrak{D}_{\infty}$ are uniform
multiplicity $\infty$ von Neumman algebra and
$\sigma(\mathfrak{C}_{\infty})=\sigma_{1},\sigma(\mathfrak{D}_{\infty})=\sigma_{2}$
and $T_{2}=T_{3}\oplus T_{4}$, where
$T_{3}\in\sigma(\mathfrak{C}_{\infty})^{\prime},T_{4}\in\sigma(\mathfrak{D}_{\infty})^{\prime}$.
Assume $\psi$ is a locally nonzero abelian analytic function on $\mathcal{D}$
and $\sigma(T)\subseteq\mathcal{D}$, then
$\psi(T)=\psi(T_{1})\oplus\psi(T_{3})\oplus\psi(T_{4})$, note that
$\psi(T_{3})$ is a compact operator and
$\sigma(\mathfrak{C}_{\infty})=\sigma_{1}$, so $\psi(T_{3})=0$. Since
$\mathfrak{D}_{\infty}$ are uniform multiplicity $\infty$ and
$\sigma(\mathfrak{D}_{\infty})=\sigma_{2}$, by Lemma 4.1, it follows that
$T_{4}$ is direct sum of polynomial compact operators. According to [8,
Theorem 2.1], there exists a sequence of mutually orthogonal projections
$\\{P_{n},Q_{m}\\}$ in $\mathfrak{A}$ with $I=\sum P_{n}+\sum Q_{m}$ so that
$T|_{P_{n}}$ is finite type spectral operator and $T|_{Q_{m}}$ is polynomial
compact operator. By [17, Theorem 3.5, 4.5], we get that $T$ is similar to a
normal operator. ∎
## References
* [1] E. A. Azoff; C. K. Fong and F. Gilfeather, A reduction theory for non-self-adjoint operator algebras. Trans. Amer. Math. Soc. 224 (1976) 351–366 (1977).
* [2] K. R. Davidson, Nest algebras, Longman group UK limited, Essex, 1988\.
* [3] K. R. Davidson, (3-WTRL) $C^{*}$-algebras by example. (English summary) Fields Institute Monographs, 6. American Mathematical Society, Providence, RI, 1996.
* [4] C. K. Fong, Operator algebras with complemented invariant subspace lattices. Indiana Univ. Math. J. 26 (1977) 1045–1056.
* [5] D. R. Farenick, B.E. Forrest and L. W. Marcoux, Amenable operators on Hilbert spaces, J. reine angew. Math. 582(2005) 201-228.
* [6] D. R. Farenick, B.E. Forrest and L. W. Marcoux, Amenable operators on Hilbert spaces, J. reine angew. Math. 602(2007) 235.
* [7] J. A. Gifford, Operator algebras with a reduction proprety, J. Aust. Math. Soc 80 (2006) 279–315.
* [8] F. Gilfeather, Operator valued roots of abelian analytic functions, Pacific J. Math. 55 (1974), 127–148.
* [9] D. W. Hadwin, An asymptotic double commutant theorem for $C^{\ast}$-algebras. Trans. Amer. Math. Soc. 244 (1978) 273–297.
* [10] B. E. Johnson. Cohomology in Banach Algebras. Mem. Amer. Math. Soc. Vol. 127 (Amer. Math. Soc., 1972).
* [11] R. G. Douglas, On operators similar to normal operators, Rev. Roum. Math. Pures Appl. 14(1969) 193-197.
* [12] S. Rosenoer, Completely reducible operator algebras and spectral synthesis, Canad. J. Math. 34 (1982), no. 5, 1025–1035.
* [13] S. Rosenoer, Completely reducible algebras containing compact operators, J. Operator Theory. 29 (1993), no. 2, 269–285.
* [14] H. Radjavi and P. Rosenthal, Invariant subspaces. Second edition. Dover Publications, Inc., Mineola, NY, 2003.
* [15] J. T. Schwartz, W*-algebras, Gordon and Breach, New York, 1967.
* [16] G. A. Willis, When the algebra generated by an operator is amenable, J. Operator Theorey. 34 (1995) 239–249.
* [17] Y. Q. Ji and L. Y. Shi, Amenable operators on Hilbert spaces, Houston Journal of Mathematics(to appear).
|
arxiv-papers
| 2010-08-31T07:43:02 |
2024-09-04T02:49:12.562413
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luo Yi Shi, Yu Jing Wu and You Qing Ji",
"submitter": "Luoyi Shi",
"url": "https://arxiv.org/abs/1008.5238"
}
|
1008.5241
|
# An example of weakly amenable and character amenable operator
Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn , YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and You Qing Ji Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA jiyq@jlu.edu.cn
(Date: June 15 2010)
# An example of weakly amenable and character amenable operator
Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn , YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and You Qing Ji Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA jiyq@jlu.edu.cn
(Date: June 15 2010)
###### Abstract.
A complete characterization of Hilbert space operators that generate weakly
amenable algebras remains open, even in the case of compact operator.
Farenick, Forrest and Marcoux proposed the question that if $T$ is a compact
weakly amenable operator on a Hilbert space $\mathfrak{H}$, then is $T$
similar to a normal operator? In this paper we demonstrate an example of
compact triangular operator on infinite-dimension Hilbert space which is a
weakly amenable and character amenable operator but is not similar to a normal
operator.
###### Key words and phrases:
Amenable; weakly amenable
###### 2000 Mathematics Subject Classification:
47C05 (46H35 47A65 47A66 47B15)
Supported by NCET(040296), NNSF of China(10971079) and the Specialized
Research Fund for the Doctoral Program of Higher Education(20050183002)
## 1\. Introduction
Let $\mathfrak{A}$ be a Banach algebra, and let $X$ be a Banach
$\mathfrak{A}$-bimodule. A derivation $D:\mathfrak{A}\rightarrow X$ is a
continuous linear map such that $D(ab)=a\cdot D(b)+D(a)\cdot b$, for all
$a,b\in\mathfrak{A}$. A derivation $D:\mathfrak{A}\rightarrow X$ is said to be
inner if there exists $x\in X$ such that $D(a)=a\cdot x-x\cdot a$ for all
$a\in\mathfrak{A}$. A Banach $\mathfrak{A}$-bimodule $X$ is said to be
commutative if $a\cdot x=x\cdot a$ for each $a\in\mathfrak{A},x\in X$. For any
Banach $\mathfrak{A}$-bimodule $X$, its dual $X^{*}$ is naturally equipped
with a Banach $\mathfrak{A}$-bimodule structure via $[a\cdot f](x)=f(x\cdot
a),[f\cdot a](x)=f(a\cdot x),a\in\mathfrak{A},x\in X,f\in X^{*}.$
We can now give the definition of amenability, weak amenability and character
amenability for Banach algebra:
###### Definition 1.1.
A Banach algebra $\mathfrak{A}$ is amenable if , for each Banach
$\mathfrak{A}$-bimodules $X$, every derivation $D:\mathfrak{A}\rightarrow
X^{*}$ is inner.
###### Definition 1.2.
A commutative Banach algebra $\mathfrak{A}$ is weakly amenable if , for each
commutative Banach $\mathfrak{A}$-bimodules $X$, every derivation
$D:\mathfrak{A}\rightarrow X$ is inner.
Let $\mathfrak{A}$ be a Banach algebra and $\sigma(\mathfrak{A})$ the spectrum
of $\mathfrak{A}$, that is, the set of all non-zero multiplicative linear
functionals on $\mathfrak{A}$. If $\varphi\in\sigma(\mathfrak{A})\cup\\{0\\}$
and if $X$ is a Banach space, then $X$ can be viewed as left or right Banach
$\mathfrak{A}$-module by the following actions. For $a\in\mathfrak{A},x\in X$:
$None$ $a\cdot x=\varphi(a)x,$ $None$ $x\cdot a=\varphi(a)x.$
If the left action of $\mathfrak{A}$ on $X$ is given by (2.1), then it is
easily verified that the right action of $\mathfrak{A}$ on the dual
$\mathfrak{A}$-module $X^{*}$ is given by $f\cdot a=\varphi(a)f,$ for all
$f\in X^{*},a\in\mathfrak{A}$. Throughout, by
$(\varphi,\mathfrak{A})$-bimodule $X$, we mean that $X$ is a Banach
$\mathfrak{A}$-bimodule for which the left module action is given by (2.1).
$(\mathfrak{A},\varphi)$-bimodule is defined similarly by (2.2). Let
$\varphi\in\sigma(\mathfrak{A})\cup\\{0\\}$, a Banach algebra $\mathfrak{A}$
is said to be left $\varphi$ amenable, if every derivation $D$ from
$\mathfrak{A}$ into the dual $A$-bimodule $X^{*}$ is inner for all
$(\varphi,A)$-bimodules $X$; $\mathfrak{A}$ is said to be right $\varphi$
amenable, if every derivation $D$ from $\mathfrak{A}$ into the dual
$A$-bimodule $X^{*}$ is inner for all $(A,\varphi)$-bimodules $X$.
$\mathfrak{A}$ is said to be left character amenable, if it is left $\varphi$
amenable for all $\varphi\in\sigma(\mathfrak{A})\cup\\{0\\}$; $\mathfrak{A}$
is said to be right character amenable, if it is right $\varphi$ amenable for
all $\varphi\in\sigma(\mathfrak{A})\cup\\{0\\}$.
###### Definition 1.3.
A Banach algebra $\mathfrak{A}$ is said to be character amenable, if it is
both left character amenable and right character amenable.
The concept of amenable Banach algebras was first introduced by B. E. Johnson
in [7]. Weak amenability was first defined by Bade, Curtis and Dales in [1,
3]. Character amenability was first defined by Sangani-Monfared [8]. Ever
since its introduction, the concepts of amenability, weak amenability and
character amenability have played an important role in research in Banach
algebras, operator algebras and harmonic analysis. We only would like to
mention the following deep results due to Willis [9] and Farenick, Forrest and
Marcoux [4], [5]:
Given a complex separable infinite-dimensional Hilbert space $\mathfrak{H}$,
we write $\mathfrak{B}(\mathfrak{H})$ for the bounded linear operators on
$\mathfrak{H}$. If $T\in\mathfrak{B}(\mathfrak{H})$, denote the norm-closure
of span$\\{T^{k}:k\in\mathbb{N}\\}$ by $\mathfrak{A}_{T}$, where $\mathbb{N}$
is the set of natural numbers. $T$ is said to be amenable (weakly amenable or
character amenable) if $\mathfrak{A}_{T}$ is amenable (respectively, weakly
amenable, character amenable).
In [9], Willis showed that:
###### Theorem 1.4.
Suppose $T$ is a compact amenable operator, then $T$ is similar to a normal
operator.
In [4], [5] Farenick, Forrest and Marcoux showed that:
###### Theorem 1.5.
Suppose $T$ is a triangular operator with respect to an orthonormal basis of
$\mathfrak{H}$, then $T$ is amenable if and only if $T$ is similar to a normal
operator whose spectrum has connected complement and empty interior.
A complete characterisation of Hilbert space operators that generate weakly
amenable algebras remains open, even in the case of compact operator. In [4],
[5] Farenick, Forrest and Marcoux proposed the following question:
###### Question 1.6.
If $T$ is a compact weakly amenable operator on $\mathfrak{H}$, then is $T$
similar to a normal operator?
It is well known that if $\mathfrak{H}$ is a finite-dimensional Hilbert space,
then $T\in\mathfrak{B}(\mathfrak{H})$ is amenable, weakly amenable or
character amenable if and only if $T$ is similar to a normal operator. The
purpose of this paper is to demonstrate an example of compact triangular
operator on an infinite-dimensional Hilbert space which is weakly amenable and
character amenable but is not similar to a normal operator.
## 2\. Compact triangular weakly amenable operator
Suppose $\sigma$ is a compact Hausdorff space, Let $C(\sigma)$ denote the
Banach algebra of all continuous functions on $\sigma$ with the supremum norm
$||f||_{\infty}=\sup_{x\in\sigma}|f(x)|$. Throughout this paper we let
$\sigma=\\{0,\lambda_{1},\lambda_{2},\cdots\\}$, where
$\\{\lambda_{n}\\}_{n=1}^{\infty}$ is a sequence of positive real numbers
which converge to zero. Let
$T=\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\ \end{array}\right),$
where $N$ is a normal operator with spectrum $\sigma$, then $T$ is an operator
on an infinite-dimensional Hilbert space. In this section, we obtain that $T$
is weakly amenable, but is not similar to a normal operator. Especially, if
let
$N=\left(\begin{array}[]{cccc}\lambda_{1}&&&\\\ &\lambda_{2}&&\\\
&&\lambda_{3}&\\\ &&&\ddots\\\ \end{array}\right),$
then $T$ is a compact triangular operator.
The following lemma is easily verified:
###### Lemma 2.1.
Suppose $\mathfrak{A}$ is a commutative Banach algebra which is generated by
the idempotent elements in $\mathfrak{A}$, then $\mathfrak{A}$ is weak
amenable.
###### Proof.
Let $\mathcal{P}$ denote the sets of the idempotent elements in
$\mathfrak{A}$. Assume $X$ commutative Banach $\mathfrak{A}$-bimodules, and
$D:\mathfrak{A}\rightarrow X$ is a derivation. For any $p\in\mathcal{P}$,
$D(p)=D(p^{2})=D(p^{3})$ and $D(p^{2})=2pD(p),D(p^{3})=3p^{2}D(p)$, so
$D(p)=0$. Since $p\in\mathcal{P}$ is arbitrary and $\mathfrak{A}$ is generated
by $\mathcal{P}$, it follows that $D(a)=0$ for all $a\in\mathfrak{A}$. That is
to say, $\mathfrak{A}$ is weak amenable. ∎
Our main result in this section will be that for any normal operator $N$ with
spectrum $\sigma$, $T=\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\
\end{array}\right)$ is weakly amenable but is not similar to a normal
operator.
###### Theorem 2.2.
Let $T=\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\
\end{array}\right),$ where $N$ is a normal operator with spectrum $\sigma$,
then $T$ is weakly amenable.
###### Proof.
By Lemma 2.1, it suffices to show that $\mathfrak{A}_{T}$ is generated by the
idempotent elements in it.
Step 1. $\mathfrak{A}_{T}=\\{\left(\begin{array}[]{cc}0&f(N)\\\
0&N^{\frac{1}{2}}f(N)\\\ \end{array}\right);f\in C(\sigma),f(0)=0\\}\triangleq
M$, where $f(N)$ denotes the functional calculus for $N$ respective to $f$.
Indeed, for any polynomial
$p(z)=\Sigma_{k=1}^{n}a_{k}z^{k}=z\Sigma_{k=0}^{n-1}a_{k+1}z^{k}\triangleq
zq(z)$, $p(T)$ has the form
$\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}q(N)\\\ 0&p(N)\\\
\end{array}\right).$
For any $A=\left(\begin{array}[]{cc}0&A_{12}\\\ 0&A_{22}\\\
\end{array}\right)\in\mathfrak{A}_{T}$, there exists a sequence of polynomials
$\\{p_{n}\\},p_{n}(0)=0$ for all $n$ such that $||p_{n}(T)-A||\rightarrow 0$.
i.e. $||p_{n}(N)-A_{22}||\rightarrow 0$ and
$||N^{\frac{1}{2}}q_{n}(N)-A_{12}||\rightarrow 0$. Therefore, there exists a
function $g$ on $\sigma$, such that
$||z^{\frac{1}{2}}q_{n}-g||_{\infty}\rightarrow 0$ and
$||p_{n}-z^{\frac{1}{2}}g||_{\infty}\rightarrow 0$. It follows that
$A=\left(\begin{array}[]{cc}0&g(N)\\\ 0&N^{\frac{1}{2}}g(N)\\\
\end{array}\right),$ and $g\in C(\sigma),g(0)=0$. That is to say,
$\mathfrak{A}_{T}\subseteq M$.
For any $f\in C(\sigma),f(0)=0$, there exists a sequence of polynomials
$\\{p_{n},p_{n}(0)=0\\}$ such that $||p_{n}-f||_{\infty}\rightarrow 0$. Let
$p_{n}=zq_{n}$, for any $n$ there exists a polynomial $r_{n},r_{n}(0)=0$ such
that $||r_{n}-z^{\frac{1}{2}}q_{n}||_{\infty}<\frac{1}{n}$. Therefore,
$||z^{\frac{1}{2}}r_{n}-f||_{\infty}\leq||z^{\frac{1}{2}}(r_{n}-z^{\frac{1}{2}}q_{n})||_{\infty}+||p_{n}-f||_{\infty}\rightarrow
0$, and $||zr_{n}-z^{\frac{1}{2}}f||_{\infty}\rightarrow 0$. It follows that
$Tr_{n}(T)\rightarrow\left(\begin{array}[]{cc}0&f(N)\\\
0&N^{\frac{1}{2}}f(N)\\\ \end{array}\right)$. That is to say,
$M\subseteq\mathfrak{A}_{T}$.
Step 2. $\mathfrak{A}_{T}$ is generated by the idempotent elements in it.
It is verity that for any $\lambda_{n}$ let
$h_{n}(z)=\begin{cases}\frac{1}{\sqrt{\lambda_{n}}}&z=\lambda_{n};\\\
0,&z\neq\lambda_{n},\end{cases}$
then $\left(\begin{array}[]{cc}0&h_{n}(N)\\\ 0&N^{\frac{1}{2}}h_{n}(N)\\\
\end{array}\right)$ is an idempotent element in $\mathfrak{A}_{T}$, and
$\mathfrak{A}_{T}$ is generated by idempotent elements
$\\{\left(\begin{array}[]{cc}0&h_{n}(N)\\\ 0&N^{\frac{1}{2}}h_{n}(N)\\\
\end{array}\right)\\}_{n=1}^{\infty}$. The proof is completed. ∎
Finally, we will obtain that $T$ is not similar to a normal operator. Indeed,
Suppose $T$ is similar to a normal operator, by the proof of [6] Theorem 2.1
and [4] Theorem 2.7, $T$ is amenable and there exists an bounded operator $B$
such that
$\left(\begin{array}[]{cc}I&B\\\ 0&I\\\
\end{array}\right)\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\
\end{array}\right)\left(\begin{array}[]{cc}I&-B\\\ 0&I\\\
\end{array}\right)=\left(\begin{array}[]{cc}0&0\\\ 0&N\\\ \end{array}\right).$
Therefore, there exists an bounded operator $B$ such that
$BN=N^{\frac{1}{2}}$, which is impossible. Hence, $T$ is not similar to a
normal operator.
###### Remark 2.3.
Theorem 2.2 shows that there exists a compact triangular operator with
infinite spectrum which is a weakly amenable operator but is not similar to a
normal operator. However, we do not know if a compact quasinilpotent operator
can be weakly amenable. It would be very interesting to know whether this
result is true. Indeed, if any compact quasinilpotent operator can not be
weakly amenable, then it is easy to get that a compact operator $T$ with
finite spectrum is weakly amenable if and only if $T$ is similar to a normal
operator.
## 3\. Compact triangular character amenable operator
Let $Q$ be the Volterra operator on infinite-dimension Hilbert space, then by
[8] Corollary 2.7 and [2] Corollary 5.11, $Q$ is a compact quasinilpotent
operator which is character amenable. However, the lattice of invariant
subspaces of $Q$ is a continuous nest. i.e. $Q$ is not a triangular operator.
In this section, we will prove that
$T=\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\ \end{array}\right)$ is
a character amenable operator, for any normal operator $N$ with spectrum
$\sigma$. Hence, there exists a compact triangular operator which is a
character amenable operator but is not similar to a normal operator.
In [8], Sangani-Monfared obtained a necessary and sufficient condition for a
Banach algebra to be character amenable:
###### Lemma 3.1.
A Banach algebra $\mathfrak{A}$ is character amenable if and only if
$\ker\phi$ has a bounded approximate identity for every
$\phi\in\sigma(\mathfrak{A})\cup\\{0\\}$.
Using Lemma 3.1, we will prove $T$ is a character amenable operator:
###### Theorem 3.2.
Let $T=\left(\begin{array}[]{cc}0&N^{\frac{1}{2}}\\\ 0&N\\\
\end{array}\right),$ then $T$ is character amenable.
###### Proof.
By Lemma 3.1, it suffices to show that $\mathfrak{A}_{T}$ and
$\mathfrak{A}_{\lambda_{n}I-T}$ have a bounded approximate identity for all
$n$.
It is verity that $\mathfrak{A}_{T}$ has a bounded approximate identity, hence
we only to need prove that $\mathfrak{A}_{\lambda_{n}I-T}$ has a bounded
approximate identity for any $n$.
For some fix $n$, assume that $f_{n}$ is a smooth function defined on
$[\lambda_{n}-\lambda_{1},\lambda_{n}]$ and satisfies that
$f_{n}(z)=\begin{cases}0&z=0;\\\
1,&z\in[\lambda_{n}-\lambda_{1},\lambda_{n}-\lambda_{n-1}]\cup[\lambda_{n}-\lambda_{n+1},\lambda_{n}].\end{cases}$
There exists a sequence of polynomials $\\{p_{k},p_{k}(0)=0\\}$ such that
$p_{k}$ and $p_{k}^{\prime}$ (the derivative of $p_{k}$ ) converge to $f_{n}$
and $f_{n}^{\prime}$ uniformly on $[\lambda_{n}-\lambda_{1},\lambda_{n}]$,
respectively . Since $\lambda_{n}I-N$ is a normal operator with spectrum
$\sigma(\lambda_{n}I-N)=\\{\lambda_{n},\lambda_{n}-\lambda_{1},\lambda_{n}-\lambda_{2},\cdots\\}$,
it follows that $f_{n}(\lambda_{n}I-N)$ is an identity of
$\mathfrak{A}_{\lambda_{n}I-N}$ and $f_{n}(\lambda_{n}I)=\lambda_{n}I$. Hence
$||(\lambda_{n}I-N)p_{k}(\lambda_{n}I-N)-(\lambda_{n}I-N)||\rightarrow 0$ and
$||p_{k}(\lambda_{n}I)-I||\rightarrow 0$, when $k\longrightarrow\infty$.
Note that if $T$ has the form
$T=\left(\begin{array}[]{cc}N_{1}&N_{2}\\\ 0&N_{3}\\\ \end{array}\right),$
with $\\{N_{i}\\}$ a collection of commuting operators, then
$T^{k}=\left(\begin{array}[]{cc}N_{1}^{k}&A_{k}N_{2}\\\ 0&N_{3}^{k}\\\
\end{array}\right),$
where $N_{1}^{k}-N_{3}^{k}=(N_{1}-N_{3})A_{k}$ for all $k\in\mathbb{N}$.
It is easy to check that
$(\lambda_{n}I-T)p_{k}(\lambda_{n}I-T)=\left(\begin{array}[]{cc}\lambda_{n}p_{k}(\lambda_{n}I)&-q_{k}(N)N^{\frac{1}{2}}\\\
0&(\lambda_{n}I-N)p_{k}(\lambda_{n}I-N)\\\ \end{array}\right),$
where $\\{q_{k}\\}$ is a sequence of polynomials which satisfy the equation
$\lambda_{n}p_{k}(\lambda_{n})-(\lambda_{n}-z)p_{k}(\lambda_{n}-z)=zq_{k}(z)$.
Note that
$q_{k}(z)-p_{k}(\lambda_{n}-z)=\frac{\lambda_{n}p_{k}(\lambda_{n})-\lambda_{n}p_{k}(\lambda_{n}-z)}{z}=\lambda_{n}p_{k}^{\prime}(\xi_{k,n,z})$
for some $\xi_{k,n,z}\in[\lambda_{n}-\lambda_{1},\lambda_{n}]$ and for all
$z\in\sigma$. Hence $\\{q_{k}\\}$ is bounded on $\sigma$.
Since
$||Nq_{k}(N)-N||=||\lambda_{n}p_{k}(\lambda_{n}I)-(\lambda_{n}I-N)p_{k}(\lambda_{n}I-N)-N||\rightarrow
0$, it follows that $\\{q_{k}(N)\\}$ is a bounded approximate identity for
$\mathfrak{A}_{N}$. Hence
$||N^{\frac{1}{2}}q_{k}(N)-N^{\frac{1}{2}}||\rightarrow 0$. Therefore
$\\{p_{k}(\lambda_{n}I-T)\\}$ is a bounded approximate identity for
$\mathfrak{A}_{\lambda_{n}I-T}$. ∎
###### Remark 3.3.
Theorem 3.2 shows that there exists a compact triangular operator with
infinite spectrum which is a character amenable operator but is not similar to
a normal operator. Moreover, by Lemma 3.1 we can describe character amenable
operator with finite spectrum: If $T\in\mathfrak{B}(\mathfrak{H})$ with finite
spectrum $\sigma(T)=\\{\delta_{1},\delta_{2},\cdots\delta_{n}\\}$, then $T$ is
similar to
$\left(\begin{array}[]{cccc}\delta_{1}I+Q_{1}&&&\\\ &\delta_{2}I+Q_{2}&&\\\
&&\ddots&\\\ &&&\delta_{n}I+Q_{n}\\\ \end{array}\right),$
where $Q_{k}$ is a quasinilpotent operator for $1\leq k\leq n$. By Lemma 3.1,
$T$ is character amenable if and only if $\mathfrak{A}_{Q_{k}}$ has a bounded
approximate identity for $1\leq k\leq n$.
## References
* [1] W. G. Bade, P. C. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. 55 (1987) 359–377.
* [2] K. R. Davidson, Nest algebras, Longman group UK limited, Essex, 1988.
* [3] H. G. Dales, Banach Algebras and Automatic Continuity (Oxford, 2000).
* [4] D. R. Farenick, B.E. Forrest and L. W. Marcoux, Amenable operators on Hilbert spaces, J. reine angew. Math. 582 (2005) 201-228.
* [5] D. R. Farenick, B.E. Forrest and L. W. Marcoux, Amenable operators on Hilbert spaces, J. reine angew. Math. 602 (2007) 235.
* [6] J. A. Gifford, Operator algebras with a reduction property, J. Aust. Math. Soc. 80 (2006) 297-315.
* [7] B. E. Johnson. Cohomology in Banach Algebras. Mem. Amer. Math. Soc. Vol. 127 (Amer. Math. Soc., 1972).
* [8] Monfared, Mehdi Sangani, Character amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 697–706.
* [9] G. A. Willis, When the algebra generated by an operator is amenable, J. Operator Theorey. 34 (1995) 239–249.
|
arxiv-papers
| 2010-08-31T07:47:40 |
2024-09-04T02:49:12.568180
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luo Yi Shi, YU Jing Wu and You Qing Ji",
"submitter": "Luoyi Shi",
"url": "https://arxiv.org/abs/1008.5241"
}
|
1008.5249
|
# Cocycle perturbation on Banach algebra
YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn
# Cocycle perturbation on Banach algebra
YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn
###### Abstract.
Let $\alpha$ be a flow on a Banach algebra $\mathfrak{B}$, and $t\longmapsto
u_{t}$ a continuous function on $\mathbb{R}$ into the group of invertible
elements of $\mathfrak{B}$ such that
$u_{s}\alpha_{s}(u_{t})=u_{s+t},s,t\in\mathbb{R}$. Then
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ is also a flow on
$\mathfrak{B}$. $\beta$ is said to be a cocycle perturbation of $\alpha$. We
show that if $\alpha,\beta$ are two flows on nest algebra (or quasi-triangular
algebra), then $\beta$ is a cocycle perturbation of $\alpha$. And the flows on
nest algebra (or quasi-triangular algebra) are all uniformly continuous.
###### Key words and phrases:
cocycle perturbation, inner perturbation, nest algebra, quasi-triangular
algebra.
###### 2000 Mathematics Subject Classification:
47D03(46H99, 46K50, 46L57)
Supported by NCET(040296), NNSF of China(10971079)
## 1\. Introduction
In the quantum mechanics of particle systems with an infinite number of
degrees of freedom, an important problem is to study the differential equation
$\frac{d\alpha_{t}(A)}{dt}=S\alpha_{t}(A)$
under variety of circumstances and assumptions. In each instance the $A$
corresponds to an observable, or state, of the system and is represented by an
element of some suitable Banach algebra $\mathfrak{B}$. And $S$ is an operator
on $\mathfrak{B}$, $\\{\alpha_{t}\\}_{t\in\mathbb{R}}$ is a group of bounded
automorphisms on $\mathfrak{B}$. The Function
$t\in\mathbb{R}\longmapsto\alpha_{t}(A)\in\mathfrak{B}$
describes the motion of $A$. The dynamics are given by solutions of the
differential equation which respect certain supplementary conditions of
continuity. Thus it is worth to study the group of bounded automorphisms on
$\mathfrak{B}$. More details see [1].
A flow $\alpha$ on $\mathfrak{B}$ is a group homomorphism of the real line
$\mathbb{R}$ into the group of bounded automorphisms on $\mathfrak{B}$ (i.e.
$t\longmapsto\alpha_{t}$) such that
$\lim_{t\rightarrow t_{0}}||\alpha_{t}(B)-\alpha_{t_{0}}(B)||=0$
for each $t_{0}\in\mathbb{R}$ and each $B\in\mathfrak{B}$. If there exists a
$h\in\mathfrak{B}$ such that $\alpha_{t}(B)=e^{th}Be^{th},\forall
B\in\mathfrak{B},t\in\mathbb{R}$, then we call $\alpha$ an inner flow. We say
that a flow $\alpha$ is uniformly continuous if
$\lim_{t\rightarrow t_{0}}||\alpha_{t}-\alpha_{t_{0}}||=0$
for each $t_{0}\in\mathbb{R}$.
If $\alpha$ is a flow on $\mathfrak{B}$ and if $u$ is a continuous map of
$\mathbb{R}$ into the group of invertible elements $G(\mathfrak{B})$ of
$\mathfrak{B}$ such that $u_{s}\alpha_{s}(u_{t})=u_{s+t},s,t\in\mathbb{R}$,
then we call $u=(u_{t})_{t\in\mathbb{R}}$ an $\alpha$-cocycle for
$(\mathfrak{B},\mathbb{R},\alpha)$. Let
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ (i.e.
$\beta_{t}(B)=u_{t}\alpha_{t}(B)u_{t}^{-1},\forall B\in\mathfrak{B}$), then
$\beta$ is also a flow on $\mathfrak{B}$, and $\beta$ is said to be a cocycle
perturbation of $\alpha$.
If $\alpha$ is a flow on $\mathfrak{B}$, let $D(\delta_{\alpha})$ be composed
of those $B\in\mathfrak{B}$ for which there exists a $A\in\mathfrak{B}$ with
the property that
$A=\lim_{t\rightarrow 0}\frac{\beta_{t}(B)-B}{t}.$
Then $\delta_{\alpha}$ is a linear operator on $D(\delta_{\alpha})$ defined by
$\delta_{\alpha}(B)=A$. We call $\delta_{\alpha}$ the infinitesimal generator
of $\alpha$. By [1,Proposition 3.1.6], $\delta_{\alpha}$ is a closed
derivation. i.e. the domain $D(\delta_{\alpha})$ is a dense subalgebra of
$\mathfrak{B}$ and $\delta_{\alpha}$ is closed as a linear operator on
$D(\delta_{\alpha})$ and satisfies
$\delta_{\alpha}(AB)=\delta_{\alpha}(A)B+A\delta_{\alpha}(B),$ for $A,B\in
D(\delta_{\alpha})$. We call $\beta$ an inner perturbation of $\alpha$, if
$\alpha,\beta$ are two flows on $\mathfrak{B}$,
$D(\delta_{\alpha})=D(\delta_{\beta})$ and there exists $h\in\mathfrak{B}$
such that $\delta_{\beta}=\delta_{\alpha}+$ad$ih$ (where $i$ is the imaginary
unit, and ad$ih(B)\triangleq i(hB-Bh),\forall B\in\mathfrak{B}$). Moreover
$D(\delta_{\alpha})=\mathfrak{B}$ if and only if $\alpha$ is uniformly
continuous. More details see [1,2].
The problem we considered is classifying cocycle of flows on Banach algebras.
Such a problem has been considered in the $C^{*}$algebra cases, notably by
Kishimoto[3-10]. We refer the reader to [3] for a detailed study of the
general results concerning cocycles and invariants for cocycle perturbation of
flows on $C^{*}$-algebras. In particular, it is shown there that let $\alpha$
be a flow on a $C^{*}$-algebra $\mathfrak{A}$, if $u=(u_{t})_{t\in\mathbb{R}}$
a $\alpha$-cocycle for $(\mathfrak{A},\mathbb{R},\alpha)$, and $u$ is
differentiable (i.e. $\lim_{t\rightarrow
t_{0}}\frac{u_{t}-u_{t_{0}}}{t-t_{0}}$ exist for any $t_{0}\in\mathbb{R}$) and
$h=-i(du_{t}/dt)|_{t=0}\in\mathfrak{A}$, then the infinitesimal generator of
the flow $\beta_{t}=Adu_{t}\circ\alpha_{t}$ is given by
$\delta_{\beta}=\delta_{\alpha}+$ad$ih$. i.e. $\beta$ is an inner perturbation
of $\alpha$. Moreover for any $\alpha$-cocycle $u=(u_{t})_{t\in\mathbb{R}}$,
there is $w\in G(\mathfrak{A})$ and a differentiable $\alpha$-cocycle
$v=(v_{t})_{t\in\mathbb{R}}$ (i.e. $v_{t}$ is an $\alpha$-cocycle and
differentiable) such that $u_{t}=wv_{t}\alpha_{t}(w^{-1})$. In section 2, we
consider the cocycle of flows on Banach algebras and obtain some similar
results to [3].
It is well-known that a flow $\alpha$ on $\mathfrak{B}$ may not be uniformly
continuous even if $\mathfrak{B}$ is a $C^{*}$algebra [1,2]. In section 3, we
study the flows on nest algebra $\tau(\mathcal{N})$ and quasi-triangular
algebra $Q\tau(\mathcal{N})=\tau(\mathcal{N})+K$ [11]. We recall that a nest
$\mathcal{N}$ is a chain of closed subspaces of a Hilbert space $\mathfrak{H}$
containing $\\{0\\}$ and $\mathfrak{H}$ which is in addition closed under
taking arbitrary intersections and closed spans. The nest algebra
$\mathcal{T(N)}$ associated with $\mathcal{N}$ is the set of all $T\in
B(\mathfrak{H})$ which leave each element of the nest invariant. For instance,
if $\mathfrak{H}$ is separable with orthonormal basis
$\\{e_{n}\\}_{n=1}^{\infty}$ and
$\mathfrak{H}_{n}$=span$\\{e_{1},\cdots,e_{n}\\}$, then
$\mathcal{N}=\\{0,\mathfrak{H}\\}\cup\\{\mathfrak{H}_{n}\\}_{n=1}^{\infty}$ is
a nest. In this case, $\mathcal{T(N)}$ is simply the set of all operators
whose matrix representation with respect to this basis is upper triangular. It
is obvious, $\tau(\mathcal{N})$ and $Q\tau(\mathcal{N})$ are typical Banach
algebras. We obtain that all of the flows on $\tau(\mathcal{N})$ (or
$Q\tau(\mathcal{N})$) are uniformly continuous. Moreover all of the flows are
cocycle perturbation to each other.
## 2\. Cocycle perturbations
Let $\mathfrak{B}$ be a Banach algebra, $\alpha$ be a flow on $\mathfrak{B}$
and $u$ an $\alpha$-cocycle, then
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ is a cocycle perturbation
of $\alpha$. In this section, we obtain that $\beta$ is an inner perturbation
of $\alpha$ if and only if $u$ is differentiable (Theorem 2.4.). Moreover for
any $\alpha$-cocycle $u$ , there is a differentiable $\alpha$-cocycle $v$ and
a inventible element $w$ in $\mathfrak{B}$ such that
$u_{t}=wv_{t}\alpha_{t}(w^{-1})$ (Theorem 2.8.).
We will use the following Lemmas [1, Proposition 3.1.3, Theorem 3.1.33].
Lemma 2.1[1]: Let $\\{\alpha_{t}\\}_{t\in\mathbb{R}}$ be a flow on the Banach
algebra $\mathfrak{B}$. Then there exists an $M\geq 1$ and
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that $||\alpha_{t}||\leq
Me^{\xi|t|}.$
Lemma 2.2[1]: Let $\alpha$ be a flow on a Banach algebra $\mathfrak{B}$ with
infinitesimal generator $\delta_{\alpha}$. For each $P\in\mathfrak{B}$ define
the bounded derivation $\delta_{P}$ by $D(\delta_{P})=\mathfrak{B}$ and
$\delta_{P}(B)=i[P,B]\triangleq i(PB-BP),\forall B\in\mathfrak{B}$. Then
$\delta+\delta_{P}$ generates a flow on $\mathfrak{B}$ given by
$\alpha_{t}^{P}(B)=\alpha_{t}(B)+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}[\alpha_{t_{n}}(P),[\cdots[\alpha_{t_{1}}(P),\alpha_{t}(B)]]],\forall
B\in\mathfrak{B},t\in\mathbb{R}.$
Lemma 2.3: Let $\mathfrak{B}$ be a Banach algebra with unit $\mathbf{1}$,
$\alpha$ be a flow on $\mathfrak{B}$ and let $\delta$ denote the infinitesimal
generator of $\alpha$. Furthermore, for each $P\in\mathfrak{B}$ define
$\delta_{P}$ as in Lemma 2.2.
Then $\delta+\delta_{P}$ generates a flow on $\mathfrak{B}$ given by
$\alpha_{t}^{P}(B)=\alpha_{t}(B)+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}[\alpha_{t_{n}}(P),[\cdots[\alpha_{t_{1}}(P),\alpha_{t}(B)]]].$
Moreover, one has
$\alpha_{t}^{P}(B)=u^{P}_{t}\alpha_{t}(B){(u^{P}_{t}})^{-1},$
where $u_{t}^{p}$ is a one-parameter family of invertible elements, determined
by
$u^{P}_{t}=1+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}\alpha_{t_{n}}(P)\cdots\alpha_{t_{1}}(P)$
which satisfies the $\alpha$-cocycle relation
$u^{P}_{t+s}=u^{P}_{t}\alpha_{t}(u^{P}_{s}).$
All integrals converge in the strong topology. The integrals define norm-
convergent series of bounded operators and there exists an $M\geq 1$,
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that
$||\alpha_{t}^{P}(B)-\alpha_{t}(B)||\leq
Me^{\xi|t|}(e^{M|t|||P||}-1),||u^{P}_{t}-\mathbf{1}||\leq
Me^{\xi|t|}(e^{M|t|||P||}-1)$ .
Proof : The first statement of the proposition can be obtained from Lemma 2.2.
We just give the proof of the last statement of the Lemma.
There exists an $M\geq 1$ and $\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$)
such that $||\alpha_{t}||\leq Me^{\xi|t|},$ by Lemma 2.1. Let
$M_{t}=\left\\{\begin{array}[]{ll}Me^{\xi|t|}&\mbox{$|t|>1$;}\\\
M&\mbox{$|t|\leq 1$.}\end{array}\right.$
Next we consider $u^{P}_{t}$ defined by the series. The $n$-th term in this
series is well defined and has norm less than
$\frac{|t|^{n}}{n!}M_{t}^{n}||P||^{n}$. Thus $u^{P}_{t}$ is a norm-continuous
one-parameter family of elements of $\mathfrak{A}$ with $u^{P}_{0}=1$ and
$||u_{t}||\leq e^{|t|M_{t}||P||}$. Consequently, $u^{P}_{t}$ is invertible for
all $t\in[-t_{0},t_{0}]$ for some $t_{0}>0$ and $(u^{P}_{t})^{-1}$ is a norm-
continuous one-parameter family of elements of $\mathfrak{B}$ for all
$t\in[-t_{0},t_{0}]$.
Next one has
$\frac{du^{P}_{t}}{dt}=iu^{P}_{t}\alpha_{t}(P),$ $\lim_{t\rightarrow
0}(\frac{u^{P}_{t}-1}{t})=\frac{du^{P}_{t}}{dt}|_{t=0}=iP.$
Hence,
$\lim_{t\rightarrow 0}(\frac{(u^{P}_{t})^{-1}-1}{t})=\lim_{t\rightarrow
0}(u^{P}_{t})^{-1}\frac{1-u^{P}_{t}}{t}=-iP.$
To establish the $\alpha$-cocycle relation we first note that
$\frac{du^{P}_{t+s}}{ds}=iu^{P}_{t+s}\alpha_{t+s}(P)$
and $u^{P}_{t+s}|_{s=0}=u^{P}_{t}$.
But $\alpha_{t}(u^{P}_{s})=u^{\alpha_{t}(P)}_{s}$ and hence
$\frac{d}{ds}u^{P}_{t}\alpha_{t}(u^{P}_{s})=iu^{P}_{t}u^{\alpha_{t}(P)}_{s}\alpha_{s}(\alpha_{t}(P))=iu^{P}_{t}\alpha_{t}(u^{P}_{s})\alpha_{t+s}(P).$
Moreover, $u^{P}_{t}\alpha_{t}(u^{P}_{s})|_{s=0}=iu^{P}_{t}\alpha_{t}(p)$.
Thus $s\longmapsto u^{P}_{t+s}$ and $s\longmapsto
u^{P}_{t}\alpha_{t}(u^{P}_{s})$ satisfy the same first-order differential
equation and boundary condition for each $t\in\mathbb{R}$. Therefore, the two
functions are equal and can be obtained by iteration of the integral equation
$X_{t}(s)=u^{P}_{t}+i\int_{0}^{s}ds^{\prime}X_{t}(s^{\prime})\alpha_{t+s^{\prime}}(P).$
Hence, $u^{P}_{t+s}=u^{P}_{t}\alpha_{t}(u^{P}_{s}),t,s\in\mathbb{R}$. Since
$u^{P}_{t}$ is invertible for all $t\in[-t_{0},t_{0}]$ for some $t_{0}>0$, we
obtain that $u^{P}_{t}$ is a norm-continuous one-parameter family of
inventible elements. Thus $t\mapsto u^{P}_{t}\alpha_{t}(B)(u^{P}_{t})^{-1}$
defines a flow $\beta_{t}$ on $\mathfrak{B}$.
Let $\tilde{\delta}$ denote the infinitesimal generator of $\beta$. Next we
will prove $\tilde{\delta}=\delta+\delta_{P}$.
Choose $A\in D(\delta+\delta_{P})$, one has
$\delta(A)=\lim_{t\rightarrow 0}(\frac{\alpha_{t}(A)-A}{t}),$
$\displaystyle\tilde{\delta}(A)$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow 0}(\frac{\beta_{t}(A)-A}{t})$
$\displaystyle=$ $\displaystyle\lim_{t\rightarrow
0}(\frac{u^{P}_{t}\alpha_{t}(A)(u^{P}_{t})^{-1}-u^{P}_{t}A(u^{P}_{t})^{-1}}{t}$
$\displaystyle+$
$\displaystyle\frac{u^{P}_{t}A(u^{P}_{t})^{-1}-u^{P}_{t}A}{t}+\frac{u^{P}_{t}A-A}{t})$
$\displaystyle=$ $\displaystyle(\delta+\delta_{P})(A).$
Similarly, if $A\in D(\tilde{\delta})$, we
obtain$\tilde{\delta}(A)=(\delta+\delta_{P})(A)$. And then
$\tilde{\delta}=\delta+\delta_{P}$.
Thus one must have
$\alpha_{t}^{P}(B)=\beta_{t}(B)=u^{P}_{t}\alpha_{t}(B){(u^{P}_{t}})^{-1},$ for
any $B\in\mathfrak{B}$ by [1, Theorem 3.1.26]. Finally the estimates on
$\alpha_{t}^{P}(B)-\alpha_{t}(B)$ and $u^{P}_{t}-\mathbf{1}$ are
straightforward.
Theorem 2.4 : Let $\alpha$ be a flow on $\mathfrak{B}$,
$(u_{t})_{t\in\mathbb{R}}$ be an $\alpha$-cocycle ,
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t}$. Then $\beta$ is an inner perturbation of
$\alpha$ if and only if $u_{t}$ is is differentiable.
Proof : $\Rightarrow$ It follows immediately from Lemma 2.3.
$\Leftarrow$ If $u_{t}$ is an $\alpha$-cocycle and differentiable with
$h=-i(du_{t}/dt)|_{t=0}\in\mathfrak{B}$, then $u_{t}$ is given by
$u_{t}=1+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}\alpha_{t_{n}}(P)\cdots\alpha_{t_{1}}(P).$
Hence, $\beta$ is an inner perturbation of $\alpha$ by lemma 2.3.
Corollary 2.5 : Adopt the assumptions of the Lemma 2.3 and also assume that
$\alpha_{t}$ is an inner flow. i.e. there exists $h\in\mathfrak{B}$ such that
$\alpha_{t}(A)=e^{ith}Ae^{-ith},\forall A\in\mathfrak{B},t\in\mathbb{R}$.
Then
$\alpha^{P}_{t}(A)=\Gamma_{t}^{P}A(\Gamma_{t}^{P})^{-1},u_{t}^{P}=\Gamma_{t}^{P}e^{-ith},$
where $u_{t}^{P}$ is defined as in Lemma 2.3 and $\Gamma_{t}^{P}=e^{it(h+P)}$.
i.e. $\alpha^{P}_{t}$ is an inner flow.
Proof : If $\Gamma_{t}^{P}=e^{it(h+P)}$ and $X_{t}=\Gamma_{t}^{P}e^{-ith}$,
then
$\frac{dX_{t}}{dt}=i\Gamma_{t}^{P}Pe^{-ith}=iX_{t}\alpha_{t}(P)$
and $X_{0}=\mathbf{1}$. Thus, $X_{t}$ is the unique solution of the integral
equation
$X_{t}=\mathbf{1}+i\int_{0}^{t}dsX_{s}\alpha_{s}(P).$
This solution can be obtained by iteration and one finds $X_{t}=u_{t}^{P}$,
where $u_{t}^{P}$ is defined as in Lemma 2.3.
And$\alpha_{t}^{P}(A)=u^{P}_{t}\alpha_{t}(A){(u^{P}_{t}})^{-1}=\Gamma_{t}^{P}A(\Gamma_{t}^{P})^{-1}$.
Next we obtain that every $\alpha$-cocycle is similar to a differentiable
$\alpha$-cocycle.
Definition 2.6: Let $\alpha$ be a flow on $\mathfrak{B}$. $A\in\mathfrak{B}$
is called an analytic element for $\alpha$ if there exist a analytic function
$f:~{}~{}\mathbb{C}\rightarrow\mathfrak{B}$ such that $f(t)=\alpha_{t}(A)$ for
$t\in\mathbb{R}$.
Lemma 2.7: Let $\alpha$ be a flow on Banach algebra $\mathfrak{B}$ and $M,\xi$
are constants such that $||\alpha_{t}||\leq Me^{\xi|t|}$. For
$A\in\mathfrak{B}$, define
$A_{n}=\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-nt^{2}-\xi
t}dt,~{}~{}~{}n=1,2,\cdots.$
Then each $A_{n}$ is an entire analytic element for $\alpha_{t}$,
$||A_{n}||<M||A||$ for all n; and $A_{n}\rightarrow A$ in the weak topology as
$n\rightarrow\infty$. In particular, the $\alpha$ analytic elements form a
normal-dense subspace of $\mathfrak{B}$.
Proof :
$f_{n}(z)=\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-n(t-z)^{2}-\xi(t-z)}dt$
is well defined for all $z\in\mathbb{C}$, since $t\mapsto
e^{-n(t-z)^{2}}\in\mathbf{L}^{1}(\mathbb{R})$ for each $z\in\mathbb{C}$.
For $z=s\in\mathbb{R}$, we have
$\displaystyle f_{n}(s)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-n(t-s)^{2}-\xi(t-s)}dt$
$\displaystyle=$
$\displaystyle\sqrt{\frac{n}{\pi}}\int\alpha_{t+s}(A)e^{-nt^{2}-\xi
t}dt=\alpha_{s}(A_{n}).$
But for $\eta\in\mathfrak{B}^{*}$ we have
$\eta(f_{n}(z))=\sqrt{\frac{n}{\pi}}\int\eta(\alpha_{t}(A))e^{-n(t-z)^{2}-\xi(t-z)}dt.$
Since $|\eta(\alpha_{t}(A))|\leq M||\eta||||A||$, it follows from the Lebesgue
dominated convergence theorem that $t\mapsto\eta(f_{n}(z))$ is analytic. Hence
each $A_{n}$ is analytic for $\alpha_{t}(A)$.
Next, one derives the estimate
$||A_{n}||\leq M||A||\sqrt{\frac{n}{\pi}}\int e^{-n(t)^{2}}dt\leq M||A||.$
Next note that
$\int e^{-nt^{2}-\xi t}dt=e^{\frac{\xi^{2}}{4n^{2}}}\sqrt{\frac{\pi}{n}}.$
Hence
$\eta(A_{n}-A)=\sqrt{\frac{n}{\pi}}\int e^{-nt^{2}-\xi
t}\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)dt$
for all $\eta\in\mathfrak{B}^{*}$. But for any $\varepsilon>0$ we may choose
$\delta>0$ such that $t<|\delta|$ implies
$|\eta(\alpha_{t}(A)-A)|<\varepsilon$. Further, we can choose $N$ large enough
that $\frac{\xi}{2N}<\frac{\delta}{2}$. It follows that if $n>N$ we have
$\displaystyle|\eta(A_{n}-A)|$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$
On the other hand,
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-A)|dt$ $\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(A)(1-e^{-\frac{\xi^{2}}{4n^{2}}})|dt$ $\displaystyle<$
$\displaystyle\varepsilon
e^{\frac{\xi^{2}}{4n^{2}}}+|1-e^{-\frac{\xi^{2}}{4n^{2}}}|\cdot||\eta||\cdot||A||Me^{\frac{\xi^{2}}{4n^{2}}}.$
And
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}||\eta||\cdot||A||M\int_{|t|\geq\delta}e^{-nt^{2}}dt$
$\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}e^{-\frac{\xi^{2}}{4n^{2}}}||\eta||\cdot||A||\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}dt.$
So $|\eta(A_{n}-A)|\rightarrow 0$ when $n\rightarrow\infty$, for any
$\eta\in\mathfrak{B}^{*}$. Finally note that the norm closure and the weak
closure of convex set are the same, hence the $\alpha$ analytic elements form
a normal-dense subspace of $\mathfrak{B}$.
Theorem 2.8: If $u$ is an $\alpha$-cocycle for $\mathfrak{B}$. Then, given
$\varepsilon>0$, there is a differentiable $\alpha$-cocvcle $v$ and a $w\in
G(\mathfrak{B})$ such that
$||w-\mathbf{1}||<\varepsilon,u_{t}=wv_{t}\alpha_{t}(w^{-1}).$
Proof : The proof is similar to [3,Lemma 1.1] and is omitted.
Given two flows $\alpha$ and $\beta$ on a unitary Banach algebra
$\mathfrak{B}$ we say that $\beta$ is a conjugate to $\alpha$ if there is a
bounded automorphism $\sigma$ of $\mathfrak{B}$ such that
$\beta=\sigma\alpha\sigma^{-1}$. Conjugate, cocycle perturbation and inner
perturbation define three equivalence relations. We say that $\beta$ is
cocycle-conjugate to $\alpha$ if there is a bounded automorphism $\sigma$ of
$\mathfrak{B}$ such that $\beta$ is a cocycle perturbation of
$\sigma\alpha\sigma^{-1}$. This also defines an equivalence relation among the
flows.
We say that $\alpha$ is approximately inner if there is a sequence
$\\{h_{n}\\}$ in $\mathfrak{B}$ such that $\alpha_{t}=$ limAd $e^{th_{n}}$,
i.e., $\alpha_{t}(A)=$$\lim$Ad$e^{th_{n}}(A)$ for every $t\in\mathbb{R}$ and
$A\in\mathfrak{B}$, or equivalently, uniformly continuous in $t$ on every
compact subset of $\mathbb{R}$ and every $A\in\mathfrak{B}$. A flow on a
Banach algebra $\mathfrak{B}$ is said to be asymptotically inner if there is a
continuous function $h$ of $\mathbb{R}_{+}$ into $\mathfrak{B}$ such that
$\lim_{s\rightarrow\infty}\max_{|t\leq
1|}\parallel\alpha_{t}(A)-$Ad$e^{th(s)}(A)\parallel=0$ for any
$A\in\mathfrak{B}$.
Corollary 2.9: Let $\alpha$ and $\beta$ be two flows on Banach algebra
$\mathfrak{B}$. Then the following conditions are equivalent:
(i). $\beta$ is cocycle-conjugate to $\alpha$.
(ii). $\beta$ is an inner perturbation of $\sigma\alpha\sigma^{-1}$ for some
automorphism $\sigma$ of $\mathfrak{B}$, where $\sigma\alpha\sigma^{-1}$ is
the action $t\mapsto\sigma\alpha_{t}\sigma^{-1}$.
If one of the above conditions are satisfied and $\alpha$ is
inner(approximately or asymptotically inner), then so is $\beta$.
Proof: The first statement of the proposition can be obtained from
[3,Corollary 1.3]. We just give the proof of the last statement of the
proposition.
First we prove that if $\beta$ is an inner perturbation of $\alpha$, i.e.
$\delta_{\beta}=\delta\alpha+i$ad$P$, then it follows that if $\alpha$ is
inner(approximately or asymptotically inner), then so is $\beta$.
(a) If $\alpha$ is inner,then so is $\beta$ by Corollary 2.5.
(b) If $\alpha$ is approximately inner, then there is a sequence $\\{h_{n}\\}$
in $\mathfrak{B}$ such that
$\lim_{n\rightarrow\infty}\max_{|t|\leq
1}||\alpha_{t}(A)-Ade^{th_{n}}(A)||=0.$
$\alpha_{n,t}\triangleq Ade^{th_{n}}$, we construct $\alpha_{n}$-cocycle
$u_{n,t}$(resp. $u$) for $\alpha_{n,t}$(resp. $\alpha$) such that
$\frac{d}{dt}u_{n,t}|_{t=0}=iP$ by Lemma 2.3. Then since
$\beta_{t}=Adu_{t}\circ\alpha_{t}$ and $u_{n,t}\rightarrow u_{t}$, we obtain
that $Adu_{n,t}\circ\alpha_{n,t}\rightarrow\beta_{t}$. Besides,
$Adu_{n,t}\circ\alpha_{n,t}$ is inner by Corollary 2.5. Then $\beta$ is
approximately inner.
(c) If $\alpha$ is asymptotically inner, then there is a continuous function
$h$ of $\mathbb{R}_{+}$ into $\mathfrak{B}$ such that
$\lim_{s\rightarrow\infty}\max_{|t\leq
1|}\parallel\alpha_{t}(A)-Ade^{th(s)}(A)\parallel=0$
for any $A\in\mathfrak{B}$.
$\alpha_{s,t}\triangleq Ade^{th(s)}$, we construct $\alpha_{s}$-cocycle
$u_{s,t}$(resp. $u$) for $\alpha_{s,t}$(resp. $\alpha$) such that
$\frac{d}{dt}u_{s,t}|_{t=0}=iP$ by Lemma 2.3. Then since
$\beta_{t}=Adu_{t}\circ\alpha_{t}$ and $u_{s,t}\rightarrow u_{t}$, we obtain
that $Adu_{s,t}\circ\alpha_{s,t}\rightarrow\beta_{t}$. Besides,
$Adu_{s,t}\circ\alpha_{s,t}$ is inner and
$Adu_{s,t}\circ\alpha_{s,t}=Ade^{t(h(s)+P)}$ by Corollary 2.5. Then $\beta$ is
approximately inner.
Next we shall prove that if $\beta$ is conjugate to $\alpha$, i.e. there is a
bounded automorphism $\sigma$ of $\mathfrak{B}$ such that
$\beta=\sigma\alpha\sigma^{-1}$. Then if $\alpha$ is inner(approximately or
asymptotically inner), then so is $\beta$.
(a’) If $\alpha$ is inner, i.e. $\alpha_{t}(A)=e^{th}Ae^{-th}$. Then
$\beta_{t}(A)=\sigma(\alpha_{t}(\sigma(A)))=e^{t\sigma(h)}Ae^{-t\sigma(h)}$
i.e. $\beta$ is inner.
(b’) If $\alpha$ is approximately inner, then there exits a sequence
$\\{h_{n}\\}$ in $\mathfrak{A}$ such that
$\lim_{n\longrightarrow\infty}\max_{|t|\leq
1}||\alpha_{t}(A)-Ade^{th_{n}}(A)||=0.$
Because $\beta$ is conjugate to $\alpha$, i.e.
$\beta=\sigma\alpha\sigma^{-1}$, $\sigma$ is the bounded automorphism of
$\mathfrak{A}$, so there exists an $M>0$ such that $||\sigma||\leq
M,||\sigma^{-1}||\leq M$.
Therefore,
$||\beta_{t}(A)-Ade^{t\sigma(h_{n})}(A)||=||\sigma^{-1}(\beta_{t}(\sigma(A)))-\sigma^{-1}(e^{t\sigma(h_{n})}(\sigma(A)))||$
i.e.$\alpha$ is approximately inner.
(c’) If $\alpha$ is asymptotically inner, a similar argument shows that
$\beta$ is asymptotically inner.
Finally, if $\beta$ is cocycle-conjugate to $\alpha$, then there is a bounded
automorphism $\sigma$ of $\mathfrak{B}$ such that $\beta$ is a cocycle
perturbation of $\sigma\alpha\sigma^{-1}$. If $\alpha$ is inner(approximately
or asymptotically inner), then so is $\sigma\alpha\sigma^{-1}$, then so is
$\beta$.
## 3\. Cocycle perturbations on nest algebra
and quasi-triangular algebra
In this section $\mathfrak{H}$ denotes a Hilbert space, $\mathfrak{B}$ denotes
a nest algebra (or a quasi-triangular algebra ) on $\mathfrak{H}$. $id$
denotes the identity automorphism on $\mathfrak{B}$ .
Let us mention briefly some well-known results on this subject.
Proposition 3.1[11]: If $\sigma$ is an automorphism on $\mathfrak{B}$ and
$||\sigma-id||<1$,then there is an invertible element $T$ in $\mathfrak{B}$
such that $\sigma(A)=TAT^{-1}$ and $||T-I||<4||\alpha-id||$.
Proposition 3.2: Let $\alpha$ be a flow on $\mathfrak{B}$ and
$||\alpha_{t}-id||=O(t)$. Then there is a $P\in\mathfrak{B}$ such that
$\alpha_{t}(A)=e^{tP}Ae^{-tP}$, $\forall A\in\mathfrak{B}$.
Proof: When $||\alpha_{t}-id||=O(t)$, there is an $\varepsilon>0$ so that
$|t|<\varepsilon$ implies $||\alpha_{t}-id||<1$. By Proposition 3.1, there are
invertible elements $\\{T_{t}\\}$ in $\mathfrak{B}$ such that
$\alpha_{t}(A)=T_{t}AT_{t}^{-1}$ and $||T_{t}-I||<4||\alpha_{t}-id||$. By [1,
Proposition 3.1.1] $t^{-1}||\alpha_{t}-id||$ is bounded. So
$P_{n}=n(T_{\frac{1}{n}}-I)$ is bounded. So there is a weak∗ convergent sub-
net $\\{P_{n_{r}}\\}$ with limit $P$ in $\mathfrak{B}$. Let $\delta_{\alpha}$
denote the infinitesimal generator of $\alpha$, one obtains
$\delta_{\alpha}(A)=\lim_{r\rightarrow\infty}n_{r}(\alpha_{\frac{1}{n_{r}}}-id)AT_{\frac{1}{n_{r}}}=\lim_{r\rightarrow\infty}n_{r}(T_{\frac{1}{n_{r}}}A-AT_{\frac{1}{n_{r}}})=\lim_{r\rightarrow\infty}P_{n_{r}}A-AP_{n_{r}}=PA-
AP$
for $\forall A\in\mathfrak{B}$. Then by [1, Proposition 3.1.1]
$\alpha_{t}(A)=e^{tP}Ae^{-tP}$.
Proposition 3.3: Let $\alpha,\beta$ be two flows on $\mathfrak{B}$ and
$||\alpha_{t}-\beta_{t}||=O(t)$, then $\alpha$ is an inner perturbation of
$\beta$.
Proof: There exists an $M\geq 1$ and
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that
$||\alpha_{t}||\leq Me^{\xi|t|},||\beta_{t}||\leq Me^{\xi|t|}$
Since $||\alpha_{t}\circ\beta_{-t}-id||\leq
Me^{\xi|t|}||\alpha_{t}-\beta_{t}||=O(t)$, we get that there is a
$P\in\mathfrak{B}$ such that $\alpha_{t}(A)=e^{tP}\beta_{t}(A)e^{-tP}$, for
$A\in\mathfrak{B}$ by Proposition 3.2. Finally
$\delta_{\alpha}=\delta_{\beta}+$ad$i(-iP)$ is straightforward. i.e. $\alpha$
is an inner perturbation of $\beta$.
Remark : By the definition of cocycle perturbation, it is obvious that if
$\beta$ is an cocycle perturbation of $\alpha$, then
$||\alpha_{t}-\beta_{t}||=O(t)$. Hence, cocycle perturbation and inner
perturbation are equivalent on $\mathfrak{B}$ by Lemma 2.3 and Proposition
3.3. From the next theorem we obtain that every flow on nest algebra or quasi-
triangular algebra are uniformly continuous.
Theorem 3.4: Every flow $\mathfrak{B}$ are uniformly continuous. And any two
flows on $\mathfrak{B}$ are cocycle perturbation to each other.
Proof: In fact, if there is a flow $\alpha$ on $\mathfrak{B}$ which is not
uniformly continuous, let $\delta_{\alpha}$ be the infinitesimal generator of
$\alpha$. Then there is a $v\in G(\mathfrak{B})$ such that $v\notin
D(\delta)$. Let $v_{t}=v^{-1}\alpha_{t}(v)$,
$\beta_{t}(A)=v_{t}\alpha_{t}(A)(v_{t})^{-1},t\in\mathbb{R},A\in\mathfrak{B}$.
Then $\beta$ is an cocycle perturbation of $\alpha$. And $v_{t}$ is not
differentiable.
By Proposition 3.3 and Remark, $\beta$ is a inner perturbation of $\alpha$
i.e. there is differentiable $\alpha$-cocycle
$\\{u_{t}\\}\subseteq\mathfrak{B}$ such that
$\beta_{t}(A)=u_{t}\alpha_{t}(A){(u_{t}})^{-1}$.
But the commutant of $\mathfrak{B}$ is trivial [11]. Hence it follows that
$v_{t}=\lambda_{t}u_{t},t\in\mathbb{R}$, where $\lambda_{t}\in\mathbb{C}$.
$u_{t},v_{t}$ are $\alpha$-cocycle, it follow that
$\lambda_{t+s}=\lambda_{t}\lambda_{s}$ and $\lambda_{t}\rightarrow 1$, for
$t\rightarrow 0$. Therefore $\lambda_{t}$ is differentiable and $v_{t}$ is
differentiable. This is contrary to $v_{t}$ is not differentiable.
Finally if $\alpha,\beta$ are flows on $\mathfrak{B}$, then $\alpha,\beta$ are
uniformly continuous. So $\alpha,\beta$ are inner perturbation to
$\gamma_{t}\equiv id$. Because the inner perturbation defines an equivalence
relation on the flows, so $\alpha$ is an inner perturbation of $\beta$, and
$\alpha$ is a cocycle perturbation of $\beta$.
## References
* [1] Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics, I, Springer, 1979.
* [2] Sakai, S., Operator algebras in dynamical systems, Cambridge Univ. Press, Cambridge 1991.
* [3] Kishimoto, A., Locally representable one-parameter automorphism groups of AF algebras and KMS states, Rep. Math. Phys. 45 (2000), 333–356.
* [4] Kishimoto, A., UHF flows and the flip automorphism. Rev. Math. Phys. 13 (2001), no. 9, 1163–1181
* [5] Kishimoto, A., Examples of one-parameter automorphism groups of UHF algebra. Commun. Math. Phys. 216 (2001), 395–428
* [6] Kishimoto, A., Approximately inner flows on seperable $C^{*}$-algebras. Rev. Math. Phs. 14 (2002), 1065–1094.
* [7] Kishimoto, A., Approximate AF flows. J. Evol. Equ. 5 (2005), 153–184.
* [8] Kishimoto, A., The one-cocycle property for shifts. Ergod. Th. And. Dynam. Sys. 25 (2005), 823–859.
* [9] Kishimoto, A., Multiplier cocycles of a flow on a $C^{*}$-algebra. J. Funct. Anal. 235 (2006), 271–296.
* [10] Kishimoto, A., Lifting of an asymptotically inner flow for a separable $C^{*}$-algebra. Operator Algebras: The Abel Symposium 2004, 233–247, Abel Symp., 1, Springer, Berlin, 2006.
* [11] Davidson K. R., Nest algebras, Longman group UK limited, Essex, 1988.
|
arxiv-papers
| 2010-08-31T08:13:08 |
2024-09-04T02:49:12.573354
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Jing Wu and Luo Yi Shi",
"submitter": "Luoyi Shi",
"url": "https://arxiv.org/abs/1008.5249"
}
|
1008.5252
|
# Generalized notions of character amenability
Luo Yi Shi Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA sluoyi@yahoo.cn , YU Jing Wu Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA wuyujing111@yahoo.cn and You Qing Ji Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA jiyq@jlu.edu.cn
###### Abstract.
In this paper the concepts of character contractibility, approximate character
amenability (contractibility) and uniform approximate character amenability
(contractibility) are introduced. We are concerned with the relations among
the generalized concepts of character amenability for Banach algebra. We prove
that approximate character amenability and approximate character
contractibility are the same properties, as are uniform approximate character
amenability and character amenability, as are uniform approximate character
contractibility and character contractibility. For commutative Banach algebra,
we prove that character contractibility and contractibility are the same
properties. Moreover, general theory for those concepts is developed.
###### Key words and phrases:
Character amenability; Approximately inner; Derivation
###### 2000 Mathematics Subject Classification:
46H20(46H25 47B47)
## 1\. Introduction
The concept of amenability for Banach algebras was first introduced by B. E.
Johnson in [14]. Suppose that $A$ is a Banach algebra and that $E$ is a Banach
$A$-bimodule, then $E^{*}$, the dual of $E$, has a natural Banach $A$-bimodule
structure defined by
$(a\cdot f)(x)=f(x\cdot a),(f\cdot a)(x)=f(a\cdot x),a\in A,x\in E,f\in
E^{*}.$
Such a Banach $A$-bimodule $E^{*}$ is called a dual $A$-bimodule. A derivation
$D:A\rightarrow E^{*}$ is a continuous linear map such that $D(ab)=a\cdot
D(b)+D(a)\cdot b$ for all $a,b\in A$. Given $f\in E^{*}$, the inner derivation
$\delta_{f}:A\rightarrow E^{*}$, is defined by $\delta_{f}(a)=a\cdot f-f\cdot
a$. According to Johnson’s original definition, a Banach algebra $A$ is
amenable if every derivation from $A$ into the dual $A$-bimodule $E^{*}$ is
inner for all Banach $A$-bimodules $E$. As a complement to this notion, a
Banach algebra $A$ is contractible if every derivation from $A$ into every
Banach $A$-bimodule is inner [4, 12].
Ever since its introduction, the concept of amenability has occupied an
important place in the research of Banach algebras, operator algebras and
harmonic analysis. For example, an early result of Johnson [20] shows that the
amenability of the group algebra $L^{1}(G)$, for G a locally compact group, is
equivalent to the amenability of the underlying group G. Results of Connes and
Haagerup show that a $C^{*}$-algebra is amenable if and only if it is nuclear
[20]. However it has been realized that amenability is essentially a
finiteness condition, and in many instances is too restrictive. As for
contractibility, it is even conjectured in [12] that a contractible Banach
algebra must be finite dimensional (see also [21]). For this reason by
relaxing some of the constrains in the definition of amenability new concepts
have been introduced. The most notable are the concepts of Connes amenability
[11, 13], weak amenability [2, 5] and character amenability [17, 18]. More
recently, F. Ghahramani and R. J. Loy have introduced and studied the concepts
of approximate amenability (contractibility) and uniform approximate
amenability (contractibility) for Banach algebras [9, 10]. In this paper we
introduce the generalized concepts of character amenability (see Definition
1.2-1.7). We are concerned with the relations among those concepts and shall
develop general theory for them.
###### Definition 1.1.
A derivation $D:A\rightarrow E$ is approximately inner, if there exists a net
$\\{\xi_{i}\\}\subset E$ such that
$D(a)=\lim\limits_{i}(a\cdot\xi_{i}-\xi_{i}\cdot a)$ for all $a\in A$, the
limit being in norm.
Note that $\\{\xi_{i}\\}$ in the above is not necessarily bounded. The
stronger assumption, that $D$ is in the uniform closure of the inner
derivations, has been well studied in the $C^{*}$-algebra case with the
restriction to the single Banach $A$-bimodule $E=A$ (see [1, 15]). The case of
semigroup algebras is considered in [3] for the Banach $A$-bimodule $A^{*}$.
Let $A$ be a Banach algebra and $\sigma(A)$ be the set of all non-zero
multiplicative linear functionals on $A$. If $\varphi\in\sigma(A)\cup\\{0\\}$
and $E$ is an arbitrary Banach space, then $E$ can be viewed as a Banach left
or right $A$-module by the following actions. For $a\in A,x\in E$:
$None$ $a\cdot x=\varphi(a)x,$ $None$ $x\cdot a=\varphi(a)x.$
If the left action of $A$ on $E$ is given by (2.1), then it is easily verified
that the right action of $A$ on the dual $A$-module $E^{*}$ is given by
$f\cdot a=\varphi(a)f$ for all $f\in E^{*},a\in A$. Throughout, by a
$(\varphi,A)$-bimodule $E$, we mean that $E$ is a Banach $A$-bimodule for
which the left module action is given by (2.1). $(A,\varphi)$-bimodule is
defined similarly by (2.2).
###### Definition 1.2.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
approximately $\varphi$-amenable, if every derivation $D$ from $A$ into the
dual $A$-bimodule $E^{*}$ is approximately inner for all
$(\varphi,A)$-bimodules $E$.
###### Definition 1.3.
A Banach algebra $A$ is approximately right character amenable, if for every
$\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$, every
derivation $D$ from $A$ into the dual $A$-bimodule $E^{*}$ is approximately
inner.
Approximately left character amenability is defined similarly by considering
$(A,\varphi)$-bimodules $E$.
$A$ is approximately character amenable if it is both approximately left and
right character amenable.
###### Definition 1.4.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
$\varphi$-contractible, if every derivation $D:A\rightarrow E$ is inner, for
all $(A,\varphi)$-bimodules $E$.
###### Definition 1.5.
A Banach algebra $A$ is left character contractible, if for every
$\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$, every
derivation $D:A\rightarrow E$ is inner.
Right character contractibility is defined similarly by considering
$(A,\varphi)$-bimodules $E$.
$A$ is character contractible if it is both left and right character
contractible.
###### Definition 1.6.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
approximately $\varphi$-contractible, if every derivation $D:A\rightarrow E$
is approximately inner for all $(A,\varphi)$-bimodules $E$.
###### Definition 1.7.
A Banach algebra $A$ is approximately left character contractible, if for
every $\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$,
every derivation $D:A\rightarrow E$ is approximately inner.
Approximately right character contractible is defined similarly by considering
$(A,\varphi)$-bimodules $E$.
$A$ is approximately character contractible if it is both approximately left
and right character contractible.
Any statement about approximate left character amenability (contractibility)
turns into an analogous statement about approximate right character
amenability (contractibility, respectively) by simply replacing $A$ by its
opposite algebra. Approximate right character amenability (contractibility) of
$A$ is equivalent to approximate $\varphi$-amenability
($\varphi$-contractibility) for all $\varphi\in\sigma(A)$ together with
approximate $0$-amenability ($0$-contractibility, respectively).
The qualifier uniform on the above definitions will indicate that the
convergence of the net is uniform over the unit ball of $A$. Similarly $w^{*}$
will indicate that convergence is in the appropriate $w^{*}$-topology. Let
$CC$ denote character contractibility, $UACC$ denote uniform approximate
character contractibility, $ACC$ denote approximate character contractibility,
$CA$ denote character amenability, $UACA$ denote uniform approximate character
amenability and $ACA$ denote approximate character amenability. Clearly the
relations among the various character amenability are as follows:
$Contractibility\Rightarrow CC\Rightarrow UACC\Rightarrow ACC$
$\Downarrow\ \ \ \ \not\Uparrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \Downarrow\ \ \ \ \
\ \ \ \ \ \ \Downarrow\ \ \ \ \ \ \ \ \Downarrow$
$Amenability\Rightarrow CA\Rightarrow UACA\Rightarrow ACA$
In this paper, we prove that $ACA\Leftrightarrow ACC$ (Theorem 5.2),
$UACA\Leftrightarrow CA$ (Theorem 5.7), $UACC\Leftrightarrow CC$ (Theorem
5.9). Moreover, in Section 6, Example 1 shows that $ACA\nRightarrow UACA$.
Example 2 shows that $CA\nRightarrow Amenability$ and $CA\nRightarrow CC$.
Thus $ACC\nRightarrow UACC$. For commutative Banach algebra, we obtain that
$Contractibility\Leftrightarrow CC$ (Theorem 5.11). For non-commutative Banach
algebra, Example 3 shows that $CC\nRightarrow Amenability$.
In this paper, the second dual $A^{**}$ of a Banach algebra $A$ will always be
equipped with the first Arens product [7] which is defined as follows. For
$a,b\in A,f\in A^{*}$ and $m,n\in A^{**}$, the elements $f\cdot a$ and $m\cdot
f$ of $A^{*}$ and $mn\in A^{**}$ are defined by
$(f\cdot a)(b)=f(ab),(m\cdot f)(b)=m(f\cdot b),mn(f)=m(n\cdot f),$
respectively. With this multiplication, $A^{**}$ is a Banach algebra and $A$
is a subalgebra of $A^{**}$. Moreover, for all $m,n\in A^{**}$ and
$\varphi\in\sigma(A)$, $(mn)(\varphi)=m(\varphi)n(\varphi)$. Consequently,
each $\varphi\in\sigma(A)$ extends uniquely to some element $\varphi^{**}$ of
$\sigma(A^{**})$. The kernel of $\varphi^{**}$, $\ker(\varphi^{**})$, contains
$\ker\varphi$ in the same sense that $A^{**}$ naturally contains $A$. Since
each of these ideals has codimension 1, the theory of second dual shows that
$\ker\varphi$ is $w^{*}$-dense in $\ker(\varphi^{**})$ and that
$\ker\varphi^{**}=(\ker\varphi)^{**}$. For further details the reader is
referred to [7].
The organization of the paper is as follows. In Section 2 we characterize
(uniform) approximate character amenability in three different ways. In
Section 3 we are concerned with hereditary properties of (uniform) approximate
character amenability.
In Section 4 we characterize character contractibility and (uniform)
approximate character contractibility in two different ways.
Section 5 is devoted to the relations among generalized notions of character-
amenability. We prove that approximate character amenability and approximate
character contractibility are the same properties, as are uniform approximate
character amenability (contractibility) and character amenability
(contractibility, respectively). For commutative Banach algebra, character
contractibility and contractibility are the same properties.
Section 6 gives three examples. The first example shows that there exists a
Banach algebra which is approximately character amenable but not uniformly
approximately character amenable. The second example shows that there exists a
Banach algebra which is character amenable but neither amenable nor character
contractible. The last example shows that there exists a Banach algebra which
is character contractible but not amenable.
## 2\. Characterization of approximately character amenability
In this section, we first characterize (uniform) approximate
$\varphi$-amenability in three different ways and then characterize (uniform)
approximate character amenability in these different ways.
Suppose that $A$ is a Banach algebra, we make $E=A$ into a Banach $A$-bimodule
as usual by $a\cdot b=ab,b\cdot a=ba,$ for all $a\in A,b\in E$. Then
$A^{*},A^{**}$ are dual $A$-bimodules and the module actions are given by
$(a\cdot f)(b)=f(ba),(f\cdot a)(b)=f(ab);(a\cdot m)(f)=m(f\cdot a),(m\cdot
a)(f)=m(a\cdot f)$ for all $a,b\in A,f\in A^{*},m\in A^{**}$. If we take
$A^{**}$ with the first Arens product, then $am=a\cdot m,ma=m\cdot a$ for all
$a\in A,m\in A^{**}$. Let $\varphi\in\sigma(A)$, a net
$\\{m_{\alpha}\\}\subset A^{**}$ is called an approximate $\varphi$-mean, if
$m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$. The
following proposition characterizes approximate $\varphi$-amenable in terms of
approximately $\varphi$-mean. The corresponding result characterizing
$\varphi$-amenability of a Banach algebra was obtained in [16, Theorem 1.1].
###### Proposition 2.1.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following are
equivalent:
$(i)$ $A$ is (uniformly) approximately $\varphi$-amenable;
$(ii)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$ (uniformly
on the unit ball of $A$, respectively);
$(iii)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$ (uniformly
on the unit ball of $A$, respectively);
$(iv)$ Give $(\ker\varphi)^{**}$ a dual $A$-bimodule structure by taking the
right action to be $m\cdot a=\varphi(a)m$ for $m\in A^{**}$ and taking the
left action to be the natural one. Then any continuous derivation
$D:A\rightarrow(\ker\varphi)^{**}$is (uniformly, respectively) approximately
inner.
Proof. $(i)\Rightarrow(iv)$ and $(ii)\Rightarrow(iii)$ is clear. Therefore, in
order to establish the proposition it suffices to show the implications
$(i)\Rightarrow(ii)$, $(iii)\Rightarrow(i)$ and $(iv)\Rightarrow(ii)$.
$(i)\Rightarrow(ii)$ We first define an action of $A$ on $E=A^{*}$ by
$a*f=\varphi(a)f,f*a=f\cdot a,~{}a\in A,f\in E,$
then $E$ is $(\varphi,A)$-bimodule and $A^{**}$ is a dual $A$-bimodule and
module actions are defined by $a*m=a\cdot m,m*a=\varphi(a)m$, for all $a\in
A,m\in A^{**}$.
We know that $\varphi\in A^{*}$, and $a\cdot\varphi=\varphi\cdot
a=\varphi(a)\varphi$. Therefore
$\mathbb{C}\varphi=\\{\lambda\varphi,\lambda\in\mathbb{C}\\}$ is a closed
submodule of $A^{*}$ and $A^{*}/\mathbb{C}\varphi$ is a $(\varphi,A)$-bimodule
for which the module actions are given by $a\cdot[f]=\varphi(a)[f],[f]\cdot
a=[f*a]$, for all $a\in A,[f]\in A^{*}/\mathbb{C}\varphi$.
Choose any $m\in A^{**}$ with $m(\varphi)=1$, and define a derivation
$D:A\rightarrow A^{**}$ by $D(a)=a\cdot m-\varphi(a)m$, then $D(a)\in\\{n\in
A^{**},n(\varphi)=0\\}=\\{\mathbb{C}\varphi\\}^{\bot}\cong(A^{*}/\mathbb{C}\varphi)^{*}$.
Since $A$ is approximately $\varphi$-amenable, it follows that there exists a
net $\\{n_{\alpha}\\}\subset\\{\mathbb{C}\varphi\\}^{\bot}$ such that
$D(a)=\lim\limits_{\alpha}(a\cdot n_{\alpha}-\varphi(a)n_{\alpha})$. Set
$m_{\alpha}=m-n_{\alpha}$, then $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.
$(iii)\Rightarrow(i)$ Let $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$. Let $E$ be a
$(\varphi,A)$-bimodule. Also, let $D:A\rightarrow E^{*}$ be a continuous
derivation, and let $D^{{}^{\prime}}=D^{*}|_{E}:E\rightarrow A^{*}$ and
$g_{\alpha}=(D^{{}^{\prime}})^{*}(m_{\alpha})\in E^{*}$. Then, for all $a,b\in
A$ and $x\in E$,
$\displaystyle D^{{}^{\prime}}(x\cdot a)(b)$ $\displaystyle=$ $\displaystyle
D(b)(x\cdot a)$ $\displaystyle=$ $\displaystyle D(ab)(x)-D(a)(b\cdot x)$
$\displaystyle=$ $\displaystyle(D^{{}^{\prime}}(x)\cdot
a)(b)-D(a)(x)\varphi(b),$
and hence $D^{{}^{\prime}}(x\cdot a)=D^{{}^{\prime}}(x)\cdot
a-D(a)(x)\varphi.$ This implies that
$\displaystyle(a\cdot g_{\alpha})(x)$ $\displaystyle=$ $\displaystyle
g_{\alpha}(x\cdot a)$ $\displaystyle=$
$\displaystyle(D^{{}^{\prime}})^{*}(m_{\alpha})(x\cdot a)$ $\displaystyle=$
$\displaystyle m_{\alpha}(D^{{}^{\prime}}(x\cdot a))$ $\displaystyle=$
$\displaystyle m_{\alpha}(D^{{}^{\prime}}(x)\cdot
a)-D(a)(x)m_{\alpha}(\varphi)$ $\displaystyle=$ $\displaystyle(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))-D(a)(x)m_{\alpha}(\varphi).$
It follows that
$\displaystyle||(a\cdot g_{\alpha})(x)-\varphi(a)g_{\alpha}(x)+D(a)(x)||$
$\displaystyle\leq$ $\displaystyle||\varphi(a)g_{\alpha}(x)-(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))||$ $\displaystyle+$
$\displaystyle||D(a)(x)-D(a)(x)m_{\alpha}(\varphi)||.$
Hence,
$\displaystyle\lim\limits_{\alpha}||(a\cdot
g_{\alpha})(x)-\varphi(a)g_{\alpha}(x)+D(a)(x)||$ $\displaystyle\leq$
$\displaystyle\lim\limits_{\alpha}\\{||\varphi(a)m_{\alpha}(D^{{}^{\prime}}(x))-(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))||$ $\displaystyle+$
$\displaystyle||\varphi(a)g_{\alpha}(x)-\varphi(a)m_{\alpha}(D^{{}^{\prime}}(x))||$
$\displaystyle+$ $\displaystyle||D(a)(x)-D(a)(x)m_{\alpha}(\varphi)||\\}.$
Thus, for each $a\in A$, $D(a)=\lim\limits_{\alpha}\varphi(a)g_{\alpha}-a\cdot
g_{\alpha}$. Combining this with the equation $g_{\alpha}\cdot
a=\varphi(a)g_{\alpha}$, we obtain $D(a)=\lim\limits_{\alpha}g_{\alpha}\cdot
a-a\cdot g_{\alpha}$ for all $a\in A$. Set $f_{\alpha}=-g_{\alpha}$, then
$D(a)=\lim\limits_{\alpha}a\cdot f_{\alpha}-f_{\alpha}\cdot a$ for all $a\in
A$. Since $D$ was arbitrary, it follows that $A$ is approximately
$\varphi$-amenable.
$(iv)\Rightarrow(ii)$ Choose $b\in A$ with $\varphi(b)=1$. Then
$Da=ab-\varphi(a)b$, $a\in A$, defines a derivation from A into
$(\ker\varphi)^{**}$. By $(iv)$, D is approximately inner, it follows that
there exists a net $\\{n_{\alpha}\\}\subset(\ker\varphi)^{**}$ such that
$D(a)=\lim\limits_{\alpha}a\cdot n_{\alpha}-\varphi(a)n_{\alpha}$ for all
$a\in A$.
Set $m_{\alpha}=b-n_{\alpha}$. Then $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.
The proof in the case of uniform approximate $\varphi$-amenability is similar.
$\blacksquare$
###### Proposition 2.2.
For a Banach algebra $A$ and $\varphi\in\sigma(A)$, the following are
equivalent:
$(i)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$ ;
$(ii)$ There exists a net $\\{n_{\beta}\\}\subset A$ such that
$\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$.
Proof. It suffices to show that $(ii)\Rightarrow(i)$.
Suppose that $(ii)$ holds. Take $\varepsilon>0$ and finite sets $F\subset A$,
$\Phi\subset A^{*}$. Then there exists $\alpha$ such that
$|(a\cdot
m_{\alpha}-\varphi(a)m_{\alpha})(f)|<\frac{\varepsilon}{3},|m_{\alpha}(\varphi)|>1-\varepsilon,a\in
F,f\in\Phi.$
By Goldstine’s theorem, there exists $b_{\alpha}\in A$ such that
$|f(b_{\alpha})-m_{\alpha}(f)|<\frac{\varepsilon}{3K},f\in\Phi\cup\Phi\cdot
F\cup\\{\varphi\\},$
where $K=\sup\\{|\varphi(a)|,a\in F\\}$.
Thus, for any $f\in\Phi$ and $a\in F$,
$\displaystyle|f(ab_{\alpha}-\varphi(a)b_{\alpha})|$ $\displaystyle\leq$
$\displaystyle|f(ab_{\alpha})-(a\cdot m_{\alpha})(f)|$ $\displaystyle+$
$\displaystyle|(a\cdot m_{\alpha})(f)-\varphi(a)m_{\alpha}(f)|$
$\displaystyle+$
$\displaystyle|\varphi(a)m_{\alpha}(f)-\varphi(a)f(b_{\alpha})|$
$\displaystyle\leq$ $\displaystyle|(f\cdot a)(b_{\alpha})-m_{\alpha}(f\cdot
a)|$ $\displaystyle+$ $\displaystyle|(a\cdot
m_{\alpha})(f)-\varphi(a)m_{\alpha}(f)|$ $\displaystyle+$
$\displaystyle|\varphi(a)m_{\alpha}(f)-\varphi(a)f(b_{\alpha})|$
$\displaystyle<$ $\displaystyle\varepsilon.$
Then there exists a net $\\{b_{\lambda}\\}\subset A$ such that for every $a\in
A$, $ab_{\lambda}-\varphi(a)b_{\lambda}\rightarrow
0,\varphi(b_{\lambda})\rightarrow 1$ weakly in $A$.
Finally, for each finite set $F\subset A$, say
$F=\\{a_{1},a_{2},\cdots,a_{n}\\}$,
$(a_{1}b_{\lambda}-\varphi(a_{1})b_{\lambda},a_{2}b_{\lambda}-\varphi(a_{2})b_{\lambda},\cdots,a_{n}b_{\lambda}-\varphi(a_{n})b_{\lambda},\varphi(b_{\lambda}))\rightarrow(0,0,\cdots,0,1)$
weakly in $A^{n}\oplus\mathbb{C}.$ Thus
$(0,0,\cdots,0,1)\in\overline{co}^{weak}\\{(a_{1}b_{\lambda}-\varphi(a_{1})b_{\lambda},a_{2}b_{\lambda}-\varphi(a_{2})b_{\lambda},\cdots,a_{n}b_{\lambda}-\varphi(a_{n})b_{\lambda},\varphi(b_{\lambda}))\\}$.
The Hahn-Banach theorem now gives that for each $\varepsilon>0$, there exists
$u_{\varepsilon,F}\in co\\{b_{\lambda}\\}$, such that
$||au_{\varepsilon,F}-\varphi(a)u_{\varepsilon,F}||<\varepsilon,|\varphi(u_{\varepsilon,F})-1|<\varepsilon$
for all $a\in F$. Therefore, there exists a net $\\{n_{\beta}\\}\subset A$
such that $\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$.
$\blacksquare$
Let $A$ be a Banach algebra, $\\{e_{\alpha}\\}$ be a net of $A$. We call
$\\{e_{\alpha}\\}$ a right approximate identity for $A$, if
$||ae_{\alpha}-a||\rightarrow 0$ for all $a\in A$. Left (two-sided)
approximate identity for $A$ is defined similarly. We call $\\{e_{\alpha}\\}$
a bounded right (left, two-sided, respectively) approximate identity for $A$,
if it is a bounded net. The next proposition characterizes approximate
$\varphi$-amenability in terms of the existence of right approximate identity
for $\ker\varphi$. The corresponding result characterizing
$\varphi$-amenability of a Banach algebra was obtained in [16, Proposition
2.1].
###### Lemma 2.3.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. If the ideal
$I_{\varphi}=\ker\varphi$ has a right approximate identity, then $A$ is
approximately $\varphi$-amenable.
Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$. Then
$a_{0}=u_{0}^{2}-u_{0}\in I_{\varphi}$. Let $\\{b_{\alpha}\\}$ be a right
approximate identity for $I_{\varphi}$.
Set $m_{\alpha}=u_{0}-u_{0}b_{\alpha}\in A$. Then, for any $b\in I_{\varphi}$,
$||b(u_{0}-u_{0}b_{\alpha})||=||bu_{0}-bu_{0}b_{\alpha}||\rightarrow 0.$
Furthermore,
$||u_{0}(u_{0}-u_{0}b_{\alpha})-(u_{0}-u_{0}b_{\alpha})||=||u_{0}^{2}-u_{0}^{2}b_{\alpha}-u_{0}+u_{0}b_{\alpha}||=||a_{0}-a_{0}b_{\alpha}||\rightarrow
0.$
It follows that, $\varphi(m_{\alpha})=1$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$. Thus
$A$ is approximately $\varphi$-amenable by Proposition 2.1/2.2. $\blacksquare$
###### Lemma 2.4.
Suppose that $A$ is approximately $\varphi$-amenable for some
$\varphi\in\sigma(A)$ and that $A$ has a right approximate identity. Then
$I_{\varphi}$ has a right approximate identity.
Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$ and
$A=\mathbb{C}u_{0}\oplus I_{\varphi}$. Let
$n_{\beta}=\lambda_{\beta}u_{0}+b_{\beta}$ be a right approximate identity for
$A$, where $b_{\beta}\in I_{\varphi}$ and $\lambda_{\beta}\rightarrow 1$.
Since $A$ is approximately $\varphi$-amenable, it follows from Proposition
2.1/2.2 that there exists a net
$m_{\alpha}=\lambda_{\alpha}u_{0}+b_{\alpha}\in A$ such that
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, where $b_{\alpha}\in
I_{\varphi}$ and $\lambda_{\alpha}\rightarrow 1$.
Set $e_{\alpha,\beta}=b_{\beta}-b_{\alpha}$. Notice that
$||be_{\alpha,\beta}-b||\rightarrow 0,$ for any $b\in I_{\varphi}$. In fact,
for any $b\in I_{\varphi}$, $||bn_{\beta}-b||\rightarrow 0$ and
$||bm_{\alpha}||\rightarrow 0$. It follows that
$||bu_{0}+bb_{\beta}-b||\rightarrow 0$ and $||bu_{0}+bb_{\alpha}||\rightarrow
0$. Thus $||be_{\alpha,\beta}-b||\rightarrow 0$ for all $b\in I_{\varphi}$.
$\blacksquare$
The following result follows immediately from Lemma 2.3/2.4 and we omit its
proof.
###### Proposition 2.5.
Let $A$ be a Banach algebra with a right approximate identity and
$\varphi\in\sigma(A)$. Then $A$ is approximate $\varphi$-amenable if and only
if $I_{\varphi}$ has a right approximate identity.
If $A$ is a Banach algebra and $A\widehat{\otimes}A$ denotes the projective
product [20], then the corresponding diagonal operator is defined as
$\triangle:A\widehat{\otimes}A\rightarrow A,a\otimes b\rightarrow ab.$
Then $A\widehat{\otimes}A$ becomes a Banach $A$-bimodule through
$a\cdot(b\otimes c)=ab\otimes c,(b\otimes c)\cdot a=b\otimes ca$
for all $a,b,c\in A$. By [20, Theorem 2.2.4], $A$ is amenable if and only if
there is a net $\\{m_{\alpha}\\}\subset A\widehat{\otimes}A$ such that $a\cdot
m_{\alpha}-m_{\alpha}\cdot a\rightarrow 0,a\triangle m_{\alpha}\rightarrow a$
for all $a\in A$. Such a net is called an approximate diagonal for $A$. The
following proposition characterizes approximate $\varphi$-amenability in terms
of approximate diagonal for $A$.
###### Proposition 2.6.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following are
equivalent:
$(i)$ $A$ is (uniformly) approximately $\varphi$-amenable;
$(ii)$ There exists a net $\\{M_{\alpha}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$ and $||a\cdot
M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$ (uniformly on
the unit ball of $A$, respectively);
$(iii)$ There exists a net $\\{M_{\alpha}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)\rightarrow 1$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).
Proof. $(i)\Rightarrow(ii)$ Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$.
Then $A=\mathbb{C}u_{0}\oplus I_{\varphi}$, where $I_{\varphi}=\ker\varphi$.
We define an action of $A$ on $X=A\widehat{\otimes}A$ by
$a\cdot(b\otimes c)=ab\otimes c,(b\otimes c)\cdot a=\varphi(a)b\otimes
c,~{}~{}a,b,c\in A.$
Then the dual $A$-bimodule $(A\widehat{\otimes}A)^{*}$ is a
$(\varphi,A)$-bimodule.
We know that $\varphi\otimes\varphi\in(A\otimes A)^{*}$, and
$a\cdot(\varphi\otimes\varphi)=(\varphi\otimes\varphi)\cdot
a=\varphi(a)\varphi\otimes\varphi$. Therefore
$\mathbb{C}\varphi\otimes\varphi$ is a closed submodule of $(A\otimes A)^{*}$
and $(A\otimes A)^{*}/\mathbb{C}\varphi\otimes\varphi$ is a
$(\varphi,A)$-bimodule for which the module actions are given by
$a\cdot[f]=\varphi(a)[f],[f]\cdot a=[f*a]$, for all $a\in A,[f]\in(A\otimes
A)^{*}/\mathbb{C}\varphi\otimes\varphi$.
Moreover, define a derivation $D:A\rightarrow(A\widehat{\otimes}A)^{**}$ given
by $D(a)=au_{0}\otimes u_{0}-\varphi(a)u_{0}\otimes u_{0}$. Then
$D(a)\in\ker(\varphi\otimes\varphi)\subset\\{\mathbb{C}\varphi\otimes\varphi\\}^{\bot}\cong[(A\otimes
A)^{*}/\mathbb{C}\varphi\otimes\varphi]^{*}$. Since $A$ is approximately
$\varphi$-amenable, it follows that there exists a net
$\\{N_{\alpha}\\}\subset\\{\mathbb{C}\varphi\otimes\varphi\\}^{\bot}$ such
that $D(a)=\lim\limits_{\alpha}a\cdot N_{\alpha}-\varphi(a)N_{\alpha}$ for all
$a\in A$.
Set $M_{\alpha}=u_{0}\otimes u_{0}-N_{\alpha}$, then
$(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$ and $||a\cdot
M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$.
$(ii)\Rightarrow(iii)$ It is clear.
$(iii)\Rightarrow(i)$ Now suppose that $(iii)$ holds. Let $E$ be a
$(\varphi,A)$-bimodule and let $D:A\rightarrow E^{*}$ be a derivation. For
each $\alpha$, set $f_{\alpha}(x)=M_{\alpha}(\mu_{x})$, where for $a,b\in
A,x\in E,\mu_{x}(a\otimes b)=(D(a)\cdot b)(x)$. Then, with
$(m_{\alpha}^{\nu})\subset A\widehat{\otimes}A$ converging $w^{*}$ to
$M_{\alpha}$, and noting that for $m\in A\widehat{\otimes}A$,
$\mu_{x\cdot a}(m)=(\mu_{x}\cdot a)(m)-(D(a)\cdot\bigtriangleup(m))(x),$
we have
$\displaystyle(a\cdot f_{\alpha})(x)$ $\displaystyle=$ $\displaystyle
f_{\alpha}(x\cdot a)$ $\displaystyle=$ $\displaystyle M_{\alpha}(\mu_{x\cdot
a})$ $\displaystyle=$ $\displaystyle\lim\limits_{\nu}\mu_{x\cdot
a}(m_{\alpha}^{\nu})$ $\displaystyle=$ $\displaystyle M_{\alpha}(\mu_{x}\cdot
a)-\lim\limits_{\nu}[D(a)\cdot\bigtriangleup(m_{\alpha}^{\nu})](x)$
$\displaystyle=$ $\displaystyle(a\cdot
M_{\alpha})(\mu_{x})-\lim\limits_{\nu}\varphi(\bigtriangleup(m_{\alpha}^{\nu}))D(a)(x)$
$\displaystyle=$ $\displaystyle(a\cdot
M_{\alpha})(\mu_{x})-(\bigtriangleup^{**}M_{\alpha})(\varphi)D(a)(x).$
Thus, for $a\in A$ and $x\in E$,
$\displaystyle||D(a)(x)-(f_{\alpha}\cdot a-a\cdot f_{\alpha})(x)||$
$\displaystyle\leq$
$\displaystyle||D(a)(x)-(\bigtriangleup^{**}M_{\alpha})(\varphi)D(a)(x)||$
$\displaystyle+$ $\displaystyle||(a\cdot M_{\alpha})(\mu_{x})-(f_{\alpha}\cdot
a)(x)||$ $\displaystyle\leq$
$\displaystyle|1-(\bigtriangleup^{**}M_{\alpha})(\varphi)|\cdot||x||\cdot||D(a)||$
$\displaystyle+$ $\displaystyle||(a\cdot
M_{\alpha})-\varphi(a)M_{\alpha}||\cdot||\mu_{x}||,$
whence $D(a)=\lim\limits_{\alpha}(f_{\alpha}\cdot a-a\cdot f_{\alpha})$ for
all $a\in A$. Set $g_{\alpha}=-f_{\alpha}$, then
$D(a)=\lim\limits_{\alpha}(a\cdot g_{\alpha}-g_{\alpha}\cdot a)$ for all $a\in
A$. Since $D$ was arbitrary, it follows that $A$ is approximately
$\varphi$-amenable.
The proof in the case of uniformly approximately $\varphi$-amenable is
similar. $\blacksquare$
The following proposition characterizes (uniform) approximate $0$-amenability
in terms of right approximate identity for $A$.
###### Proposition 2.7.
Banach algebra $A$ is (uniform) approximate $0$-amenability if and only if $A$
has a (bounded, respectively) right approximate identity.
Proof. Suppose that $E$ is a $(\varphi,A)$-bimodule and $D:A\rightarrow E^{*}$
is a derivation. Then $D(b)\cdot a=0$, for all $a,b\in A$. If
$\\{e_{\alpha}\\}$ is (bounded) right approximate identity for $A$, then
$D(a)=\lim\limits_{\alpha}D(ae_{\alpha})=\lim\limits_{\alpha}a\cdot
D(e_{\alpha}).$ This implies that $A$ is (uniformly) approximately
$0$-amenable.
The converse is clear from [9, Lemma 2.1,Lemma 2.2, Theorem 4.2].
$\blacksquare$
Note that (uniform) approximate right character amenability of $A$ is
equivalent to (uniform, respective) approximate $\varphi$-amenability for all
$\varphi\in\sigma(A)$ together with (uniform, respectively) approximate
$0$-amenability. Any statement about (uniform) approximate right character
amenability turns into an analogous statement about (uniform, respectively)
approximate left character amenability by simply replacing $A$ by its opposite
algebra. Then standard arguments of Proposition 2.1, 2.5, 2.6 and 2.7 apply,
we have the following theorems.
###### Theorem 2.8.
For a Banach algebra $A$ the following are equivalent:
$(i)$ $A$ is (uniformly) approximately character amenable;
$(ii)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)=1,m_{\alpha}^{{}^{\prime}}(\varphi)=1(or~{}m_{\alpha}(\varphi)\rightarrow
1,m_{\alpha}^{{}^{\prime}}(\varphi)\rightarrow 1)$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively);
$(iii)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A$, such that
$\varphi(m_{\alpha})=1,\varphi(m_{\alpha}^{{}^{\prime}})=1$ (or
$\varphi(m_{\alpha})\rightarrow 1,\varphi(m_{\alpha}^{{}^{\prime}})\rightarrow
1)$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow
0$ for all $a\in A$ (uniformly on the unit ball of $A$, respectively);
$(iv)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$,
$(\bigtriangleup^{**}M_{\alpha}^{{}^{\prime}})(\varphi)=1(or~{}(\bigtriangleup^{**}M_{\alpha})(\varphi)\rightarrow
1,(\bigtriangleup^{**}M_{\alpha}^{{}^{\prime}})(\varphi)\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).
###### Theorem 2.9.
A Banach algebra $A$ is left (right) approximately character amenable if and
only if $\ker\varphi$ has a left (right, respectively) approximate identity
for every $\varphi\in\sigma(A)\cup\\{0\\}$.
## 3\. Hereditary properties of approximate character amenability
In this section, we are concerned with hereditary properties of (uniform)
approximate character amenability.
###### Proposition 3.1.
Suppose that $A$ is (uniformly) approximately character amenable
(contractible) and $\Phi:A\rightarrow B$ is a continuous epimorphism. Then $B$
is (uniformly, respectively) approximately character amenable (contractible,
respectively).
Proof. The standard argument, [9, Proposition 2.2] applies. $\blacksquare$
###### Proposition 3.2.
Suppose that $A$ is (uniformly) approximately character amenable
(contractible), and $J$ is a closed two-sided ideal of $A$. Then $A/J$ is
(uniformly, respectively) approximately character amenable (contractible,
respectively). If $J$ is character amenable (contractible) and $A/J$ is
(uniformly) approximately character amenable (contractible), then $A$ is
(uniformly, respectively)approximately character amenable (contractible,
respectively).
Proof. The standard argument, [19, Proposition 1.30] applies. $\blacksquare$
###### Proposition 3.3.
Let $A$ be a Banach algebra and $J$ a weakly complemented ideal of $A$. Let
$\varphi\in\sigma(A)$ satisfy $\varphi|_{J}\neq 0$. If $A$ is (uniformly)
approximately $\varphi$-amenable, then J is (uniformly, respectively)
approximately $\varphi|_{J}$-amenable.
Proof. Since $A$ is approximately $\varphi$-amenable, it follows from
Proposition 2.1 that there exists a net $\\{m_{\alpha}\\}\subset A^{**}$, such
that $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.
Since $J$ weakly complemented of $A$, there exist a closed subspace $X$ of
$A^{*}$ such that $A^{*}=J^{\bot}\oplus X$. That is to say, there exists $K>0$
such that for any $F\in A^{*}$, $F=x_{F}+y_{F}$, where $x_{F}\in
J^{\bot},y_{F}\in X$, and $||x_{F}||\leq K||F||,||y_{F}||\leq K||F||$. If, in
addition, $a\in J$, then $x_{F}\cdot a=0$. Thus
$||\varphi(a)m_{\alpha}(x_{F})||\rightarrow 0$ for all $a\in J,$ and uniformly
for $F\in A^{*}$ and $||F||\leq 1$. Choose $a=u_{0}\in J$ with
$\varphi(u_{0})=1$, then $||m_{\alpha}(x_{F})||\rightarrow 0$ uniformly for
$||F||\leq 1$.
Set $n_{\alpha}(f)=m_{\alpha}(y_{F})$ for $f\in J^{*}$, where $F$ is any
extension of $f$. Notice that $||a\cdot
n_{\alpha}-\varphi(a)n_{\alpha}||\rightarrow 0,$ and
$n_{\alpha}(\varphi|_{J})\rightarrow 1$ for all $a\in J$. To see this, for
$a,b\in J,f\in J^{*}$ and $F$ is a extension of $f$,
$(y_{F}\cdot a)(b)=y_{F}(ab)=f(ab),~{}y_{F\cdot a}(b)=f(ab).$
It follows that there is a $x\in J^{\bot}$ such that $y_{F}\cdot a=x+y_{F\cdot
a}$ and $||y_{F\cdot a}||\leq K||y_{F}\cdot a||,||x||\leq K||y_{F}\cdot a||$.
Then, for any $f\in J^{*}$ and $F$ an extension of $f$,
$\displaystyle|a\cdot n_{\alpha}(f)-\varphi(a)n_{\alpha}(f)|$ $\displaystyle=$
$\displaystyle|n_{\alpha}(f\cdot a)-\varphi(a)m_{\alpha}(y_{F})|$
$\displaystyle=$ $\displaystyle|m_{\alpha}(y_{F\cdot
a})-\varphi(a)m_{\alpha}(y_{F})|$ $\displaystyle\leq$
$\displaystyle|m_{\alpha}(y_{F}\cdot
a)-\varphi(a)m_{\alpha}(y_{F})|+|m_{\alpha}(x)|$ $\displaystyle\leq$
$\displaystyle||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\cdot||y_{F}||+|m_{\alpha}(x)|.$
It follows that $||a\cdot n_{\alpha}-\varphi(a)n_{\alpha}||\rightarrow 0$ and
$n_{\alpha}(\varphi|_{J})=m_{\alpha}(\varphi)-m_{\alpha}(x_{\varphi})\rightarrow
1$ for all $a\in J$. Then, by Proposition 2.1, $J$ is approximately
$\varphi|_{J}$-amenable.
The proof in the case of uniform approximate $\varphi$-amenability is similar.
$\blacksquare$
###### Lemma 3.4.
Let $A$ be a Banach algebra and $J$ is an ideal of $A$, which has a right or
left approximate identity. Then every $\varphi\in\sigma(J)$ can be extended to
a $\widetilde{\varphi}$ in $\sigma(A)$.
Proof. Assume that $\\{e_{\alpha}\\}$ is right approximate identity for $J$
and $\varphi\in\sigma(J)$. Choose $u_{0}\in J$ such that $\varphi(u_{0})=1$,
then $J=\mathbb{C}u_{0}\oplus I_{\varphi}$, and $u_{0}a-u_{0}\in I_{\varphi}$
for all $a\in J$, where $I_{\varphi}=\ker\varphi$.
Set $\widetilde{\varphi}(a)=\varphi(u_{0}a)$ for all $a\in A$. Then
$\widetilde{\varphi}|_{J}=\varphi$, and
$\widetilde{\varphi}(a_{1}a_{2})=\varphi(u_{0}a_{1}a_{2})=\widetilde{\varphi}(a_{1})\widetilde{\varphi}(a_{2})$,
since
$u_{0}a_{1}a_{2}-u_{0}a_{1}u_{0}a_{2}=(u_{0}a_{1}-u_{0}a_{1}u_{0})a_{2}=\lim\limits_{\alpha}(u_{0}a_{1}-u_{0}a_{1}u_{0})e_{\alpha}a_{2}\in
I_{\varphi}$. It follows that $\widetilde{\varphi}\in\sigma(A)$.
The proof in the case of a left approximate identity is similar.
$\blacksquare$
###### Theorem 3.5.
Let $A$ be a Banach algebra and $J$ a weakly complemented ideal of $A$ with
(bounded) left and right approximate identities. Suppose that $A$ is
(uniformly) approximately character amenable. Then $J$ is (uniformly,
respectively)approximately character amenable.
Proof. Clearly $J$ is (uniformly) approximately $\varphi$-amenable, for any
$\varphi\in\sigma(J)\cup\\{0\\}$, by Proposition 2.7, 3.3 and Lemma 3.4. That
is to say, $J$ is (uniformly) approximately right character amenable. The
proof in the case of (uniform) approximate left character amenability is
similar. Thus $J$ is (uniformly) approximately character amenable.
$\blacksquare$
By [18, Theorem 2.6] a Banach algebra $A$ without a unit is character amenable
if and only if $A^{\sharp}$ is character amenable. We obtain a similar result
for approximate character amenability.
###### Lemma 3.6.
Let $A$ be a Banach algebra without identity and let $A^{\sharp}$ denote the
unitization of $A$ by adjoining an identity $e$. Let
$\varphi\in\sigma(A)\cup\\{0\\}$ and let $\varphi_{e}$ be the unique extension
of $\varphi$ to an element of $\sigma(A^{\sharp})$. Then $A^{\sharp}$ is
approximately $\varphi_{e}$-amenable if $A$ is approximately
$\varphi$-amenable.
Proof. Assume that $A$ is approximately $\varphi$-amenable, where
$\varphi\in\sigma(A)$. Then the standard argument of [16, Lemma 3.2] applies,
so that $A^{\sharp}$ is approximately $\varphi_{e}$-amenable.
If $\varphi=0$, $A^{\sharp}=\mathbb{C}e\oplus A$ and $\varphi_{e}(A)=0$,
$\varphi_{e}(e)=1$. Since $A$ is approximately $0$-amenable, it follows from
Proposition 2.7 that $A$ has a right approximate identity $(b_{\alpha})$.
Set $m_{\alpha}=e-b_{\alpha}$, then
$||am_{\alpha}-\varphi_{e}(a)m_{\alpha}||\rightarrow 0$ and
$\varphi_{e}(m_{\alpha})=1$, for all $a\in A^{\sharp}$. Thus $A^{\sharp}$ is
approximately $\varphi_{e}$-amenable, by Proposition 2.1/2.2. $\blacksquare$
###### Theorem 3.7.
Let $A$ be a Banach algebra without a unit. Then $A$ is approximately
character amenable if and only if $A^{\sharp}$ is approximately character
amenable.
Proof. Assume $A$ is approximately character amenable, the argument of Lemma
3.6 applies, $A^{\sharp}$ is approximately character amenable.
For the converse, assume that $A^{\sharp}$ is approximately character
amenable, then $A^{\sharp}$ is both approximately right and left character
amenable. We shall show that $A$ has both right and left approximately
identities.
Define $\varphi\in\sigma(A^{\sharp})$ by $\varphi(A)=0,\varphi(e)=1$. Since
$A^{\sharp}$ is right approximately character amenable, Proposition 2.1/2.2
gives a net $(m_{\alpha})$ in $A^{\sharp}$ such that
$\varphi(m_{\alpha})\rightarrow 1$ and $||bm_{\alpha}||\rightarrow 0$ for all
$b\in A$. Writing $m_{\alpha}=\lambda_{\alpha}e+b_{\alpha}$ where
$\lambda_{\alpha}\in\mathbb{C},b_{\alpha}\in A$, it follows that
$||bb_{\alpha}+b||\rightarrow 0$ and hence $A$ has a right approximately
identity. Similarly, $A$ has a left approximately identity. Then, by Theorem
3.5, $A$ is approximately character amenable. $\blacksquare$
## 4\. Characterization of (approximate) character contractibility
In this section, we first characterize $\varphi$-contractibility, approximate
$\varphi$-contractibility and uniform approximate $\varphi$-contractibility in
two different ways and then characterize character contractibility,
approximate character contractibility and uniform approximate character
contractibility in the same different ways.
###### Proposition 4.1.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following
conditions are equivalent:
$(i)$ $A$ is $\varphi$-contractible;
$(ii)$ There exists $m\in A$ such that $\varphi(m)=1$ and $am=\varphi(a)m$ for
all $a\in A$.
Proof. $(i)\Rightarrow(ii)$ Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$.
Then $A=\mathbb{C}u_{0}\oplus I_{\varphi}$, where $I_{\varphi}=\ker\varphi$.
We define an action of $A$ on $X=A$ by $a\cdot x=ax,x\cdot a=\varphi(a)x,a\in
A,x\in X$. Then $X$ is a $(A,\varphi)$-bimodule.
Moreover, define a derivation $D:A\rightarrow A$ given by
$D(a)=au_{0}-\varphi(a)u_{0}$. Then $D(a)\in\ker\varphi$ and $\ker\varphi$ is
a submodule of $X$. Since $A$ is $\varphi$-contractible, it follows that there
exists $n\in\ker\varphi$ such that $D(a)=a\cdot n-n\cdot a$. Set $m=u_{0}-n$.
Then $\varphi(m)=1$ and $am=\varphi(a)m$ for all $a\in A$ .
$(ii)\Rightarrow(i)$ $m\in A$ is such that $\varphi(m)=1$ and $am=\varphi(a)m$
for all $a\in A$. Let $X$ be a $(A,\varphi)$-bimodule, and let $D:A\rightarrow
X$ be a derivation. Set $x=D(m)$, then
$\displaystyle a\cdot D(m)$ $\displaystyle=$ $\displaystyle D(am)-D(a)\cdot m$
$\displaystyle=$ $\displaystyle D(am)-\varphi(m)D(a)$ $\displaystyle=$
$\displaystyle\varphi(a)D(m)-D(a).$
It follows that $D(a)=\varphi(a)D(m)-a\cdot D(m)=D(m)\cdot a-a\cdot D(m)$ for
all $a\in A$. Since $D$ was arbitrary, it follows that $A$ is
$\varphi$-contractible. $\blacksquare$
###### Proposition 4.2.
For a Banach algebra $A$ the following are equivalent:
$(i)$ $A$ is approximately $\varphi$-contractible;
(ii) There exists a net $\\{m_{\alpha}\\}\subset A$ such that
$\varphi(m_{\alpha})=1$ and $||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow
0$ for all $a\in A$;
$(iii)$ There exists a net $\\{m_{\alpha}\\}\subset A$ such that
$\varphi(m_{\alpha})\rightarrow 1$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.
Proof. The proof is a minor modification of the proof of the analogous
statement in Proposition 4.1. $\blacksquare$
###### Lemma 4.3.
Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. If the ideal
$I_{\varphi}=\ker\varphi$ has a right identity, then $A$ is
$\varphi$-contractible.
Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$. Then
$A=\mathbb{C}u_{0}\oplus I_{\varphi}$ and there exists $a_{0}\in I_{\varphi}$
such that $u_{0}^{2}-u_{0}=a_{0}$. Let $b_{0}$ be a right identity for
$I_{\varphi}$. Set $m=u_{0}-u_{0}b_{0}\in A$. Then, for any $b\in
I_{\varphi}$, $b(u_{0}-u_{0}b_{0})=bu_{0}-bu_{0}b_{0}=0.$ Furthermore,
$u_{0}(u_{0}-u_{0}b_{0})-(u_{0}-u_{0}b_{0})$
$=u_{0}^{2}-u_{0}^{2}b_{0}-u_{0}+u_{0}b_{0}$ $=a_{0}-a_{0}b_{0}=0.$
It follows that $am-\varphi(a)m=0$ for all $a\in A$ and $\varphi(m)=1$. Thus
$A$ is $\varphi$-contractible by Proposition 4.1. $\blacksquare$
###### Lemma 4.4.
Suppose that $A$ is $\varphi$-contractible for some $\varphi\in\sigma(A)$ and
that $A$ has a right identity. Then $I_{\varphi}=\ker\varphi$ has a right
identity.
Proof. Assume that $A=\mathbb{C}u_{0}\oplus I_{\varphi}$ for some $u_{0}\in
A$, and $n=u_{0}+b_{0}$ be a right identity for $A$, where $b_{0}\in
I_{\varphi}$. Since $A$ is $\varphi$-contractible, it follows from Proposition
4.1 that there exist an $m=u_{0}+b_{1}\in A$ such that $am-\varphi(a)m=0$,
where $b_{1}\in I_{\varphi}$. Set $e=b_{0}-b_{1}$. Then $be-b=0,$ for any
$b\in I_{\varphi}$. In fact, for any $b\in I_{\varphi}$, $bn-b=0$, and $bm=0$.
It follows that $bu_{0}+bb_{0}-b=0$ and $bu_{0}+bb_{1}=0$. Thus $be-b=0$ for
all $b\in I_{\varphi}$. $\blacksquare$
The following result follows immediately from Lemma 4.3/4.4 and we omit its
proof.
###### Proposition 4.5.
Let $A$ be a Banach algebra with a right identity and $\varphi\in\sigma(A)$.
Then $A$ is $\varphi$-contractible if and only if $I_{\varphi}$ has a right
identity.
For $0$-contractibility and (uniform) approximate $0$-contractibility we have
the following parallel results to approximate $0$-amenability, the proofs of
Proposition 4.6 and Proposition 4.7 are minor modifications of the proof of
the analogous statements in Proposition 2.7 and will be omitted.
###### Proposition 4.6.
Banach algebra $A$ is $0$-contractible if and only if $A$ has a right
identity.
###### Proposition 4.7.
Banach algebra $A$ is approximately $0$-contractible if and only if $A$ has a
right approximate identity.
###### Proposition 4.8.
Banach algebra $A$ is uniformly approximately $0$-contractible if and only if
$A$ has a right identity.
Proof. It suffices to show that if $A$ is uniformly approximately
$0$-contractible, then $A$ has a right identity. Define an action of $A$ on
$E=A$ by $a*x=ax,x*a=0,~{}a\in A,x\in E,$ then $E$ is $(A,0)$-bimodule. The
natural injection $a\mapsto a:A\rightarrow A$ is a derivation. Thus there is a
net $\\{e_{\alpha}\\}$ in $A$ such that $ae_{\alpha}\rightarrow a$ uniformly
for $||a||\leq 1$. Let $R_{b}$ denote right multiplication by $b\in A$. Then
there is $e_{\lambda}\in\\{e_{\alpha}\\}$ with $||R_{e_{\lambda}}a-a||<||a||$
for all $a\in A$. Thus $R_{e_{\lambda}}$ is invertible. It follows that, there
is $e\in A$ such that $ee_{\lambda}=e_{\lambda}$, whence $(ae-a)e_{\lambda}=0$
for all $a\in A$. Then $e$ is a right identity of $A$ by injectivity of
$R_{e_{\lambda}}$. $\blacksquare$
Finally, we characterize character contractibility and (uniform) approximate
character contractibility, whose proofs are minor modifications of the proofs
of the analogous statements in Theorem 2.8/2.9 and will be omitted.
###### Theorem 4.9.
For a Banach algebra $A$ the following are equivalent:
$(i)$ $A$ is character contractible;
$(ii)$ $A$ has an identity and for any $\varphi\in\sigma(A)$, there exists
$m_{1},m_{2}\in A$ such that $m_{i}(\varphi)=1,(i=1,2)$ and
$am_{1}-\varphi(a)m=0$, $m_{2}a-\varphi(a)m_{2}=0$ for all $a\in A$;
$(iii)$ $A$ has an identity and for any $\varphi\in\sigma(A)$, there exists
bounded nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)$
such that $\varphi(\bigtriangleup M_{\alpha})=1$, $\varphi(\bigtriangleup
M_{\alpha}^{{}^{\prime}})=1(or~{}\varphi(\bigtriangleup M_{\alpha})\rightarrow
1,\varphi(\bigtriangleup M_{\alpha}^{{}^{\prime}})\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$.
###### Theorem 4.10.
Let $A$ be a Banach algebra. Then $A$ is character contractible if and only if
$\ker\varphi$ has an identity for every $\varphi\in\sigma(A)\cup\\{0\\}$.
###### Theorem 4.11.
For a Banach algebra $A$ the following are equivalent:
$(i)$ $A$ is (uniformly) approximately character contractible;
$(ii)$ $A$ has (an identity, respectively) both right and left approximate
identities and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A$ such that
$m_{\alpha}(\varphi)=1,m_{\alpha}^{{}^{\prime}}(\varphi)=1(or~{}m_{\alpha}(\varphi)\rightarrow
1,m_{\alpha}^{{}^{\prime}}(\varphi)\rightarrow 1)$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow
0$ for all $a\in A$ (uniformly on the unit ball of $A$, respectively);
$(iii)$ $A$ has (an identity, respectively) both right and left approximate
identities and for any $\varphi\in\sigma(A)$, there exist nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)$
such that $\varphi(\bigtriangleup M_{\alpha})=1$, $\varphi(\bigtriangleup
M_{\alpha}^{{}^{\prime}})=1(or~{}\varphi(\bigtriangleup M_{\alpha})\rightarrow
1,\varphi(\bigtriangleup M_{\alpha}^{{}^{\prime}})\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).
## 5\. Relations between generalized notions of character amenability
In this section, we are concerned with relations among generalized notions of
character amenability (contractibility). Firstly, we prove that
$w^{*}$-approximate $\varphi$-amenability, approximate $\varphi$-amenability
and approximate $\varphi$-contractibility are the same properties. We then
prove that approximate character amenability and approximate character
contractibility are the same properties, as are uniform approximate character
amenability (contractibility) and character amenability (contractibility,
respectively). Moreover, we obtain that character contractibility and
contractibility are the same properties for commutative Banach algebra.
###### Lemma 5.1.
For a Banach algebra $A$ and $\varphi\in\sigma(A)$, the following are
equivalent:
$(i)$ A is approximately $\varphi$-contractible;
$(ii)$ A is approximately $\varphi$-amenable;
$(iii)$ A is $w^{*}$-approximately $\varphi$-amenable.
Proof. It suffices to show that $(iii)\Rightarrow(i)$. Suppose that $(iii)$
holds. By the standard argument of Proposition 2.1, it follows that there is a
net $\\{m_{\alpha}\\}\subset A^{**}$ such that $(a\cdot
m_{\alpha}-\varphi(a)m_{\alpha})(f)\rightarrow
0,m_{\alpha}(\varphi)\rightarrow 1$ for all $a\in A,f\in A^{*}$. It follows
from the proof of Proposition 2.2 that there exists a net
$\\{n_{\beta}\\}\subset A$ such that $\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$. Thus A is
approximately $\varphi$-contractible by Proposition 4.2. $\blacksquare$
###### Theorem 5.2.
A Banach algebra $A$ is approximately character contractible if and only if
$A$ is approximately character amenable.
Proof. The standard argument of Proposition 2.7/4.7 and Lemma 5.1 applies.
$\blacksquare$
Recall that a Banach algebra $A$ is uniformly approximately
$\varphi$-amenable, if for every $(\varphi,A)$-bimodule $E$, every derivation
$D$ from $A$ into the dual $A$-bimodule $E^{*}$ may be approximated uniformly
on the unit ball of A by inner derivations. Clearly any $\varphi$-amenable
($\varphi$-contractible) Banach algebra is uniformly approximately
$\varphi$-amenable ($\varphi$-contractible, respectively). In the following
theorems we prove that the converse is also true. That is to say, uniform
approximate character amenability is equivalent to character amenability, and
uniform approximate character contractibility is equivalent to character
contractibility.
###### Lemma 5.3.
[18, Corollary 2.7] Let $A$ be a Banach algebra, if $\ker\varphi$ has both
bounded left and right approximate identities for every
$\varphi\in\sigma(A)\cup\\{0\\}$. Then $A$ is character amenable.
###### Proposition 5.4.
Let $A$ be unital Banach algebra, then $A$ is uniformly approximately
character amenable if and only if it is character amenable.
Proof. It suffices to show that if $A$ is uniformly approximately character
amenable then $A$ is character amenable.
Take $\varphi\in\sigma(A)$. Since $A$ is uniformly approximately right
character amenable it follows from Proposition 2.1 that there is a net
$\\{m_{\alpha}\\}\subset A^{**}$, such that $m_{\alpha}(\varphi)=1$ and
$||a\cdot m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ uniformly for
$||a||\leq 1$. Note that $A=\mathbb{C}e\oplus I_{\varphi}$, where
$I_{\varphi}=\ker\varphi$, and let $J(\varphi)=\\{n\in
A^{**},n(\varphi)=0\\}$. $J(\varphi)$ is a $w^{*}$-closed ideal of $A^{**}$,
and $J(\varphi)$ can be canonically identified with the second dual
$(I_{\varphi})^{**}$. We have $||a\cdot m_{\alpha}||\rightarrow 0$ uniformly
on the unit ball of $I_{\varphi}$.
Set $n_{\alpha}=e-m_{\alpha}$. Then $n_{\alpha}(\varphi)=0$, that is,
$n_{\alpha}\in J(\varphi)$. Moreover, for any $a\in I_{\varphi}$,
$|(an_{\alpha}-a)(f)|=|(a\cdot n_{\alpha}-a)(f)|=|(a\cdot
m_{\alpha})(f)|\leq||a\cdot m_{\alpha}||\cdot||f||$. It follows that
$||an_{\alpha}-a||\rightarrow 0$ uniformly on the unit ball of $I_{\varphi}$.
Now take $s\in(I_{\varphi})^{**}$, then, there is a net $(s_{i})\subset
I_{\varphi}$ such that $||s_{i}||\leq||s||$ and $s_{i}\rightarrow
s(w^{*}~{}in~{}i)$. Thus $s_{i}n_{\alpha}-s_{i}\rightarrow
sn_{\alpha}-s(w^{*}~{}in~{}i)$ and
$||sn_{\alpha}-s||\leq\sup_{i}||s_{i}n_{\alpha}-s_{i}||$. It follows that
$||an_{\alpha}-a||\rightarrow 0$ uniformly for $a\in(I_{\varphi})^{**}$ and
$||a||\leq 1$.
Thus there is a sequence $(n_{k})\subset(I_{\varphi})^{**}$ and
$\varepsilon_{k}\rightarrow 0$ such that
$||an_{k}-a||\leq\varepsilon_{k}||a||,(a\in(I_{\varphi})^{**}).$
Thus, the multiplication operator
$R_{n_{k}}:(I_{\varphi})^{**}\rightarrow(I_{\varphi})^{**}$ defined by
$R_{n_{k}}(s)=sn_{k}$ satisfies $||R_{n_{k}}-id_{(I_{\varphi})^{**}}||<1$ for
$k$ sufficiently large. Take such $k$, so that $R_{n_{k}}$ is invertible. By
surjectivity, there is $x\in(I_{\varphi})^{**}$ such that $xn_{k}=n_{k}$. Then
for each $y\in(I_{\varphi})^{**}$ we have $(yx-y)n_{k}=0$. From the
injectivity of $R_{n_{k}}$ this implies $yx=y$ for all
$y\in(I_{\varphi})^{**}$. So $(I_{\varphi})^{**}$ has a right identity, then
$I_{\varphi}$ has a bounded right approximately identity.
On the other hand, since $A$ is uniformly approximately left character
amenable. We can deduce that $I_{\varphi}$ has a bounded left approximately
identity. Since $\varphi$ is arbitrary, it follows from Lemma 5.3 that $A$ is
character amenable. $\blacksquare$
###### Corollary 5.5.
If $A^{\sharp}$ is uniformly approximately character amenable, then $A$ has
bounded right and left approximate identity.
Proof. Choose $\varphi\in\sigma(A^{\sharp})$ such that
$\varphi(A)=0,\varphi(e)=1$, the standard argument of Proposition 5.4 applies.
$\blacksquare$
###### Lemma 5.6.
Let $A$ be a Banach algebra without a unit. Then $A$ is uniformly
approximately character amenable if and only if $A^{\sharp}$ is uniformly
approximately character amenable.
Proof. Assume $A$ is uniformly approximately character amenable, the standard
argument of Lemma 3.6 applies, $A^{\sharp}$ is uniformly approximately
character amenable.
For the inverse, assume $A^{\sharp}$ is uniformly approximately character
amenable. It follows from Corollary 5.5 that $A$ has bounded left and right
approximately identity. Thus $A$ is uniformly approximately character amenable
by Theorem 3.5. $\blacksquare$
Note that $A$ is (uniformly approximately) character amenable if and only if
its unitization $A^{\sharp}$ is (uniformly approximately, respectively)
character amenable [18, Theorem 2.6] and Lemma 5.6, then, by Proposition 5.4,
we have the following theorem.
###### Theorem 5.7.
A Banach algebra $A$ is uniformly approximately character amenable if and only
if it is character amenable.
###### Corollary 5.8.
If a finite-dimensional Banach algebra is uniformly approximately character
amenable, then it is character amenable.
For uniform approximate character contractibility and character
contractibility , we have the following parallel result whose proof is a minor
modification of the proof of the analogous statements in Proposition 5.4 and
will be omitted.
###### Theorem 5.9.
A Banach algebra $A$ is uniformly approximately character contractible if and
only if it is character contractible.
###### Corollary 5.10.
If a finite-dimensional Banach algebra is approximately character
contractible, then it is character contractible.
Now, we shall conclude this section by proving the following result.
###### Theorem 5.11.
Let $A$ be a commutative Banach algebra. Then $A$ is character contractible if
and only if $A$ is isomorphic to $\mathbb{C}^{n}$.
Proof. It suffices to show that if $A$ is character contractible then $A$ is
contractible.
Let $\mathfrak{M}$ be the maximal ideal space of $A$. It follows from Theorem
4.10 that, given $\varphi\in\mathfrak{M}\cup\\{0\\}$, there exists
$E_{\varphi}\in\ker\varphi$ such that $E_{\varphi}$ is an identity for
$\ker\varphi$. Arbitrarily choose $\varphi_{1},\varphi_{2}\in\mathfrak{M}$. If
$\varphi_{1}\neq\varphi_{2}$, then $\varphi_{1}(E\varphi_{2})=1$ and
$\varphi_{2}(E\varphi_{1})=1$. It follows that each point of $\mathfrak{M}$ is
isolated, so that $\mathfrak{M}$ is finite since $\mathfrak{M}$ is compact.
Hence $\mathfrak{M}=\\{\varphi_{1},\varphi_{2},\cdots,\varphi_{n}\\}$ and
$E_{\varphi_{i}}$ is the identity for $\ker\varphi_{i}$ for all $1\leq i\leq
n$. Hence, $E_{0}=E_{\varphi_{1}}\cdot E_{\varphi_{2}}\cdot\cdots\cdot
E_{\varphi_{n}}$ is an identity for
Rad$(A)=\ker\varphi_{1}\cap\cdots\cap\ker\varphi_{n}$. This is possible only
if Rad$(A)=\\{0\\}$, so that $A$ is semisimple. It follows that
$A\cong\mathbb{C}^{n}$, then, by [20], $A$ is contractible. $\blacksquare$
## 6\. Examples
In this section, we give three examples. The first example shows that there
exists a Banach algebra which is approximately character amenable but not
uniformly approximately character amenable. The second example shows that
there exists a Banach algebra which is character amenable but not character
contractible. The last example shows that there exists a Banach algebra which
is character contractible but not amenable.
Given a family $(A_{\alpha})_{\alpha\in\Lambda}$ of Banach algebras defined
their $l^{\infty}$ direct sum as
$l^{\infty}(A_{\alpha})=\\{(a_{\alpha}):a_{\alpha}\in
A_{\alpha},||(a_{\alpha})||=\sup||a_{\alpha}||<\infty\\}.$
Further, set
$c_{0}(A_{\alpha})=\\{(a_{\alpha}):(a_{\alpha})\in
l^{\infty}(A_{\alpha}),||a_{\alpha}||\rightarrow 0\\}.$
Example 1. There exists an approximately character amenable Banach algebra
which is not uniformly approximately character amenable.
For each $n\in\mathbb{N}$, take $A_{n}=\mathbb{C}^{n}$, each with the
corresponding $l^{1}$ norm. Then $A_{n}$ has an identity
$e_{n}=(1,1,\cdots,1)$ of norm $n$; clearly each $A_{n}$ is amenable, so
$c_{0}(A_{n}^{\sharp})$ is approximately amenable by [9, Example 6.1], then it
is also approximately character amenable. If $f_{\alpha}$ is a left
approximate identity for $c_{0}(A_{n}^{\sharp})$ then, given $n$, there is
$\alpha$ such that $||f_{\alpha}e_{n}-e_{n}||\leq 1$. Hence we have
$||f_{\alpha}||\geq||f_{\alpha}e_{n}||\geq n-1$ and then $\\{f_{\alpha}\\}$ is
unbounded.
But $c_{0}(A_{n}^{\sharp})$ is not uniformly approximately character amenable.
Indeed, if it were, Theorem 2.8 would imply that $c_{0}(A_{n}^{\sharp})$ had
bounded left and right approximate identity.
Example 2. There exists a character amenable Banach algebra which is neither
amenable nor character contractible.
Let $V$ be the Volterra operator, $A_{V}$ be the Banach algebra generated by
$V$ and $A$ be the Banach algebra generated by $V$ and $e$. Then
$A=\mathbb{C}e\oplus A_{V}$, and $A_{V}$ has a bounded two sided approximate
identity [6, Theorem 5.10]. And $\sigma(A)=\\{\varphi\\},$ here
$\varphi(A_{V})=0,\varphi(e)=1$. Thus $A$ is a character amenable by Lemma
5.3. However, $A$ is not amenable by [8].
Moreover, $A$ is not character contractible. Indeed, if it were, Proposition
4.1 implies that there exists an $m=e+b\in A$ such that
$a(e+b)=\varphi(a)(e+b)$ for all $a\in A$. If, in addition, $a\in A_{V}$ and
$a\neq 0$, then $a(e+b)=0$; since $e+b$ is invertible, we obtain $a=0$, a
contradiction.
Example 3. Let $A=B(H)$, where $H$ is an infinite-dimensional Hilbert space.
Then $A$ has an identity and $A$ is character contractible by Theorem 4.10.
But $A$ is not amenable by [20].
## References
* [1] R. J. Archbold, On the norm of an inner derivation of a $C^{*}$-algebra, Math. Proc. Cambridge Philos. Soc. 84 (1978) 273–291.
* [2] W. G. Bade, P. C. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987) 359–377.
* [3] S. Bowling, J. Duncan, First order cohomology of Banach semigroup algebras, Semigroup Forum. 56 (1) (1998) 130–145.
* [4] P. C. Curtis and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. 40 (1989) 89–104.
* [5] H. G. Dales, Banach Algebras and Automatic Continuity (Oxford, 2000).
* [6] K. R. Davidson, Nest algebras, Longman group UK limited, Essex, 1988\.
* [7] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 19 (1979) 309–325.
* [8] D. R. Farenick, B. E. Forrest and L. W. Marcoux, Amenable operators on Hilbert spaces, J. reine angew. Math. 582 (2005) 201–228.
* [9] F. Ghahramani and R. J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (2004) 229–260.
* [10] F. Ghahramani, R. J. Loy and Y. Zhang, Generalized notions of amenability. II, J. Funct. Anal. 254 (2008) 1776–1810.
* [11] A. YA. Helemskii, Homological essence of amenability in the sense of A. Connes: the injectivity of the predual bimodule. Mat. Sb. 180 (1989), 1680 C1690; Math. USSR-Sb. 68 (1991) 555–566.
* [12] A. YA. Helemskii, Banach and Locally Convex Algebras (Oxford University Press, 1993).
* [13] B. E. Johnson, R. V. Kadison and J. Ringrose, Cohomology of operator algebras. III. Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972) 73–96.
* [14] B. E. Johnson, Cohomology in Banach Algebras, Mem. Amer. Math. Soc. Vol. 127 (Amer. Math. Soc., 1972).
* [15] R. V. Kadison, E. C. Lance and J. R. Ringrose, Derivations and automorphisms of operator algebras. II, J. Funct. Anal. 1 (1967) 204–221.
* [16] E. Kaniuth, A. T. Lau and J. Pym, On $\varphi$-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85–96.
* [17] E. Kaniuth, A. T. Lau and J. Pym, On character amenability of Banach algebras, J. Math. Anal. Appl. 344 (2008) 942–955.
* [18] M. S. Monfared, Character amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 697–706.
* [19] A. L. T. Paterson, Amenability, American Mathematical Society, Providence, RI, 1988.
* [20] V. Runde, Lectures on amenability, in: Lecture Notes in Mathematics, Vol. 1774, Springer, Berlin, 2002.
* [21] Y. Zhang, Maximal ideals and the structure of contractible and amenable Banach algebras, Bull. Austral. Math. Soc. 62 (2000) 221–226.
|
arxiv-papers
| 2010-08-31T08:22:04 |
2024-09-04T02:49:12.579371
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luo Yi Shi, Yu Jing Wu and You Qing Ji",
"submitter": "Luoyi Shi",
"url": "https://arxiv.org/abs/1008.5252"
}
|
1008.5253
|
# Entanglement and nonlocality of one- and two-mode combination squeezed
state††thanks: Work supported by a grant from the Key Programs Foundation of
Ministry of Education of China (No. 210115) and the Research Foundation of the
Education Department of Jiangxi Province of China (No. GJJ10097).
Li-yun Hu1, Xue-xiang Xu1,2, Qin Guo1, and Hong-yi Fan2
1College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Corresponding author. E-mail: hlyun2008@126.com.
###### Abstract
We investigate the entanglement and nonlocality properties of one- and two-
mode combination squeezed vacuum state (OTCSS, with two-parameter $\lambda$
and $\gamma$) by analyzing the logarithmic negativity and the Bell’s
inequality. It is found that this state exhibits larger entanglement than that
of the usual two-mode squeezed vacuum state (TSVS), and that in a certain
regime of $\lambda$, the violation of Bell’s inequality becomes more obvious,
which indicates that the nonlocality of OTCSS can be stronger than that of
TSVS. As an application of OTCSS, the quantum teleportaion is examined, which
shows that there is a region spanned by $\lambda$ and $\gamma$in which the
fidelity of OTCSS channel is larger than that of TSVS.
Keywords: Entanglement, nonlocality, IWOP technique, teleportation
PACS number(s): 42.50.Dv, 03.65.Wj, 03.67.Mn
## 1 Introduction
Entanglement between quantum systems plays a key role in quantum information
processing, such as quantum teleportation, dense coding, and quantum cloning.
In recent years, various entangled states have brought considerable attention
and interests of physicists because of their potential uses in quantum
communication [1, 2]. For instance, the two-mode squeezed state is a typical
entangled state of continuous variable and exhibits quantum entanglement
between the idle-mode and the signal-mode in a frequency domain manifestly.
Theoretically, the two-mode squeezed state is constructed by the two-mode
squeezing operator
$S=\exp[\lambda(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})]$ [3, 4, 5] acting
on the two-mode vacuum state $\left|00\right\rangle$,
$S\left|00\right\rangle=\text{sech}\lambda\exp\left[-a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda\right]\left|00\right\rangle,$
(1)
where $\lambda$ is a squeezing parameter, the disentangling of $S$ can be
obtained by using SU(1,1) Lie algebra,
$[a_{1}a_{2},a_{1}^{\dagger}a_{2}^{\dagger}]=a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1,$
or by using the entangled state representation $\left|\eta\right\rangle$ [6,
7, 8], which was constructed according to the idea of Einstein, Podolsky and
Rosen in their argument that quantum mechanics is incomplete [9].
Using the relation between Bosonic operators and the coordinate $Q_{i},$
momentum $P_{i},$ $Q_{i}=(a_{i}+a_{i}^{\dagger})/\sqrt{2},\
P_{i}=(a_{i}-a_{i}^{\dagger})/(\sqrt{2}\mathtt{i}),$ and introducing the two-
mode quadrature operators of light field, $x_{1}=(Q_{1}+Q_{2})/2,$
$x_{2}=(P_{1}+P_{2})/2,$ the variances of $x_{1}$ and $x_{2}$ in the state
$S\left|00\right\rangle$ are in the standard form
$\left\langle
00\right|S^{\dagger}x_{2}^{2}S\left|00\right\rangle=\frac{1}{4}e^{-2\lambda},\text{
}\left\langle
00\right|S^{\dagger}x_{1}^{2}S\left|00\right\rangle=\frac{1}{4}e^{2\lambda},$
(2)
thus we get the standard squeezing for the two quadrature:
$x_{1}\rightarrow\frac{1}{2}e^{\lambda}x_{1},$
$x_{2}\rightarrow\frac{1}{2}e^{-\lambda}x_{2}$. On the other hand, the two-
mode squeezing operator can also be recast into the form
$S=\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\right].$ Then
some interesting questions naturally rise: what is the property of the
following operator
$V=\exp\left[-\mathtt{i}\left(\lambda_{1}Q_{1}P_{2}+\lambda_{2}Q_{2}P_{1}\right)\right],$
(3)
with two parameters $\lambda_{1}=\lambda e^{\gamma},\lambda_{2}=\lambda
e^{-\gamma},\lambda>0$? What is the normally ordered expansion of $V$ and what
is the state $V\left|00\right\rangle$? What are the entanglement and
nonlocality properties of $V\left|00\right\rangle?$ When $\gamma=0,$ Eq.(3)
just reduces to the usual two-mode squeezing operator $S$. Thus we can
consider $V$ as a generalized two-mode squeezing operator and
$V\left|00\right\rangle$ as one- and two-mode combination squeezed vacuum
state (OTCSS).
In this paper, we investigate entanglement properties and quantum nonlocality
of $V\left|00\right\rangle$ in terms of logarithmic negativity and the Bell’s
inequality, respectively. Subsequently, we consider its application in the
field of quantum teleportation by using the characteristic-function formula.
It is shown that this state exhibits larger entanglement than that of the
usual two-mode squeezed vacuum state (TSVS); and in a certain smaller regime
of $\lambda$, that the nonlocality of this state can be stronger than that of
TSVS due to the presence of $\gamma.$ In addition, application to quantum
teleportation with OTCSS is also considered, which shows that there is a
region spanned by $\lambda$ and $\gamma$ in which the fidelity of OTCSS
channel is larger than that of TSVS.
Our paper is arranged as follows. In section 2, we derive the normal ordering
form of one- and two-mode combination squeezing operator by using the
technique of integration within an ordered product (IWOP) of operators. In
section 3, using the Weyl ordering form of single-mode Wigner operator and the
order-invariance of Weyl ordered operators under similar transformations, we
derive analytically the Wigner function of $V\left|00\right\rangle$. Sections
4 and 5 are devoted to investigating the entanglement properties and the
nonlocal properties OTCSS by using the Bell’s inequality and the logarithmic
negativity, respectively. An application to quantum teleportation with OTCSS
is involved in section 6. We end with the main conclusions of our work.
## 2 The normal ordering form of $V$ and fluctuations in
$V\left|00\right\rangle$
In order to know $V\left|00\right\rangle,$ we need to derive the normal
ordering form of the unitary operator $V$ by virtue of the IWOP technique [10,
11, 12]. Using the Baker-Hausdorff formula,
$e^{A}Be^{-A}=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots,$
(4)
and noticing that
$\displaystyle
i\left[\lambda_{1}Q_{1}P_{2}+\lambda_{2}Q_{2}P_{1},Q_{1}\right]$
$\displaystyle=\lambda_{2}Q_{2},\text{ }$ (5) $\displaystyle
i\left[\lambda_{1}Q_{1}P_{2}+\lambda_{2}Q_{2}P_{1},Q_{2}\right]$
$\displaystyle=\lambda_{1}Q_{1},$ (6) $\displaystyle
i\left[\lambda_{1}Q_{1}P_{2}+\lambda_{2}Q_{2}P_{1},P_{1}\right]$
$\displaystyle=-\lambda_{1}P_{2},$ (7) $\displaystyle
i\left[\lambda_{1}Q_{1}P_{2}+\lambda_{2}Q_{2}P_{1},P_{2}\right]$
$\displaystyle=-\lambda_{2}P_{1},$ (8)
we have
$\displaystyle V^{-1}Q_{1}V$
$\displaystyle=Q_{1}\cosh\lambda+Q_{2}e^{-\gamma}\sinh\lambda,$ (9)
$\displaystyle V^{-1}Q_{2}V$
$\displaystyle=Q_{2}\cosh\lambda+Q_{1}e^{\gamma}\sinh\lambda,$ (10)
$\displaystyle V^{-1}P_{1}V$ $\displaystyle=P_{1}\cosh\lambda-
P_{2}e^{\gamma}\sinh\lambda,$ (11) $\displaystyle V^{-1}P_{2}V$
$\displaystyle=P_{2}\cosh\lambda-P_{1}e^{-\gamma}\sinh\lambda.$ (12)
Thus, in order to keep the eigenvalues invariant under the $V$ transformation,
i..e.,
$V^{-1}Q_{k}V\left|q_{1}q_{2}\right\rangle^{\prime}=q_{k}\left|q_{1}q_{2}\right\rangle^{\prime},(k=1,2),$
(13)
the base vector must be changed to
$\left|q_{1}q_{2}\right\rangle^{\prime}=V^{-1}\left|q_{1}q_{2}\right\rangle=\left|\Lambda^{-1}\left(\begin{array}[c]{c}q_{1}\\\
q_{2}\end{array}\right)\right\rangle,\Lambda=\left(\begin{array}[c]{cc}\cosh\lambda&e^{-\gamma}\sinh\lambda\\\
e^{\gamma}\sinh\lambda&\cosh\lambda\end{array}\right),$ (14)
where
$\left|q_{1}q_{2}\right\rangle=\left|q_{1}\right\rangle\otimes\left|q_{2}\right\rangle$,
and $\left|q_{k}\right\rangle$ is the coordinate eigenstate,
$\left|q_{k}\right\rangle=\pi^{-1/4}\exp[-\frac{1}{2}q^{2}+\sqrt{2}qa^{{\dagger}}-\frac{1}{2}a^{{\dagger}2}]\left|0\right\rangle.$
(15)
Using the completeness raltion
$\int_{-\infty}^{\infty}dq_{1}dq_{2}\left|q_{1},q_{2}\right\rangle\left\langle
q_{1},q_{2}\right|=1$, we have
$V^{-1}=\int_{-\infty}^{\infty}dq_{1}dq_{2}\left|\Lambda^{-1}\left(\begin{array}[c]{c}q_{1}\\\
q_{2}\end{array}\right)\right\rangle\left\langle q_{1},q_{2}\right|,$ (16)
which leads to
$V=\int_{-\infty}^{\infty}dq_{1}dq_{2}\left|\Lambda\left(\begin{array}[c]{c}q_{1}\\\
q_{2}\end{array}\right)\right\rangle\left\langle q_{1},q_{2}\right|.$ (17)
Actually, one can check (17) by $V^{-1}V=VV^{-1}=1.$ Further using the vacuum
projector $\left|00\right\rangle\left\langle
00\right|=\colon\exp[-a^{{\dagger}}a-b^{{\dagger}}b]\colon(\colon\colon$
denoting normal ordering$)$, as well as the IWOP technique, we can put $V$
into the normal ordering form [13],
$\displaystyle V$
$\displaystyle=\frac{2}{\sqrt{L}}\exp\left\\{\frac{1}{L}\left[\left(b^{{\dagger}2}-a^{{\dagger}2}\right)\sinh^{2}\lambda\sinh
2\gamma+2a^{{\dagger}}b^{{\dagger}}\sinh 2\lambda\cosh\gamma\right]\right\\}$
$\displaystyle\colon\exp\left\\{\frac{4}{L}\left[\left(a^{{\dagger}}a+b^{{\dagger}}b\right)\cosh\lambda+\left(b^{{\dagger}}a-a^{{\dagger}}b\right)\sinh\lambda\sinh\gamma\right]-a^{{\dagger}}a-b^{{\dagger}}b\right\\}\colon$
$\displaystyle\exp\left\\{\frac{1}{L}\left[\left(b^{2}-a^{2}\right)\sinh^{2}\lambda\sinh
2\gamma-2a^{{\dagger}}b^{{\dagger}}\sinh 2\lambda\cosh\gamma\right]\right\\},$
(18)
where $L=4\left(1+\sinh^{2}\gamma\tanh^{2}\lambda\right)\cosh^{2}\lambda.$ Eq.
(18) is just the normal ordering form of $V$. It is obviously to see that when
$\gamma=0$, Eq.(18) just reduces to the usual two-mode squeezing operator.
Operating $V$ on the two-mode vacuum state $\left|00\right\rangle$, we obtain
the squeezed vacuum state,
$V\left|00\right\rangle=\frac{2}{\sqrt{L}}\exp\left\\{\frac{1}{L}\left[\left(b^{{\dagger}2}-a^{{\dagger}2}\right)\sinh^{2}\lambda\sinh
2\gamma+2a^{{\dagger}}b^{{\dagger}}\sinh
2\lambda\cosh\gamma\right]\right\\}\left|00\right\rangle.$ (19)
On the other hand, by using the transformations Eqs.(9)-(12), one can derive
the variances of $x_{1}$ and $x_{2}$ in the state $V\left|00\right\rangle$
[13]
$\displaystyle\left\langle\left(\Delta x_{1}\right)^{2}\right\rangle$
$\displaystyle=\frac{1}{4}\left(\cosh
2\lambda+2\sinh^{2}\lambda\sinh^{2}\gamma+\sinh 2\lambda\cosh\gamma\right),$
(20) $\displaystyle\left\langle\left(\Delta x_{2}\right)^{2}\right\rangle$
$\displaystyle=\frac{1}{4}\left(\cosh
2\lambda+2\sinh^{2}\lambda\sinh^{2}\gamma-\sinh 2\lambda\cosh\gamma\right),$
(21)
which indicate that the variances are not only dependent on parameter
$\lambda$, but also on parameter $\gamma.$ When $\gamma=0,$ Eqs.(20) and (21)
reduce to $\left\langle\left(\Delta
x_{1}\right)^{2}\right\rangle=\frac{1}{4}e^{2\lambda},$ and
$\left\langle\left(\Delta
x_{2}\right)^{2}\right\rangle=\frac{1}{4}e^{-2\lambda},$ corresponding to the
usual TSVS. In particular, by modulating the two parameters ($\lambda$ and
$\gamma$), we can realize that
$\left\langle\left(\Delta
x_{1}\right)^{2}\right\rangle>\frac{1}{4}e^{2\lambda},\left\langle\left(\Delta
x_{2}\right)^{2}\right\rangle<\frac{1}{4}e^{-2\lambda},$ (22)
whose condition is given by
$0<\tanh\lambda<\frac{1}{1+\cosh\gamma},\lambda>0,$ (23)
which mean that the OTCSS may exhibit stronger squeezing in one quadrature
than that of the TSVS while exhibiting weaker squeezing in another quadrature
when the condition (23) satisfied. Then, can the OTCSS exhibits stronger
nonlocality or more observable violation of Bell’s inequality? In the
following, we pay our attention to these two aspects.
## 3 Wigner function of $V\left|00\right\rangle$
Wigner distribution functions [14, 15, 16] of quantum states are widely
studied in quantum statistics and quantum optics. Now we derive the expression
of the Wigner function of $V\left|00\right\rangle.$ Here we take a new method
to do it. Recalling that in Ref.[17, 18, 19] we have introduced the Weyl
ordering form of single-mode Wigner operator
$\Delta_{1}\left(q_{1},p_{1}\right)$,
$\Delta_{1}\left(q_{1},p_{1}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(24)
its normal ordering form is
$\Delta_{1}\left(q_{1},p_{1}\right)=\frac{1}{\pi}\colon\exp\left[-\left(q_{1}-Q_{1}\right)^{2}-\left(p_{1}-P_{1}\right)^{2}\right]\colon,$
(25)
where the symbols $\colon\colon$ and
$\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ denote the normal
ordering and the Weyl ordering, respectively. Note that the order of Bose
operators $a_{1}$ and $a_{1}^{\dagger}$ within a normally ordered product and
a Weyl ordered product can be permuted. That is to say, even though
$[a_{1},a_{1}^{\dagger}]=1$, we can have $\colon
a_{1}a_{1}^{\dagger}\colon=\colon a_{1}^{\dagger}a_{1}\colon$
and$\genfrac{}{}{0.0pt}{}{:}{:}a_{1}a_{1}^{\dagger}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}a_{1}^{\dagger}a_{1}\genfrac{}{}{0.0pt}{}{:}{:}.$
For one- and two-mode combination squeezed vacuum state
$V\left|00\right\rangle$, its Wigner function is given by
$W\left(q_{1},p_{1};q_{2},p_{2}\right)=\mathtt{tr}\left[V\left|00\right\rangle\left\langle
00\right|V^{-1}\Delta_{1}\left(q_{1},p_{1}\right)\Delta_{2}\left(q_{2},p_{2}\right)\right]=\left\langle
00\right|U\left|00\right\rangle,$ (26)
where
$U=V^{-1}\Delta_{1}\left(q_{1},p_{1}\right)\Delta_{2}\left(q_{2},p_{2}\right)V.$
Further using Eq.(24), noticing that the Weyl ordering has a remarkable
property, i.e., the order-invariance of Weyl ordered operators under similar
transformations [17, 18, 19], which means
$V^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}V=\genfrac{}{}{0.0pt}{}{:}{:}V^{-1}\left(\circ\circ\circ\right)V\genfrac{}{}{0.0pt}{}{:}{:},$
(27)
as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did
not exist, thus $U$ can be cast into the following form (see appendix A),
$\displaystyle U$
$\displaystyle=V^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\delta\left(q_{2}-Q_{2}\right)\delta\left(p_{2}-P_{2}\right)\genfrac{}{}{0.0pt}{}{:}{:}V$
$\displaystyle=\Delta_{1}\left(q_{1}\cosh\lambda-
q_{2}e^{-\gamma}\sinh\lambda,p_{1}\cosh\lambda+p_{2}e^{\gamma}\sinh\lambda\right)$
$\displaystyle\times\Delta_{2}\left(q_{2}\cosh\lambda-
q_{1}e^{\gamma}\sinh\lambda,p_{2}\cosh\lambda+p_{1}e^{-\gamma}\sinh\lambda\right).$
(28)
So the Wigner function of $V\left|00\right\rangle$ is given by
$W\left(q_{1},p_{1};q_{2},p_{2}\right)=\frac{1}{\pi^{2}}\exp\left\\{-m_{1}\left(q_{1}^{2}+p_{2}^{2}\right)\allowbreak-
m_{2}\left(p_{1}^{2}+q_{2}^{2}\right)+2\left(q_{1}q_{2}\allowbreak-
p_{1}p_{2}\right)m_{3}\right\\},$ (29)
where
$m_{1}=\cosh^{2}\lambda+e^{2\gamma}\sinh^{2}\lambda,\text{
}m_{2}=\cosh^{2}\lambda+e^{-2\gamma}\allowbreak\sinh^{2}\lambda,\text{
}m_{3}=\cosh\gamma\sinh 2\lambda.$
In particular, when $\gamma=0$, Eq.(29) becomes
$\displaystyle W\left(q_{1},p_{1};q_{2},p_{2}\right)$
$\displaystyle=\frac{1}{\pi^{2}}\exp\left\\{-\left(p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\right)\cosh
2\lambda\allowbreak\right.$
$\displaystyle\left.+2\left(q_{1}q_{2}-p_{1}p_{2}\right)\sinh
2\lambda\right\\},$ (30)
which is just the Wigner function of the usual TSVS.
## 4 Entanglement properties of $V\left|00\right\rangle$
In this section, we consider the entanglement properties of
$V\left|00\right\rangle$. It is well known that a two-mode Gaussian state can
be completely characterized by its first and second statistical moments and
the covariance matrix of elements $\sigma$. In general, the first statistical
moments can be adjusted by local displacements without affecting entanglement,
thus they they will can be set to be zero without loss of generality and the
behavior of the covariance matrix $\sigma$ is all important for the study of
entanglement. There are several quantitative measurements of quantum
entanglement proposed [20, 21, 22]. For a two-mode Gaussian state, the
entanglement is best characterized by the logarithmic negativity
$E_{\mathcal{N}}$, a quantity evaluated in terms of the symplectic eigenvalues
of $\sigma$ [23, 24].
In order to evaluate the entanglement of $V\left|00\right\rangle,$ we reform
Eq.(29) as follows in terms of phase space quadrature variables,
$W\left(q_{1},p_{1};q_{2},p_{2}\right)=\frac{1}{\pi^{2}}\exp\left[-\frac{1}{2}\left(\begin{array}[c]{cccc}q_{1}&p_{1}&q_{2}&p_{2}\end{array}\right)\sigma^{-1}\left(\begin{array}[c]{cccc}q_{1}&p_{1}&q_{2}&p_{2}\end{array}\right)^{T}\right],$
(31)
where the covariance matrix $\sigma$ of this OTCSS is [25]
$\sigma=\left(\begin{array}[c]{cc}u&w\\\ w^{T}&v\end{array}\right),\text{
}u=\frac{1}{2}\left(\begin{array}[c]{cc}m_{2}&0\\\
0&m_{1}\end{array}\right),v=\frac{1}{2}\left(\begin{array}[c]{cc}m_{1}&0\\\
0&m_{2}\end{array}\right),w=\frac{1}{2}\left(\begin{array}[c]{cc}m_{3}&0\\\
0&-m_{3}\end{array}\right).$ (32)
In particular, when $\gamma=0$, $m_{1}=m_{2}=\cosh 2\lambda,$ $m_{3}=\sinh
2\lambda$, Eq.(32) just reduces to the so-called standard form of covariance
matrix for TSVS [24].
The condition for entanglement of a Gaussian state is derived from the
partially transposed density matrix (PPT criterion) [24], according to the
smallest symplectic eigenvalue $\tilde{n}_{s}$ of the partially transposed
state, $\tilde{n}_{s}<\frac{1}{2},$ i.e., $\tilde{n}_{s}\geqslant\frac{1}{2}$
means the a two-mode Gaussian state is separable, where $\tilde{n}_{s}$ is
defined as
$\tilde{n}_{s}=\min\left[\tilde{n}_{+},\tilde{n}_{-}\right],$ (33)
and $\tilde{n}_{\pm}\ $is given by [26]
$\tilde{n}_{\pm}=\sqrt{\frac{\tilde{\Delta}\left(\sigma\right)\pm(\tilde{\Delta}\left(\sigma\right)^{2}-4\det\sigma)^{1/2}}{2}},$
(34)
where
$\tilde{\Delta}\left(\sigma\right)=\Delta\left(\tilde{\sigma}\right)=\det
u+\det v-2\det w.$
Using Eqs.(32)-(34), the corresponding sympletic eigenvalues $\tilde{n}_{\pm}$
are then given by
$\tilde{n}_{\pm}=\frac{1}{2}\left(\sqrt{m_{1}m_{2}}\pm m_{3}\right).$ (35)
One the other hand, the corresponding quantification of entanglement is given
by the logarithmic negativity $E_{\mathcal{N}}$ defined as [20, 26, 27],
$E_{\mathcal{N}}=\max\left[0,-\ln 2\tilde{n}_{s}\right].$ (36)
From Eqs.(32), (35) and (36), one can clearly see that the logarithmic
negativity $E_{\mathcal{N}}$ is dependent on $\lambda$ and $\gamma.$ In figure
1, we plot the logarithmic negativity $E_{\mathcal{N}}$ as a function of
parameters $\lambda$ and $\gamma$. From Fig.1, we clearly see a new feature,
i.e., in presence of parameter $\gamma$, the logarithmic negativity becomes
larger than that of tha usual squeezed state ($\gamma=0$).
Figure 1: (Color online) The logarithmic negativity $E_{\mathcal{N}}$as a
function of parameters $\lambda$and $\gamma$.
## 5 Violations of Bell’s inequality for $V\left|00\right\rangle$
We now turn our attention to the nonlocal properties of
$V\left|00\right\rangle$ in terms of the Bell’s inequality. For a two-mode
continuous variable system, the Bell’s inequality is given, using correlations
between parity measurement, by
$\left|\mathcal{B}\right|\equiv\left|\left\langle\hat{\Pi}^{ab}\left(\alpha^{\prime},\beta^{\prime}\right)+\hat{\Pi}^{ab}\left(\alpha,\beta^{\prime}\right)+\hat{\Pi}^{ab}\left(\alpha^{\prime},\beta\right)-\hat{\Pi}^{ab}\left(\alpha,\beta\right)\right\rangle\right|\leqslant
2,$ (37)
where $B$ is the Bell function, and the superscripts $a$ and $b$ denote the
modes and $\hat{\Pi}^{ab}\left(\alpha,\beta\right)$ is the displaced parity
operator (Wigner operator)[28] defined as
$\displaystyle\hat{\Pi}^{ab}\left(\alpha,\beta\right)$
$\displaystyle\equiv\hat{\Pi}^{a}\left(\alpha\right)\hat{\Pi}^{b}\left(\beta\right)=D\left(\alpha\right)D\left(\beta\right)(-1)^{a^{{\dagger}}a+b^{{\dagger}}b}D^{{\dagger}}\left(\beta\right)D^{{\dagger}}\left(\alpha\right)$
$\displaystyle=\pi^{2}\Delta_{a}\left(\alpha\right)\Delta_{b}\left(\beta\right),$
(38)
where $\alpha=(q_{1}+ip_{1})/\sqrt{2},\beta=(q_{2}+ip_{2})/\sqrt{2}.$ The
expectation value of this displaced parity operator is just proportional to
the two-mode Wigner function, i.e.,
$\Pi\left(\alpha,\beta\right)=\mathtt{tr}\left[\rho\hat{\Pi}^{ab}\left(\alpha,\beta\right)\right]=\pi^{2}W\left(\alpha,\beta\right),$
(39)
which shows that the connection between this displaced parity operator and
Wigner function provides an equivalent definition [29].
The Bell function is measured for any of four combinations of
$\alpha=0,\sqrt{J}e^{i\varphi}$ and $\beta=0,\sqrt{J}e^{i\theta}$, where
$J(=\left|\alpha\right|^{2}=\left|\beta\right|^{2})$ is a positive constant
characterizing the magnitude of the displacement. From these quantities we
construct the combination [30]
$\mathcal{B}\equiv\pi^{2}\left[W\left(0,0\right)+W\left(\sqrt{J}e^{i\varphi},0\right)+W\left(0,\sqrt{J}e^{i\theta}\right)-W\left(\sqrt{J}e^{i\varphi},\sqrt{J}e^{i\theta}\right)\right].$
(40)
In particular, when $\varphi=0$, $\theta=\pi,$ Eq.(40) just reduces to Eq.(7)
in Ref.[28]. Then we can test Bell’s inequality
$-2\leqslant\mathcal{B}\leqslant 2$ by means of the two-mode Wigner function
measurement. Recently, a generalized quasiprobability function is proposed to
test quantum nonlocality [31], which includes two-type of Bell-inequality by
using the Wigner function [32] and the $Q-$function [30] as its limiting
cases.
By noticing that
$\alpha=(q_{1}+ip_{1})/\sqrt{2},\beta=(q_{2}+ip_{2})/\sqrt{2}$, and
$\left(\begin{array}[c]{cccc}q_{1}&p_{1}&q_{2}&p_{2}\end{array}\right)N^{-1}=\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right),N=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}1&i&0&0\\\
1&-i&0&0\\\ 0&0&1&i\\\ 0&0&1&-i\end{array}\right),$ we can put Eq.(31) into
another form
$W\left(\alpha;\beta\right)=\frac{1}{\pi^{2}}\exp\left[-\frac{1}{2}\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)M\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)^{T}\right],$
(41)
where $\bar{M}$ is a $4\times 4$ Hermitian matrix
$M=\left(\begin{array}[c]{cccc}m_{1}-m_{2}&m_{1}+m_{2}&-2m_{3}&0\\\
m_{1}+m_{2}&m_{1}-m_{2}&0&-2m_{3}\\\ -2m_{3}&0&m_{2}-m_{1}&m_{1}+m_{2}\\\
0&-2m_{3}&m_{1}+m_{2}&m_{2}-m_{1}\end{array}\right).$ (42)
Substituting Eq.(41) into Eq.(40) we have
$\displaystyle\mathcal{B}$
$\displaystyle=1+\exp\left[-2J\cosh^{2}\lambda-2J\left(e^{2\gamma}\cos^{2}\varphi+e^{-2\gamma}\sin^{2}\varphi\allowbreak\right)\sinh^{2}\lambda\right]$
$\displaystyle\text{ \ \ \ \
}+\exp\left[-2J\cosh^{2}\lambda-2J\left(e^{2\gamma}\sin^{2}\theta+e^{-2\gamma}\cos^{2}\theta\allowbreak\right)\sinh^{2}\lambda\right]$
$\displaystyle\text{ \ \ \ \
}-\exp\left\\{-4J\cosh^{2}\lambda-2J\left(\cos^{2}\varphi+\sin^{2}\theta\right)e^{2\gamma}\sinh^{2}\lambda\right.$
$\displaystyle\ \ \ \ \ \ \
\left.-2J\left(\sin^{2}\varphi+\cos^{2}\theta\right)e^{-2\gamma}\allowbreak\sinh^{2}\lambda+4J\cos\left(\theta+\varphi\right)\cosh\gamma\sinh
2\lambda\right\\},$ (43)
Thus we can say that $V\left|00\right\rangle$ is quantum mechanically nonlocal
as $\left|\mathcal{B}\right|>2,$ and the nonlocality is stronger with the
increase of $\left|\mathcal{B}\right|$. From Eq.(43) one can see that the
degree of nonlocality not only depends on the coherent amplitude $J$, on the
phases $\theta$ and $\varphi$, but also on the parameter $\gamma$. In
particular, when $\varphi=0$, $\theta=\pi,$ Eq.(43) just reduces to
$\displaystyle\mathcal{B}$
$\displaystyle=1+\exp\left[-2J\left(\cosh^{2}\lambda+e^{2\gamma}\sinh^{2}\lambda\right)\right]$
$\displaystyle\text{ \ \ \ \
}+\exp\left[-2J\left(\cosh^{2}\lambda+e^{-2\gamma}\sinh^{2}\lambda\right)\right]$
$\displaystyle\text{ \ \ \ \ }-\exp\left[-4J\left(\cosh^{2}\lambda+\cosh
2\gamma\sinh^{2}\lambda\right)-4J\allowbreak\cosh\gamma\sinh 2\lambda\right],$
(44)
which further becomes Eq.(7) in Ref.[28] with $\gamma=0$.
Figure 2: (Color online) Plot of the Bell function $\mathcal{B}$as a function
of parameters $\lambda$and $J$, for $\gamma=0,\theta=\pi,\varphi=0$. Only
values exceeding the bound imposed by local theories are shown.
In figure 2, we plot Bell function in the space spanned by parameters $J$ and
$\lambda$ with $\gamma=0$ (corresponding to the usual squeezed vacuum state).
From Fig. 2, one can clearly see that the result (43) violates the upper bound
imposed by local theories. With the increase of $\lambda$, the violation of
Bell’s inequality becomes more observable for smaller $J$.
Figure 3: (Color online) Plot of the Bell function $\mathcal{B}$as a function
of parameters $\lambda$ and $\gamma,$for given
$\left|\alpha\right|=\left|\beta\right|=0.05$and $\theta=\pi,\varphi=0.$Only
values exceeding the bound imposed by local theories are shown. Figure 4:
(Color online) Plot of the Bell function $\mathcal{B}$as a function of
parameters $J$ and $\gamma$,for given $\theta=\pi,\varphi=0,$and
$\lambda=0.1$. Only values exceeding the bound imposed by local theories are
shown.
As depicted in Fig. 3, the Bell function also violates the upper bound in the
space spanned by parameters $\gamma$ and $\lambda$ with given $\alpha,\beta$
values. From Fig.3, one can see that for a given small $\gamma,$ the violation
of Bell’s inequality becomes more observable with the increase of $\lambda$;
while for a large $\gamma$, the Bell function is not always monotone for an
increasing $\lambda$; In a certain smaller regime of $\lambda$, it is found
that the violation of Bell’s inequality becomes more observable with
increasing $\gamma$, which indicates that the nonlocality of
$V\left|00\right\rangle$ is enhanced due to the presence of $\gamma$ (also see
Fig.4). In addition, for a certain larger regime of $\lambda$, the maximum
value of $\mathcal{B}$ becomes smaller with the increase of $\gamma$.
On the other hand, from the expression we see that the degree of nonlocality
depends on the coherent amplitude $J$, and on the squeezed parameters
$\lambda$ and $\gamma$, and on the phases $\varphi$, $\theta$. We have plotted
the Bell function $\mathcal{B}$ as a function of the phases $\varphi$,
$\theta$ and with fixed $J=0.01$ and several different $\lambda,\gamma$, as
shown in Fig. 5. One can clearly see from Fig. 5 that $\mathcal{B}$ is always
greater than zero and for a smaller $\lambda$, the variable $\mathcal{B}$
reaches its maxmum value for $\varphi=0,\theta=\pi$ or $\varphi=\pi,\theta=0$
(see Fig.5(a),(b)); while for a larger $\lambda$, the phases $\varphi$,
$\theta$ corresponding to the maxmum value of $\mathcal{B}$ are different from
those above and vary as $\gamma$ parameter.
Figure 5: (Color online) Plot of the Bell function $\mathcal{B}$ as a function
of parameters $\theta$ and $\varphi$,for given $J=0.01,$and (a) $\lambda=0.5$,
$\gamma=1;$(b) $\lambda=0.5$, $\gamma=2;$(c) $\lambda=1$, $\gamma=1;$(a)
$\lambda=1$, $\gamma=2.$
## 6 Application to quantum teleportation with $V\left|00\right\rangle$
In quantum teleportation (QT), an unknown state is transmitted from a sender
(Alice) to a receiver (Bob) via a quantum channel with the aid of some
classical information. This process may be regarded as sending and extracting
quantum information via the quantum channel. QT was firstly proposed by
Barnnett et al in the discrete variable regime [33] followed by experimental
demonstration [34, 35]. For the continuous variables (CVs) case, the
theoretical analysis of teleportation was firstly made by Vaidman [36]. The
role of teleportation in the CV quantum information is analyzed in the review
Ref.[37].
Recently, a CV teleportation protocol has been given in terms of the
characteristic functions (CFs) of the quantum states involved (input, source
and teleported (output) states) [38]. By using the Weyl expansion of density
operator, it is shown that the CF $\chi_{out}\left(\beta\right)$ of the output
state has a remarkably factorized form
$\chi_{out}\left(\beta\right)=\chi_{in}\left(\beta\right)\chi_{E}\left(\beta^{\ast},\beta\right),$
(45)
where $\chi_{in}\left(\beta\right)$ and
$\chi_{E}\left(\beta^{\ast},\beta\right)$ are the CFs of the input state and
the entangled source, respectively,
$\chi_{in}\left(\alpha_{1}\right)=$tr$\left[D_{1}\left(\alpha_{1}\right)\rho\right]$
and
$\chi_{E}\left(\alpha_{1},\alpha_{2}\right)=$tr$\left[D_{1}\left(\alpha_{1}\right)D_{2}\left(\alpha_{2}\right)\rho\right]$,
$D_{i}\left(\alpha_{i}\right)$ is the displacement operator corresponding to
mode $i$, and $\rho$ is the density operator associated to the state.
In order to measure the success probability of a teleportation protocol, it is
convenient to use the fidelity of teleportation
$\mathcal{F=}$tr$\left(\rho_{in}\rho_{out}\right)$, an overlap between the
input state $\rho_{in}$ and the output (teleported) state $\rho_{out}$, which
can, in the CF form, be expressed as
$\mathcal{F=}\int\frac{d^{2}\eta}{\pi}\chi_{in}\left(\eta\right)\chi_{out}\left(-\eta\right).$
(46)
Substituting Eq.(45) into (46) yields
$\mathcal{F=}\int\frac{d^{2}\eta}{\pi}\left|\chi_{in}\left(\eta\right)\right|^{2}\chi_{E}\left(-\eta^{\ast},-\eta\right).$
(47)
In the following we use Eq.(47) to analyze the efficiency of teleportation for
$V\left|00\right\rangle$ as a quantum channel.
Let us first consider Braunstein and Kimble protocol [39] of QT for single-
mode coherent states $\left|\beta\right\rangle$, whose CF reads
$\chi_{\text{coh}}\left(\alpha\right)=\exp\left[-\frac{1}{2}\left|\alpha\right|^{2}+\alpha\beta^{\ast}-\alpha^{\ast}\beta\right].$
(48)
For the OTCSS, its CF is given by (see Appendix B)
$\chi\left(\alpha;\beta\right)=\exp\left[-\frac{1}{2}\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)\frac{1}{4}M\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)^{T}\right],$
(49)
where $M$ is defined in Eq.(42). Upon substituting Eqs.(48) and (49) into
(47), we worked out the fidelity for teleporting a coherent state based on the
OTCSS (19),
$\mathcal{F}=\frac{1}{1-f},$ (50)
where $f=\cosh\gamma\sinh 2\lambda-\cosh^{2}\lambda-\cosh
2\gamma\sinh^{2}\lambda.$ Eq.(50) indicates that the fidelity is only
dependent on the parameters $\lambda$ and $\gamma$, and is independent of
amplitude of the coherent state. In particular, when $\gamma=0$, Eq.(50) just
reduces to $\mathcal{F=}(1+\tanh\lambda)/2$ [40].
Next we consider to teleport the single-mode squeezed vacuum state,
$\exp\left[r/2\left(a^{2}-a^{{\dagger}2}\right)\right]\left|0\right\rangle,$
whose CF reads
$\chi_{sq}\left(\alpha\right)=\exp\left[-\frac{1}{2}\left|\alpha\right|^{2}\cosh
2r-\frac{1}{4}\left(\alpha^{2}+\alpha^{\ast 2}\right)\sinh 2r\right],$ (51)
substituting Eqs.(51),(49) into Eq.(47) yields the fidelity,
$\mathcal{F}\left(r\right)\mathcal{=}\frac{1}{\sqrt{\allowbreak f^{2}-2f\cosh
2r+1}}.$ (52)
In order to compare the fidelity obtained by the TSVS channel and the OTCSS
channel, we plot figure for the difference fidelity
($\mathcal{F}\left(r\right)-\mathcal{F}\left(0\right)$) as a function of
parameters ($\lambda$ and $\gamma$) in Fig.6. From Fig.6, one can see that
there is a region spanned by $\lambda$ and $\gamma$ in which the fidelity of
OTCSS channel is larger than the other one and the difference fidelity becomes
smaller with the increase of $r$.
Figure 6: (Color online) Plot of the fidelity $\mathcal{F}$ as a function of
parameters $\lambda$ and $\gamma$,(a) the initial coherent state, with
$r=0;$(b) the initial squeezed vacuum state with $r=1.$
## 7 Conclusion
In conclusion, we have introduced a one- and two-mode combination squeezed
state (OTCSS) which can be considered as a generalized two-mode squeezed state
and investigated the entanglement properties and quantum noncocality of this
state in terms of logarithmic negativity and the Bell’s inequality,
respectively. It is shown that this state presents larger entanglement than
that of the usual two-mode squeezed vacuum state (TSVS); In a certain smaller
regime of $\lambda$, it is found that the violation of Bell’s inequality
becomes more observable with increasing $\gamma$, which indicates that the
nonlocality of $V\left|00\right\rangle$ can be stronger than that of TSVS due
to the presence of $\gamma.$ In addition, application to quantum teleportation
with OTCSS is also considered, which shows that there is a region spanned by
$\lambda$ and $\gamma$ in which the fidelity of OTCSS channel is larger than
that of TSVS.
ACKNOWLEDGEMENT: Work supported by a grant from the Key Programs Foundation of
Ministry of Education of China (No. ) and the Research Foundation of the
Education Department of Jiangxi Province of China (No. GJJ10097).
Appendix A: Derivation of Eq.(29)
Using the order-invariance of Weyl ordered operators under similar
transformations (27) and Eqs.(9)-(12), we have
$\displaystyle U$
$\displaystyle=V^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\delta\left(q_{2}-Q_{2}\right)\delta\left(p_{2}-P_{2}\right)\genfrac{}{}{0.0pt}{}{:}{:}V$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\cosh\lambda-
Q_{2}e^{-\gamma}\sinh\lambda\right)\delta\left(q_{2}-Q_{2}\cosh\lambda-
Q_{1}e^{\gamma}\sinh\lambda\right)$
$\displaystyle\times\delta\left(p_{1}-P_{1}\cosh\lambda+P_{2}e^{\gamma}\sinh\lambda\right)\delta\left(p_{2}-P_{2}\cosh\lambda+P_{1}e^{-\gamma}\sinh\lambda\right)\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\left(\begin{array}[c]{c}q_{1}\\\
q_{2}\end{array}\right)-\left(\begin{array}[c]{cc}\cosh\lambda&e^{-\gamma}\sinh\lambda\\\
e^{\gamma}\sinh\lambda&\cosh\lambda\end{array}\right)\left(\begin{array}[c]{c}Q_{1}\\\
Q_{2}\end{array}\right)\right)$ (59)
$\displaystyle\times\delta\left(\left(\begin{array}[c]{c}p_{1}\\\
p_{2}\end{array}\right)-\left(\begin{array}[c]{cc}\cosh\lambda&-e^{\gamma}\sinh\lambda\\\
-e^{-\gamma}\sinh\lambda&\cosh\lambda\end{array}\right)\left(\begin{array}[c]{c}P_{1}\\\
P_{2}\end{array}\right)\right)\genfrac{}{}{0.0pt}{}{:}{:}$ (66)
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\left(\begin{array}[c]{cc}\cosh\lambda&-e^{-\gamma}\sinh\lambda\\\
-e^{\gamma}\sinh\lambda&\cosh\lambda\end{array}\right)\left(\begin{array}[c]{c}q_{1}\\\
q_{2}\end{array}\right)-\left(\begin{array}[c]{c}Q_{1}\\\
Q_{2}\end{array}\right)\right)$ (73)
$\displaystyle\times\delta\left(\left(\begin{array}[c]{cc}\cosh\lambda&e^{\gamma}\sinh\lambda\\\
e^{-\gamma}\sinh\lambda&\cosh\lambda\end{array}\right)\left(\begin{array}[c]{c}p_{1}\\\
p_{2}\end{array}\right)-\left(\begin{array}[c]{c}P_{1}\\\
P_{2}\end{array}\right)\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (A1)
which indicates that (comparing with Eq.(24))
$\displaystyle U$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}\cosh\lambda-
q_{2}e^{-\gamma}\sinh\lambda-
Q_{1}\right)\delta\left(p_{1}\cosh\lambda+p_{2}e^{\gamma}\sinh\lambda-
P_{1}\right)$ $\displaystyle\times\delta\left(q_{2}\cosh\lambda-
q_{1}e^{\gamma}\sinh\lambda-
Q_{2}\right)\delta\left(p_{2}\cosh\lambda+p_{1}e^{-\gamma}\sinh\lambda-
P_{2}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\text{Eq.(\ref{25})}.$
(A2)
Thus the Wigner function of $V\left|00\right\rangle$ is
$\displaystyle W\left(q_{1},p_{1};q_{2},p_{2}\right)$
$\displaystyle=\frac{1}{\pi^{2}}\exp\left\\{-\left(q_{1}\cosh\lambda-
q_{2}e^{-\gamma}\sinh\lambda\right)^{2}-\left(q_{2}\cosh\lambda-
q_{1}e^{\gamma}\sinh\lambda\right)^{2}\right.$
$\displaystyle\left.-\left(p_{1}\cosh\lambda+p_{2}e^{\gamma}\sinh\lambda\right)^{2}-\left(p_{2}\cosh\lambda+p_{1}e^{-\gamma}\sinh\lambda\right)^{2}\right\\}$
$\displaystyle=(\ref{25}).$ (A3)
Appendix B: Derivation of characteristic function of $V\left|00\right\rangle$
For two-mode quantum state $V\left|00\right\rangle$, its characteristic
function is given by
$\displaystyle\chi\left(q_{1},p_{1};q_{2},p_{2}\right)$
$\displaystyle=\mathtt{tr}\left[V\left|00\right\rangle\left\langle
00\right|V^{-1}D_{1}\left(q_{1},p_{1}\right)D_{2}\left(q_{2},p_{2}\right)\right]$
$\displaystyle=\left\langle
00\right|V^{-1}D_{1}\left(q_{1},p_{1}\right)D_{2}\left(q_{2},p_{2}\right)V\left|00\right\rangle,$
(B1)
where $D_{i}\left(q_{i},p_{i}\right)$ $\left(i=1,2\right)$ are the
displacement operators, defined by
$D_{i}\left(q_{i},p_{i}\right)=\exp\left[\mathtt{i}\left(p_{i}Q_{i}-q_{i}P_{i}\right)\right].$
Noticing that the Weyl ordering of $D_{i}\left(q_{i},p_{i}\right)$ is itself,
$D_{i}\left(q_{i},p_{i}\right)=\genfrac{}{}{0.0pt}{}{:}{:}D_{i}\left(q_{i},p_{i}\right)\genfrac{}{}{0.0pt}{}{:}{:}$,
and that a remarkable property of the invariance of Weyl ordered operators
under similar transformations (27), we have
$\displaystyle
V^{-1}D_{1}\left(q_{1},p_{1}\right)D_{2}\left(q_{2},p_{2}\right)V$
$\displaystyle=V^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[i\left(p_{1}Q_{1}-q_{1}P_{1}\right)\right]\exp\left[i\left(p_{2}Q_{2}-q_{2}P_{2}\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}V$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[i\left(p_{1}\left(Q_{1}\cosh\lambda+Q_{2}e^{-\gamma}\sinh\lambda\right)-q_{1}\left(P_{1}\cosh\lambda-
P_{2}e^{\gamma}\sinh\lambda\right)\right)\right]$
$\displaystyle\times\exp\left[i\left(p_{2}\left(Q_{2}\cosh\lambda+Q_{1}e^{\gamma}\sinh\lambda\right)-q_{2}\left(P_{2}\cosh\lambda-
P_{1}e^{-\gamma}\sinh\lambda\right)\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[i\left(p_{1}^{\prime}Q_{1}-q_{1}^{\prime}P_{1}\right)\right]\exp\left[\allowbreak
i\left(p_{2}^{\prime}Q_{2}-q_{2}^{\prime}P_{2}\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=D_{1}\left(q_{1}^{\prime},p_{1}^{\prime}\right)D_{2}\left(q_{2}^{\prime},p_{2}^{\prime}\right),$
(B2)
where
$\displaystyle q_{1}^{\prime}$ $\displaystyle=q_{1}\cosh\lambda-
q_{2}e^{-\gamma}\sinh\lambda,\text{
}p_{1}^{\prime}=p_{1}\cosh\lambda+p_{2}\allowbreak e^{\gamma}\sinh\lambda,$
$\displaystyle q_{2}^{\prime}$ $\displaystyle=q_{2}\cosh\lambda-
q_{1}e^{\gamma}\sinh\lambda,\text{
}p_{2}^{\prime}=p_{2}\cosh\lambda+p_{1}e^{-\gamma}\sinh\lambda.$ (B3)
Then substituting Eqs.(B2), (B3) into Eq.(B1) we can directly obtain the CF of
$V\left|00\right\rangle,$
$\displaystyle\chi\left(q_{1},p_{1};q_{2},p_{2}\right)$
$\displaystyle=\left\langle
00\right|D_{1}\left(q_{1}^{\prime},p_{1}^{\prime}\right)D_{2}\left(q_{2}^{\prime},p_{2}^{\prime}\right)\left|00\right\rangle$
$\displaystyle=\exp\left[-\frac{1}{2}\left(\begin{array}[c]{cccc}q_{1}&p_{1}&q_{2}&p_{2}\end{array}\right)\sigma\left(\begin{array}[c]{cccc}q_{1}&p_{1}&q_{2}&p_{2}\end{array}\right)^{T}\right],$
(B4)
or
$\chi\left(\alpha;\beta\right)=\exp\left[-\frac{1}{2}\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)\bar{\sigma}\left(\begin{array}[c]{cccc}\alpha^{\ast}&\alpha&\beta^{\ast}&\beta\end{array}\right)^{T}\right],$
(B5)
where
$\sigma=\frac{1}{2}\left(\begin{array}[c]{cccc}m_{1}&0&-m_{3}&0\\\
0&m_{2}&0&m_{3}\\\ -m_{3}&0&m_{2}&0\\\
0&m_{3}&0&m_{1}\end{array}\right),\bar{\sigma}=N\sigma N^{T}=\frac{1}{4}M.$
(B6)
When $\gamma=0,$ $\left(m_{1}+m_{2}\right)\rightarrow 2\cosh
2\lambda,m_{3}\rightarrow\sinh 2\lambda,$we have
$\chi\left(\alpha;\beta\right)=\exp\left[-\frac{1}{2}\allowbreak\left(\left|\alpha\right|^{2}+\left|\beta\right|^{2}\right)\cosh
2\lambda+\frac{1}{2}\left(\alpha^{\ast}\beta^{\ast}+\alpha\beta\right)\sinh
2\lambda\allowbreak\right],$ (B7)
which is just the CF of the usual TSVS.
## References
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|
arxiv-papers
| 2010-08-31T08:25:08 |
2024-09-04T02:49:12.586610
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu, Xue-xiang Xu, Qin Guo, and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1008.5253"
}
|
1008.5256
|
# Photon-subtracted squeezed thermal state: nonclassicality and
decoherence††thanks: Work supported by the National Natural Science Foundation
of China under grant 10775097 and 10874174, and the Research Foundation of the
Education Department of Jiangxi Province of China (No. GJJ10097).
Li-yun Hu1,2, Xue-xiang Xu1,2, Zi-sheng Wang1,2 and Xue-fen Xu3
1College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Key Laboratory of Optoelectronic and Telecommunication of Jiangxi, Nanchang,
Jiangxi 330022, China
3School of Mathematics and Physics, Jiangsu Teachers University of Technology,
Changzhou, Jiangsu 213001, China Corresponding author. E-mail:
hlyun2008@126.com.
###### Abstract
We investigate nonclassical properties of the field states generated by
subtracting any number photon from the squeezed thermal state (STS). It is
found that the normalization factor of photon-subtracted STS (PSSTS) is a
Legendre polynomial of squeezing parameter ${\small r}$ and average photon
number $\bar{n}$ of thermal state. Expressions of several quasi-probability
distributions of PSSTS are derived analytically. Furthermore, the
nonclassicality is discussed in terms of the negativity of Wigner function
(WF). It is shown that the WF of single PSSTS always has negative values if
$\bar{n}<\sinh^{2}r$ at the phase space center. The decoherence effect on
PSSTS is then included by analytically deriving the time evolution of WF. The
results show that the WF of single PSSTS has negative value if $2\kappa
t<\ln\\{1-(2\bar{n}+1)(\bar{n}-\sinh^{2}r)$/$[(2\mathfrak{N}+1)(\bar{n}\cosh
2r+\sinh^{2}r)]\\}$, which is dependent not only on average number
$\mathfrak{N}$ of environment, but also on $\bar{n}$ and $r$.
Keywords: Nonclassicality, decoherence, Photon-subtraction, Squeezed thermal
state
PACS number(s): 42.50.Dv, 03.65.Wj, 03.67.Mn
## 1 Introduction
Nonclassical Gaussian states play an important role in quantum information
processing with continuous variables, such as teleportation, dense coding, and
quantum cloning. In a quantum optics laboratory, Gaussian states have been
generated but there is some limitation in using them for various tasks of
quantum information procession [1]. For example, when a two-mode squeezed
vacuum state (a Gaussian state) with low squeezing is used as an entangled
resource to realize quantum teleportation, the average fidelity is just more
$\left(8\pm 2\right)\%$ than the classical limits. On the other hand, it is
possible to generate and manipulate various nonclassical optical fields by
subtracting or adding photons from/to traditional quantum states or Gaussian
states, which are useful ways to conditionally manipulate nonclassical state
of optical field [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Recently, subtracting or
adding photon states have received more attention from both experimentalists
and theoreticians [12, 13, 14, 15, 16, 17]. One of reasons is that photon
subtraction can be applied to improve entanglement between Gaussian states
[18, 19], loophole-free tests of Bell’s inequality [20, 21], and quantum
computing [22]. Thus the photon subtraction (a non-Gaussian operation) can
satisfy the requirement of quantum information protocols for long-distance
communication. Nevertheless, the photon addition and subtraction have been
successfully demonstrated experimentally for probing quantum commutation rules
by Parigi et al. [23, 24]. In fact, they have implemented simple alternated
sequences of photon creation (addition) and annihilation (subtraction) on a
thermal field and observed the noncommutativity of the creation and
annihilation operators.
In addition, Olivares et al. [25] theoretically discussed the relation between
the photon subtracted squeezed vacuum (PSSV), as an output state passing
through a beamsplitter, and two parameters (the transmissivity of beamsplitter
and the photodetection quantum efficiency). Then the case of two-mode photon-
subtraction is also further discussed in the presence of noise [26, 27].
Kitagawa et al [28] investigated the degree of entanglement for non-Gaussian
mixed (pure) states generated by photon subtraction from two-mode squeezed
vacuum states with on-off photon detectors. For the single PSSV, furthermore,
its nonclassical properties and decoherence was investigated theoretically in
two different decoherent channels (amplitude decay and phase damping) by
Biswas and Agarwal [5]. They indicated that the WF losses its non-Gaussian
nature and becomes Gaussian at long times in amplitude decay case. Recently,
it is found that consecutive applications of photon subtraction (or
subtracting a well-defined number of photons) from a squeezed vacuum state
result in the generation of a squeezed superpositions of coherent state (SSCS)
with nearly the perfect fidelity regardless of the number of photons
subtracted [6]. The amplitude of the SSCS increases as the number of the
subtracted photons gets larger.
It is interesting in noticing that single-mode displaced-squeezed thermal
state can be considered as a generalized Gaussian state, which has received
more attention [29, 30, 31, 32, 33]. For example, phase estimations for
squeezed thermal states (STSs) and displaced thermal states are presented
[29], which shows that a larger temperature can enhance the estimation
fidelity for the former. Another example is, for Gaussian squeezed states of
light, that a scheme is also presented experimentally to measure its
squeezing, purity and entanglement [32, 33]. To our knowledge, however, the
investigation of photon subtraction from STS (even for single photon
subtraction case) has not been previously addressed (especially when this
state interacts with its surrounding environment). In addition, the exact
threshold value of the decay time has not been explicity given. In this paper,
we focus on any number photon-subtracted single-mode STS (PSSTS), which is
optically produced single-mode non-Gaussian states, and explore theoretically
its nonclassical properties and decoherence in a thermal channel by deriving
analytically some expressions, such as normalized constant, photon-number
distribution and Wigner function (WF). For single PSSTS, it is shown that the
WF of single PSSTS always has the negative values under the condition of
$\bar{n}<\sinh^{2}r$ at the phase space center ($\bar{n}$ and $r$ are an
average number of thermal state and a squeezing parameter, respectively), and
that the threshold value of the decay time is dependent not only on the
average number of environment, but also on $\bar{n}$ and $r$.
In section II, we introduce the single-mode PSSTS, where the normalized factor
turns out to be a Legendre polynomial with a remarkable result. In Sec. III,
the nonclassical properties of the PSSTS, such as Mandel’s $Q$-parameter, and
distribution of photon number (related to a Legendre polynomial), are
calculated analytically and then be discussed in details. In Sec. IV, the
explicitly analytical expressions of quasiprobability distributions for PSSTS,
such as P-distribution, Q-function and WF of the PSSTS, are derived by using
the Weyl ordered operators’ invariance under a similar transformations. Then
we derive an explicitly analytical expression of time evolution of WF for the
arbitrary PSSTS in the thermal channel and discuss the loss of nonclassicality
in reference of the negativity of WF in Sec. V. It is found that the threshold
value of decay time corresponding to the transition of WF from partial
negative to completely positive definite is obtained at the center of the
phase space, which is not only dependent on the average number $\mathfrak{N}$
of environment, but also on $\bar{n}$ and $r$. We show that the WF for single
PSSTS has always negative value if the decay time $\kappa
t<\frac{1}{2}\ln\\{1-(2\bar{n}+1)(\bar{n}-\sinh^{2}r)/[(2\mathfrak{N}+1)(\bar{n}\cosh
2r+\sinh^{2}r)]\\}$ (see Eq.(51) below), where $\kappa$ denotes a dissipative
coefficient of interacting with the environment. Sec. VI is devoted to
calculating the fidelity between the PSSTS and the STS. It is shown that the
fidelity decreases monotonously with the increment of both photon-subtraction
number $m$ and the squeezing parameter $r$. We end with the main conclusions
of our work.
## 2 Photon-subtraction squeezed thermal state
At first, let’s introduce the photon-subtraction squeezed thermal state
(PSSTS). For a squeezed thermal field, its density operator is
$\rho_{s}=S(r)\rho_{c}S^{\dagger}(r),$ (1)
where $S(r)=\exp[r(a^{\dagger 2}-a^{2})/2]=\exp[-$i$r(QP+PQ)/2]$ is the
squeezing operator [34, 35] with squeezing parameter $r$, here the coordinate
operators $Q=(a+a^{\dagger})/\sqrt{2}$ and the momentum operators
$P=(a-a^{\dagger})/(\sqrt{2}\mathtt{i)}$ $(\left[a,a^{\dagger}\right]=1)$ are
introduced as functions of create and annihilation operators $a^{\dagger}$ and
$a$, respectively, and $\rho_{c}$ is a density operator of thermal state,
$\rho_{c}=(1-e^{\sigma})e^{\sigma a^{\dagger}a},\text{
}\sigma=-\frac{\hbar\omega}{kT},$ (2)
where $k$ is a Boltzmann constant, and the temperature $T$ is qualified to be
a density operator of thermal (chaotic) field with tr$\rho_{c}=1$. Using the
operator identity [37, 38]
$e^{\sigma
a^{\dagger}a}=\colon\exp[(e^{\sigma}-1)a^{\dagger}a]\colon=\frac{2}{e^{\sigma}+1}\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{\frac{e^{\sigma}-1}{e^{\sigma}+1}\left(Q^{2}+P^{2}\right)\right\\}\genfrac{}{}{0.0pt}{}{\colon}{\colon},$
(3)
where these two symbols $\colon$ $\colon$ and
$\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$denote normal ordering
and Weyl ordering, respectively, and using the Weyl ordering invariance under
similarity transformations [37, 38], which means that
$S\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}S^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}S\left(\circ\circ\circ\right)S^{-1}\genfrac{}{}{0.0pt}{}{:}{:},$
(4)
as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did
not exist, so $S$ can pass through it, as well as the technique of integration
within an ordered product of operators (IWOP), one can convert $\rho_{s}$ to
its normally ordered Gaussian form [38] (see Appendix A), i.e.,
$\rho_{s}=\frac{1}{\tau_{1}\tau_{2}}\colon\exp\left\\{-\frac{Q^{2}}{2\tau_{1}^{2}}-\frac{P^{2}}{2\tau_{2}^{2}}\right\\}\colon,$
(5)
where
$2\tau_{1}^{2}=(2\bar{n}+1)e^{2r}+1,2\tau_{2}^{2}=(2\bar{n}+1)e^{-2r}+1,$ (6)
which leads to the following relations,
$\displaystyle\tau_{1}^{2}-\tau_{2}^{2}$ $\displaystyle=(2\bar{n}+1)\sinh 2r,$
(7) $\displaystyle\tau_{1}^{2}+\tau_{2}^{2}$ $\displaystyle=(2\bar{n}+1)\cosh
2r+1,$ (8) $\displaystyle\tau_{1}^{2}\tau_{2}^{2}$
$\displaystyle=\bar{n}^{2}+\left(2\bar{n}+1\right)\allowbreak\cosh^{2}r,$ (9)
and
$\bar{n}=\mathtt{tr}\left(\rho_{c}a^{\dagger}a\right)=(e^{-\sigma}-1)^{-1}$[39]
denotes the average photon number of thermal (chaotic) field $\rho_{c}$ in Eq.
(2). The form in Eq.(5) is similar to the bivariate normal distribution in
statistics, which is useful for us to further derive the marginal
distributions of $\rho_{s}$.
Theoretically, the PSSTS can be obtained by repeatedly operating the photon
annihilation operator $a$ on a squeezed thermal state, so its density operator
is given by
$\rho=C_{m}^{-1}a^{m}\rho_{s}a^{{\dagger}m},$ (10)
where $m$ is the subtracted photon number (a non-negative integer), and
$C_{m}$ is a normalized constant with (see Appendix B)
$C_{m}=\mathtt{Tr}(a^{m}\rho_{s}a^{{\dagger}m})=m!D^{m/2}P_{m}\left(B/\sqrt{D}\right),$
(11)
which indicates that $C_{m}$ is just related to Legendre polynomial
$P_{m}\left(x\right)$ (see Appendix B (B10)), and
$\displaystyle B$ $\displaystyle=\frac{1}{2}\left[\left(2\bar{n}+1\right)\cosh
2r-1\right],$ (12) $\displaystyle D$
$\displaystyle=\bar{n}^{2}-\left(2\bar{n}+1\right)\sinh^{2}r.$ (13)
It is noted that, for the case of no-photon-subtraction with $m=0$, $C_{0}=1$
as expected. Under the case of $m$-photon-subtraction thermal state with
$B=\bar{n}$, $D=\bar{n}^{2},$ and $P_{m}\left(1\right)=1$,
$C_{m}=m!\bar{n}^{m}.$ The same result as Eq.(24) can be found in Ref.[40].
Here we should point out that, as Agarwal et al introduced the excitations on
a coherent state by repeated application of the photon creation operator on
the coherent state [41], we introduce theoretically the PSSTS (10). In
realistic situations, one the other hand, the photon subtraction would be done
by on/off detector and the tapping beam splitters with a non-unity
transmittance, which leads to a generated mixed state. For various schemes for
generating photon subtraction, one can refer to Refs.[1, 28, 42].
## 3 Nonclassical properties of PSSTS
### 3.1 Mandel’s $Q$-parameter
The analytical expression of $C_{m}$ is of importance for further
investigating the properties of PSSTS. For instance, one can easily calculate
$\displaystyle\left\langle a^{{\dagger}}a\right\rangle$
$\displaystyle=\mathtt{Tr}(C_{m}^{-1}a^{m+1}\rho_{s}a^{{\dagger}m+1})=\frac{C_{m+1}}{C_{m}},$
(14) $\displaystyle\left\langle a^{{\dagger}2}a^{2}\right\rangle$
$\displaystyle=\mathtt{Tr}(C_{m}^{-1}a^{m+2}\rho_{s}a^{{\dagger}m+2})=\frac{C_{m+2}}{C_{m}},$
(15)
thus the Mandel’s $Q$-parameter is given by
$Q_{M}=\frac{\left\langle a^{{\dagger}2}a^{2}\right\rangle}{\left\langle
a^{{\dagger}}a\right\rangle}-\left\langle
a^{{\dagger}}a\right\rangle=\frac{C_{m+2}}{C_{m+1}}-\frac{C_{m+1}}{C_{m}},$
(16)
which measures the deviation of the variance of the photon number distribution
of the field state under consideration from the Poissonian distribution of the
coherent state. If $Q_{M}=0$ we say the field has Poissonian photon
statistics, while for $Q_{M}>0$ ($Q_{M}<0$), the field has super-(sub-)
Poissonian photon statistics. It is well-known that the negativity of the
$Q_{M}$-parameter refers to sub-Poissonian statistics of the state. But a
state may be nonclassical even though $Q_{M}$ is positive as pointed out in
Ref.[43]. This case is true for the present state. In fact, if $Q_{M}$ is
positive, it does not mean that the state is classical. In such cases, we have
to use other parameters to test the non-classicality [43]. From Fig.1, one can
see clearly that for odd number $m$, $Q_{M}$ becomes negative when the
squeezing parameter $r$ is less than a certain threshold value which decreases
as $m$ increases. Differently from the case of odd number $m$, $Q_{M}$ is
always positive for even number $m$. It is necessary to emphasize that the
Wigner function (WF) has negative region for all $r,$ and thus the PSSTS is
nonclassical. In addition, when the average photon number $\bar{n}$ is larger
than a certain threshold value, $Q_{M}$ is also always positive. Without loss
of generality, thus, we consider only the (ideal) PSSTS in a thermal channel
in our following work.
Figure 1: (Color online) The $Q$-parameter as the function of squeezing
parameter $r$ for different $m=0,1,2,3,4,19,20.$
### 3.2 Photon-number distribution of PSSTS
Next we discuss the photon-number distribution (PND) of PSSTS. Noticing
$a^{{\dagger}m}\left|n\right\rangle=\sqrt{(m+n)!/n!}\left|m+n\right\rangle$
and using the un-normalized coherent state
$\left|\alpha\right\rangle=\exp[\alpha
a^{{\dagger}}]\left|0\right\rangle$,[44, 45] leading to
$\left|n\right\rangle=\frac{1}{\sqrt{n!}}\frac{\mathtt{d}^{n}}{\mathtt{d}\alpha^{n}}\left|\alpha\right\rangle\left|{}_{\alpha=0}\right.,$
$\left(\left\langle\beta\right.\left|\alpha\right\rangle=e^{\alpha\beta^{\ast}}\right)$,
as well as the normal ordering form of $\rho_{s}$ in Eq. (5), the probability
of finding $n$ photons in the field is given by
$\displaystyle\mathcal{P}(n)$ $\displaystyle=\left\langle
n\right|\rho\left|n\right\rangle=C_{m}^{-1}\left\langle
n\right|a^{m}\rho_{s}a^{{\dagger}m}\left|n\right\rangle$
$\displaystyle=\frac{(m+n)!}{n!C_{m}}\left\langle
m+n\right|\rho_{s}\left|m+n\right\rangle$
$\displaystyle=\frac{C_{m}^{-1}}{n!\tau_{1}\tau_{2}}\frac{d^{m+n}}{d\beta^{\ast
m+n}}\frac{d^{m+n}}{d\alpha^{m+n}}\left.\left\langle\beta\right|\colon\exp\left\\{-\frac{\left(a+a^{{\dagger}}\right)^{2}}{4\tau_{1}^{2}}+\frac{\left(a-a^{{\dagger}}\right)^{2}}{4\tau_{2}^{2}}\right\\}\colon\left|\alpha\right\rangle\right|_{\alpha=\beta^{\ast}=0}$
$\displaystyle=\frac{C_{m}^{-1}}{n!\tau_{1}\tau_{2}}\frac{d^{m+n}}{d\beta^{\ast
m+n}}\frac{d^{m+n}}{d\alpha^{m+n}}\left.\exp\left\\{-\frac{\left(\alpha+\beta^{\ast}\right)^{2}}{4\tau_{1}^{2}}+\frac{\left(\alpha-\beta^{\ast}\right)^{2}}{4\tau_{2}^{2}}+\alpha\beta^{\ast}\right\\}\right|_{\alpha=\beta^{\ast}=0}$
$\displaystyle=\frac{C_{m}^{-1}}{n!\tau_{1}\tau_{2}}\frac{d^{2m+2n}}{d\beta^{\ast
m+n}d\alpha^{m+n}}\left.\exp\left\\{A_{1}\alpha\beta^{\ast}+A_{2}\left(\alpha^{2}+\beta^{\ast
2}\right)\right\\}\right|_{\alpha=\beta^{\ast}=0},$ (17)
where $A_{1}$ and $A_{2}$ are defined by
$\displaystyle A_{1}$
$\displaystyle=1-\frac{1}{2\tau_{2}^{2}}-\frac{1}{2\tau_{1}^{2}}=\frac{\allowbreak\bar{n}\left(\bar{n}+1\right)}{\bar{n}^{2}+\left(2\bar{n}+1\right)\allowbreak\cosh^{2}r},\text{
}$ (18) $\displaystyle A_{2}$
$\displaystyle=\frac{1}{4\tau_{2}^{2}}-\frac{1}{4\tau_{1}^{2}}=\frac{(2\bar{n}+1)\sinh
2r}{4\left(\bar{n}^{2}+\left(2\bar{n}+1\right)\allowbreak\cosh^{2}r\right)}.$
(19)
In a similar way to deriving Eq.(B11), we finally obtain
$\displaystyle\mathcal{P}(n)$
$\displaystyle=\frac{C_{m}^{-1}}{n!\tau_{1}\tau_{2}}\frac{d^{2m+2n}}{d\beta^{\ast
m+n}d\alpha^{m+n}}\left.\exp\left\\{-\left(i\sqrt{A_{2}}\alpha\right)^{2}-\left(-i\sqrt{A_{2}}\beta^{\ast}\right)^{2}+\frac{A_{1}}{A_{2}}\left(i\sqrt{A_{2}}\alpha\right)\left(-i\sqrt{A_{2}}\beta^{\ast}\right)\right\\}\right|_{\alpha=\beta^{\ast}=0}$
$\displaystyle=\frac{\left(A_{2}\right)^{m+n}C_{m}^{-1}}{n!\tau_{1}\tau_{2}}\frac{d^{2m+2n}}{d\beta^{\ast
m+n}d\alpha^{m+n}}\left.\exp\left\\{-\alpha^{2}-\beta^{\ast
2}+\frac{A_{1}}{A_{2}}\alpha\beta^{\ast}\right\\}\right|_{\alpha=\beta^{\ast}=0}$
$\displaystyle=\frac{\left(m+n\right)!}{n!\tau_{1}\tau_{2}C_{m}}E^{(m+n)/2}P_{m+n}\left(A_{1}/\sqrt{E}\right),$
(20)
where $P_{m+n}\left(x\right)$ is Legendre polynomial in (B10), and
$E=A_{1}^{2}-4A_{2}^{2}=\frac{\bar{n}^{2}-\left(2\bar{n}+1\right)\sinh^{2}r}{\bar{n}^{2}+\left(2\bar{n}+1\right)\allowbreak\cosh^{2}r}.$
(21)
In particular, when $m=0$ $(C_{0}=1),$ Eq.(20) reduces to
$\mathcal{P}(n)=\frac{E^{n/2}}{\tau_{1}\tau_{2}}P_{n}\left(A_{1}/\sqrt{E}\right),$
(22)
which is just the PND of STS which seems a new result; while for $r=0$
($\tau_{1}\tau_{2}=\bar{n}+1,A_{1}=\bar{n}/(\bar{n}+1)$,
$E=\bar{n}^{2}/(\bar{n}+1)^{2}$, $P_{m}\left(1\right)=1$,
$C_{m}=m!\bar{n}^{m},$), Eq.(20) becomes
$\mathcal{P}(n)=\frac{\left(m+n\right)!}{m!n!\bar{n}^{m}}\frac{\bar{n}^{m+n}}{\left(\bar{n}+1\right)^{m+n+1}},$
(23)
which is the PND of $m-$photon-subtracted thermal state which also seems a new
result, and the PND ($\bar{n}^{n}/\left(\bar{n}+1\right)^{n+1}$) of thermal
state without photon-subtraction [46, 47].
In Fig.2, the PND is shown for different values ($\bar{n};r$) and $m$, from
which we can see that by subtracting photons, we have been able to move the
peak from zero photons to nonzero photons (see Fig.2 (a)-(c)). The position of
peak depends on how many photons are annihilated and how much the state is
squeezed initially. In addition, the PND mainly shifts to the bigger number
states and becomes more “flat” and “wide” with the increasing parameter $r$
and the average photon-number of thermal field $\rho_{c}$ (see Fig.2 (b) and
(d)).
Figure 2: (Color online) Photon-number distributions of PASSTS with n̄=1 for
(a) r=0.3,m=0; (b) r=0.3,m=1; (c) r=0.3,m=5; and (d) r=0.8;m=1.
## 4 Quasiprobability distributions of PSSTS
In this section, several quasiprobability distributions of PSSTS are derived
in order to provide a convenient way for studying the nonclassical properties
of fields.
### 4.1 P distribution
We first calculate the Glauber-Sudarshan P distribution function [48] of
PSSTS. For this purpose, we start from the anti-normal ordering form of $\rho$
in Eq. (10). Recalling the integration formula converting an operator
$\hat{O}$ into its anti-normal ordering form[49] with anti-normal ordering
$\vdots$ $\vdots$, i.e.,
$\hat{O}=\int\frac{\mathtt{d}^{2}\beta}{\pi}\vdots\left\langle-\beta\right|\hat{O}\left|\beta\right\rangle\exp\left(\left|\beta\right|^{2}+\beta^{\ast}a-\beta
a^{{\dagger}}+a^{{\dagger}}a\right)\vdots,$ (24)
where $\left|\beta\right\rangle$ is a coherent state, one can obtain the anti-
normal ordering form of the squeezed thermal state $\rho_{s}$ by substituting
Eq.(5) into Eq. (24) and using the integration formula (B7). Such as
$\displaystyle\rho_{s}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\int\frac{\mathtt{d}^{2}\beta}{\pi}\vdots\exp\left\\{-A_{1}\left|\beta\right|^{2}-\beta
a^{{\dagger}}+\beta^{\ast}a+A_{2}\left(\beta^{2}+\beta^{\ast
2}\right)+a^{{\dagger}}a\right\\}\vdots$
$\displaystyle=\frac{1}{\sqrt{D}}\vdots\exp\left[\left(1-\frac{A_{1}}{E}\right)a^{{\dagger}}a+\frac{A_{2}}{E}\left(a^{{\dagger}2}+a^{2}\right)\right]\vdots$
$\displaystyle=\frac{1}{\sqrt{D}}\vdots\exp\left[\frac{2-\tau_{1}^{2}-\tau_{2}^{2}}{2D}a^{{\dagger}}a+\frac{\tau_{1}^{2}-\tau_{2}^{2}}{4D}\left(a^{{\dagger}2}+a^{2}\right)\right]\vdots$
(25)
thus the anti-normal ordering form of $\rho$ in Eq. (10) is
$\rho=\frac{C_{m}^{-1}}{\sqrt{D}}\vdots
a^{m}\exp\left[\frac{2-\tau_{1}^{2}-\tau_{2}^{2}}{2D}a^{{\dagger}}a+\frac{\tau_{1}^{2}-\tau_{2}^{2}}{4D}\left(a^{{\dagger}2}+a^{2}\right)\right]a^{{\dagger}m}\vdots,$
(26)
which leads to the P-function $P\left(\alpha\right)$ of PSSTS,
$P\left(\alpha\right)=C_{m}^{-1}\left|\alpha\right|^{2m}P_{0}\left(\alpha\right),$
(27)
where $P_{0}\left(\alpha\right)$ is the P-function of STS,
$P_{0}\left(\alpha\right)=\frac{1}{\sqrt{D}}\exp\left[\frac{2-\tau_{1}^{2}-\tau_{2}^{2}}{2D}\left|\alpha\right|^{2}+\frac{\tau_{1}^{2}-\tau_{2}^{2}}{4D}\left(\alpha^{\ast
2}+\alpha^{2}\right)\right].$ (28)
It is interesting in noticing that when $r=0$, Eq.(28) becomes
$P\left(\alpha\right)=\left|\alpha\right|^{2m}e^{-\left|\alpha\right|^{2}/\bar{n}}/(\bar{n}C_{m}),$
which is just the P-function of $m-$photon-subtraction thermal state which
seems a new result. From Eq.(27) one can see that the P-representation of
density operator $\rho$ can be expanded as
$\rho=C_{m}^{-1}\int\frac{d^{2}z}{\pi}\left|z\right|^{2m}P_{0}\left(z\right)\left|z\right\rangle\left\langle
z\right|,$ (29)
which is a non-Gaussian function due to the presence of $\left|z\right|^{2m}.$
### 4.2 Q-function
The Q-function is the absolute magnitude squared of the projection of a state
of the field onto a coherent state $\left\langle\alpha\right|$, defined by
$Q\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi}\left\langle\alpha\right|\rho\left|\alpha\right\rangle.$
(30)
Substituting Eq.(29) into (30), we can obtain
$Q\left(\alpha,\alpha^{\ast}\right)=R_{m}\left(\alpha,\alpha^{\ast}\right)Q_{0}\left(\alpha,\alpha^{\ast}\right),$
(31)
where $Q_{0}\left(\alpha,\alpha^{\ast}\right)$ is the $Q$-function of STS,
$Q_{0}\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi\allowbreak\tau_{1}\tau_{2}}\exp\left[-\frac{\tau_{1}^{2}+\tau_{2}^{2}}{2\allowbreak\tau_{1}^{2}\tau_{2}^{2}}\left|\alpha\right|^{2}+\frac{\tau_{1}^{2}-\tau_{2}^{2}}{4\allowbreak\tau_{1}^{2}\tau_{2}^{2}}\left(\alpha^{\ast
2}+\alpha^{2}\right)\right],$ (32)
which seems a new result not reported before, and
$R_{m}\left(\alpha,\alpha^{\ast}\right)$ is a factor generated from the
photon-subtraction, i.e.,
$R_{m}\left(\alpha,\alpha^{\ast}\right)=C_{m}^{-1}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}M^{m}\left(2O\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(-i\sqrt{M}(O\alpha^{\ast}+\alpha))\right|^{2},$
(33)
where $M=[(2\bar{n}+1)\sinh
2r]/[4\left(\bar{n}^{2}+\left(2\bar{n}+1\right)\allowbreak\cosh^{2}r\right)],$
and $O=2\bar{n}(\bar{n}+1)/[\left(2\bar{n}+1\right)\sinh 2r]$. Eq.(31)
indicates that the $Q$-function of PSSTS is also a non-Gaussian type due to
the presence of $R_{m}\left(\alpha,\alpha^{\ast}\right)$ and always positive
since $O>0$. In particular, when
$m=0,R_{m}\left(\alpha,\alpha^{\ast}\right)=1,$ thus
$Q\left(\alpha,\alpha^{\ast}\right)=Q_{0}\left(\alpha,\alpha^{\ast}\right)$,
as expected.
### 4.3 Wigner function
Next, the P-function is applied to deduce the WF of PSSTS. The partial
negativity of WF is indeed a good indication of the highly nonclassical
character of the state. Therefore it is worth of obtaining the WF for any
states. The WF $W\left(\alpha,\alpha^{\ast}\right)$ associated with a quantum
state can be derived as follows[36]:
$W\left(\alpha,\alpha^{\ast}\right)=\text{tr}[\rho\Delta\left(\alpha,\alpha^{\ast}\right)],\text{
}\alpha=\left(q+\mathtt{i}p\right)/\sqrt{2},$ (34)
where $\Delta\left(\alpha,\alpha^{\ast}\right)$ is Wigner operator, whose
coherent state representation is
$\Delta\left(\alpha,\alpha^{\ast}\right)=e^{2\left|\alpha\right|^{2}}\int\frac{\mathtt{d}^{2}\beta}{\pi^{2}}\left|\beta\right\rangle\left\langle-\beta\right|e^{2\left(\alpha\beta^{\ast}-\alpha^{\ast}\beta\right)},$
(35)
where $\left|\beta\right\rangle=\exp(-\left|\beta\right|^{2}/2+\beta
a^{{\dagger}})\left|0\right\rangle$ is the coherent state. Using the vacuum
projector $\left|0\right\rangle\left\langle 0\right|=\colon
e^{-a^{{\dagger}}a}\colon$, and the IWOP technique [37] one can put Eq.(35)
into its normal ordering form,
$\Delta\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi}\colon\exp\left[-2\left(a^{{\dagger}}-\alpha^{\ast}\right)\left(a-\alpha\right)\right]\colon.$
(36)
Thus substituting Eqs.(28), (36) and (29) into Eq.(34), we can finally obtain
the WF of PSSTS (see Appendix C),
$W\left(\alpha,\alpha^{\ast}\right)=F_{m}\left(\alpha,\alpha^{\ast}\right)W_{0}\left(\alpha,\alpha^{\ast}\right),$
(37)
where $W_{0}\left(\alpha,\alpha^{\ast}\right)$ is the WF of STS,
$W_{0}\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi\allowbreak\left(\allowbreak
2\bar{n}+1\right)\allowbreak}\exp\left[-\frac{2\cosh
2r}{2\bar{n}+1}\left|\alpha\right|^{2}+\frac{\sinh 2r}{\allowbreak
2\bar{n}+1}\left(\alpha^{2}+\alpha^{\ast}{}^{2}\right)\right],$ (38)
and
$F_{m}\left(\alpha,\alpha^{\ast}\right)=\frac{\left(m!\right)^{2}C_{m}^{-1}\sinh^{m}2r}{2^{2m}\left(\allowbreak
2\bar{n}+1\right)^{m}}\sum_{l=0}^{m}\frac{2^{2l}\left(\bar{n}-\sinh^{2}r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}\sinh^{l}2r}\left|H_{m-l}\left(\bar{\beta}\right)\right|^{2},$
(39)
where $\bar{\beta}=[2\alpha^{\ast}(\bar{n}-\sinh^{2}r)+\alpha\sinh
2r]/\\{i[(2\bar{n}+1)\sinh 2r]^{1/2}\\}.$ Eq.(37) is the analytical expression
of WF for PSSTS, related to single-variable Hermite polynomials. It is obvious
that there does not exist negative region for WF in phase space when
$\bar{n}>\sinh^{2}r$ which is agreement with Eq.(28) in Ref.[50]. In
particular, when $m=0,$ $F_{0}\left(\alpha,\alpha^{\ast}\right)=1,$ Eq.(37)
becomes
$W\left(\alpha,\alpha^{\ast}\right)=W_{0}\left(\alpha,\alpha^{\ast}\right)$;
while for $r=0$, note $C_{m}=m!\bar{n}^{m}$,
$W_{0}\left(\alpha,\alpha^{\ast}\right)=e^{-2\left|\alpha\right|^{2}/\allowbreak\left(\allowbreak
2\bar{n}+1\right)}/\allowbreak[\pi\left(\allowbreak 2\bar{n}+1\right)]$
(Eq.(30) in Ref.[40]) and
$F_{m}\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\left(2\bar{n}+1\right)^{m}}L_{m}\left(-\frac{4\bar{n}}{2\bar{n}+1}\left|\alpha\right|^{2}\right)$,
Eq.(37) reduces to
$W\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi\allowbreak\allowbreak\left(2\bar{n}+1\right)^{m+1}}e^{-\frac{2\left|\alpha\right|^{2}}{2\bar{n}+1}}L_{m}\left(-\frac{4\bar{n}\left|\alpha\right|^{2}}{2\bar{n}+1}\right),$
(40)
which corresponds to the WF of $m$-photon subtracted thermal state [40], and
can be checked directly from Eq.(C6). In addition, for $m=1,$ (single-photon-
subtracted squeezed thermal state (SPSSTS)), $C_{1}=B$ (12), the special WF of
SPSSTS is
$W_{1}\left(\alpha,\alpha^{\ast}\right)=F_{1}\left(\alpha,\alpha^{\ast}\right)W_{0}\left(\alpha,\alpha^{\ast}\right),$
(41)
where
$\displaystyle F_{1}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{1}{\left(\allowbreak
2\bar{n}+1\right)B}\left(\left(2\bar{n}+1\right)\left|\bar{\alpha}\right|^{2}+\bar{n}-\sinh^{2}r\right)$
$\displaystyle=\frac{\left|2\alpha^{\ast}\left(\bar{n}-\sinh^{2}r\right)+\alpha\sinh
2r\right|^{2}}{\left(\allowbreak
2\bar{n}+1\right)^{2}B}+\frac{\bar{n}-\sinh^{2}r}{\left(\allowbreak
2\bar{n}+1\right)B}.$ (42)
Noting $B>0$, thus from Eq.(41) one can see that when the factor
$F_{1}\left(\alpha,\alpha^{\ast}\right)<0,$ the WF of SPSSTS has its negative
distribution in phase space. This indicates that the WF of SPSSTS always has
the negative values under the condition: $\bar{n}<\sinh^{2}r$ at the phase
space center $\alpha=0,$ which is similar to the case of single-photon-
subtracted squeezed vacuum [5, 7].
Figure 3: (Color online) Wigner function distributions ${\small
W}\left(\alpha,\alpha^{\ast}\right)$of PASSTS for different ($\bar{n},r$) and
$m$ values (a) $\bar{n}=0.1,r=0.5,m=1;$(b) $\bar{n}=0.1,r=0.5,m=2;$(c)
$\bar{n}=0.2,r=0.5,m=1;$(d) $\bar{n}=0.1,r=0.8,m=1.$
Using Eq.(37), the WFs of PSSTS are depicted in Fig.3 for several different
values of $\bar{n},r$ and $m$ in phase space. It is easy to see that the the
WF is non-Gaussian in phase space. As an evidence of nonclassicality of the
state, squeezing in one of the quadratures is clear in the plots (see
Figs.3(a) and 3(d)). In addition, we can clearly see that there is some
negative region of WF, which is another evidence of nonclassicality of the
state, and that the negative region of WF gradually disappears as the
$\bar{n}$ (or the temperature) increases for given $r$ and $m$ (see Fig.3(a)
and (c)). Furthermore, for a larger squeezing, the WF shows a smaller minimum
negative value at the center of phase space (Figs.3(a) and 3(d)). For two-
photon subtracted case, the WF presents two positive peaks and two negative
peaks, different from the case of single-photon subtracted case.
## 5 Decoherence of PSSTS in thermal environment
When the $m$-PSSTS evolves in the thermal channel, the evolution of the
density matrix can be described by master equation [51]
$\frac{d\rho}{dt}=\kappa\left(\mathfrak{N}+1\right)\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a\right)+\kappa\mathfrak{N}\left(2a^{\dagger}\rho
a-aa^{\dagger}\rho-\rho aa^{\dagger}\right),$ (43)
where $\kappa$ represents the dissipative coefficient and $\mathfrak{N}$
denotes the average thermal photon number of the environment. When
$\mathfrak{N}=0,$ Eq.(43) reduces to the master equation describing the
photon-loss channel. The evolution of the WF is governed by the following
integration equation [52],
$W\left(\zeta,\zeta^{\ast},t\right)=\frac{2}{\left(2\mathfrak{N}+1\right)\mathcal{T}}\int\frac{d^{2}\alpha}{\pi}W\left(\alpha,\alpha^{\ast},0\right)e^{-2\frac{\allowbreak\left|\zeta-\alpha
e^{-\kappa t}\right|^{2}}{\left(2\mathfrak{N}+1\right)\mathcal{T}}},$ (44)
where $W\left(\alpha,\alpha^{\ast},0\right)$ is the WF of the initial state,
and $\mathcal{T}=1-e^{-2\kappa t}$. Eq.(44) is just the evolution formula of
WF in thermal channel. Thus the WF at evolving time may be obtained by
performing the integration with an initial value.
Substituting Eqs.(37)-(39) into (44), and using Eq.(B7) we finally obtain the
evolution of WF for PSSTS in thermal environment (see Appendix D, in a similar
way to deriving Eq.(11)),
$W\left(\zeta,\zeta^{\ast},t\right)=F_{m}\left(\zeta,\zeta^{\ast},t\right)W_{0}\left(\zeta,\zeta^{\ast},t\right),$
(45)
where $W_{0}\left(\zeta,\zeta^{\ast},t\right)$ is the WF of squeezed thermal
state in thermal channel,
$\displaystyle W_{0}\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{1/\allowbreak\left(\allowbreak
2\bar{n}+1\right)}{\pi\left(2\mathfrak{N}+1\right)\mathcal{T}\sqrt{G}}\exp\left[-\Delta_{2}\left|\zeta\right|^{2}+\frac{\allowbreak
g_{2}g_{3}^{2}}{4G}\left(\zeta^{2}+\zeta^{\ast 2}\right)\right],$ (46)
$\displaystyle F_{m}\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=C_{m}^{-1}\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\chi^{l}\Delta_{1}^{m-l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}\left[\omega/(2i\sqrt{\Delta_{1}})\right]\right|^{2},$
(47)
and
$g_{0}=\frac{\cosh 2r}{2\bar{n}+1},\text{
}g_{1}=\frac{\bar{n}-\sinh^{2}r}{2\bar{n}+1},\text{ }g_{2}=\frac{\sinh
2r}{2\bar{n}+1},\text{ }g_{3}=\frac{2e^{-\kappa
t}}{\left(2\mathfrak{N}+1\right)\mathcal{T}},$ (48)
as well as
$\displaystyle G$ $\displaystyle=\left(g_{0}+g_{3}e^{-\kappa
t}/2\right)^{2}-g_{2}^{2},$ $\displaystyle\omega$
$\displaystyle=\frac{2e^{-\kappa
t}}{2\mathfrak{N}\mathcal{T}+1}\left(\chi\zeta+2\Delta_{1}\zeta\allowbreak^{\ast}\right),$
$\displaystyle\Delta_{1}$
$\displaystyle=\frac{g_{2}}{4G}\left(1+g_{3}e^{-\kappa t}/2\right)^{2},$ (49)
$\displaystyle\Delta_{2}$
$\displaystyle=\frac{2\allowbreak}{\left(2\mathfrak{N}+1\right)\mathcal{T}}-\allowbreak\frac{g_{3}^{2}}{2G}\left(g_{0}+g_{3}e^{-\kappa
t}/2\right),$ $\displaystyle\chi$ $\displaystyle=\frac{1+g_{3}e^{-\kappa
t}/2}{2G}\left[g_{0}\allowbreak+g_{1}g_{3}e^{-\kappa
t}-1/\left(2\bar{n}+1\right)^{2}\allowbreak\right].$
It is noted that, at the initial time ($t=0$),
$\left(2\mathfrak{N}+1\right)T\sqrt{G}\rightarrow 1,\Delta_{2}\rightarrow
2g_{0},\frac{\allowbreak g_{2}g_{3}^{2}}{4G}\rightarrow\allowbreak
g_{2},\Delta_{1}\rightarrow\frac{1}{4}g_{2},\chi\rightarrow\allowbreak g_{1},$
$\omega\rightarrow 2g_{1}\zeta+g_{2}\zeta\allowbreak^{\ast},$ Eqs.(46) and
(47) just reduce to Eqs.(38) and (39), respectively, i.e., the WF of PSSTS. In
addition, for the case of $m=1$, corresponding to the case of SPSSTS, Eq.(47)
just becomes
$F_{1}\left(\zeta,\zeta^{\ast},t\right)=C_{1}^{-1}\left(\left|\omega\right|^{2}+\chi\right),$
(50)
from which one can see that when the factor
$F_{1}\left(\zeta,\zeta^{\ast},t\right)<0,$ the WF of SPSSTS in thermal
channel has its negative distribution in phase space. At the phase space
center $\zeta=0,$ the WF of SPSSTS always has the negative values when
$\chi<0$, leading to the following condition:
$\kappa t<\kappa
t_{c}=\frac{1}{2}\ln\left[1-\frac{2\bar{n}+1}{2\mathfrak{N}+1}\frac{\bar{n}-\sinh^{2}r}{\bar{n}\cosh
2r+\sinh^{2}r}\right],$ (51)
which implies that the threshold value $\kappa t_{c}$ is dependent not only on
the average number $\mathfrak{N}$ of environment, but also on the average
number $\bar{n}$ of thermal state and the squeezing parameter $r$ (a result
different from other discussions about the threshold value $\kappa t_{c}$ in
thermal channel [5, 7]). The WF of SPSSTS is always positive in the whole
phase space when $\kappa t\ $exceeds the threshold value $\kappa t_{c}$.
Actually, Eq.(51) is also true for the case with any number ($m$) photon-
subtraction (see Eq.(47)). From Eq.(51) one can clarify how the thermal noise
$\left(\bar{n},\mathfrak{N}\right)$ shortens the threshold value of the decay
time.
Using Eq. (45) we present the time-evolution of WF at different times scales
in Fig.4. From Fig.4, one can see clearly that the partial negative region of
WF gradually diminishes. At long times $\kappa t\rightarrow\infty$, one has
$\omega\rightarrow 0,\chi\rightarrow\bar{n}\cosh
2r+\sinh^{2}r,\Delta_{1}\rightarrow\frac{1}{4}\left(2\bar{n}+1\right)\allowbreak\sinh
2r$ and $H_{m}(0)=\left(-1\right)^{k}m!\delta_{m,2k}/k!.$ Thus
$W\left(\zeta,\zeta^{\ast},\infty\right)=\frac{1}{\pi\left(2\mathfrak{N}+1\right)}e^{-\frac{2\allowbreak\left|\zeta\right|^{2}}{2\mathfrak{N}+1}},$
(52)
which is independent of photon-subtraction number $m$ and corresponds to
thermal states with mean thermal photon number $\mathfrak{N}$. This implies
that the system reduces to thermal state after a long time interaction with
the environment. Eq.(52) denotes a Gaussian distribution. Thus the thermal
noise causes the absence of the partial negative of the WF if the decay time
$\kappa t$ exceeds a threshold value. In addition, from Fig.4, it is found
that the SPSSTS is similar to a Schrodinger cat state.
Figure 4: (Color online) Wigner function ${\small
W}\left(\alpha,\alpha^{\ast},t\right)$of SPSSTS for $r=0.3,\bar{n}=0.05$ and
different $r$and $\kappa t$: (a) $\kappa t=0.05;$(b) $\kappa t=0.1;$(c)
$\kappa t=0.2;$(d) $\kappa t=0.5.$
## 6 Fidelity as a non-Gaussianity measure for PSSTS
Recently, some quantitative measures to assess non-Gaussianity are proposed
[53, 54]. A non-Gaussianity measure may serve as a guideline to quantify the
non-Gaussian states. Therefore, it is of interest to evaluate the degree of
the resulting non-Gaussianity and assess this operation as a resource to
obtain non-Gaussian states starting from Gaussian ones. Here, we examine the
fidelity between the PSSTS $\rho_{s}$ and the STS $\rho$. Since the STS can be
considered as a generalized Gaussian state, the fidelity may be seen as a non-
Gaussianity measure able to quantify the non-Gaussian character of a quantum
state. In order to quantify the non-Gaussian character of the PSSTS, we
introduce the fidelity by defining
$\mathfrak{F}=\mathtt{tr}\left(\rho_{s}\rho\right)/\mathtt{tr}\left(\rho_{s}^{2}\right),$
(53)
where $\rho_{s}$ and $\rho$ are the squeezed thermal state (a generalized
Gaussian state) and the PSSTS, respectively. Obviously, when photon-
subtraction number $m=0,$ leading to $\rho=\rho_{s}$, then $\mathfrak{F}=1$
which means that $\rho$ is a Gaussian state described by $\rho_{s}$.
Using Eqs.(1) and (2), one has
$\mathtt{tr}\left(\rho_{s}^{2}\right)=\mathtt{tr}\left(\rho_{c}^{2}\right)=\frac{1}{2\bar{n}+1}.$
(54)
On the other hand, the fidelity ($\mathtt{tr}\left(\rho_{s}\rho\right)$) can
then be calculated as the overlap between the two WFs:
$\mathtt{tr}\left(\rho_{s}\rho\right)=4\pi\int d^{2}\alpha
W_{0}\left(\alpha,\alpha^{\ast}\right)W_{\rho}\left(\alpha,\alpha^{\ast}\right),$
(55)
where $W_{0}\left(\alpha,\alpha^{\ast}\right)$ is the WF of squeezed thermal
state $\rho_{s}$. Using Eq.(37) we may express Eq.(55) as
$\mathtt{tr}\left(\rho_{s}\rho\right)=4\pi\int
F_{m}\left(\alpha,\alpha^{\ast}\right)W_{0}^{2}\left(\alpha,\alpha^{\ast}\right)d^{2}\alpha.$
(56)
Then employing Eqs.(38) and (C6), similarly to Eq.(11), Eq. (56) may rewritten
as (see Appendix E)
$\mathtt{tr}\left(\rho_{s}\rho\right)=\frac{m!B_{2}^{m/2}}{\allowbreak\allowbreak\left(2\bar{n}+1\right)C_{m}}P_{m}\left(B_{1}/\sqrt{B_{2}}\right),$
(57)
where $P_{m}\left(x\right)$ is the Legendre polynomial with
$B_{1}=\frac{\allowbreak\bar{n}\left(\bar{n}+1\right)}{2\bar{n}+1}\cosh
2r,\text{
}B_{2}=\frac{\bar{n}^{2}\left(\bar{n}+1\right)^{2}}{\left(2\bar{n}+1\right)^{2}}-\sinh^{2}r\cosh^{2}r.$
(58)
Thus the fidelity (53) for the PSSTS is given by
$\mathfrak{F}=\frac{m!}{C_{m}}B_{2}^{m/2}P_{m}\left(B_{1}/\sqrt{B_{2}}\right)=\left(\frac{B_{2}}{D}\right)^{m/2}\frac{P_{m}\left(B_{1}/\sqrt{B_{2}}\right)}{P_{m}\left(B/\sqrt{D}\right)},$
(59)
which is an analytical expression for the fidelity between PSSTS and SSTS. We
see that when $m=0$ (the case of no photon-subtraction), $\mathfrak{F}=1$;
while for $m=1$ (the case of SPSSTS), Eq.(59) reduces to
$\mathfrak{F=}\frac{\allowbreak\bar{n}\left(\bar{n}+1\right)\cosh
2r}{\left(2\bar{n}+1\right)\left(\sinh^{2}r+\bar{n}\cosh 2r\right)}.$ (60)
In Fig.5, we plot the fidelity between PSSTS and STS as the function of
squeezing parameter $r$ for different photon-subtraction number $m.$ From Fig.
5 one can see that the fidelity decreases monotonously with the increment of
both photon-subtraction number $m$ and the squeezing parameter $r$, as
expected.
Figure 5: (Color online) The fidelity between PSSTS and squeezed thermal state
as the function of squeezing parameter $r$ for different photon-subtraction
number $m=0,1,2,3,4,19,20.(\bar{n}=0.2)$. The cases of ${\small m=}$19 and 20
are not identical, but they are almost overlap each other, which can not be
seen clearly from figure due to the use of thick style for line.
## 7 Conclusions and Remarks
In summary, we investigate the nonclassicality photon-subtracted squeezed
thermal state (PSSTS) and its decoherence in thermal channel with average
thermal photon number $\mathfrak{N}$ and dissipative coefficient $\kappa$. For
arbitrary number PSSTS, we have, for the first time, obtained an analytical
express for the normalization factor, which turns out to be a Legendre
polynomial of squeezing parameter $r$ and average photon number $\bar{n}$ of
thermal state, a remarkable result. Based on Legendre polynomials’ behavior
the nonclassical properties of the field, such as Mandel’s $Q$-parameter and
photon number distribution, are also derived analytically, Furthermore, the
nonclassicality of PSSTS is discussed in terms of the negativity of WF after
deriving the explicit expression of WF, which implies the highly nonclassical
properties of quantum states. It is shown that the WF of single PSSTS always
has negative values if $\bar{n}<\sinh^{2}r$ at the phase space center. Then
the decoherence of PSSTS in thermal channel is also demonstrated according to
the compact expression for the WF. It is found that the threshold value of the
decay time corresponding to the transition of the WF from partial negative to
completely positive definite is obtained at the center of the phase space,
which is dependent not only on the average number $\mathfrak{N}$ of
environment, but also on the average number $\bar{n}$ of thermal state and the
squeezing parameter $r$. We show that the WF for single PSSTS has always
negative value if the decay time $\kappa
t<\frac{1}{2}\ln\\{1-(2\bar{n}+1)(\bar{n}-\sinh^{2}r)/[(2\mathfrak{N}+1)(\bar{n}\cosh
2r+\sinh^{2}r)]\\}$. A non-Gaussianity measure may serve as a guideline to
quantify them for the class of non-Gaussian states, where the fidelity
decreases monotonously with the increment of both photon-subtraction number
$m$ and the squeezing parameter $r$.
In addition, Mandel’s $Q$ parameter does not always indicate a negative value
for non-classical state. In fact, for the photon-subtracted squeezed states by
even number, this parameter is positive. Thus the negativity of Q parameter is
a sufficient condition to distinguish non-classical state from classical one.
While for photon subtracted squeezed states by odd number, the negativity of
single photon subtracted case is noticeable. To compare further non-
classicality of quantum states for different number subtracted case, the
measures based on the volume of the negative part of the Wigner function[55],
on the nonclassical depth [56] and on the entanglement potential [57] may be
other alternative methods. Non-classical state introduced in this work will
maybe used in combination with other non-classical states such as entangled
states.
On the other hand, we should mention that for a photon-subtracted squeezed
state generated with some realistic probability, its non-classicality, in
particular, its non-Gaussianity would not be always superior to the input
Gaussian state. For example, photon-subtracted two-mode squeezed vacuum state
has more entanglement than initial two-mode squeezed state in not so strong
squeezing parameter; while for strong squeezing region, its superiority
disappears [28]. Entanglement evaluation investigation for photon-subtracted
two-mode squeezed thermal state is a future problem.
Acknowledgments Work supported by the the National Natural Science Foundation
of China under Grant Nos.10775097, and a grant from the Key Programs
Foundation of Ministry of Education of China (No. 210115), and the Research
Foundation of the Education Department of Jiangxi Province of China (No.
GJJ10097).
APPENDIX A: Derivation of Eq.(5)
Using the operator identity (3) and noticing the single-mode squeezing
operator yields the transformations,
$S\left(r\right)QS^{{\dagger}}\left(r\right)=e^{-r}Q,\text{ \
}S\left(r\right)PS^{{\dagger}}\left(r\right)=e^{r}P,$ (A1)
one has
$\rho_{s}=(1-e^{\sigma})S(r)e^{\sigma
a^{\dagger}a}S^{\dagger}(r)=\frac{2(1-e^{\sigma})}{e^{\sigma}+1}\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{\frac{e^{\sigma}-1}{e^{\sigma}+1}\left(e^{-2r}Q^{2}+e^{2r}P^{2}\right)\right\\}\genfrac{}{}{0.0pt}{}{\colon}{\colon},$
(A2)
which is still in Weyl ordering, in deriving (A2) we have used the Weyl
ordering invariance under similarity transformations (4). According to the
definition of Weyl correspondence rule [58], i.e., the classical Weyl function
$f\left(q,p\right)$ of operator $\rho_{s}$ can be given by replacing the $Q$
and by $q$ and $p$ in its Weyl ordered form, respectively,
$f\left(q,p\right)=\frac{2(1-e^{\sigma})}{e^{\sigma}+1}\exp\left\\{\frac{e^{\sigma}-1}{e^{\sigma}+1}\left(e^{-2r}q^{2}+e^{2r}p^{2}\right)\right\\},$
(A3)
then using the relation between $\rho_{s}$ and Wigner operator
$\Delta\left(q,p\right),$ i.e., operator $\rho_{s}$ can be expanded in terms
of $\Delta\left(q,p\right)$,
$\rho_{s}=\int_{-\infty}^{\infty}dqdpf\left(q,p\right)\Delta\left(q,p\right),$
(A4)
where the normal ordering form of $\Delta\left(q,p\right)$ is given by
$\Delta\left(q,p\right)=\frac{1}{\pi}\colon\exp\left[-\left(q-Q\right)^{2}-\left(p-P\right)^{2}\right]\colon$
(A5)
one can see that
$\displaystyle\rho_{s}$
$\displaystyle=\frac{2(1-e^{\sigma})}{\pi\left(e^{\sigma}+1\right)}\int_{-\infty}^{\infty}dqdp\exp\left\\{\frac{e^{\sigma}-1}{e^{\sigma}+1}\left(e^{-2r}q^{2}+e^{2r}p^{2}\right)\right\\}$
$\displaystyle\times\colon\exp\left[-\left(q-Q\right)^{2}-\left(p-P\right)^{2}\right]\colon$
$\displaystyle=\text{Eq.(\ref{f5}).}$ (A6)
thus we complete the proof of Eq.(5).
APPENDIX B: Deduction of Eq.(11)
Using the completeness relation and $\rho_{s}^{\prime}s$ normal ordering form
in (5), as well as the overlap of coherent state,
$\left\langle\beta\right|\left.\alpha\right\rangle=\exp\left[-\frac{1}{2}\left|\alpha\right|^{2}-\frac{1}{2}\left|\beta\right|^{2}+\beta^{\ast}\alpha\right],$
(B1)
we have
$\displaystyle C_{m}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\mathtt{Tr}\left\\{a^{m}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left|\alpha\right\rangle\left\langle\alpha\right|\colon\exp\left[-\frac{Q^{2}}{2\tau_{1}^{2}}-\frac{P^{2}}{2\tau_{2}^{2}}\right]\colon\left|\beta\right\rangle\left\langle\beta\right|a^{{\dagger}m}\right\\}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\alpha^{m}\beta^{\ast
m}\exp\left[-\left|\alpha\right|^{2}-\left|\beta\right|^{2}+\beta^{\ast}\alpha+A_{1}\beta\alpha^{\ast}+A_{2}\left(\beta^{2}+\alpha^{\ast
2}\right)\right]$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\frac{\partial^{2m}}{\partial
k^{m}\partial s^{m}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\exp\left[-\left|\alpha\right|^{2}+\left(\beta^{\ast}+k\right)\alpha+A_{1}\beta\alpha^{\ast}+A_{2}\alpha^{\ast
2}\right]$
$\displaystyle\times\left.\exp\left[-\left|\beta\right|^{2}+\beta^{\ast}s+A_{2}\beta^{2}\right]\right|_{s=k=0}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}}\frac{\partial^{2m}}{\partial
k^{m}\partial
s^{m}}e^{A_{2}k^{2}}\int\frac{d^{2}\beta}{\pi}\exp\left[-\left(1-A_{1}\right)\left|\beta\right|^{2}+kA_{1}\beta+\left(s+2A_{2}k\right)\beta^{\ast}+A_{2}\left(\beta^{2}+\beta^{\ast
2}\right)\right]_{s=k=0}$
$\displaystyle=\frac{1}{\tau_{1}\tau_{2}\sqrt{A_{3}}}\frac{\partial^{2m}}{\partial
k^{m}\partial s^{m}}\exp\left[\left(k^{2}+s^{2}\right)A+Bks\right]_{s=k=0},$
(B2)
where
$\displaystyle A_{1}$
$\displaystyle=1-\frac{1}{2\tau_{2}^{2}}-\frac{1}{2\tau_{1}^{2}},A_{2}=\frac{1}{4\tau_{2}^{2}}-\frac{1}{4\tau_{1}^{2}}>0,$
(B3) $\displaystyle A_{3}$
$\displaystyle=\left(1-A_{1}\right)^{2}-4A_{2}^{2}=\frac{1}{\tau_{1}^{2}\tau_{2}^{2}},$
(B4) $\displaystyle A$
$\displaystyle=\frac{A_{2}}{A_{3}}=\allowbreak\frac{1}{4}\left(\tau_{1}^{2}-\tau_{2}^{2}\right)=\allowbreak\frac{2\bar{n}+1}{4}\sinh
2r>0,$ (B5) $\displaystyle B$
$\displaystyle=\frac{A_{1}-A_{1}^{2}+4A_{2}^{2}}{A_{3}}=\allowbreak\frac{1}{2}\left(\tau_{1}^{2}+\tau_{2}^{2}\right)-1$
$\displaystyle=\frac{1}{2}\left[\left(2\bar{n}+1\right)\cosh 2r-1\right]>0,$
(B6)
and using the integration formula [59]
$\int\frac{d^{2}z}{\pi}\exp\left\\{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}+fz^{2}+gz^{\ast
2}\right\\}=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left\\{\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right\\},$
(B7)
whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0$ and$\
\mathtt{Re}\left(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g}\right)<0$ and noting
that
$\displaystyle\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+2x\tau
t\right)\right|_{t,\tau=0}$
$\displaystyle=\sum_{n,l,k=0}^{\infty}\frac{\left(-\right)^{n+l}}{n!l!k!}\left(2x\right)^{k}\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\tau^{2n+k}t^{2l+k}\right|_{t,\tau=0}$
$\displaystyle=2^{m}m!\sum_{n=0}^{\left[m/2\right]}\frac{m!}{2^{2n}\left(n!\right)^{2}\left(m-2n\right)!}x^{m-2n},$
(B8)
one rewritten Eq. (B2) as
$\displaystyle C_{m}$
$\displaystyle=\left(-A\right)^{m}\frac{\partial^{2m}}{\partial k^{m}\partial
s^{m}}\exp\left[-k^{2}-s^{2}-\frac{B}{A}ks\right]_{s=k=0}$
$\displaystyle=\left(-A\right)^{m}2^{m}m!\sum_{n=0}^{\left[m/2\right]}\frac{m!\left(\frac{-B}{2A}\right)^{m-2n}}{2^{2n}\left(n!\right)^{2}\left(m-2n\right)!}.$
(B9)
Recalling the newly found expression of Legendre polynomial (its equivalence
to the well-known Legendre polynomial’s ($P_{m}\left(x\right)$) expression is
[60]
$x^{m}\sum_{{l}=0}^{\left[m/2\right]}\frac{m!}{2^{2{l}}\left({l}!\right)^{2}\left(m-2{l}\right)!}\left(1-\frac{1}{x^{2}}\right)^{{l}}=P_{m}\left(x\right),$
(B10)
we derive the compact form for $C_{m}$,
$\displaystyle C_{m}$
$\displaystyle=m!B^{m}\sum_{n=0}^{\left[m/2\right]}\frac{m!}{2^{2n}\left(n!\right)^{2}\left(m-2n\right)!}\left(\frac{4A^{2}}{B^{2}}\right)^{n}$
$\displaystyle=m!D^{m/2}P_{m}\left(B/\sqrt{D}\right),$ (B11)
where
$D=\allowbreak B^{2}-4A^{2}=\bar{n}^{2}-\left(2\bar{n}+1\right)\sinh^{2}r.$
(B12)
Eq.(B11) indicates that the normalization factor $C_{m}$ is just related to
Legendre polynomial.
Combining Eqs.(B8) and (B10), On the other hand, one can derive a new formula
for Legendre polynomial, i.e.,
$\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+\frac{2x\tau
t}{\sqrt{x^{2}-1}}\right)\right|_{t,\tau=0}=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}P_{m}\left(x\right).$
(B13)
APPENDIX C: Derivation of WF (11) for PSSTS
According to Eqs.(34), (36) and (29), we have
$\displaystyle W\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=C_{m}^{-1}\text{tr}[\int\frac{d^{2}z}{\pi}\left|z\right|^{2m}P_{0}\left(z\right)\left|z\right\rangle\left\langle
z\right|\Delta\left(\alpha,\alpha^{\ast}\right)]$
$\displaystyle=\frac{C_{m}^{-1}}{\pi}\int\frac{d^{2}z}{\pi}\left|z\right|^{2m}P_{0}\left(z\right)\exp\left[-2\left(z^{\ast}-\alpha^{\ast}\right)\left(z-\alpha\right)\right]$
$\displaystyle=\frac{C_{m}^{-1}e^{-2\left|\alpha\right|^{2}}}{\pi\sqrt{D}}\int\frac{d^{2}z}{\pi}\left|z\right|^{2m}\exp\left[-g\left|z\right|^{2}+2\alpha^{\ast}z+2\alpha
z^{\ast}+\frac{\tau_{-}}{4D}\left(z^{\ast 2}+z^{2}\right)\right]$
$\displaystyle=\frac{C_{m}^{-1}e^{-2\left|\alpha\right|^{2}}}{\pi\sqrt{D}}\frac{\partial^{2m}}{\partial
k^{m}\partial
t^{m}}\int\frac{d^{2}z}{\pi}\exp\left[-g\left|z\right|^{2}+\left(2\alpha+k\right)z^{\ast}+\left(2\alpha^{\ast}+t\right)z+\frac{\tau_{-}}{4D}\left(z^{\ast
2}+z^{2}\right)\right]_{k=t=0},$ (C1)
where
$g=\frac{\tau_{+}-2}{2D}+2=\frac{\allowbreak\left(2\bar{n}+1\right)}{D}\left(\bar{n}-\sinh^{2}r\right),$
(C2)
which leads to
$g^{2}-\frac{\tau_{-}^{2}}{4D^{2}}=\frac{\allowbreak\left(2\bar{n}+1\right)^{2}}{D}.$
(C3)
Then using the integration formula (B7), we can write Eq.(C1) as following
form,
$\displaystyle W\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{C_{m}^{-1}e^{-2\left|\alpha\right|^{2}}}{\pi\allowbreak\left(2\bar{n}+1\right)}\frac{\partial^{2m}}{\partial
k^{m}\partial
t^{m}}\exp\left[g_{1}\left(2\alpha+k\right)\left(2\alpha^{\ast}+t\right)\right.$
$\displaystyle\left.+\frac{g_{2}}{4}\left(\left(2\alpha+k\right)^{2}+\left(2\alpha^{\ast}+t\right)^{2}\right)\right]_{k=t=0}$
$\displaystyle=F_{m}\left(\alpha,\alpha^{\ast}\right)W_{0}\left(\alpha,\alpha^{\ast}\right),$
(C4)
where $W_{0}\left(\alpha,\alpha^{\ast}\right)$ is the WF of squeezed thermal
state defined in Eq.(38), and
$\bar{\alpha}=2g_{1}\alpha^{\ast}+g_{2}\alpha,\text{
}g_{1}=\frac{\bar{n}-\sinh^{2}r}{2\bar{n}+1},\text{ }g_{2}=\frac{\sinh
2r}{2\bar{n}+1},$ (C5)
as well as
$F_{m}\left(\alpha,\alpha^{\ast}\right)=C_{m}^{-1}\frac{\partial^{2m}}{\partial
k^{m}\partial t^{m}}\exp\left[\bar{\alpha}k+\bar{\alpha}^{\ast}\allowbreak
t+\frac{g_{2}}{4}\left(k^{2}+t^{2}\right)+g_{1}kt\right]_{k=t=0}.$ (C6)
Further expanding the exponential term $kt$ included in (C6) into sum series,
and using the generating function of single-variable Hermite polynomials,
$H_{n}(x)=\left.\frac{\partial^{n}}{\partial
t^{n}}\exp\left(2xt-t^{2}\right)\right|_{t=0},$ (C7)
which leads to
$\left.\frac{\partial^{n}}{\partial
t^{n}}\exp\left(At+Bt^{2}\right)\right|_{t=0}=\left(i\sqrt{B}\right)^{n}H_{n}\left[A/(2i\sqrt{B})\right]=\left(-i\sqrt{B}\right)^{n}H_{n}\left[A/(-2i\sqrt{B})\right],$
(C8)
thus we can see
$\displaystyle F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=C_{m}^{-1}\sum_{l=0}^{\infty}\frac{g_{1}^{l}}{l!}\frac{\partial^{2l}}{\partial\bar{\alpha}^{l}\partial\bar{\alpha}^{\ast
l}}\frac{\partial^{2m}}{\partial k^{m}\partial
t^{m}}\exp\left[\bar{\alpha}k+\bar{\alpha}^{\ast}\allowbreak
t+\frac{g_{2}}{4}\left(k^{2}+t^{2}\right)\right]_{k=t=0}$
$\displaystyle=\frac{C_{m}^{-1}}{2^{2m}}g_{2}^{m}\sum_{l=0}^{\infty}\frac{g_{1}^{l}}{l!}\frac{\partial^{2l}}{\partial\bar{\alpha}^{l}\partial\bar{\alpha}^{\ast
l}}H_{m}(\bar{\beta})H_{m}(\bar{\beta}^{\ast}),$ (C9)
where
$\bar{\beta}=\frac{\sqrt{\allowbreak 2\bar{n}+1}}{i\sqrt{\sinh
2r}}\bar{\alpha}=\frac{2\alpha^{\ast}\left(\bar{n}-\sinh^{2}r\right)+\alpha\sinh
2r}{i\sqrt{\left(2\bar{n}+1\right)\sinh 2r}}.$ (C10)
Then using the recurrence relation of $H_{n}(x),$
$\frac{\mathtt{d}}{\mathtt{d}x^{l}}H_{n}(x)=\frac{2^{l}n!}{\left(n-l\right)!}H_{n-l}(x),$
(C11)
Eq.(C9) becomes
$\displaystyle F_{m}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{C_{m}^{-1}}{2^{2m}}g_{2}^{m}\sum_{l=0}^{\infty}\frac{1}{l!}\left(\frac{\bar{n}-\sinh^{2}r}{\sinh
2r}\right)^{l}\frac{\partial^{2l}}{\partial\bar{\beta}^{l}\partial\bar{\beta}^{\ast
l}}H_{m}(\bar{\beta})H_{m}(\bar{\beta}^{\ast})$
$\displaystyle=\frac{\left(m!\right)^{2}g_{2}^{m}}{2^{2m}C_{m}}\sum_{l=0}^{m}\frac{2^{2l}\left(\bar{n}-\sinh^{2}r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}\sinh^{l}2r}\left|H_{m-l}(\bar{\beta})\right|^{2}=\text{Eq.}(\ref{f29}).$
(C12)
Thus we complete the derivation of WF Eq.(11) by combing Eqs. (C4) and (C12).
APPENDIX D: Derivation of (45)
Substituting Eqs.(37)-(39) into (44), we have
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{2C_{m}^{-1}/\allowbreak\left(\allowbreak
2\bar{n}+1\right)}{\pi\left(2\mathfrak{N}+1\right)T}\exp\left[\frac{-2\allowbreak\left|\zeta\right|^{2}}{\left(2\mathfrak{N}+1\right)T}\right]\frac{\partial^{2m}}{\partial
k^{m}\partial\tau^{m}}\exp\left[g_{1}k\tau+\frac{g_{2}}{4}\left(k^{2}+\tau^{2}\right)\right]$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\exp\left[-\left(2g_{0}+g_{3}e^{-\kappa
t}\right)\left|\alpha\right|^{2}+\left(2\allowbreak\tau
g_{1}+kg_{2}+g_{3}\zeta^{\ast}\right)\alpha\right.$
$\displaystyle+\left.\left(2kg_{1}+\allowbreak\tau
g_{2}+g_{3}\zeta\right)\alpha^{\ast}+g_{2}\left(\alpha^{2}+\alpha^{\ast}{}^{2}\right)\right]_{k=\tau=0}$
$\displaystyle=\frac{C_{m}^{-1}/\allowbreak\left(\allowbreak
2\bar{n}+1\right)}{\pi\left(2\mathfrak{N}+1\right)T\sqrt{G}}\exp\left[-\Delta_{2}\left|\zeta\right|^{2}+\frac{\allowbreak
g_{2}g_{3}^{2}}{4G}\left(\zeta^{2}+\zeta^{\ast 2}\right)\right]$
$\displaystyle\times\frac{\partial^{2m}}{\partial
k^{m}\partial\tau^{m}}\exp\left[\chi
k\tau+\omega\allowbreak^{\ast}k+\omega\tau+\Delta_{1}\left(k^{2}+\tau^{2}\right)\right]_{k=\tau=0},$
(D1)
where $T=(1-e^{-2\kappa t},$ ($g_{0},g_{1},g_{2}$, $g_{3})$ and
($\chi,\omega,G,\Delta_{1},\Delta_{2})$ are defined in Eqs.(48) and (49),
respectively. In a similar way to deriving Eq.(11), we can further put Eq.(D1)
into Eqs.(45)-(47).
APPENDIX E: Derivation of (57)
Then employing Eqs.(38) and (C6) as well as the integration formula (B7), we
can treat the integration in a similar way to deriving Eq.(11),
$\displaystyle\mathtt{tr}\left(\rho_{s}\rho\right)$
$\displaystyle=\frac{4C_{m}^{-1}}{\allowbreak\left(\allowbreak
2\bar{n}+1\right)\allowbreak^{2}}\frac{\partial^{2m}}{\partial k^{m}\partial
t^{m}}\exp\left[\frac{g_{2}}{4}\left(k^{2}+t^{2}\right)+g_{1}kt\right]$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\exp\left[-4g_{0}\left|\alpha\right|^{2}+\left(kg_{2}+2tg_{1}\right)\alpha+\left(2kg_{1}+tg_{2}\right)\alpha^{\ast}+2g_{2}\left(\alpha^{2}+\alpha^{\ast}{}^{2}\right)\right]_{k=t=0}$
$\displaystyle=\frac{C_{m}^{-1}}{\allowbreak\left(\allowbreak
2\bar{n}+1\right)\allowbreak^{2}\sqrt{g_{0}^{2}-g_{2}^{2}}}\frac{\partial^{2m}}{\partial
k^{m}\partial
t^{m}}\exp\left[\frac{g_{2}}{4}\left(k^{2}+t^{2}\right)+g_{1}kt\right]$
$\displaystyle\times\exp\left.\left[\frac{g_{2}\left(4g_{1}^{2}+4g_{0}g_{1}+g_{2}^{2}\right)\allowbreak}{8\left(g_{0}^{2}-g_{2}^{2}\right)}\left(k^{2}+t^{2}\right)+\frac{4g_{0}g_{1}^{2}+4g_{1}g_{2}^{2}+g_{0}g_{2}^{2}}{4\left(g_{0}^{2}-g_{2}^{2}\right)}kt\right]\right|_{k=t=0}$
$\displaystyle=\frac{C_{m}^{-1}}{\allowbreak\left(\allowbreak
2\bar{n}+1\right)\allowbreak^{2}\sqrt{g_{0}^{2}-g_{2}^{2}}}\left.\frac{\partial^{2m}}{\partial
k^{m}\partial
t^{m}}\exp\left[B_{2}^{\prime}\left(k^{2}+t^{2}\right)+\allowbreak
B_{1}^{\prime}kt\right]\right|_{k=t=0},$ (E1)
where $g_{0}^{2}-g_{2}^{2}=\frac{1}{\left(2\bar{n}+1\right)^{2}}$ and
$\displaystyle B_{1}$
$\displaystyle=\frac{1}{4}\frac{g_{0}}{g_{0}^{2}-g_{2}^{2}}\left(4g_{1}^{2}+4g_{0}g_{1}+g_{2}^{2}\right)$
$\displaystyle=\frac{\allowbreak\bar{n}\left(\bar{n}+1\right)}{2\bar{n}+1}\cosh
2r=g_{0}\allowbreak\bar{n}\left(\bar{n}+1\right),$ (E2) $\displaystyle
B_{2}^{\prime}$
$\displaystyle=\frac{1}{8}\frac{g_{2}\left(2g_{0}^{2}+4g_{0}g_{1}+4g_{1}^{2}-g_{2}^{2}\right)}{g_{0}^{2}-g_{2}^{2}}$
$\displaystyle=\allowbreak\frac{2\bar{n}^{2}+2\bar{n}+1}{4\left(2\bar{n}+1\right)}\sinh
2r=\frac{g_{2}}{4}\left(2\bar{n}^{2}+2\bar{n}+1\right).$ (E3)
Similarly to deriving Eq.(B11), we have
$\left.\frac{\partial^{2m}}{\partial k^{m}\partial
t^{m}}\exp\left[B_{2}^{\prime}\left(k^{2}+t^{2}\right)+\allowbreak
B_{1}kt\right]\right|_{k=t=0}=m!B_{2}^{m/2}P_{m}\left(B_{1}/\sqrt{B_{2}}\right),$
(E4)
and $B_{2}\equiv B_{1}^{2}-4B_{2}^{\prime 2}$ given in Eq.(58), which leads to
Eq.(57).
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* [58] H. Weyl, Z. Phys. 46, 1 (1927).
* [59] R. R. Puri, Mathematical Methods of Quantum Optics, (Springer-Verlag Berlin Heidelberg New York, pp.269 (A.29), 2001).
* [60] H. Y. Fan, X. G. Meng, and J. S. Wang, Commun. Theor. Phys. 46, 845 (2006).
|
arxiv-papers
| 2010-08-31T08:40:10 |
2024-09-04T02:49:12.593747
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu, Xue-xiang Xu, Zi-sheng Wang, and Xue-fen Xu",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1008.5256"
}
|
1008.5346
|
# Comment on “Reviewing the evidence for two-proton emission from the high-
spin isomer in 94Ag”
I. Mukha CSIC – IFIC Universidad de Valencia, E-46071 Valencia, Spain E.
Roeckl GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt,
Germany H. Grawe GSI Helmholtzzentrum für Schwerionenforschung, D-64291
Darmstadt, Germany S.L. Tabor Florida State University, FL-32306
Tallahassee, USA
###### Abstract
A recent publication [D.G. Jenkins, Phys. Rev. C 80, 054303 (2009)] claims to
discredit the experimental observation of two-proton decay of the (21+) high-
spin isomer in 94Ag [I. Mukha _et al.,_ Nature (London) 439, 298 (2006)]. Its
conclusion, which would require a reestablishment of the two-proton emission,
is made on the basis of unwarranted assumptions by Jenkins concerning the data
analysis of the original work. We provide proof that these assumptions do not
correspond to reality, and that therefore the conclusion of the paper is
misleading.
radioactivity; one-proton, two-proton decays of high-spin isomer 94Ag(21+)
###### pacs:
23.20.Lv; 27.60.+j; 23.50.+z
The recent publication of Jenkins jenkins09 presents a very negative view of
the experiment reporting the two-proton (2p) radioactivity from the (21+)
high-spin isomer 94Ag mukh_ag2p . The author challenges the unambiguous
signature for 2p emission given in mukh_ag2p , namely the observation of 5
known $\gamma$ rays from the lowest states in the 2p-decay daughter 92Rh.
These $\gamma$ rays were registered in 4-fold coincidence demanded between two
silicon (Si) charged-particle and two germanium (Ge) $\gamma$-ray detectors
(Si1+Si2+$\gamma_{1}$+$\gamma_{2}$). About 50000 decays of the (21+) isomer in
94Ag were investigated in this way. In particular, Jenkins claims that
spurious peaks from Compton-scattered $\gamma$ rays associated with the
dominant background from 94Ag $\beta$ decays could have been misidentified as
92Rh $\gamma$ rays. Such a claim is based on the author’s unwarranted
assumption that the processes of $\gamma$-ray Compton scattering between
adjacent Ge crystals were not reduced in the analyzed $\gamma$–$\gamma$
coincidences. However, the author has apparently overlooked that such a
suppression procedure was applied as a standard routine during the data
analysis, as was mentioned in one of the preceding publications on 94Ag
$\beta$ decay, see page 28 in Ref. plett_ag_beta_g . This is surprising as
Jenkins refers to previous publications on the same experiment considering
different decay branches of isomers in 94Ag: (i) $\beta$-delayed $\gamma$-ray
emission plett_ag_beta_g , (ii) $\beta$-delayed proton emission mukh_ag_beta_p
, and (iii) single-proton radioactivity mukh_ag1p . As the author correctly
points out, this series of papers reports on successively weaker decay
branches from the (21+) isomer in 94Ag, obtained by analyzing and re-analyzing
the same data set. The most complete description of the analysis used, the
data obtained, and the calibrations are given in the two earlier regular
papers plett_ag_beta_g ; mukh_ag_beta_p . In particular, one of the basic
routines used for reducing Compton scattering effects in $\gamma$–$\gamma$
coincidence events excluded double hits in adjacent Ge crystals while they
were accepted in the other crystals plett_ag_beta_g (in addition to Ref.
mukh_ag2p , we must bring that the coincidence events with the sum energy of
two $\gamma$ rays amounting to 511$\pm$1.5 keV were excluded for all crystals
due to an association with positron annihilations). This procedure has indeed
reduced the Compton-scattered events, as one may conclude from the cross-check
$\gamma$-ray spectrum shown in Fig. 1(c) in Ref. mukh_ag2p . This spectrum was
obtained by applying the same conditions as those used for projecting the
$\gamma$-ray spectrum with 92Rh evidence displayed in Fig. 1(d) of Ref.
mukh_ag2p , except that the Si1+Si2 sum-energy gates were chosen differently,
i.e. covering the ranges of 1.2–1.6 and 1.8–1.95 MeV, respectively. If the
effect of Compton scattering mocks-up 92Rh $\gamma$-rays as claimed by Jenkins
then it should produce the same $\gamma$-ray peaks in both spectra mentioned
above, which is clearly not true. We are surprised that this straightforward
cross-check published in mukh_ag2p has escaped the attention of Jenkins.
Another result of mukh_ag2p questioned by Jenkins is the discrete 1.9-MeV
Si1+Si2 sum-energy peak observed in coincidence with two $\gamma$ rays from
excited states of the 2p-decay daughters 92Rh. The alternative interpretation
by Jenkins is that the reported 1.9 MeV peak was not produced by 2p but by
electron-positron pairs generated by $\gamma$ decay of the 2.86 MeV state in
94Rh which was assumed to be present in a sufficient amount due to $\beta$
decay of 94Pd. Besides ignoring the fact that 94Pd was greatly suppressed by
using a cooling trap in the ISOL ion source and by selecting a short
collection-transport cycle of the tape station, this interpretation contains a
fancy qualitative assumption how two $\gamma$ rays de-exciting 92Rh states can
be mocked-up by 4 511 keV $\gamma$-rays following annihilations of two
positrons from 94Pd decay. In particular, it assumes that $\gamma$ rays from
92Rh can be simulated by Compton-scattered 511-keV photons. Such a claim is
not consistent with the above-mentioned fact that the data analysis used in
mukh_ag2p has excluded $\gamma$–$\gamma$ hits with sum energy of 511 keV. In
addition, Jenkins has ignored the cross-check Si1+Si2 spectrum shown in Fig.
1(b) of Ref. mukh_ag2p which has been projected by shifting coincident
$\gamma$ gates by $\pm$3 keV from the nominal 92Rh values. According to the
interpretation of Jenkins, this spectrum should reveal the 1.9 MeV peak as
well, in contrast to the real data. In this Comment, we put aside questions
and problems of interpretation of the observed 2p decay. We believe that
improved measurements of the 94mAg decay rather than wild guesses about the
existing data will help in understanding of a physics behind this phenomenon.
In conclusion, the Ref. jenkins09 attempts to discredit the observed 2p decay
of the (21+) high-spin isomer in 94Ag by using wrong unwarranted assumptions
about the data analysis applied in Ref. mukh_ag2p , and its unfounded
speculations contradict the two cross-check spectra given along with the data
in mukh_ag2p .
## References
* (1) D.G. Jenkins, Phys. Rev. C 80, 054303 (2009).
* (2) I. Mukha _et al.,_ Nature (London) 439, 298 (2006).
* (3) C. Plettner _et al.,_ Nucl. Phys. A 733, 20 (2004).
* (4) I. Mukha _et al.,_ Phys. Rev. C 70, 044311 (2004).
* (5) I. Mukha _et al.,_ Phys. Rev. Lett. 95, 022501 (2005).
|
arxiv-papers
| 2010-08-31T15:58:34 |
2024-09-04T02:49:12.604139
|
{
"license": "Public Domain",
"authors": "I. Mukha, E. Roeckl, H.Grawe and S. Tabor",
"submitter": "Ivan Mukha",
"url": "https://arxiv.org/abs/1008.5346"
}
|
1009.0105
|
# Unveiling the tachyon dynamics in the Carrollian limit
C. Escamilla-Rivera celia˙escamilla@ehu.es Fisika Teorikoaren eta Zientziaren
Historia Saila, Zientzia eta Teknologia Fakultatea, Euskal Herriko
Unibertsitatea, 644 Posta Kutxatila, 48080, Bilbao, Spain. G. Garcia-Jimenez
ggarcia@fcfm.buap.mx Facultad de Ciencia Fisico Matematicas de la Universidad
Autonoma de Puebla, P.O. Box 1364, 72000, Puebla, Mexico. O. Obregon
obregon@fisica.ugto.mx Departamento de Fisica de la Universidad de Guanajuato,
C.P 37150, Leon, Guanajuato, Mexico.
###### Abstract
We briefly study the dynamics at classical level of the Carrollian limit, with
vanishing speed of light and no possible propagation of signals, for a simply
effective action in a flat space with a open string tachyon as scalar field.
The canonical analysis of the theory indicates that the equation of motion is
of Dirac type contrary to non-relativistic case where the equation is of
Schrodinger type. The ultimate intention is to analize the latter case with
electromagnetic fluxes finding that in this case the open string tachyon
cannot be interpreted as time.
###### pacs:
11.25.Sq,04.20.Fy,02.30.Mv
## I Introduction
In past years the role of the tachyon in certain string theories has been
explored and this has resulted in a better understanding of the D-brane
decaying process Sen:2002qa ,Gorini:2003wa . The basic idea is that the usual
open string vacuum is unstable, but there exists a stable vacuum with zero
energy density which is stable, which a tachyon field $T(x)$ naturally moves
to. Nevertheless it seems that aspects of this process can be compared with a
simple effective field theory models. In this case maybe the simplest model
was proposed by Sen Sen:2002qa . This success of effective action methods,
together with the difficulties of other approaches described encourages one to
pursue this further and to attempt a exact description of the cosmology of
tachyon rolling Gibbons:2002md .
Moreover, in the case where there are electromagnetic fluxes, the tachyon
field is on the same footing as a transverse scalar in the Dirac-Born-Infeld
action for a brane Gibbons:2002tv . In this case we look for a solution with a
constant electromagnetic field and find that the condensed state at
$V(T)\rightarrow 0$ is given by ${\dot{T}}^{2}+{E}^{2}=1$, where $\dot{T}$
means derivative with respect to the dimensionless time of tachyon field and
$E=|\vec{E}|$. To understand the dynamics it is convenient to do the
Hamiltonian formulation of the theory.
The present manuscript is organized as follows. In Sec. II we review the role
of the open string tachyon in the field theory and how this scalar field takes
place in the decaying process.
In Sec. III we describe what we have called the Carrollian limit mechanism for
open string states. Since this entails familiarity with Carroll group, I
planned to include also the Galilean group and the differences between them.
In Sec. IV we discuss some aspects when this theory is coupled to gravity.
In Sec. V we use the low energy effective action of the open string tachyon
and take the two possible limits: first the Galileo limit (when
$c\rightarrow\infty$), i.e the contravariant metric
$\eta^{\mu\nu}=(-c^{-2},1,1,1$) is well defined, contrary to the Carrollian
limit (when $c\rightarrow 0$). For this case we obtain in the Hamiltonian
formulation a Dirac type equation.
In section VI we use again the effective action and consider the case in which
$F_{\mu\nu}\neq 0$ to find that the tachyon is accelerated and emits radiation
in the direction of the electromagnetic field.
## II Open string tachyon in field theory
To understand clearly the tachyon dynamics we take into consideration a real
scalar field $\phi$ in a flat space-time. The Lagrangian of this theory is
given by
$\displaystyle
L=-\frac{1}{2}{\left(\partial\phi\right)}^{2}+\frac{1}{2}V\left(\phi\right),$
(1)
where $V\left(\phi\right)$ is the scalar field potential. In perturbation
theory usually we expand the potential of the form
$\displaystyle
V\left(\phi\right)=V_{0}+{\lambda}_{1}\phi+{\lambda}_{2}{\phi}^{2}+{\lambda}_{3}{\phi}^{3}+\ldots,$
(2)
and assume that ${\phi}^{n}$ becomes small, as the system evolves in time for
large $n$. We also know that ${\lambda}_{2}=m^{2}$, i.e this is a mass term.
In this expansion we have two interesting cases: a)
$V^{\prime\prime}\left(\phi=0\right)={\lambda}_{2}>0$, i.e ${m}^{2}>0$, the
theory has a real mass spectrum. In this case, the solutions of ${\phi}^{n}$
decrease for large $n$ over time and therefore the perturbation theory is
valid. And, b) $V^{\prime\prime}\left(\phi=0\right)={\lambda}_{2}<0$, i.e
${m}^{2}<0$, and the theory has a imaginary mass spectrum, i.e a tachyon. In
this case, the solutions of ${\phi}^{n}$ grow to infinity for large $n$ over
time and as a consequence the perturbation theory is no longer valid. The
latter case indicates that the theory is unstable around $\phi=0$. The usual
way to solve this is to find a critical point (stable) $\phi={\phi}_{0}$,
where the perturbation theory must be valid and obtain a real mass spectrum.
Sen found a clear way to study the tachyons in certain string theories similar
to previous case Sen:2002qa . He suggested that at a effective theory level
(low energy) the tachyons indicate the instability of the system and
correspond to decaying processes systems of open string with branes.
If we configure the system to a initial time so that the tachyon have a
initial amplitude in $T=0$ we have an unstable state and
$V\left(T=0\right)>0$. Any small perturbation would allow to the tachyon
potential descends and reach any of the two minimum. In theory these two
minimum are stable under small perturbations in the field $\phi$ and its mass
spectrum is real.
On the other hand, as proposed by Sen, Gibbons suggested analyze the coupling
to gravity and considering the resulting cosmology Gibbons:2002md
,Gibbons:2003gb . In this case he found that Sen’s action is defined with a
covariant metric $\eta_{\mu\nu}$ and then the limit of the theory is correct
when $c\rightarrow 0$, because in this case there exists a regular metric,
this limit is the so-called Carrollian limit. Of course, Gibbons took this
into consideration for classical cases and geometric level (collapse of cones
of light).
## III How does the Carrollian limit work?
The Carroll limit is defined as the limit when $c\rightarrow 0$, where $c$, as
we know, is the speed of light, which in this context is seen as a parameter.
In this limit, the resulting space is called Carroll space-time and the
symmetries of this space define a transformation group called the Carroll
group. Then, given a theory that incorporates the speed of light as a
parameter (i.e a relativistic theory) it is possible to make this limit, also
called contraction, and obtain new properties very different from what we
originally had. A well known example of this contraction is the case of the
Poincare group, in which it is possible to get the Galilean group through the
limit $c\rightarrow\infty$. The latter limit is physically interpreted as the
unreal limit of the theory. However, from a geometric point of view, we can
see that given the line element
$\displaystyle ds^{2}=-c^{2}dt^{2}+dx_{i}^{2},$ (3)
we introduce the covariant metric
$\displaystyle\eta_{\mu\nu}=(-c^{2},1,1,1),$ (4)
where $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$. The inverse matrix is just the
contravariant matrix
$\displaystyle\eta^{\mu\nu}=(-c^{-2},1,1,1).$ (5)
The remarkable thing is that in the limit $c\rightarrow\infty$, the
contravariant metric (5) is well defined and the covariant not, while in the
limit $c\rightarrow 0$, the opposite happens. The first case defines a
structure called the Newton-Cartan and the second defines a Carroll space-
time.
## IV Coupling to gravity
Follow the common wisdom and assume that the relavant action in a flat space
is
$\displaystyle S=\int{d^{4}xL},$ (6)
where the Lagrangian density has the form of Born-Infeld
$\displaystyle L=-V(T)\sqrt{-detA_{\mu\nu}},\quad
A_{\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}\partial_{\nu}T.$ (7)
With this in mind, the natural way to introduce the gravitational field is by
hand,
$\displaystyle
S=-\int{d^{4}xV(T)\sqrt{-g}\sqrt{1+g^{\mu\nu}\partial_{\mu}T\partial_{\nu}T}}.$
(8)
The term inside the root is the metric associated to the open string sector
(i.e only tachyonic matter)
$\displaystyle G_{\mu\nu}=g_{\mu\nu}+\partial_{\mu}T\partial_{\nu}T.$ (9)
For the case when the open string tachyon $T$ depends on time,
$\displaystyle G_{\mu\nu}=diag(-1+{\dot{T}}^{2},1,1,1).$ (10)
As explained before, the tachyon condensate takes place in the limit when its
velocity tends to one, so (10) can be rewritten as
$\displaystyle G_{\mu\nu}\rightarrow diag(0,1,1,1).$ (11)
Here the covariant metric is well defined and therefore the tachyon condensate
naturally gives us a Carroll spacetime.
## V Sen’s action in two limits
The open string tachyon can be described by an effective action where the flat
space has a Lagrangian given by
$\displaystyle
L=-V(T)\sqrt{-det(\eta_{\mu\nu}+\partial_{\mu}T\partial_{\nu}T)},$ (12)
where $V(T)$ is the tachyon potential and has a positive maximum at the origin
and a minimum at $T=T_{0}$. At this point the potential vanishes. The equation
(12) reproduces correctly the asymptotic behaviour $T\rightarrow\pm\infty$ for
the energy density and pressure obtain by Sen, and therefore, it is a good
model to describe the effective theory Gibbons:2002md .
For a homogeneous tachyon $T=T(t)$ the equation (12) has the form
$\displaystyle L=-cV(T)\sqrt{1-\frac{{\dot{T}}^{2}}{c^{2}}}.$ (13)
We known that for the Galileo group $c\rightarrow\infty$, if we expand the
root and take this limit our Lagrangian can be written as
$\displaystyle L\cong-
cV(T)\left(1-\frac{{\dot{T}}^{2}}{2c^{2}}+\ldots\right).$ (14)
On the other hand, if we take the Carrollian limit $c\rightarrow 0$ over (13)
the expansion of the Lagrangian is now
$\displaystyle L\cong-iV(T)\dot{T}+\ldots.$ (15)
Calculating the canonical momentum associated with the tachyon we get
$\displaystyle\Pi_{T}\equiv\frac{\partial L}{\partial\dot{T}}=-iV(T).$ (16)
From here we have the following constraint
$\displaystyle\Phi_{T}=\Pi_{T}+iV(T)\approx 0,$ (17)
where the notation $\approx$ means weakly zero in Dirac’s language. Due to the
constraint (17), the canonical Hamiltonian is zero and therefore the
Hamiltonian of the theory (total Hamiltonian) is given by the product of an
arbitrary function (Lagrange’s multiplier) and the constraint (17). If we
impose at quantum level the total Hamiltonian and therefore the constraint
then we see that the Hamiltonian is not hermitian if the potential is real.
So, the dynamics are defined at quantum level only if the potential is pure
imaginary, i.e when exists the creation of tachyons. At classical level, the
solution for the tachyon is formally given by the temporal integral of the
Lagrange’s multiplier. As we can easily see by calculating the equation of
Hamilton using the total Hamiltonian we can interpret the tachyon as time only
if the Lagrange’s multiplier is a constant.
## VI Inclusion of fluxes
We now turn our attention to the case in which $F_{\mu\nu}\neq 0$. Gibbons got
the following Lagrangian for the tachyon condensation $V(T)\rightarrow 0$,
$\displaystyle
L=-V(T)\sqrt{-det(\eta_{\mu\nu}+{\partial}_{\mu}T{\partial}_{\nu}T)+F_{\mu\nu}}.$
(18)
In this last equation we only added to equation (12) the electromagnetic term.
If $E=|\vec{E}|$ is a constant, then ${\dot{T}}^{2}+{\vec{E}}^{2}\rightarrow
1$, when $T\rightarrow\infty$. In the literature we only found the effects of
the electric field $\vec{E}$. In the same line, our intention is to discuss
what happens in the case when the magnetic field $\vec{B}$ exists? The above
expresion changes in the following way: the matrix obtain for this case is
$\displaystyle
G_{\mu\nu}=\left(\begin{array}[]{cccc}-1+\frac{{\dot{T}}^{2}}{c^{2}}&\lambda
E_{1}&\lambda E_{2}&\lambda E_{3}\\\ -\lambda E_{1}&1&\lambda cB_{3}&-\lambda
cB_{2}\\\ -\lambda E_{2}&-\lambda cB_{3}&1&\lambda cB_{1}\\\ -\lambda
B_{3}&\lambda cB_{2}&-\lambda cB_{1}&1\end{array}\right),$ (19)
where ${\lambda}^{2}=c^{-2}$. After lengthy but otherwise straightforward
calculations, we can write the Lagrangian
$\displaystyle
L=-V(T)\sqrt{1-\frac{{\dot{T}}^{2}\left(1+c^{2}{\vec{B}}^{2}\right)}{c^{2}}+{\vec{B}}^{2}-\frac{{\vec{E}}^{2}+{\left(\vec{E}\cdot\vec{B}\right)}^{2}}{c^{2}}}.$
(20)
In the limit of tachyon condensation we have
$\displaystyle{\dot{T}}^{2}\left(1+{\vec{B}}^{2}\right)+\left({\vec{E}}^{2}-{\vec{B}}^{2}\right)+{\left(\vec{E}\cdot\vec{B}\right)}^{2}=1,$
(21)
where we consider $c=1$ for simplicity. In the case proposed by Gibbons
($\vec{B}\approx 0$) we note that the tachyon is accelerated and therefore
emits radiation and the propagation is in the direction of the electric field
$\vec{E}$. The allowed range for tachyon velocity is then
$\left[0,\sqrt{1-\vec{E}^{2}}\right]$.
In our case, the propagation of radiation occurs in the component of
electromagnetic field, but if we consider $\vec{E}\approx 0$, may imply that
$\dot{T}\rightarrow 1$, in other words, the condensate is not affected in the
presence of magnetic fields. This suggests that the tachyon does not interact
with this field.
It should be remarkable, however, that under the presence of a uniform
electromagnetic field, the open string tachyon cannot be interpreted as time
in the sense of a Schrodinger type equation GarciaCompean:2005zn because as
we can see from (21) the tachyon does not decouple from the electromagnetic
field.
###### Acknowledgements.
C. Escamilla-Rivera would like to thank J. Socorro-Garcia for very helpful
discussions. This work was supported by CONACyT, Fundacion Pablo Garcia and
FUNDEC Mexico.
## References
* (1) A. Sen, Time and Tachyon, Int. J. Mod. Phys. A18(2003) 4869-4888, hep-th/0209122.
* (2) V. Gorinni, A. Kamenshchik, U. Moschella, V. Pasquier Tachyons, Scalar Fields and Cosmology, Phys. Rev. D69 (2004)123512, hep-th/0311111.
* (3) G.W. Gibbons Cosmological Evolution of the Rolling Tachyon, arXiv:hep-th/0204008v2 (2002).
* (4) G.W. Gibbons Thoughts on Tachyon Cosmology, arXiv:hep-th/0301117v1 (2003).
* (5) H. Garc a Compe n, G. Garc a Jim nez, O. Obreg n and C. Ram rez Tachyon driven quantum cosmology in string theory, Phys. Rev. D 71, 063517 (2005).
* (6) Gibbons, Gary and Hashimoto, Koji and Yi, Piljin Tachyon condensates, Carrollian contraction of Lorentz group, and fundamental strings, hep-th/0209034 (2002).
|
arxiv-papers
| 2010-09-01T07:46:42 |
2024-09-04T02:49:12.614425
|
{
"license": "Public Domain",
"authors": "Celia Escamilla-Rivera, G. Garcia-Jimenez and O. Obregon",
"submitter": "Celia Escamilla-Rivera",
"url": "https://arxiv.org/abs/1009.0105"
}
|
1009.0387
|
# Structure in the 3D Galaxy Distribution:
I. Methods and Example Results
M.J. Way11affiliation: NASA Goddard Institute for Space Studies, 2880
Broadway, New York, NY, 10025, USA 22affiliation: Department of Astronomy and
Space Physics, Uppsala, Sweden , P.R. Gazis and Jeffrey D. Scargle NASA Ames
Research Center, Space Science Division, Moffett Field, CA 94035, USA
Michael.J.Way@nasa.gov, PGazis@sbcglobal.net, Jeffrey.D.Scargle@nasa.gov
Methods: data analysis – Galaxies: clusters: general – Cosmology: observations
– large-scale structure of Universe
## 1 Abstract
Three methods for detecting and characterizing structure in point data, such
as that generated by redshift surveys, are described: classification using
self-organizing maps, segmentation using Bayesian blocks, and density
estimation using adaptive kernels. The first two methods are new, and allow
detection and characterization of structures of arbitrary shape and at a wide
range of spatial scales. These methods should elucidate not only clusters, but
also the more distributed, wide-ranging filaments and sheets, and further
allow the possibility of detecting and characterizing an even broader class of
shapes. The methods are demonstrated and compared in application to three data
sets: a carefully selected volume-limited sample from the Sloan Digital Sky
Survey (SDSS) redshift data, a similarly selected sample from the Millennium
Simulation, and a set of points independently drawn from a uniform probability
distribution – a so-called Poisson distribution. We demonstrate a few of the
many ways in which these methods elucidate large scale structure in the
distribution of galaxies in the nearby Universe.
## 2 Introduction and Historical Background
By the mid-1700s telescopes began to be used to catalog large areas of the
night sky. It quickly became clear that the distribution of objects is not
homogeneous. Wright (1750) was the first to note that our Sun appears to
reside in a disk of stars while Messier (1781) was probably the first to
detect a cluster of galaxies. Of the 103 objects in Messier’s catalog 13 are
actually part of the Virgo cluster. Of course there was no distinction between
galactic and extra-galactic nebulae at this early stage, but an overall
inhomogeneity was obvious. In his larger catalog Herschel (1784) discovered
the Coma cluster along with voids and other congregations of matter. By 1847
his son John Herschel was able to use his larger catalog of 4,000 nebular
objects (Herschel, 1847) to quantify the inhomogeneity for the first time
using counts-in-cells (15$\arcmin$ in Right Ascension by 3$\arcdeg$
Declination) confirming Messier’s discovery of Virgo with the addition of
several other clusters and even superclusters of galaxies as we understand
them today. Huggins (1864) measurements of nebular spectra would open the door
to categorizing these strange objects, but not until 1925 would it be
confirmed that the Spiral Nebulae were in fact external to the Milky Way
(Hubble, 1925) and their distribution on the night sky better understood.
Using the Shapley-Ames, Harvard and Hubble surveys of galaxies in the early
1930s Shapley (1933); Bok (1934); Hubble (1936) and Mowbray (1938) essentially
demonstrated that galaxies to at least 18th magnitude are not randomly
distributed. Also around this period Hubble (1934) used galaxy counts-in-cells
to find for the first time that the distribution of galaxies is log-normal.
By the 1950s the Lick Catalog of galaxy counts (reaching over 1 million and
superseding all previous catalogs in scale) could be used to statistically
characterize the galaxy distribution. Neyman & Scott (1952, 1959) assumed that
“Galaxies occur only in clusters” and built a multi-parameter model to
characterize the distribution of galaxies. Then for the first time a number of
authors attempted to use the 2-pt correlation function to characterize the
galaxy distribution (Limber, 1953, 1954; Layzer, 1956; Limber, 1957; Neyman,
1962) using the Lick survey. According to Saslaw (2000), at about the same
time “His (Gamow, 1954) was probably the first claim that quantitative details
of the observed galaxy distribution (Rubin, 1954) supported a specific
physical theory of cosmogony.”
Characterizing clusters of galaxies from the National Geographic Society –
Palomar Observatory Sky Survey (POSS) Abell (1958) used counts in equal-area
cells to show that galaxies are more strongly clustered than a Poisson111For
reasons described below in §4, we prefer to call such random processes as
_uniformly and independently distributed_ , more directly indicating their
fundamental nature. However, the term _Poisson_ is entrenched in much of the
literature. distribution. He found the maximum clustering scale to be about 45
Mpc (Ho=100km/s/Mpc), the scale for superclusters. Zwicky (1957) also used the
POSS survey but came to the conclusion that clustering stops at the scale of
clusters of galaxies and is uniform above that scale. But it was clear from
other observations that there are superclusters of galaxies (de Vaucouleurs,
1953, 1958) present in the local universe.
Using the new Lick Observatory catalog of Shane & Wirtanen (1967) for galaxies
brighter than m=19, Totsuji & Kihara (1969) realized for the first time that
the two-point correlation function for the spatial distribution of galaxies
follows the power-law
$g(r)=(r_{o}/r)^{s}\ ,$ (1)
where r is the distance between galaxies, ro = 4.7 Mpc, and the index $s$ was
estimated to be about $1.8$. The results were later confirmed by other groups
using the same survey (e.g. Groth & Peebles, 1977) with very similar results
($s=1.77$ instead of $1.8$, but with the same $r_{o}$). Both Martinez & Saar
(2001) and Saslaw (2000) do a nice job of reviewing the progress of the use of
correlation functions for galaxy distributions. Szapudi & Szalay (1998) is one
of the later developments proposing Landy-Szalay (Landy & Szalay, 1993)
estimators for higher order correlation functions. They claim that it is the
most natural estimator (see e.g. Peebles & Hauser, 1974).
Turner & Gott (1976) used positions and magnitudes from 1087 galaxies from the
Catalog of Galaxies and Clusters of Galaxies (Zwicky et al., 1961) and applied
a well defined, objective group identification procedure in contrast to the
somewhat subjective criteria used previously (e.g. Holmberg, 1937; Reiz, 1941;
de Vaucouleurs, 1975; Sandage & Tammann, 1975; Gregory & Thompson, 1978).
Later these workers applied the same methodology to a small N-body simulation
(Turner et al., 1979). In essence they attempted to estimate the surface
density of galaxies with volume density enhancements $\geq$10, as suggested by
de Vaucouleurs (1975) at that time. They admitted their catalog would have
contamination from foreground and background objects since they did not have
redshift information. Nonetheless they assigned 737 galaxies to 103 separate
groups and 350 to the field (see Figure 2 in Turner & Gott, 1976). The largest
group contained 238 members, including Virgo cluster members.
Oort (1983) reviews some of the earliest results on large-scale structure
analyses, but also points out the problem with using the increasingly popular
correlation function (e.g. Peebles, 1980) to characterize all structures in
the universe. “The correlation function has proved to be extremely useful in
providing such a unified description of the clumpiness. However, it is not
suitable for describing the very long filamentary or flat structures that we
encounter in superclusters, nor does it describe the large voids between these
superclusters.”
The deficiencies of the correlation function led to the use of methods like
percolation analysis and Minimal Spanning Trees in the 1980s. For example,
Zeldovich et al. (1982); Shandarin (1983); Einasto et al. (1984) were some of
the first to attempt to quantify galaxy clustering using percolation analysis.
These groups had the belief that it could appropriately quantify the pancake
and filamentary structures of the universe in models of structure formation
(e.g. Zeldovich, 1970). However, Dekel & West (1985) pointed out a number of
problems with using percolation analysis and stated that they are in fact not
sensitive to the “pancake” structures expected from the calculations of
Zeldovich (1970). They recommended a volume limited sample an order of
magnitude denser than the then state-of-the-art Center for Astrophysics survey
(Huchra et al., 1983); but even after more dense samples were obtained the
validity of the method as a tool for analyzing observational data remained in
doubt. On the other hand, it was utilized for comparing N-body simulations
with observational data and Poisson (uniform) distributions. More recent
percolation work (Pandey & Bharadwaj, 2005) has used the SDSS Data Release One
(Abazajian et al., 2003) in a 2-D projection to demonstrate that filaments are
the dominant pattern in the galaxy distribution.
One now understands the limitations of second-order statistical quantities,
such as correlation functions and power spectra, by noting that they discard
phase information. As percolation analysis demonstrated the application of
more powerful techniques allowing the identification of sheet and filamentary
structure in the large scale structure of the universe, at nearly the same
time the Minimal Spanning Tree (MST) took hold as a filament-finding
algorithm. The MST is a pattern recognition technique borrowed from graph
theory which gives an objective measure of the connectedness of a set of
points. Barrow et al. (1985) were the first to apply the MST to galaxy
clustering using the 2-D catalog of Zwicky et al. (1961), the 3-D catalog of
the Center for Astrophysics Redshift Survey (Huchra et al., 1983, hereafter
CFA), and the N-body simulations of Gott et al. (1979). These authors
demonstrated how markedly different both the observational data and N-body
simulations are from a Poisson distribution. Advances in the MST technique
have been applied to Large-Scale Structure analysis by a number of other
groups in subsequent years (Pearson & Coles, 1995; Krzewina & Saslaw, 1996;
Ueda & Itoh, 1997; Doroshkevich et al., 2004; Colberg, 2007). The percolation
and MST methods are related to Friends-of-Friends (FoF) techniques, which were
first applied to the 3-D CFA survey by Press & Davis (1982); Huchra & Geller
(1982) and later to simulation data by Croft & Efstathiou (1994) and even
larger samples of galaxies to obtain catalogs of groups (Ramella et al.,
1997). The FoF technique has even been expanded for use with photometric
redshift surveys of galaxies (Botzler et al., 2004). There are additional ways
to use the Nth nearest neighbor distances to estimate the underlying density
field (e.g. Gomez et al., 1998; Dressler, 1980). Another approach is to use
all N nearest neighbors (Ivezić et al., 2005) within a Bayesian probability
framework.
It should surprise no one that wavelets, used to characterize structure in
large galaxy catalogs, were applied in other 2-D (e.g. Slezak et al., 1990)
cases, and in the 3-D case (e.g. Slezak et al., 1993). What is surprising is
that they have not been utilized more extensively in the largest modern
redshift surveys of galaxies (e.g. Martinez et al., 2005). Paredes et al.
(1995) have done a nice job of comparing the relative merits of MST, FoFs and
wavelets as cluster finding algorithms, although there have been significant
developments since.
By the late 1980s and early 1990s there was interest in attempting to measure
the topology of Large Scale Structure from observational data and various
models (Gott et al., 1986; Hamilton et al., 1986; Gott et al., 1987; Park &
Gott, 1991; Beaky et al., 1992). This was done using the genus statistic which
is related to the fourth Minkowski functional (Stoyan et al., 1985). These
kinds of measures should give an idea of the topological connectedness of a
systems of points after they have been smoothed by some kind of filter. In the
end this method allowed one to distinguish among different galaxy
distributions by obtaining the genus, using isodensity surfaces at different
density levels. These clearly require some kind of smoothing, but the choice
of levels at which to apply smoothing is not obvious. This is important
because over-smoothing tends to create a positive genus, while under-smoothing
creates a negative one. Nonetheless these problems have not stopped groups
from applying these techniques to the largest available redshift surveys of
galaxies available at the moment, such as QDOT, CfA2, PSCz, 2dFGRS, and the
SDSS (Moore et al., 1992; Vogeley et al., 1994; Canavezes, et al., 1998; James
et al., 2007; Gott et al., 2009). Sheth et al. (2003) used Minkowski
Functionals combined with percolation analysis to compare the supercluster-
void network in three cosmological models and that of the present epoch. Some
of the latest studies (Gott et al., 2009; Choi et al., 2010) seem to confirm a
sponge like topology, and is consistent with the Gaussian random phase initial
conditions expected from inflation. Recent work (Aragón-Calvo et al., 2010;
Zhang et al., 2010) has attempted to calculate Minkowski Functionals using
Delaunay Tessellation to calculate the isodensity surfaces to try and get
around the smoothing problem mentioned above.
Voronoi tesselation was applied for the first time to study the structure of
the universe with the pioneering works of Matsuda & Shima (1984) and Icke &
van de Weygaert (1987). This was extended to 3-D distributions by Yoshioka &
Ikeuchi (1989) and van de Weygaert (1994). In the meantime Voronoi
tessellation-based methods have been used to study the clustering of galaxies
by many for differing purposes (e.g. Coles, 1990; Ikeuchi & Turner, 1991; Kim
et al., 1999; Ramella et al., 1999, 2001; Pizarro et al., 2006; Aragón-Calvo
et al., 2007). For example, Ebeling & Wiedenmann (1993), used a high-density
selection in the distribution of Voronoi volumes, coupled with the adjacency
information, to develop a method for source detection in 2D point maps. This
approach has been adapted into analysis toolkits for Chandra X-ray source
identification; see _e.g._ Diehl & Statler (2006) for details. Melnyk, Elyiv &
Vavilova (2006) applied a similar threshold method to study the distribution
of 7,000 local supercluster galaxies. See Elyiv, Melnyk & Vavilova (2009) for
discussion of an extension of Voronoi tessellation to more complex neighbor
relationships. See Cappellari (2009) regarding various applications. Two of
our methods utilize this procedure, and details are found below in §4 and
§5.2.
The pace of development of innovative methods for charactering large scale
structure has not much diminished in recent years. Two recent methods first
generate a continuous density field from the 3-D point distribution and then
identify structures via similar means. Aragón-Calvo et al. (2007) use the
“Delaunay Tessellation Field Estimator” (Schaap & van de Weygaert, 2000;
Schaap, 2007) and then rescale using isotropic Gaussian filters to create the
continuous field, while Bond et al. (2009) use a fixed-width Gaussian kernel
to estimate the density field. They both then compute the matrix of second
spatial derivatives to yield the so-called Hessian matrix. The eigenvalues and
eigenfunctions of this continuous matrix are evaluated at the locations of the
galaxies yielding clouds of points in what Bond et al. (2009) call
$\lambda-$space. Bond et al. (2009) demonstrate the relationship between the
shapes of these clouds and the morphology of the corresponding structures –
clusters, sheets, and filaments in particular. Aragón-Calvo et al. (2007) use
what they term the “Multiscale Morphology Filter” which “looks to synthesize
global structures by identifying local structures on a variety of scales and
assembling them into a single scale independent map”. Aragón-Calvo et al.
(2007); Jones et al. (2010) convincingly demonstrate the abilities of their
technique via toy models, complex N-body simulations and the SDSS. The Bond et
al. (2009) technique is unlike adaptive smoothing (e.g. Stein, 1997), because
Bond et al. (2009) smooth separately on a series of length scales, with the
goal of characterizing the spatial structures more accurately. Choi et al.
(2010) use a Hessian approach to compare the length of filaments found at a
redshift of $\sim$ 0.8 to 33 lower-redshift subsamples from the SDSS to find
that the length scales have not changed very much over this range of
redshifts. van de Weygaert & Schaap (2009) review in excellent detail the use
of density estimation in “The Cosmic Web” via the “Delaunay Tessellation Field
Estimator”. After submission two other papers (Sousbie, 2010; Sousbie, Pichon
& Kawahara, 2010) using DTFE as a density estimator were submitted which
characterize the cosmic web and filamentary structure using a method from
computational topology called Morse theory.
Recently Hahn et al. (2007a, b) have developed a classification scheme
designed to distinguish between dark matter halos in four structures;
clusters, filaments, sheets and voids, in N-body simulations of the universe.
The scheme relies upon the dynamical differences of the four different
structures quantified by an application of the Zeldovich (1970) approximation
to the evolved density field which allows one to determine their asymptotic
dynamics. There is one free parameter that acts as a smoothing parameter for
the density field. Nonetheless they claim to be capable of quantifying the
redshift evolution of dark matter halo properties of mass and environmment.
This is comparable to work by a number of authors in recent years (e.g. Lemson
& Kauffmann, 1999; Sheth & Tormen, 2004; Croton et al., 2007).
While characterizing the clustering of galaxies was the initial focus of many
researchers void characterization in 3-D simulations and surveys has also been
of interest. Recently Colberg et al. (2008) assembled 13 different void-
finding algorithms and for the first time tested them all on a single data set
– the Millennium Simulation (Springel et al., 2005). They claim that the
results agree very well with each other. Since then two other interesting
approaches with zero or few free parameters have appeared. Platen et al.
(2007) have utilized the watershed transform to develop what they term the
“watershed void finder” to find voids in 3-D distributions in a “relatively”
parameter free way (also see Sousbie, Colombi & Pichon (2009)). Neyrinck et
al. (2005); Neyrinck (2008) have used Voronoi tesselation to develop a
relatively parameter free “halo-finding” algorithm called VOBOZ (VOronoi BOund
Zones) and another to find voids and subvoids called ZOBOV (ZOnes Bordering On
Voidness) “without any free parameters or assumptions about shape”.
Regardless of method, clusters and voids were clearly visible in the first
large area redshift survey: The Center for Astrophysics Redshift Survey
(Huchra et al., 1983) and explicitly described in Davis et al. (1982). Davis
et al. (1982) also discuss the discrepancies between their observational data
and N-body simulations22220,000 points, 150Mpc on a side via Efstathiou &
Eastwood (1981) at the time: “We also present redshift-space maps generated
from N-body simulations, which very roughly match the density and amplitude of
the galaxy clustering, but fail to match the frothy nature of the actual
distribution”.
Giovanelli & Haynes (1991) has an excellent summary of the largest redshift
surveys up to 1991, by which time there were approximately 30,000 galaxies
with measured redshifts. Surveys up to 1990 were mainly done with single slit
spectrographs in the optical or 21-cm H I line surveys of spirals and gas-rich
dwarfs, both measuring one galaxy at a time. Since that time the number of
measured galaxy redshifts has increased by orders of magnitude because of
advances in large format CCD technology in combination with multi-fiber and
multi-object spectrographs. One of the first of these new surveys was the Las
Campanas Redshift Survey (LCRS Shectman et al., 1996) which collected over
23,000 redshifts in 6 years. As one can surmise from the above historical
survey of methods, it was expected that a large variety of techniques would be
applied in rapid fashion by a large number of groups. For example,
Doroshkevich et al. (1996) applied a “core sampling technique” (Buryak et al.,
1994) to find the characteristic scales for large scale structure in the LCRS.
A few years later Doroshkevich et al. (2001) combined inertia tensor and
minimal spanning tree analysis to three-dimensional data to confirm their
earlier LCRS results and determine cluster dimensions.
The next large redshift survey completed was the Two Degree Field Galaxy
Redshift Survey (Colless et al., 2001), which collected approximately 250,000
galaxy redshifts. The state of the art at present is the Sloan Digital Sky
Survey (York et al., 2000) with over 1 million measured redshifts thus far,
with more on the way.
The availability of these new large-area low-redshift surveys has greatly
enhanced prospects for an objective quantitative description of so-called
large scale structure (LSS) as delineated by optical and other observations of
galaxies. In addition to the intrinsic importance of assessing large scale
structure itself, links between structure and galaxy morphology or color have
provided much of the inspiration for a explosion of interest in large-scale
observational surveys.
In fact there are several near-future large-area surveys of the sky which will
allow one to test the predictions of general relativity for the growth of
structures in the universe and its consistency with the history of cosmic
expansion (e.g. Stril et al., 2010; Rapetti et al., 2009). A sampling of these
surveys include the Large Synoptic Survey Telescope (LSST) (Ivezic et al.,
2008), PanStarrs (Kaiser et al., 2002), and BigBOSS (Schlegel et al., 2009).
One of the oldest uses of large scale structure analysis is in the area of the
environmental effects on galaxy formation and evolution. Starting from the
time of Hubble (1936) astronomers have found that the properties of galaxies
are dependent upon conditions in their surroundings. Since then a large and
varied research effort has explored the dependence of galaxy color,
morphology, and star formation history on local density, using ever larger
samples of galaxies (e.g. Oemler, 1974; Butcher & Oemler, 1978; Dressler,
1980; Postman & Geller, 1984; Santiago & Strauss, 1992; Zehavi et al., 2002;
Hogg et al., 2003; Kauffmann et al., 2004; Croton et al., 2005; Blanton et
al., 2006; Blanton & Berlind, 2007; Zehavi et al., 2010).
Part of the present work differs from the tessellation procedures referenced
above by combining Voronoi cells into contiguous sets, called _blocks_ , using
a statistically principled method called _Bayesian blocks_ (Scargle, 1998,
2002; Scargle et al., 2008). The blocks are collected into contiguous sets to
form structures meant to model the shapes of clusters and other large scale
entities. Since no constraints – such as spherical symmetry, convexity, or
even simple-connectivity – are imposed on the derived structures, our results
are useful for detecting and characterizing complex structures such as
filaments, sheets, and irregular clusters, not just classical galaxy clusters.
This approach is consonant with the notions of the _Cosmic Web_ and _Voronoi
Foam_ (van de Weygaert, 2003; van de Weygaert & Aragón-Calvo, 2009). Although
we leave analysis of the detection efficiency for such complex structures to
the next paper in this series, the flexibility of the _Bayesian blocks_
representation of the density field allows such structural features to be
detected and characterized
Our approach to density estimation is outlined in Section 3, the data sets
used are described in Section 4, density and structure estimation methods in
Section 5, results in Section 6, and conclusions in Section 7.
## 3 Basic Approach: Density Estimation plus Structure Analysis
The approach here is the commonly adopted one of treating galaxies as mass
points,333Throughout, the terms galaxy and point will be used more or less
interchangeably using positional and redshift data from surveys to determine
locations of these points in three-dimensional space. As described below the
subsequent structure analysis flows from the coordinates of the points
themselves, and by determining the properties of a postulated underlying
continuous field.
Several factors impose limits on this approach. First, note that the data are
inherently four, not three, dimensional: distant galaxies are placed by the
data where they were a look-back time prior to now, not where they are now.
Interpretation of any data analysis results must account for this lack of co-
temporality.
Next, there is an inevitable positional uncertainty due to random
observational errors in the basic data and systematic effects arising in the
transformation from redshift to spatial coordinates. For example, see the
discussion of redshift distortion in §18.2 of Saslaw (2000).
And finally note that there are fundamental limitations on the information
that can be extracted from coordinates of a set of points. One can carry out
statistical analysis directly on the discrete data points, for example by
studying multiple-point correlation function estimators, the distribution of
nearest neighbor distances, the related minimal spanning trees, and the like.
Another, more or less complementary approach, is to postulate the existence of
an underlying continuum field, and regard the points as samples related in
some way to the field. However, the meaning of such a continuum is problematic
in general, especially at small spatial scales – _e.g._ less than that
characterizing galaxy nearest neighbor separations.
One such continuum scheme is to regard the field as an estimate of the density
of points (say in units of galaxies per cubic parsec), smoothed on scales at
least as large as the typical distance between points, and very much larger
than the sizes of the galaxies, which are after all treated as points of zero
size. Excellent overviews of the mathematical aspects of multivariate
densities and their estimation from point data are to be found in Silverman
(1986); Scott (1992). Discussions of this concept in relation to the large-
scale structure of the Universe are found in Martinez & Saar (2001); Saslaw
(2000); Dekel & Ostriker (1999).
A different, but related, scheme interprets the field as a probability
distribution, and treats the galaxies as points drawn from it in the usual
statistical sense. More specficially, this process can best be viewed as a
doubly-stochastic process, sometime called a Cox process. The spatial
dependence of the galaxy formation is described by process 1, reflecting the
evolution of the initial density fluctuations into a formation rate parameter
in a probability distribution locally defined in space-time. Process 2
represents the random sampling from the rate determined by process 1. That is
to say, the actual appearance of a galaxy in the data is a second random
process, independent of the first, reflecting the appearance of a galaxy at a
given point in space-time. Indeed, one could separate the galaxy formation and
observational detection aspects into two separate, independent processes, if
such a triply stochastic representation should prove useful. A mathematical
introduction to the basics of such random processes can be found in Papoulis
(1965), and excellent overviews of the mathematics of the corresponding theory
and estimation methods are Snyder (1991); Daley & Vere-Jones (2002); Andersen
et al. (1992); Kutoyants (1998); Preparata & Shamos (1985); de Berg et al.
(1997).
Both of the above approaches have to deal with difficult problems related to
the fact that the points are not independently distributed with respect to
both processes 1 and 2, due to the physics of the underlying formation,
evolution, and clustering processes and observational effects (such as the
“fiber collision” problem described below). These and other issues are well
described in a large literature (e.g. Martinez & Saar, 2001).
All of the algorithms used in this paper have some relation to density
estimation from points. But some go farther. For example, spatial Voronoi or
Delaunay tessellations extract information about relations between galaxies –
in terms of quantities such as local galaxy density gradients, nearest
neighbor distances (where, importantly, the number of nearest neighbors is not
fixed, but rather determined by the data themselves), the distributions of
these distances, and information about connectivity within the galactic
network that forms the skeleton of the Cosmic Web.
## 4 The Data
We have applied our three techniques (based on adaptive kernel smoothing,
self-organizing maps, and Bayesian blocks), to three individual datasets (one
observed, one simulated, and one a simulated purely random distribution).
Figure 1: Views of the SDSS DR7 data. Left: Positions of galaxies in the
Volume Limited (VL) selected SDSS DR7 catalog showing the boundary points that
are removed. Middle: The full SDSS DR7 and the volume limited sub-sample
selection. Right: Redshift histograms of the full SDSS DR7 and Volume Limited
samples.
Dataset 1 is a volume limited sample drawn from the SDSS DR7 (Abazajian et
al., 2009, DR7) Main Galaxy Sample (MGS) Catalog (Strauss et al., 2002) which
contains a redshift for each galaxy. The dataset was drawn from the DR7 in the
same manner that Cowan & Ivezić (2008, hereafter CI08) generated their sample
from the SDSS data release 5 (Adelman-McCarthy et al., 2007). We chose to use
the DR7 sample because the sample is larger and essentially geographically
contiguous in the north galactic cap region. Rather than use the standard SDSS
_casjobs_ interface to obtain the actual data444http://casjobs.sdss.org the
New York University Value Added Galaxy Catalog (NYU-VAGC) (Blanton et al.,
2005) was utilized. The NYU-VAGC includes the k-corrections for all galaxies
from the MGS spectroscopic survey. This makes generating the volume limited
sample rather trivial. Figure 1 shows the selection of the volume limited
subset of the NYU-VAGC sample, after a selection of apparent magnitude in
r$<$18 which mimics the MGS properly. Figure 1 also shows the respective
redshift distributions of the Magnitude Limited and Volume Limited Samples.
The MGS sample is obtained from the SDSS via the primtarget flag:
primtarget=TARGET_GALAXY (p.primtarget & 0x00000040 $>$ 0). The photometric
quality is constrained via the three flags !BRIGHT and !BLENDED and
!SATURATED: ((flags & 0x8) = 0) and ((flags & 0x2) = 0) and ((flags & 0x40000)
= 0), respectively. All redshifts are required to have an SDSS defined
redshift confidence better than 0.95 (zConf$>$0.95) and there should be no
redshift estimation warning errors (zWarning=0). Our sample contains 561,421
galaxies at this stage. An example of what the query would look like in
casjobs is given in Appendix A. The query shown does not include the absolute
magnitudes or k-corrections, as these were obtained from the NYU-VAGC catalog.
The SDSS also has a fiber collision issue which will play a role for density
estimation. In essence, fibers cannot be placed closer than 55” to each other.
However, overlap of repeated plates in some areas means that in fact redshifts
have been measured for both galaxies in many pairs separated by less than 55”.
To eliminate bias and ensure a homogeneous sample, we removed a randomly
chosen member of each such pair.
Our volume limited data set was drawn from the 561,421 galaxies in the NYU-
VAGC DR7 data set above. The largest contiguous region in the South Galactic
Cap was chosen and then a redshift/color cut of z$<0.12$ and $M_{R}<-20.0751$
was applied yielding 146,112 galaxies (see Figure 1). These samples were then
processed as follows:
1. 1.
Generate angular (2D) separation information: Find each galaxy’s 6 nearest
neighbors on the sky. We verified that this process guarantees identification
of all neighbors within 55”. Deleting randomly chosen members in these close
pairs eliminated 6,314 galaxies from the sample.
2. 2.
From redshifts and sky coordinates generate 3D Cartesian coordinates, in
redshift units, for each remaining galaxy.
3. 3.
Generate 3D nearest neighbor information by calculating distances to the 12
nearest neighbors. This number was chosen for convenience, to avoid
statistical issues that might be associated with a smaller number of
neighbors. This neighbor information was used only in the self-organizing map
approach.
4. 4.
Generate the Voronoi tessellation of the remaining set of galaxies. This
yields the cell vertices associated with each galaxy, from which one finds the
identities of the variable number of near neighbors in the Voronoi-Delaunay
sense.
5. 5.
Calculate from the tessellation information a set of derived parameters,
including the cell volume V and radius $R_{Voronoi}$, defined as $({3V\over
4\pi})^{1/3}$; the distance $d_{CM}$ between each galaxy and the center of its
cell; and an ‘elongation’ measure equal to the ratio between the maximum and
minimum dimension of the cell (See Appendix B).
6. 6.
Normalize the nearest neighbor distances and the Voronoi radius
($R_{Voronoi}$) by the radius $d_{uniform}=3.2\times 10^{-3}$ associated with
a uniform density distribution. This information was used in both the self-
organizing map (SOM) and Bayesian block (BB) approaches. Scale also the offset
distance $d_{CM}$ by $R_{Voronoi}$.
7. 7.
Flag questionable samples: Apply a set of tests to eliminate Voronoi cells
that appear to be distorted by boundary effects. These tests are described in
detail in a discussion of the ‘Boundary Problem’ in section 5.2.2. 5807 points
are removed which is about 4% of the initial volume limited sample of 146,112.
After the removal of the boundary points and those within 55” of each other we
are left with 133,991 points.
Combining these derived data (nearest neighbor distances and characteristics
of Voronoi cells) with attributes taken directly from the survey data
(positions, photometry data, etc.) yielded a unified set of attributes for
each galaxy as described in Appendix B below.
Dataset 2 is a volume limited sample drawn from the Millennium Simulation
(Springel et al., 2005, hereafter MS). We follow the same recipe for creating
our sample as is done by CI08 to make it comparable to the SDSS sample. After
a redshift and magnitude cut to mimic the SDSS Main Galaxy Sample ($r<$18 and
0.005$<z<$0.25) there are 509,877 galaxies. Another redshift and absolute
magnitude cut is made to mimic the SDSS volume limited sample described above
($R<$-20.0751 and $z<$0.120). This leaves 171,388 galaxies in our simulated
volume limited sample. See Figure 2 for a representation of these samples.
Figure 2: The data from the full Millennium Simulation displayed as in Figure
1: Left: Positions of galaxies in the Volume Limited (VL) selected Millennium
Simulation catalog showing the boundary points that are removed. Middle: The
full Millennium Simulation and the volume limited sub-sample selection. Right:
Redshift histograms of the full Millennium Simulation and Volume Limited
samples.
Dataset 3 is a set of randomly distributed points that mimics the SDSS DR7
Volume Limited sample above. We took a cube of space enclosing a volume
equivalent to the SDSS DR7 Volume Limited sample. We then filled this cube
with points drawn independently from a spatially uniform probability
distribution. It is common to call this a Poisson distribution, because the
number of such independent and uniformly distributed points in a predefined
volume of size $V$ obeys the Poisson distribution, $N(n)=(\lambda
V)^{n}e^{-\lambda V}/n!$, where $\lambda$ is the event rate per unit volume.
It can be confusing to use the same term for this auxiliary distribution as
for the overall spatial distribution. We therefore prefer to call the random
process based on its essential nature: independent, or for the case where the
rate parameter $\lambda$ is constant, independent and uniform. (Indeed, the
“Poisson” nature of this distribution is merely an incidental consequence of
these properties.) The number of points was chosen such that, after removing
pairs just as with the SDSS fiber collision criterion (none closer than 55”),
there remained a number of galaxies (144,700) close to that in the SDSS DR7
Volume limited sample. Note that this sample differs from the others in two
separate ways: the uniformity of the distribution and its simple, geometrical
boundary. For the most part the former is the more important consideration.
## 5 Structure Estimation Methods
As described in the Sections 2 and 3, analysis of large scale structure is not
a simple matter, especially if one wishes to invoke an underlying continuum.
Here we describe the various methods we have used, each of which explores a
different aspect of the distribution of galaxies on various scales.
### 5.1 Kernel Density Estimation
Kernel Density Estimation is probably the most widely used non-parametric
density estimator in use today. For this reason several groups have used 3D
kernel density estimation in recent years to study the large scale structure
of the Universe from redshift surveys (e.g. Connolly et al., 2000; Balogh et
al., 2004), and we include such an analysis in order to compare the results of
our two newer methods to this well known approach.
The underlying idea of 3D kernel density estimation (KDE) is simple: construct
a 3D profile (or kernel) centered at each data point, and sum the
contributions of these kernels for all of the data points. The kernels and
their sums are evaluated at a grid of 3D points, typically arranged in a
uniform rectangular grid. What needs to be specified are: the shape of the
kernel (Gaussian and Epanechnikov kernels are commonly used) and its
width555Sometimes called bandwidth, although strictly speaking this term
refers to the frequency domain. (this can be fixed or adaptive to the
underlying distribution) and amplitude, plus the locations of the grid
elements.
Since our other two methods are effectively adaptive (although the adaptivity
is implemented differently), we use an adaptive-bandwidth Gaussian kernel to
calculate the density. To describe it as simply and transparently as possible
we first explain the 1D univariate case and then 3D. In 1D one first starts by
estimating the density with a fixed bandwidth ($h$) where the Gaussian kernel
($K$) is given by Equation 2. Equation 3 is then the density estimate ($p$)
for the 1-D fixed bandwidth case where the points are given by $x_{i}$. To
estimate the variable or adaptive 1D KDE one allows the bandwidth to vary from
point to point. Let $d_{i,j}$ represent the distance from point $x_{i}$ to the
kth nearest point in the set making up the other $n-1$ data points. Equation 4
represents the 1D variable KDE where one sees that the window width of the
kernel at point $x_{i}$ is proportional to $d_{i,j}$ such that regions with
sparser data points will have flatter kernels. Hence the new adaptive
bandwidth could be represented as $h_{i}=h\times d_{i,j}$. This estimation
method is based on the approach laid out by Silverman (1986).
In the 3D case one has to find an initial estimate of the density for each
point, normally by using the fixed bandwidth 3D KDE shown in Equation 5. One
then must build a local bandwidth term $\lambda_{i}$ at each point. These
should have unit (geometric) mean and be multipled by the global bandwidth
$h$. In this case $h$ is the overall smoothing and $\lambda_{i}$ adjusts the
bandwidth at each point to “adapt” to the density of the data. The 3D adaptive
density estimate is given by Equation 6.
However, multi-dimensional multi-bandwidth KDE on large data sets can be
computationally expensive. In order to deal with a large number of points
(e.g. 100,000) in a reasonable time Gray & Moore (2003a, b) have devised an
efficient “Dual Tree” algorithm. The algorithm also gives gives an error
within a user specified tolerance at any evaluated point. Rather than code the
algorithm ourselves we utilized a package of MatLab666© The Mathworks, Inc.;
http://www.mathworks.com routines based on the Kernel Density Estimation
Toolbox of Ihler777http://www.ics.uci.edu/$\sim$ihler/code which has
implemented the dual tree algorithm of Gray & Moore (2003a, b). We made some
small modifications to allow the code to run on 64-bit platforms so that one
could evaluate the largest of our data sets.
$K=e^{-\frac{(x-x_{i})^{2}}{2h^{2}}}$ (2)
$p(x)=\frac{1}{nh}\sum_{i=1}^{n}K(\frac{x-x_{i}}{h})$ (3)
$p(x)=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{hd_{i,j}}K(\frac{x-x_{i}}{hd_{i,j}})$
(4)
$p(x)=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{V_{h}}K(\frac{x-x_{i}}{h})$ (5)
$p(x)=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{V_{h}\lambda_{i}}K(\frac{x-x_{i}}{h\lambda_{i}})$
(6)
The Kernel Density Estimation (KDE) method gives an almost continuous
distribution of densities. In order to make easier comparisons between this
and the two other methods to be discussed below we have translated the
continuous distribution of densities into discrete classes. This was done by
collecting the base-10 logarithms of the densities into a small number of
bins. For the SDSS DR7, Millennium Simulation, and uniform random data sets
this led to 11, 13, and 10 KDE logarithmic density classes, respectively,
chosen to approximately match the SOM-based class structure.
### 5.2 Tessellation
Tessellation is a natural partitioning scheme for analysis of the distribution
of points in a space of any dimension. We have found it exceptionally useful
for this study of the spatial distribution of galaxies. Accordingly, two of
our structure analysis procedures (Bayesian blocks and self-organizing maps)
use as building blocks the elements of the Voronoi tessellation of 3D space
defined by the galaxy positions, as described in the following subsection.
#### 5.2.1 Voronoi Tessellation
Tessellation divides the data space into sub-volumes, here called cells. The
first four of the following are properties of tessellation in general, while
the last two are specific to Voronoi tessellation in three dimensions (Okabi
et al., 2000):
1. 1.
$N$ data points generate $N$ cells.
2. 2.
The cells and data points are in a one-to-one correspondence.
3. 3.
The union of all $N$ cells is the whole data space.
4. 4.
The intersection of any pair of cells is empty (no cell overlap).
5. 5.
A cell comprises that part of the data space closer to its data point than to
any other.
6. 6.
The cell boundaries are flat 2D polygons.
7. 7.
Computation of the tessellation yields a data structure containing the
following information:
1. (a)
An estimate of the local point density: $V^{-1}$, where $V$ is the cell
volume.
2. (b)
The 3D vector from cell centroid to data point estimates the local density
gradient, in both magnitude and direction.
3. (c)
Information on nearest neighbors is encoded in the vertices of the bounding
polygons. One can define two cells to be _adjacent_ in three ways, depending
on whether they share at least one vertex, edge, or face; in this order, each
definition is included in the next.
In regions of high density, a small volume is apportioned among many points,
so the cells are small. In low density regions, where points are few and far
between, the opposite is true: the cells are large. This is the key inverse
relationship between density and cell size (_cf._ item 5 in the list in §4),
supplemented by the gradient information
Each cell is that part of the data space dominated by the corresponding data
point (item 5); in Voronoi tessellation, this means in the sense of being
closer to it than to any other data point. Items 3 and 4 together mean that
the tessellation is a partition of the data space. The subsidiary information
in item 7 exemplifies the way in which both point and local information are
conveniently represented in the tessellation construct. Our Bayesian block and
self-organizing map schemes make direct use of this information in different
ways, as described in later sections. In the former case density and
geometrical information alone is used to gather cells into connected sets,
called blocks, to represent the underlying density structure. In the latter
case incorporation of other subsidiary information allows the SOM
representation to describe more general characteristics of the large-scale
structure.
In both cases, the adjacency information encoded in cell faces, edges and
vertices is rather like a list of nearest neighbors – where the number of
neighbors is not pre-set, and in fact is part of the information extracted
from the raw data. Further, the density gradient information mentioned above
can be utilized for analysis and for visualization purposes. A handy density
visualization scheme depicts each cell as a frustum with the Voronoi cell as
the base with straight vertical sides, and capped by a copy of the Voronoi
cell at a height $\rho_{i}=n_{i}/V_{i}$ where the number of points (often 1)
is divided by the cell volume. This fast and convenient density representation
involves no loss of information by binning or smoothing, but therefore has a
discontinuous and ragged appearance. Display issues limit this device to data
spaces of dimension 1 or 2 (and therefore it is not used here); nevertheless
this construct is useful for computing subsidiary quantities such as widths of
structures, local mean density gradient, _etc._ In short Voronoi tessellation
yields a convenient data representation that enables many useful local,
intermediate, or global quantities to be computed.
There are many excellent, fast algorithms for tessellating spaces of any
dimension. We used the Matlab routine Qhull (Barber et al., 1996) which is
computationally efficient and returns adjacency and other auxiliary
information in a convenient form. Without any further computations, the
Voronoi cells express considerable statistical information about the point
distribution. For example, Figure 3 shows the distribution functions of the
local densities computed as the reciprocal of the volumes of the Voronoi cells
for the three cases: the SDSS DR7 data, the Millennium Simulation data, and
the uniform data. These distributions characterize the dynamic range of the
cell sizes. As expected, the cells in the uniform case have a relatively
narrow distribution centered around the mean cell size, while in the other
cases a broader range reflects the presence of structure on a wider range of
scales. The degree to which the distribution for the case of the MS data is
similar to that for the DR7 data confirms the correctness of this aspect of
the simulations. While the log densities are approximately normally
distributed, the density distributions themselves have long tails that render
the (log) of the mean value a misleading central measure.
Figure 3: Distribution functions of the logarithm of local densities, computed
as the reciprocals of the volumes of each galaxy’s Voronoi cell. In both
panels: dark line = SDSS DR7, medium line = Millennium Simulation, light line
= spatially uniform random distribution. Left: unbinned cumulative
distributions. Right: differential distributions. All distances (r) used to
calculate the volumes are in redshift units (z): $r(z)=3\times 10^{3}h^{-1}z$
Mpc. The units of volume for the random uniform case are chosen so that the
mean is unity (indicated by the vertical line at log(cell density)=0).
Figure 4 compares the distributions of the number of neighbors of each cell. A
neighbor of a cell is defined to be any cell sharing one or more Voronoi
vertices with the given cell. In this case the distributions of the actual DR7
data and the MS simulation data are nearly indistinguishable, whereas that of
the random data is distinctively different.
Figure 4: Normalized distribution functions of the number of Voronoi neighbors
of individual galaxies. In both panels: dark line = SDSS DR7, medium line =
Millennium Simulation, light line = spatially uniform random distribution.
Left: unbinned cumulative distributions, normalized to unit total fraction.
Right: differential distributions.
Figure 5 depicts the distribution functions of the logarithm of the average
distance to the Voronoi neighbors of each galaxy. As expected, the actual and
simulated galaxy data shows much more dispersion than does that for the
randomized case.
Figure 5: Distribution functions of (log) mean distances to Voronoi neighbors.
In both panels: dark line = SDSS DR7, medium line = Millennium Simulation,
light line = spatially uniform random distribution. Left: unbinned cumulative
distributions, normalized to unit total fraction. Right: differential
distributions.
#### 5.2.2 The Voronoi Cell Boundary Problem
For points lying sufficiently deep within the main population Voronoi
tessellation is a stable and well-understood procedure that gives meaningful
results. For galaxies near an edge of the sample space the situation becomes
problematic. Some cell vertices for these points characteristically lie
unrealistically far beyond the sampled region. Such outsized cells are an
artifact due entirely to the sampling and not to the actual galaxy
distribution. For this reason and other difficulties, such as vertices
formally assigned to lie at infinity, the reliability, or even the meaning, of
the tessellation as a density estimation tool breaks down near the edges of
the volume populated by the data points. This is the Voronoi Tessellation
‘Boundary Problem’.
It is possible to attempt to fix the problem, either by modifying the Voronoi
tessellation procedure itself or by modifications to the data set. One
possibility would be to construct replacement data cells, truncated to finite
volumes, as surrogates for the offending cells. However, unless the edges of
the sample space are well defined and smooth, procedures of this sort tend to
be arbitrary, and can introduce problems of their own. For a data set bounded
by complex boundaries with irregularly-shaped indentations and projections
there is no simple way to distinguish every cell that suffers from the
Boundary Problem from those that do not without eliminating a larger than
necessary number of points. Note that after the submission of our paper a
similar study to ours was also submitted (Sousbie, Pichon & Kawahara, 2010).
They deal with the boundary problem in the SDSS in a relatively simple manner
by defining boundary points as those that “belong to a pixel with at least one
completely empty neighbor”. While we agree that this method is simple and
effective, we believe it removes too many non-boundary points and given the
already small size of our volume limited sample we did not feel this would be
appropriate.
Regardless, it is possible to devise a set of ad hoc criteria that will
identify all of the worst case situations without excluding a prohibitive
number of ‘good’ samples. These criteria were obtained by studying the
distributions of various parameters of the Voronoi cells, in order to set
corresponding thresholds.
We evaluated a wide range of different parameters by using the complete data
set and subsets of the data that filled simple convex shapes. This was used to
help determine which parameters tended to assume extreme values for samples at
a boundary without excluding an unacceptable number (N$<$1–200) of the samples
well inside the data volume (what we call the ‘interior region’). The boundary
points were identified by the extent of their Voronoi cells with respect to
the edge. The parameters most sensitive to the position of a sample with
respect to a boundary were $R_{Voronoi}$, $d_{CM}$, and the normalized
distance from the center of a Voronoi cell to its furthest apex, $R_{Max}$. We
used these three parameters in conjunction to obtain the best performance. We
evaluated our criteria for a range of different thresholds to verify that the
results were comparatively insensitive to the values of these thresholds. The
final values used are listed in Table 1.
The choice of the ‘interior region’ mentioned above is described as follows:
1. 1.
One desires a region deep enough inside the full sample region such that one
is certain that no sample in this interior region will suffer from the
’boundary problem’. To ensure this one has to be certain that even samples
with extremely large Voronoi volumes have volumes that lie inside the full
sample region.
2. 2.
An ‘interior region’ is chosen with a boundary that lies 10$\times
d_{uniform}$ inside the boundary of the full sample region. Recall that
$d_{uniform}=3.2\times 10^{-3}$ in units of redshift.
3. 3.
To extend outside the full sample region a point in this ‘interior region’
would have to have at least one dimension of its Voronoi volume greater than
10$\times d_{uniform}$ in length. If the volume was shaped as a very thin
slice (which is unlikely) it could reach to the boundary, but our own tests
showed that this did not take place in our data sets. Regardless, this means
that the volume would be roughly $(10\times d_{uniform})^{3}$ and our tests
show that the number of samples with volumes that size or larger in our
interior region is extremely small: N $<$1–200 as mentioned above.
4. 4.
One can conclude that an interior region with a boundary 10$\times
d_{uniform}$ inside the boundary of the full data set cannot contain a
significant number of points that suffer from the boundary problem.
The number of affected boundary data points was small (our selection criteria
flagged 5807 of 146112 points or $\sim$4% of the population), so we simply
mark them to exclude them from any further analysis.
Table 1: Boundary Tests | SDSS | Millennium Simulation | Uniform
---|---|---|---
Attribute | Threshold | Number11Number that failed this test. | Threshold | Number11Number that failed this test. | Threshold | Number11Number that failed this test.
$R_{Voronoi}$ | 0.0040 | 4147 | 0.0040 | 4904 | 0.0040 | 3556
$d_{CM}$ | 0.0023 | 4515 | 0.0023 | 3475 | 0.0023 | 5001
$R_{Max}$ | 0.0067 | 5566 | 0.0067 | 6022 | 0.0067 | 6636
Union22Number that failed one or more of the 3 tests. | - | 5807 | - | 6178 | - | 6649
Fraction33Fraction of samples that failed one or more of the 3 tests. | - | 0.0415 | - | 0.0398 | - | 0.0480
### 5.3 3D Bayesian Blocks using Voronoi tessellation
This section describes the modeling procedure we used for the 3D galaxy
distribution using the Bayesian blocks algorithm. In a nutshell, we partition
the data space with a set of surfaces enclosing 3D solids. A constant density
is assigned to each solid which is equal to the number of galaxies within it
divided by its volume. This partitioning is implemented via an optimization
procedure designed to express spatial density variations that are real, and at
the same time suppress statistical fluctuations that are not real. The former
is regarded as the true signal and the latter as noise (especially that due to
the presence of small numbers of points). Of course these two goals cannot be
achieved perfectly. The corresponding signal-to-noise tradeoff is mediated by
the model fitness function (detailed below in §5.3.2). As in 2D there are an
infinite number of ways to partition a given volume. However, allowing only
partitions whose elements are collections of the polyhedra defined through the
Voronoi tessellation of the data points, as described in §5.2, yields a
completely tractable, finite, combinatorial optimization problem.
In summary, the goal of finding the optimal piece-wise constant model is
achieved with the Bayesian block algorithm. Optimality is in the sense of
maximizing a measure of goodness-of-fit of models of this kind. The basic
elements, _i.e._ the Voronoi cells, are determined using standard
computational geometry algorithms. In the next subsections we describe how the
cells are collected together into density levels, and how the cells within a
level are collected together to form connected blocks. The assembly of blocks
into meaningful structures (such as clusters, sheets, filaments, or other
structures) will be described only briefly, as details will appear in a
separate paper.
#### 5.3.1 Levels
The segmentation process described above begins by collecting the galaxies
into levels – i.e. sets forming a hierarchy ordered by density (galaxies per
unit volume). The goal is to find the best piecewise constant model described
above (§5.3). This optimization is implemented with an algorithm Jackson et
al. (2010) that maximizes goodness-of-fit for piecewise constant models. This
procedure for optimal segmentation of a data space of any dimension is an
extension of a one-dimensional algorithm Jackson et al. (2005) that in turn is
an exact, dynamic programming based version of the approximate algorithm in
Scargle (1998).
In general a set of 3D, or even 2D, data cells cannot be ordered in a way that
allows implementation of the basic idea behind the 1D algorithm.
Extension to higher dimension Scargle (2002); Jackson et al. (2010) is
achieved by discarding the condition that the elements of the partition of the
data space be connected sets of cells. That is to say, the levels are
generalized to be arbitrary subsets of the cells in the tessellated data
space. Since relaxing this constraint slightly changes the fundamental problem
and results in a larger search space, it would seem to be counterproductive.
It turns out that the resulting simplicity of the problem outweighs the
enlargement of the search space. Without the contiguity constraint the actual
locations of the cells are irrelevant to the model. Accordingly all orderings
of the cells are equivalent. It is convenient to sort them in a 1D array
ordered by cell volume. Now if the fitness function satisfies a simple
convexity condition each level in the optimal 3D partition contains all the
cells in an interval in the ordered 1D cell array, and only those cells. It is
this “intermediate density” order property that allows the 1D algorithm to
find the optimal partition of the original 3D data. The convexity condition
referred to is that the fitness function is convex as a function of the number
of galaxies in the block and also of block volume, and has nothing to do with
convexity of the block or level structures. See Jackson et al. (2010) for
details.
One problem results from this approach: the partition elements, here called
_levels_ in analogy with the contour levels in topographical maps, are
typically fragmented into a number of disconnected parts – much as
cartographic contours for the same level can be disconnected. The next section
describes our treatment of this issue: in a nutshell identify the parts of
each level that are indeed connected, and use these as the building blocks for
large scale structure.
#### 5.3.2 Blocks
The innovation of our approach, compared to previous Voronoi tessellation
methods is that neighboring cells are collected together into levels and
blocks (structures within which the galaxy density is modeled as constant) in
a statistically principled way. A block is a set of cells constrained to be
connected, but not restricted to have any particular shape properties such as
convexity or simple connectivity. Various abstract definitions of
connectedness are used in topology, but with finite spaces the basic ideas are
simple: a connected set consists of one piece, not two or more disconnected
pieces; a simply connected set additionally has no holes. More formally, in a
connected set any pair of cells in the block can be joined by a path
consisting of an ordered list in which each successive pair of cells are
touching. This is sometimes called path-connected. In a simply connected set,
the same is true, but in addition there are no cases where a pair of cells is
joined by two or more paths that cannot be smoothly distorted into each other.
Since the blocks represent coherent structures of sensibly constant galaxy
density, it is natural to associate them with astrophysically meaningful
structures. Without implying any assumption about structural evolution or
gravitational binding, we assume that our blocks do correspond to coherent
structures in the galaxy distribution.
As presaged in the previous section one ramification needs to be discussed: A
given optimal level may well consist of a set of _disconnected fragments_ –
sets of one or more cells spread throughout the data space and not touching
each other. To the extent that a partition’s levels are not connected, it does
not solve the constrained optimization problem originally posed.
If it turns out that each level has only one such component (_i.e._ is simply
connected), then _de facto_ we have solved the original problem. The levels
would then be regarded as the connected blocks that we originally sought. But
if not, then what? If some levels consist of two or more fragments detached
from each other, it is easy enough to identify these fragments and re-label
them as separate blocks. One can consider the resulting partition an
approximate solution (to the constrained problem) or as an exact solution of a
related problem of equal or greater astrophysical interest (the unconstrained
problem). The analog presented by topographical maps, with contour lines
indicating loci of constant altitude, may serve to clarify. Suppose that the
altitude values are assigned based on some statistical measure, and not fixed
at even multiples or the like. Then there would be two choices, namely to
constrain or not constrain distinct closed contours to be assigned the same
value. That is to say, use a global vs. a local statistical measure to
determined contour values. The results presented below incorporate this _post
facto_ re-labeling of block fragments as blocks.
To fully define the optimization problem we need to specify a quantity to be
maximized, such as a goodness-of-fit measure for the piece-wise constant block
model. That is, we maximize a measure of how well the data in a given block
are modeled as points randomly and independently distributed (with a single
constant probability density) uniformly across the block. A number of such
fitness functions were described in Scargle (1998), but here we use a maximum-
likelihood based fitness function described in Scargle et al. (2008), namely
the logarithm of the maximum likelihood for a model, of a block of volume $V$
containing $N$ points in which the event rate is constant.
Before exhibiting this fitness function, a few comments are in order regarding
the nature of the random process we are postulating for each block. Our
idealized mathematical picture is that the spatial locations of events
(galaxies) within the block have two properties:
Independence:
the occurrence of an event at any location does not affect the occurrence of
any other event at any location.
Uniform distribution:
The probability of an event occurring in any given block does not depend on
where in the block the interval lies.
Note that these conditions are stronger than the usual, weaker assumption that
the events are uncorrelated: independence implies uncorrelated, but not vice
versa. However, neither of these conditions is rigorously true. In addition to
observational issues, such as the fiber collision effect, the physical process
of galaxy formation prohibits the formation of two galaxies at the same
location. We are relying on this kind of correlation being important only at
small scales compared to those under study here. On the other hand, the
distribution of galaxies is of course not actually constant over significant
spatial regions. In this sense, we are simply forming the best piece-wise
constant (or step-function) approximation to a distribution that is presumably
continuously variable.
Hence, as in Scargle (1998) for time series data, we are led to model the
points in a block as identically and independently distributed with a single
probability that is constant across the block. As mentioned above this process
is often called a _constant rate Poisson process_ , because under it the
number of points in a fixed volume obeys the _Poisson distribution_ :
$P(N)={(\lambda V)^{N}e^{-\lambda V}\over N!}$ (7)
giving the probability $P$ that $N$ points fall in volume $V$, when the event
rate is $\lambda$ events per unit volume. The usual derivation of this formula
as the limit of repeated Bernoulli trials (see e.g. Papoulis, 1965) has led to
a common misunderstanding that it is fundamentally an approximation, but the
above equation is exact – absent correlations of the sort discussed above.
Maximizing the expression in equation (7) leads to the following maximum
likelihood fitness function for the block model of the full data interval:
$L_{max}=\Pi_{k=1}^{K}\ \ ({N_{k}\over V_{k}})^{N_{k}}e^{-N_{k}}$ (8)
where $N_{k}$ is the number of points in block k, $V_{k}$ is the volume of
block $k$, and the product is over all blocks in the model, covering the whole
observation region (Scargle et al., 2008). The corresponding logarithmic
fitness for a block, as implemented in our algorithm, is simply
$logL_{k}=N_{k}\ log{N_{k}\over V_{k}}$ (9)
for each block, and
$logL=\sum_{k=1}^{K}N_{k}\ log{N_{k}\over V_{k}}$ (10)
for the total model comprising $K$ blocks. In the last two expressions a term
proportional to $N_{k}$ is dropped because, when summed over $k$, it
contributes an unimportant constant to the fitness of the full model. Note
that these likelihood expressions depend on only the _sufficient statistics_
$N$ and $V$, and not on the actual distribution of the points within the
interval. This fact – somewhat counterintuitive, as this quantity is meant to
measure the goodness-of-fit of the assumed uniform distribution – follows
because under our model only the total number of events, and not their
locations, matters.
In the semi-Bayesian formalism of this model, the fitness function must be
augmented with a term that expresses prior information about the value for
$K$, the number of blocks. Optimization using equation (10) without such a
supplement tends to yield a large number of blocks, as many as $K\approx N$.
Specification of a _prior probability distribution_ $P(K)$ is the Bayesian
approach to this model complexity problem. A convenient choice for favoring a
small number of blocks is the geometric prior:
$P(K)\sim\gamma^{-K}\ ,$ (11)
where $\gamma$ is some constant. If the log of this prior is added to the
fitness of each block, the appropriate prior is assigned to the model for the
full interval. While it is not a smoothing parameter, its value regulates the
number of blocks, in effect influencing the apparent smoothness of the
representation. In most cases the details of the block representation do not
change much for a broad range of values of log($\gamma$), and derived
quantities (such as the sizes of structures) tend to be even less sensitive to
the adopted value of log($\gamma$). The main departure from a rigorous
Bayesian analysis is the fact that $K$, while weighted according to the prior
distribution described above, it is not explicitly marginalized, but instead
is optimized in a dynamic programming algorithm.
Figure 6 shows
Figure 6: Pictorial representation of the density values associated with the
different levels (shown in different colors) and blocks within the levels. The
base-10 logarithm of the density estimate – number of galaxies per unit volume
in redshift units cubed – is plotted against an arbitrary index ordered by
level. (The order within the levels is not meaningful. In particular, the
curved structure of the envelopes of the points is merely due to the order in
which the algorithm identifies blocks within the level.) The horizontal dashed
lines indicate the mean galaxy densities in the levels. The distribution is
truncated at the bottom-right end for display purposes.
the density levels for the DR7 data, organized by level and block. There are
three densities that can be assigned to a given galaxy (here denoted cell $n$)
1. 1.
the cell density: ${N_{cell}/V_{cell}}$
2. 2.
the block density: ${N_{block}/V_{block}}$
3. 3.
the level density: ${N_{level}/V_{level}}$
where $N_{cell}$ is the number of galaxies in a cell $n$, here always unity,
$N_{block}$ is the number of galaxies in the block containing cell $n$, and
$N_{level}$ is the number of galaxies in the level containing cell $n$. The
cell, block and level volumes are defined in an obvious and similar way. In
the figure, the ordinate is the block density of the individual blocks, and
the horizontal lines indicate the level density assigned to all of the blocks
in that level. Note the lack of overlap of block densities from one level to
the next, a result of the algorithm.
#### 5.3.3 Galaxy Structures: Sets of Blocks
Fruitful analysis of the galaxy density distribution can be carried out
directly from the blocks themselves, without regard to aggregation into
structures. Indeed, the same is true even at the level of Voronoi cells.
However, for various applications and for comparison with other work oriented
toward cataloging clusters, voids, etc., it is useful to take the aggregation
process one step farther and collect neighboring blocks with different
densities together to form structures – not just clusters in the classical
sense, but also filaments, sheets, and other coherent structures.
Of the many possible algorithmic approaches to this step, we adopt a
straightforward approach. First identify local _density maxima_ : blocks with
a higher density than any block adjacent to it. In 3D there are three ways of
defining adjacency: blocks can be deemed adjacent if they share Voronoi cell
(1) vertices, (2) edges, or (3) faces. Almost no difference in the deduced
structure results from using these progressively restrictive definitions, and
throughout we use definition (1).
Next, consider these maxima as seeds, growing into larger structures by
attachment of adjacent blocks in the next lower level in the density
hierarchy. This procedure is repeated until terminated by some stopping
condition. Three examples are: (a) stop at a fixed level in the density
hierarchy, either locally (for each structure) or globally; (b) stop when the
structure contains blocks for a fixed number of levels; and (c) stop when all
blocks belong to one cluster or another. In void analysis, one would adopt a
similar strategy beginning at the lower end of the density hierarchy. This
approach has some resemblance to that of Platen et al. (2007). In the
preliminary large-scale structure analysis reported here we adopt version (b),
taking the structures to consist of the block defining the local maxima plus
blocks from the two next lower density levels.
### 5.4 Self-organizing maps
Self-organizing maps (SOMs) (Kohonen, 1984; Ritter et al., 1992) are widely
used for unsupervised classification. They map points in the input
N-dimensional data space ${\cal R}^{N}$ into an array of cells or principal
elements (PEs) in a classification space $\cal{A}$ of reduced dimensionality
(usually one or two dimensions). The algorithm is designed to make the output
of the SOM reproduce, as much as possible, the topological structure of the
input distribution. In particular it attempts to map adjacent clusters in the
input space into adjacent PEs (or more commonly, adjacent blocks of contiguous
PEs) in the output space. A variety of measures have been proposed to evaluate
the degree to which topology is preserved by a particular mapping (Villmann et
al., 1997; Bauer & Villmann, 1997; Hsu & Halgamuge, 2003).
Used alone, SOMs serve as a means to visualize complicated relationships
between groups of points. For classification purposes, they must be combined
with some partitioning scheme that can identify regions in the output map that
correspond to different clusters in the input data. We used a modified version
of the same Bayesian Blocks algorithm described for direct cluster analysis in
§5.3 (Scargle, 1998; Scargle et al., 2008; Jackson et al., 2010) to partition
SOMs. This algorithm partitions the SOM output space into contiguous segments
(_blocks_) in a way that optimizes a fitness function which measures how
constant the values of the attributes are within each segment.
Let the array of attributes (two in our case) in principal element i of the
SOM output map be denoted $x_{i}$, and the corresponding variance measure by
$\sigma^{2}_{i}$; then the relevant average attribute for block $k$ is
$X_{k}={\sum_{i}\frac{x_{i}}{\sigma_{i}^{2}}\over\sum_{i}\frac{1}{\sigma_{i}^{2}}}\
,$ (12)
where the summations are over the $N_{k}$ PE’s in block $k$. The fitness
function for block $k$ takes the form (Gazis & Scargle, 2008)
$C_{k}=(N_{k}-1)(\ln(R)+\ln\sqrt{\pi})-(\ln(\prod_{i}\sigma_{i})+\ln(\sum_{i}\frac{1}{\sigma_{i}^{2}}))-(\sum_{i}\frac{x_{i}^{2}}{\sigma_{i}^{2}}-X_{k})\
,$ (13)
where again the sums are over the PEs in the block. The cost for the entire
partition is
$C=\sum_{k=1}^{K}C_{k}\ \ .$ (14)
In the SOM case the space to be partitioned is the map itself and the blocks
will consist of clusters of contiguous PEs. Note that this is subtly different
from the conventional Bayesian Blocks approach, in which partitioning is
performed in the original data space.
SOMs were generated using the Neuralware package, discussed at length by
Merényi (1998). This software can use a variety of neighborhood schemes and
implements the ‘conscience’ algorithm proposed by DeSieno (1988) to prevent
any particular PE from representing too much of the input data.
Classifications were performed using a $7\times 7$ array of PEs. Neighborhoods
were rectangular, and decreased in size from $5\times 5$ to $1\times 1$ during
training. Multiple classifications were performed using different values for
the range and standard deviation parameters in Equation (13) to evaluate the
sensitivity of the algorithm to these parameters. These partitionings were
also compared with the best possible partitioning and the results of a
conventional threshold-based scheme.
One advantage of SOM-based classification is that it can be performed on any
set of parameters. In principle kernel density and Bayesian Blocks methods
could be modified to include other parameters, but for a SOM this extension is
natural – essentially automatic. Care must be taken to chose parameters that
are physically meaningful. Initially we tried using the $N+1$ nearest neighbor
distances as a proxy for $N$-point correlation functions, but the results were
too sensitive to statistical fluctuations that occur when $N$ is small. Our
final classifications were performed using two parameters: a scaled Voronoi
radius, $R_{Voronoi}/d_{uniform}$, and an offset distance,
$d_{CM}/R_{Voronoi}$, where
$R_{Voronoi}=(3V_{Voronoi}/4\pi)^{1/3}\ ,$ (15)
$V_{Voronoi}$ is the volume of the Voronoi cell of that galaxy, and
$d_{uniform}$ is the average spacing between points in an independent uniform
distribution. These parameters are good proxies for the mean and gradient of
the local density, respectively. _Bagging_ (short for bootstrap aggregating)
was performed to improve accuracy and stability, avoid over-fitting, reduce
variance, and provide estimates of the uncertainty of the SOM classifications.
This standard machine learning procedure involves running the complete
analysis algorithm on data sets comprising subsamples from the actual data in
the bootstrap fashion (randomly sample with replacement). We averaged the
results of 10 such randomly selected subsets of the full data set.
The SOM-based scheme partitioned the SDSS and Millennium Simulation (MS) data
sets into six classes. The SOM based scheme partitioned the uniformly random
data set into eight classes (see Table 4), but given the non-physical nature
of these classes they were not easily defined and will not be discussed
further. Based on inspection of the SDSS and MS spatial distributions we
identified the six SOM classes as: _Cluster_ , _Cluster Gradient_ , _Strong
Gradient_ , _Field Gradient_ , _Halo_ and _Field_. We indicate these
fundamental classes, the number and identity of which are determined by the
SOM, in italics. Roman type is used for the names of the somewhat less
fundamental BB and KDE classes, derived by clumping their fine-grained
densities in order to approximately match the populations of these SOM
classes, as detailed in §6.1. This classification could be further refined
into eight sub-classes: _Dense Cluster_ , _Cluster_ , _Dense Cluster Gradient_
, _Cluster Gradient_ , _Strong Gradient_ , _Field Gradient_ , _Halo_ and
_Field_. It should be noted that this later partitioning was determined
entirely by the distribution of two attributes ($R_{Voronoi}/d_{uniform}$ and
$d_{CM}/R_{Voronoi}$) used by the SOM, and did not involve any a priori choice
of thresholds to identify particular categories. The characteristics typical
of galaxies in the classes were determined by a _post facto_ inspection of the
results and summarized in Table 2.
Table 2: Classes Identified by the SOM Algorithm, Ordered by Mean Density. Note that these class ID numbers only apply to the SDSS and MS datasets. See §5.4 for details on these classes. ID | Class | Subclass | Characteristics
---|---|---|---
1 | _Cluster_ | _Dense Cluster_ | very high density, low gradient
1 | _Cluster_ | _Cluster_ | high density, low gradient
2 | _Cluster Gradient_ | _Dense Cluster Gradient_ | very high density, moderate gradient
2 | _Cluster Gradient_ | _Cluster Gradient_ | high density, moderate gradient
3 | _Strong Gradient_ | _Strong Gradient_ | very high gradient
4 | _Field Gradient_ | _Field Gradient_ | moderate-high gradient
5 | _Halo_ | _Halo_ | moderate density, low-moderate gradient
6 | _Field_ | _Field_ | low density, low-moderate gradient
Among the different bagged data sets the boundaries of the six main classes
were almost identical; while the subclasses were less consistent their general
structure was preserved. Attempts to probe deeper into the hierarchy did not
produce stable results, which suggests that any structure that might exist at
deeper levels is ambiguous and/or poorly-determined.
The six classes identified by the SOM algorithm can be characterized as
follows. The _Cluster_ class involved regions of high density and low gradient
associated with centers of clusters. The _Halo_ and _Field_ classes involved
regions of moderate and low density respectively, with low gradient. Samples
were distributed uniformly in space, though galaxies in the _Halo_ class may
have had some tendency to be associated with the outer portions of clusters.
The _Cluster Gradient_ class involved regions of high density and moderate
gradient associated with filaments and the outer portions of clusters. The
_Strong Gradient_ and _Field Gradient_ classes involved regions of extremely
high gradient and moderate gradient, generally of high density, associated
with the portions of filaments midway between clusters. The _Field Gradient_
class involved regions of low density and moderate gradient respectively, with
moderate to low density, and were associated with filaments. This is
illustrated by Figure 7.
Figure 7: Location, in the SOM phase space, of types of galaxies identified by
the SOM algorithm: upper left = all galaxies; upper right = _Cluster_ and
_Cluster Gradient_ classes; lower left = _Strong Gradient_ and _Field
Gradient_ classes; lower right = _Halo_ and _Field_ classes.
For the classes listed in Table 2 Figure 8 presents scatter plots of the input
parameters ($R_{Voronoi}/d_{uniform}$ vs $d_{CM}/R_{Voronoi}$) of the SDSS,
Millennium Simulation (MS), and our uniform synthetic data, along with class
boundaries. The SDSS and MS data are similar, but the MS data spans a slightly
larger range of gradients, $d_{CM}/R_{Voronoi}$. There are also subtle but
significant differences in the class structure. While the SDSS and MS data
sets both contained the same classes, the _Halo_ and _Field_ classes in the MS
data contained more samples and occupied significantly larger regions in phase
space, while the three _Gradient_ classes were correspondingly smaller.
Figure 8: Locations, in the neighbor-distance/cell-volume space, of the
galaxies assigned to the various SOM classes. Left panel: SDSS DR7 data;
middle panel: Millennium Simulation data; right panel: spatially uniform
random distribution.
The class structure of the uniform data is noticeably different. Even though
the number of samples was similar, they occupy a much smaller region in phase
space, with a significantly smaller range of densities and much fewer samples
with large gradients. The distribution is sufficiently uniform that the
SOM/Bayesian Block technique does not identify any stable classes, and places
class boundaries at arbitrary locations. The figure shows a typical result
from among the bagged samples, with a large number of poorly-defined classes
that in no way resemble the well-ordered structure observed with the SDSS and
MS data.
## 6 Results
Comparison of the results of the three methods, for each of the three data
sets, is not entirely straightforward. We have identified a few simple
measures to quantify the differences. A future paper will present more
detailed quantitative comparisons. In a nutshell, a description of the results
of the three methods gives insight into (1) the similarities of the SDSS and
Millennium Simulation data sets, (2) the stark differences between them and
the uniform distribution regardless of the structure analysis method, and (3)
the similarities between the SOM and BB methods, and their differences from
the KDE method.
### 6.1 Classes: From Clusters to the Field
As discussed in §5.3 and demonstrated in Figure 6 the Bayesian Block method
yields a series of density levels. Each level contains one or more _blocks_ ,
defined as connected sets of cells each of which is disconnected from all
other blocks in the level. The galaxy density within a block is close to the
density characterizing the level as a whole, differing only via statistical
fluctuations. Obviously blocks correspond directly to structural elements of
various densities: blocks of highest density are found in cores of dense
clusters, lowest in voids or around isolated field galaxies. Blocks between
these extremes trace the intermediate structures of the Cosmic Web. But since
the multi-scale structure of the galaxy distribution is characterized by
quantities other than local density, blocks do not necessarily correspond
directly to physically meaningful structural classes. For example our way of
applying Self-organizing maps (§5.4) incorporates density gradient information
to generate a set of discrete structural classes (see Figures 7 and 8) which
may be more physically significant because their definitions are based on more
information than just density. Similarly kernel density estimation
incorporates non-local density information by virtue of adaptive smoothing.
Figure 9 depicts how the galaxies are distributed among various classes, one
row for each of the three data sets. The histograms in the first column
display the distribution of galaxies among the SOM-based classes listed in
Table 2. The other columns display, for the other two analysis methods, the
distribution of galaxies based solely on their estimated densities in bins
chosen to approximately match the resolution of the histograms in the first
column, in a way that will now be described.
Figure 9: These histograms show the number of points in each class, for the
three methods applied to the three data sets. The columns indicate the
analysis method: 1 = SOM: Self-organizing map; 2 = BB: Bayesian block; 3 =
KDE: Kernel density estimator. In first column the bins are the natural
classed yielded by the SOM; the other two are approximately matched density
bins, as described in the text. The rows indicate the data analyzed: 1 = SDSS;
2 = MS: Millennium Simulation; 3 = spatially uniform random distribution.
In this paper we compare the results of the three analysis methods only for
galaxies in the highest density classes. This is because they contain the most
easily identifiable structures – readily identified with clusters of galaxies.
More complete comparisons will be presented in a later paper. Because there is
neither a one-to-one or strictly monotonic relation between the density
classes uncovered by the three analysis methods we adopted the following
procedure. For each of the two non-SOM methods (BB and KDE), start from the
high density end and include the maximum number of the corresponding
classes888 _I.e._ the density levels described at the end of §5.1 for KDE and
in §5.3.1 for BB. such that the total number of galaxies included does not
exceed the number of galaxies in the SOM _Cluster_ class (ID number 1 in Table
2). For example, in Table 4 one sees that the SOM _Cluster_ class contains
44336 galaxies in the SDSS dataset. To reach a similar number of galaxies in
the BB method we utilize BB classes 1-4, which sum to 38293 galaxies (see
Table 3).
Table 3: Number of galaxies and classes in the SOM _Cluster_ class for the each dataset (SDSS, MS, Uniform) and algorithm (SOM, BB, KDE). The third row gives the corresponding percentage of the total volume. | SDSS | Millennium Simulation | Uniform
---|---|---|---
| SOM | BB | KDE | SOM | BB | KDE | SOM | BB | KDE
Number | 44336 | 38293 | 18286 | 60945 | 43645 | 40500 | 20008 | 10017 | 13279
Classes | 1 | 1-4 | 1-4 | 1 | 1-3 | 1-6 | 1 | 1-4 | 1-2
Volume | 12% | 6% | 9% | 16% | 6% | 13% | 8% | 7% | 55%
Similarly KDE classes 1-4 contain 18286 galaxies.
Table 4: Number of objects in each class for each dataset (SDSS,MS,Uniform) and algorithm (SOM,BB,KDE) | SDSS | Millennium Simulation | Uniform
---|---|---|---
Class | SOMaaSee Table 2 and §5.4 for a description of the SOM classes. | BB | KDE | SOM | BB | KDE | SOM | BB | KDE
1 | 44336 | 166 | 30 | 60945 | 323 | 81 | 20008 | 288 | 33
2 | 36689 | 1038 | 243 | 33075 | 24496 | 904 | 31250 | 1214 | 13246
3 | 15367 | 22724 | 3318 | 6968 | 18826 | 2478 | 7801 | 3134 | 74819
4 | 12223 | 14365 | 14695 | 12089 | 18016 | 4650 | 19279 | 5381 | 34754
5 | 16132 | 15038 | 33357 | 30674 | 17437 | 10298 | 17181 | 11848 | 6883
6 | 9244 | 16738 | 42548 | 5176 | 13353 | 22089 | 19424 | 60437 | 1777
7 | | 11380 | 29116 | | 10677 | 35988 | 10292 | 17097 | 275
8 | | 11436 | 9748 | | 9211 | 37847 | 6597 | 10692 | 39
9 | | 10304 | 916 | | 7410 | 23726 | | 6546 | 5
10 | | 7725 | 19 | | 7877 | 8634 | | 6991 | 1
11 | | 6551 | 1 | | 6296 | 1924 | | 3215 |
12 | | 3968 | | | 4771 | 280 | | 2070 |
13 | | 4800 | | | 4356 | 28 | | 1596 |
14 | | 3358 | | | 2220 | | | 940 |
15 | | 1890 | | | 1689 | | | 360 |
16 | | 1583 | | | 1039 | | | 23 |
17 | | 645 | | | 581 | | | |
18 | | 236 | | | 312 | | | |
19 | | 46 | | | 36 | | | |
Figure 10: Volume distributions for the SDSS cluster classes, in equal
logarithmic bins. The legend describes the percentage of _Cluster_ class
Voronoi volumes for each method. Figure 11: Volume histograms for the
Millennium Simulation cluster classes. The legend describes the percentage of
_Cluster_ class Voronoi volumes for each method. Figure 12: Volume histograms
for the uniform cluster classes. The legend describes the percentage of
_Cluster_ class Voronoi volumes for each method.
Figures 10, 11, and 12 compare the distributions of densities, indirectly via
histograms of Voronoi volumes, for the SDSS, MS, and uniform random data,
respectively. Each figure plots three histograms, of the Voronoi volumes of
those galaxies in the SOM selected _Cluster_ class, and in the counterpart
selections for the BB and KDE methods as just defined. The independent
variable of these histograms is a common logarithmic binning of the range of
Voronoi volumes (labeled with bin number, not to be confused with a class
identifier). Even though the KDE method does not use the Voronoi volumes in
its calculation of density we rely on the Voronoi volume associated with a
given galaxy for all three methods to make the volumes more comparable. The
legends for each of these three figures gives the percentage of total cluster
volume versus the full volume for each dataset. These numbers also appear in
Table 3.
In Figure 10 the easiest distribution to understand is that for the Bayesian
Block method. Since it uses the cell-based volumes, solely and directly, the
distribution is naturally a broad lump of small (high density) cells, with no
tail of larger (low density) ones. In other words its levels are defined
directly in terms of the volumes, as depicted in Fig 6. Both of the other
methods blend in other non-local information – the SOM explicitly through
density gradients, and KDE implicitly via its adaptive kernel – leading to the
rather long tails to the high end of the volume distributions. The KDE
distribution resides nearly midway between the SOM and BB ones, presumably
because of its implicit blend of local and non-local information. Nearly the
same pattern as seen in the SDSS is repeated for the Millennium Simulation
dataset in Figure 11 for each of the methods and the cluster volume
percentages. However, for the Uniform dataset in Figure 12 the SOM and BB
cluster classes appear very similar in volume percentage, while the
corresponding KDE classes contain many more galaxies.
### 6.2 Visualizing high density classes
A thin spatial slice (from a fixed viewing angle) of the galaxies found in the
high density classes just described in §6.1, for each method and dataset are
compared side-by-side in Figure 13.
Figure 13: Projections of the spatial locations of the main density structures
found with the three methods, using the three data sets. These are the central
plot from Figures 14 – 16, 20 – 22 and 26 – 28. As discussed in the text in
some sense these structures are clusters, but they are defined simply as
localized density peaks. From left to right: Bayesian blocks, SOM clusters,
and KDE peaks. Top to bottom: spatially uniform random distribution, SDSS DR7,
and Millennium Simulation.
This figure collects the view shown in the central panels of the $3\times 3$
plots from Figures 14-16, 20-22 and 26-28, below. The three methods identify
similar structures in the SDSS and MS data, but of course not in the uniformly
random data. In the bottom row note that the three methods select markedly
different depths of the upper end of the density distribution (_cf_. the
right-hand panel of Figure 3) but do not falsely reveal medium or large scale
structure.
The remaining figures of this section elucidate clustering associated with the
highest density regions for the three analysis methods, with sets of figures
for the SDSS, the MS, and the uniformly random data. Begin with three spatial
distributions for the SDSS data, Figures 14-16, as derived with SOM, BB, and
KDE respectively.
Figure 14: Self-organizing map analysis of the Volume Limited SDSS data. The
three rows in each column show the locations of the derived block structures
in three orthogonal projections. Column 1: The green points (found in the SOM
_Cluster_ class) are those assigned higher densities by the SOM algorithm,
while the red are all other points. For clarity the corresponding points in
thin spatial slices (indicated as gray bands in Column 1) are plotted in green
(points in the SOM _Cluster_ class) and red (non-cluster points) in Columns 2
and 3 respectively.
The rows in figure 14 show SOM-derived structures in three orthogonal
projections, the first column being the entire data-cube (see Figure 1 and
§4). The green points are galaxies in the SOM _Cluster_ class, while red
points are not. The remaining two columns differ from the first in two ways:
they show only galaxies within thin spatial slices (delineated as light gray
bands in Column 1), and they separate the cluster and non-cluster galaxies
(displayed in gray and black, respectively, in all 3 columns) to better reveal
the structures and the gross differences in the distributions of _Cluster_
galaxies and non-cluster galaxies.
Figure 15 presents the same display pattern for the BB analysis, and Figure 16
for the KDE analysis. The SOM and BB cluster classes appear to be relatively
similar, while the KDE appears markedly different from the other two, although
some structures do appear more or less the same with all three analysis
algorithms.
Figure 15: The same as Figure 14, but for the Bayesian Block (BB) Structure
analysis of the Volume Limited SDSS data. The three rows in each column show
the locations of the derived block structures in three different projections.
Column 1: The green points are those assigned higher densities by the BB
algorithm, while the red are all other points. Many of these green points
would be considered to be in high-density clusters and are what we consider to
constitute the BB _Cluster_ class. Column 2 shows the same BB structures in a
thin slice, to better visualize these results. Column 3 is the complement of
column 2: all structures not selected in the same thin slice shown in column
2. Figure 16: The same as Figure 14, but for the Kernel Density Estimation
(KDE) analysis of the Volume Limited SDSS data. The three rows in each column
show the locations of the KDE derived structures in three different
projections. Column 1: The green points are considered to be in high-density
clusters and are what we consider to constitute the KDE _Cluster_ class, while
the red are all other points. Column 2 shows the same KDE structures in a thin
slice, to better visualize these results. Column 3 is the complement of column
2: all structures not selected in the same thin slice shown in column 2.
Continuing the discussion of the SDSS, now turn to a somewhat detailed look at
the distribution of the galaxies over various classes that have been defined
above. The next three plots, Figures 17, 18, and 19, show histograms of the
classes for the three methods applied to the SDSS dataset. Figure 17 plots the
BB classes on the x-axis and the KDE ones on the y-axis. The number of KDE
objects in a given BB class for a given KDE class is shown in the
corresponding histogram bin.
Ignoring the coloring scheme for the moment, in Figure 17 one sees a clear
correlation between the density classes (indicated inversely by the class
number labels on the axes) in the KDE and BB classifications. To wit, KDE
class 1 through 4 objects (see Table 3) are found exclusively in BB classes 1
through 7 – implying that there are no KDE-class 1 through 4 objects in BB
classes 8 through 19. The coloring scheme used for the individual histograms
is intended to show how the method not plotted on either the x or y axis
distributes its cluster classes in green in the other two method classes. Non-
cluster classes are in red. For example, for this Figure 17 the method not
plotted on the x (BB) or y (KDE) axes is the SOM method. The SOM cluster class
is plotted in green and all other SOM classes are in red. Most of the SOM
cluster class objects show up in KDE classes 2–7 with a few in class 8. All of
the SOM cluster class objects appear in BB classes 1–10. None of the SOM
cluster class objects show up in the lowest density BB classes 11-19 or KDE
class 9. Clearly the overlap between the cluster classes of one method and the
non-cluster classes of others is not insignificant, in accordance with the
fact that the structural classifications carried out by the three methods are
based on different information content.
Figures 18 and 19 are identical to 17, but for the other two combinations of
variables assigned to the x- and y- axes (in both cases including the third
variable with the shown histograms).
Figure 17: For the SDSS data, this figure compares high and low-density
classes from the 3 methods. Each of the 9 sets of histograms shows the
distribution among the BB classes (horizontal axis) of those in the
corresponding KDE classes (indicated on the vertical axis). The full
distribution over the SOM clusters is not shown, but in each histogram bar the
SOM defined _Cluster_ class is in green. The SOM non-cluster classes are in
red.
Figure 18: Also for SDSS data, and similar to Figure 17, this figure compares
the high and low-density classes from the 3 methods. Each of the 6 histograms
shows the distribution among the KDE classes (horizontal axis) of those in the
corresponding SOM classes (indicated on the vertical axis). The full
distribution over the BB classes is not shown, but in each histogram bar the
high-density BB _Cluster_ classes are in green. Non-high-density BB classes
are in red. Figure 19: Also for the SDSS data, and similar to Figure 17, this
figure compares the high and low-density classes from the 3 methods. Each of
the 6 histograms shows the distribution among the BB classes (horizontal axis)
of those in the corresponding SOM classes (indicated on the vertical axis).
The full distribution over the KDE classes is not shown, but in each histogram
bar the high-density KDE _Cluster_ classes are in green. Non-high-density KDE
classes are in red.
Having discussed the example results for the actual SDSS data, we now present
an exactly parallel set of figures for the artificial data contained in the
Millennium Simulation data, as described in §4. The first three spatial
distribution plots for MS, Figures 20 – 22, are parallel to Figures 14 – 16,
discussed above for the SDSS data. These are followed by the class
distribution plots in Figures 23 – 25, parallel to those in Figures 17 – 19.
Figure 20: Similar to Figure 14, but instead the Self-organizing map (SOM)
analysis of the Volume Limited Millennium Simulation (MS) data. The three rows
in each column show the locations of the derived block structures in three
different projections. Column 1: The green points are those assigned higher
densities by the SOM algorithm (found in the SOM _Cluster_ class), while the
red are all other points. Column 2 shows the same SOM _Cluster_ structures in
a thin slice, to better visualize these results. Column 3 is the complement of
column 2: all structures not selected in the same thin slice shown in column
2. Figure 21: The same as Figure 20, but for the Bayesian Block (BB) Structure
analysis of the Volume Limited MS data. The three rows in each column show the
locations of the derived block structures in three different projections.
Column 1: The green points are those assigned higher densities by the BB
algorithm (found in the BB _Cluster_ class), while the red are all other
points. Column 2 shows the same BB _Cluster_ structures in a thin slice, to
better visualize these results. Column 3 is the complement of column 2: all
structures not selected in the same thin slice shown in column 2. Figure 22:
The same as Figure 20, but for the Kernel Density Estimation (KDE) analysis of
the Volume Limited MS data. The three rows in each column show the locations
of the derived block structures in three different projections. Column 1: The
green points are those assigned higher densities by the KDE algorithm (found
in the KDE _Cluster_ class), while the red are all other points. Column 2
shows the same KDE _Cluster_ structures in a thin slice, to better visualize
these results. Column 3 is the complement of column 2: all structures not
selected in the same thin slice shown in column 2.
Figure 23: For the Millennium Simulation (MS) data, this figure compares high
and low-density classes from the 3 methods. Each of the 12 sets of histograms
shows the distribution among the BB classes (horizontal axis) of those in the
corresponding KDE class (indicated on the vertical axis). The full
distribution over the SOM classes is not shown, but in each histogram bar the
SOM defined _Cluster_ class is in green. The SOM non-cluster classes are in
red. Figure 24: Also for MS data, and similar to Figure 23. This figure
compares high and low-density classes from the 3 methods. Each of the 6 sets
of histograms shows the distribution among the KDE classes (horizontal axis)
of those in the corresponding SOM class (indicated on the vertical axis). The
full distribution over the BB classes is not shown, but in each histogram bar
the BB defined _Cluster_ classes are in green. The BB non-cluster classes are
in red. Figure 25: Also for MS data, and similar to Figure 23. This figure
compares high and low-density classes from the 3 methods. Each of the 6 sets
of histograms shows the distribution among the BB classes (horizontal axis) of
those in the corresponding SOM class (indicated on the vertical axis). The
full distribution over the KDE classes is not shown, but in each histogram bar
the KDE defined _Cluster_ classes are in green. The KDE non-cluster classes
are in red.
Having discussed the example results for the actual SDSS data, and the
Millennium Simulation data, we now present an exactly parallel set of figures
for the artificial data contained in the uniformly and randomly distributed
data, as described in §4.
The first three spatial distribution plots for the uniformly random data
Figures 26 – 28 are parallel to Figures 14 – 16 discussed above for the SDSS
data, and Figures 20 – 22 discussed above for the MS data. These are followed
by the class distribution plots in Figures 29 – 31, parallel to those in
Figures 23 – 25, and Figures 17 – 19.
Figure 26: Self-organizing map (SOM) analysis of the spatially uniform random
distribution data. The three rows in each column show the locations of the
derived block structures in three different projections. Column 1: The green
points are those assigned higher densities by the SOM algorithm (found in the
SOM _Cluster_ class), while the red are all other points. Column 2 shows the
same SOM structures in a thin slice, to better visualize these results. Column
3 is the complement of column 2: all structures not selected in the same thin
slice shown in column 2. Figure 27: The same as Figure 26, but for the
Bayesian Block (BB) analysis of the spatially uniform random distribution
data. The three rows in each column show the locations of the derived block
structures in three different projections. Column 1: The green points are
those assigned higher densities by the BB algorithm (found in the BB _Cluster_
class), while the red are all other points. Column 2 shows the same BB
structures in a thin slice, to better visualize these results. Column 3 is the
complement of column 2: all structures not selected in the same thin slice
shown in column 2. Figure 28: The same as Figure 26, but for the Kernel
Density Estimation (KDE) analysis of the spatially uniform random distribution
data. The three rows in each column show the locations of the derived block
structures in three different projections. Column 1: The green points are
those assigned higher densities by the KDE algorithm (found in the KDE
_Cluster_ class), while the red are all other points. Column 2 shows the same
KDE high density structures in a thin slice, to better visualize these
results. Column 3 is the complement of column 2: all structures not selected
in the same thin slice shown in column 2. Figure 29: For the spatially uniform
random distribution data, this figure compares high and low-density classes
from the 3 methods. Each of the 6 sets of histograms shows the distribution
among the BB classes (horizontal axis) of those in the corresponding KDE class
(indicated on the vertical axis). The full distribution over the SOM classes
is not shown, but in each histogram bar the SOM defined _Cluster_ class is in
green. The SOM non-cluster classes are in red.
Figure 30: Also for spatially uniform random distribution data, and similar to
Figure 29, this figure compares high and low-density classes from the 3
methods. Each of the 8 sets of histograms shows the distribution among the SOM
classes (horizontal axis) of those in the corresponding KDE class (indicated
on the vertical axis). The full distribution over the BB classes is not shown,
but in each histogram bar the BB defined _Cluster_ classes are in green. The
BB non-cluster classes are in red. Figure 31: Also for the spatially uniform
random distribution data, and similar to Figure 29, this figure compares high
and low-density classes from the 3 methods. Each of the 8 sets of histograms
shows the distribution among the BB classes (horizontal axis) of those in the
corresponding SOM class (indicated on the vertical axis). The full
distribution over the KDE classes is not shown, but in each histogram bar the
KDE defined _Cluster_ classes are in green. The KDE non-cluster classes are in
red.
In all cases there is very little evidence of clustering in the uniformly
distributed points, exactly as one would expect. The average densities are
again very similar for the BB and SOM. The KDE apears to select more galaxies
for its cluster class, while explicitly avoiding the majority of galaxies at
the border; this odd behavior was not demonstrated in the other datasets, but
is likely just an edge-effect that could easily be removed.
Table 5: This table addresses how much the different methods assign galaxies (at the high density end of the distribution) to the same/different classes. Entries indicate which classes (defined by the method labeled in the second row from the top of the columns, and for the data set indicated in the first row) are contained in the cluster class for the method indicated in the left-most column. | SDSS | Millennium Simulation | Uniform
---|---|---|---
| SOM | BB | KDE | SOM | BB | KDE | SOM | BB | KDE
SOM _Cluster_ class | 1 | 1–6 | 2–7 | 1 | 1–10 | 2–10 | 1 | - | -
BB _Cluster_ classes | 1–3 | 1–4 | 2–7aaFor example, how many KDE classes are found in the BB cluster classes (1–4)? In this case KDE classes 2-7 contain BB cluster classes 1–4. | 1–3 | 1–3 | 2–9 | 1,6 | 1–4 | 2–4
KDE _Cluster_ classes | 1–3 | 1–7 | 1–4 | 1–4 | 1–7 | 1–6 | 2–6 | 2–6 | 1–2
Table 5 distills the cluster class overlap between methods into a single table
as shown in Figures 17 – 19, 23 – 25, and 29 – 31. For the most part these
summaries for the SDSS and MS cases are more alike than not, whereas those for
the uniformly random case are very different. It is clear that all three
algorithms assign high density regions to classes in somewhat different ways,
just as one would expect.
## 7 Summary and Conclusions
We have described two techniques newly applied to characterize structures in
large 3-D galaxy surveys based on Voronoi tesselation – “Bayesian Blocks” (BB)
and “Self-organizing maps” (SOM). These two new techniques were compared with
a third well known technique called Kernel Density Estimation (KDE).
The techniques were applied to three example datasets. The first was a volume
limited sub-sample of the SDSS Data Release 7. The second was a volume limited
sub-sample of the Millennium Simulation, while the 3rd was a uniform
randomized set of points similar in size to the other two. The BB and SOM
methods proved to pick similar high-density structures from the SDSS and
Millennium Simulation datasets. The KDE method generally gives rather
different results, although it was able to identify some of the same high-
density structures. The uniform randomized sample proved a challenge to all
three techniques ability to discern statistically significant high-density
concentrations – as it should have, since they don’t exist.
In future publications we plan to provide more details on the analysis
previewed here, including preparation of an all-scale structure catalog
(distinguishing from the term _large-scale structure_). Our catalog will
include features unique to our analysis approach, such as:
* •
internal comparison between structures which have been found using two
different analysis methods, but which can be reliably identified as comprising
the same physical structure, say based on spatial coincidence.
* •
measures of convexity/concavity and their distributions
* •
the sizes and directions of tri-axial ellipsoids fit to the blocks,
* •
other morphological quantities
This will allow us to further compare our self-organizing map and Bayesian
block analysis on the Sloan Digital Sky Survey data with other workers’
results including catalogs of clusters, sheets (walls), filaments, voids,
_etc._
Certainly the reader may be skeptical of any one of the three methods
abilities to distinguish between similar structures in SDSS redshift data such
as Fingers-of-God and line-of-sight filaments. However, given our ability to
obtain the “ground truth” from the original Millennium Simulation positions
($x,y,z$) and velocities ($V_{x},V_{y},V_{z}$) we believe it will be possible
characterize and distinguish structures that mimic each other in SDSS type
data sets.
We are grateful to the NASA-Ames Director’s Discretionary Fund and to Joe
Bredekamp and the NASA Applied Information Systems Research Program for
support and encouragement. We thank the Institute for Pure and Applied
Mathematics at UCLA and the Banff International Research Station for
hospitality over times where some of this work was carried out. Helpful
discussions and suggestions over the years came from Chris Henze, Creon Levit,
and Ashok Srivastava. Thanks goes to Ani Thakar and Maria Nieto-Santisteban
for their help with our many SDSS casjobs queries. Michael Blanton’s help with
using his SDSS NYU VAGC catalog were also very much appreciated. Zeljko
Ivezic, Robert Lupton, Jim Gray and Alex Szalay also provided essential help
in utilizing the SDSS. Funding for the SDSS has been provided by the Alfred P.
Sloan Foundation, the Participating Institutions, the National Aeronautics and
Space Administration, the National Science Foundation, the U.S. Department of
Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web
site is http://www.sdss.org/. The SDSS is managed by the Astrophysical
Research Consortium for the Participating Institutions. The Participating
Institutions are The University of Chicago, Fermilab, the Institute for
Advanced Study, the Japan Participation Group, The Johns Hopkins University,
Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy, the
Max-Planck-Institute for Astrophysics, New Mexico State University, University
of Pittsburgh, Princeton University, the United States Naval Observatory, and
the University of Washington. This research has made use of NASA’s
Astrophysics Data System Bibliographic Services. This research has also
utilized the viewpoints (Gazis, Levit, & Way, 2010) software package.
## Appendix A Appendix: SDSS casjobs query
Select p.ObjID, p.ra, p.dec,
p.dered_u, p.dered_g, p.dered_r, p.dered_i, p.dered_z,
p.Err_u, p.Err_g, p.Err_r, p.Err_i, p.Err_z,
s.z, s.zErr, s.zConf
FROM SpecOBJall s, PhotoObjall p
WHERE s.specobjid=p.specobjid
and s.zConf$>$0.95 and s.zWarning=0 and
(p.primtarget & 0x00000040 $>$ 0)
and ( ((flags & 0x8) = 0) and ((flags & 0x2) = 0) and ((flags & 0x40000) = 0))
## Appendix B Appendix: Catalog Attributes
Table 6: Attributes u, g, r, i, z | Apparent magnitudes from the SDSS DR7.
---|---
U, G, R, I, Z | Absolute magnitudes from the SDSS DR7.
z, zerr | Redshift and the uncertainty in redshift.
$d_{uniform}$ | Average spacing between points for a uniform distribution.
$d_{1-6}$ | Distances in units of z to the six nearest neighbors.
$R_{Voronoi}$ | (Voronoi volume)1/3 in units of z. A measure of local density.
$d_{CM}$ | Distance in z from a galaxy to the CM of its Voronoi cell.
$R_{Max}$ | Maximum distance from the point to a vertex of the Voronoi cell.
$R_{Min}$ | Minimum distance from the point to a vertex of the Voronoi cell.
$R_{Voronoi}/d_{Uniform}$ | A dimensionless measure of local density.
$R_{Max}/d_{Uniform}$ | A dimensionless measure of $R_{Max}$.
$R_{Min}/d_{Uniform}$ | A dimensionless measure of $R_{Min}$.
$d_{CM}/R_{Voronoi}$ | A dimensionless measure of the local gradient.
‘elongation’ | A simple dimensionless measure of the elongation of a Voronoi cell.
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|
arxiv-papers
| 2010-09-02T10:54:56 |
2024-09-04T02:49:12.628186
|
{
"license": "Public Domain",
"authors": "M.J. Way, P.R. Gazis, Jeffrey D. Scargle",
"submitter": "Michael Way",
"url": "https://arxiv.org/abs/1009.0387"
}
|
1009.0398
|
11institutetext: Instytut Astronomiczny, Uniwersytet Wrocławski, ul. Kopernika
11, 51-622 Wrocław, Poland
11email: daszynska@astro.uni.wroc.pl, szewczuk@astro.uni.wroc..pl
# Effect of NLTE model atmospheres on photometric amplitudes
and phases of early B-type pulsating stars
J. Daszyńska-Daszkiewicz W. Szewczuk
(Received …; accepted …)
###### Abstract
Context. Amplitudes and phases of the light variation of a pulsating star in
various photometric passbands contain information about geometry of observed
modes. Because oscillation spectra of early B-type main sequence stars do not
exhibit regular patterns, these observables are very often the only ones from
which mode identification can be derived. Moreover, these data can yield
valuable constraints on mean stellar parameters, subphotospheric convection,
microphysics and atmospheres.
Aims. We study all possible sources of inaccuracy in theoretical values of the
photometric observables, i.e. amplitude ratios and phase differences, of early
B-type main sequence pulsators. Here, we discuss effects of parameters coming
from both models of stellar atmospheres and linear nonadiabatic theory of
stellar pulsation. In particular, we evaluate for the first time the effect of
the departure from the LTE approximation.
Methods. The photometric amplitudes and phases are calculated from the semi-
analytical formula for the light variation. The atmospheric input comes from
line-blanketed, LTE and NLTE plane-parallel, hydrostatic models. To compute
the limb-darkening coefficients for NLTE models, we use the Least-Square
Method taking into account the accuracy of the flux conservation. The linear
nonadiabatic stellar pulsations were computed by means of Dziembowski code. We
consider the OPAL and OP opacity tables and two determinations of the solar
chemical mixture: GN93 and AGSS09.
Results. We present effects of NLTE atmospheres, chemical composition and
opacities on theoretical values of the photometric observables of early B-type
pulsators. To this end, we compute tables with the passband fluxes, flux
derivatives over effective temperature and gravity as well as the non-linear
limb-darkening coefficients in 12 most often used passbands, i.e. in the
Strömgern system, $uvby$, and in the Johnson-Cousins-Glass system, $UBVRIJHK$.
We make these tables public available at the Wrocław HELAS Web page ††thanks:
http://helas.astro.uni.wroc.pl.
Conclusions. In the case of radial modes, effects of uncertainties in model
atmospheres are much smaller than those coming from the pulsation theory. In
turn, for nonradial modes effects of NLTE become more important and they are
most significant for dipole modes. Therefore, if one wants to construct an
accurate seismic model and gain more constraints, all inaccuracies in the
photometric observables have to be allowed for.
###### Key Words.:
Stars: early-type – Stars: oscillations – Stars: atmospheres
## 1 Introduction
Data on the photometric and spectroscopic variations of a pulsating star bring
information on frequencies of excited modes and their geometry. The latter
property is particularly important if oscillation spectra are sparse and lack
equidistant patterns. This is the case for main sequence pulsators of early B
spectral type, i.e. $\beta$ Cephei variables. Additionally, using data on the
photometric amplitudes and phases one can constrain mean stellar parameters,
test efficiency of convection transport in the case of cooler pulsators, like
$\delta$ Scuti stars, (Daszyńska-Daszkiewicz, Dziembowski, Pamyatnykh 2003) or
test opacity data in the case of B-type pulsators (Daszyńska-Daszkiewicz,
Dziembowski, Pamyatnykh 2005).
Ones of the most popular tools to identify a pulsation mode are the amplitude
ratios and phase differences in various photometric passbands. In the zero-
rotation approximation, these observables depend on the mode degree, $\ell$,
but are independent of the azimuthal order, $m$, the intrinsic mode amplitude,
$\varepsilon$, and the inclination angle, $i$.
The semi-analytical expression for the bolometric light variation was
formulated by Dziembowski (dziemb (1977)). Then, Balona & Stobie (balst
(1979)) and Stamford & Watson (stwa (1981)) expanded it for the light
variation in the photometric passbands. They showed that modes with different
values of $\ell$ are located in separated parts on the amplitude ratio $vs.$
phase difference diagrams. Subsequently, this method has been applied to
various types of pulsating stars by Watson (watson (1988)). Cugier,
Dziembowski & Pamyatnykh (cdp (1994)) improved the method by including
nonadiabatic effects in calculations for the $\beta$ Cephei stars. Effects of
rotation on photometric observables were studied by Daszyńska-Daszkiewicz et
al. (dd2002 (2002)) for close frequency modes and by Townsend (townsend2003
(2003)) and Daszyńska-Daszkiewicz, Dziembowski & Pamyatnykh (dd2007 (2007))
for long-period g-modes. To compute values of the photometric amplitudes and
phases we need two kinds of input. The first one comes from stellar model
atmospheres and the second one from computations of stellar pulsation. Both
data contain various sources of uncertainties.
The goal of this paper is to examine for the first time an influence of NLTE
effects on theoretical values of the photometric amplitude ratios and phase
differences for early B-type main sequence pulsators. We check also effects of
metallicity and microturbulent velocity in the atmosphere. Subsequently, we
compare these effects with uncertainties coming from the linear nonadiabatic
theory of stellar pulsation. As an example, we consider main sequence models
with a mass of 10 $M_{\odot}$ and low degree modes with $\ell$=0, 1, 2. Here,
we neglect all effects of rotation on pulsation.
The structure of the paper is the following. In Section 2, we recall the
linear formula for the pulsation complex amplitude in a photometric band.
Section 3 contains description of NLTE model atmospheres and results of our
computations of the band fluxes, corresponding flux derivatives over effective
temperature and gravity, and the nonlinear limb darkening coefficients, in 12
photometric passbands: $uvbyUBVRIJHK$. Tables with these data can be
downloaded from the Wrocław HELAS web page. Moreover, we study effects of
temperature, gravity, NLTE, atmospheric metallicity and microturbulent
velocity on the above mentioned quantities. How these atmospheric
uncertainties translate into the pulsation photometric observables, i.e.
amplitude ratios and phase differences, is discussed in Section 4.
Inaccuracies in the photometric observables connected with the linear
nonadiabatic theory of stellar pulsation are presented in Section 5. We end
with conclusions in Section 6.
## 2 Light variation due to stellar pulsation
Stellar pulsations cause changes of temperature, normal to the surface element
and pressure. If pulsations are linear and all effects of rotation on
pulsation can be ignored, then the total amplitude of the light variation in
the passband $x$ can be written in the following complex form (Daszyńska-
Daszkiewicz et al. dd2002 (2002)):
$\mathcal{A}_{x}(i)=-1.086\varepsilon
Y_{\ell}^{m}(i,0)b_{\ell}^{x}(D_{1,\ell}^{x}f+D_{2,\ell}+D_{3,\ell}^{x}),$ (1)
where $\varepsilon$ is the intrinsic mode amplitude, $Y_{\ell}^{m}$ – the
spherical harmonic and $i$ – the inclination angle. The amplitude is given by
$abs(\mathcal{A}_{x})$ and phase by $arg(\mathcal{A}_{x})$. The
$D_{1,\ell}^{x}\cdot f$ product stands for temperature changes, where
$D_{1,\ell}^{x}=\frac{1}{4}\frac{\partial\log(\mathcal{F}_{x}|b_{\ell}^{x}|)}{\partial\log
T_{\rm eff}}.$ (2)
${\cal F}_{x}$ is the flux in the passband $x$ and $f$ is the nonadiabatic
complex parameter describing the amplitude of the radiative flux perturbation
to the radial displacement at the photosphere level
$\frac{\delta{\cal F}_{\rm bol}}{{\cal F}_{\rm bol}}={\rm Re}\\{\varepsilon
fY_{\ell}^{m}(\theta,\varphi){\rm e}^{-{\rm i}\omega t}\\}.$ (3)
Geometrical term, $D_{2,\ell}$, is given by
$D_{2,\ell}=(2+\ell)(1-\ell),$ (4)
and the pressure term, $D_{3,\ell}^{x}$, by
$D_{3,\ell}^{x}=-\left(2+\frac{\omega^{2}R^{3}}{GM}\right)\frac{\partial\log(\mathcal{F}_{x}|b_{\ell}^{x}|)}{\partial\log
g}.$ (5)
$b_{\ell}^{x}$ is the disc averaging factor defined by
$b_{\ell}^{x}=\int_{0}^{1}h_{x}(\mu)\mu P_{\ell}(\mu)d\mu$ (6)
where $h_{x}(\mu)$ is the limb darkening law, $P_{\ell}$ is the Legendre
polynomial and $\mu$ is a cosine of the angle between a line of sight and the
emergent intensity. Remaining parameters have their usual meaning.
In the above expressions, we can distinguish two sorts of input parameters
needed to compute theoretical values of the photometric amplitudes and phases.
The first input is derived from models of stellar atmospheres and these are
the flux derivatives over effective temperature and gravity (Eq. 2 and 5), as
well as limb-darkening and its derivatives (Eq. 2, 5 and 6). The second input
comes from the nonadiabatic theory of stellar pulsation and this is the
$f$-parameter (Eq. 1 and 3).
In this paper, we used the Warsaw-New Jersey evolutionary code and the linear
nonadiabatic pulsation code of Dziembowski (1977). We considered opacity
tables from OPAL (Iglesias & Rogers 1996) and OP (Seaton 2005) projects, and
two determinations of the solar chemical composition: GN93 by Grevesse & Noels
1993 and AGSS09, a recent one by Asplund et al. 2009. As for models of stellar
atmospheres, we considered Kurucz models (Kurucz kurucz (2004)) and TLUSTY
models (Lanz & Hubeny lanzhubeny (2007)).
## 3 NLTE line-blanketed model atmospheres
The most widely used models of stellar atmospheres are line blanketed, plane-
parallel, hydrostatic models of Kurucz (2004) computed within an approximation
of local thermodynamic equilibrium (LTE). However, in the case of atmospheres
of early B-type stars, effects of the departure from LTE and a proper
treatment of line opacity become important. Escalation of the quality of
spectro-photometric observations calls for a need of high resolution and
accurate models of stellar atmospheres.
A grid of non-LTE (NLTE) model atmospheres were computed by Lanz & Hubeny
(lanzhubenyO (2003)) and Lanz & Hubeny (lanzhubeny (2007)). These are metal
line-blanketed, plane-parallel, hydrostatic model atmospheres of O-type stars
(OSTAR2002) and of early B-type stars (BSTAR2006), respectively. In their
computations, they adopted a solar chemical mixture by Grevesse & Sauval
(grevese (1998)), helium to hydrogen abundance of He/H=0.1 by number and two
values of the microturbulent velocity, $\xi_{t}$=2 and 10 km/s. Lanz & Hubeny
(lanzhubenyO (2003), lanzhubeny (2007)) showed that in the case of OB stars, a
neglect of NLTE effects causes differences not only in spectral lines but also
in the continuum level. In the near ultraviolet (the Balmer continuum), the
LTE fluxes are up to $10\%$ higher than the NLTE counterparts. In turn, in the
far and extreme ultraviolet (the Lyman continuum), the LTE fluxes are lower
than the NLTE ones. For more details see Lanz & Hubeny (lanzhubenyO (2003),
lanzhubeny (2007)).
The grid of the OSTAR2002 models were computed for 12 values of effective
temperatures appropriate to O-type stars, i.e. between 27500 and 55000 K with
a step of 2500 K, 8 values of surface gravities from $\log g$=3.0 to 4.75 with
a step of 0.25 dex and one microturbulent velocity, $\xi_{t}$=10 km/s.
Moreover, 10 values of the atmospheric metallicity were considered,
$(Z/Z_{\odot})_{\rm atm}$=2, 1, 0.5, 0.2, 0.1, 1/30, 1/50, 1/100, 1/1000, 0.0,
where $Z$ is the metal abundance by mass and $Z_{\odot}$ is the solar value.
The grid of the BSTAR2006 models contains 16 values of effective temperatures
between 15000 and 30000 K and a step of 1000 K, 13 values of surface gravities
in the range from $\log g=1.75$ to 4.75 dex and a 0.25 dex step. In this case,
6 values of metallicity were considered, $(Z/Z_{\odot})_{\rm atm}$=2, 1, 0.5,
0.2, 0.1, 0.0, and two values of the microturbulent velocities, $\xi_{t}$=2
km/s and $\xi_{t}$=10 km/s. Models with $\xi_{t}$=10 km/s were computed for
two sets of chemical mixture but only for B-type supergiants ($\log g\leq
3.0$). The first mixture was the same as for $\xi_{t}$=2 km/s, i.e. the solar
composition, and the second one was enriched in helium and nitrogen, and
depleted in carbon.
The lower limit for gravity at a given effective temperature was determined
approximately by the Eddington limit. The TLUSTY code used by Hubeny and Lanz
(1995) becomes unstable near this limit.
### 3.1 The passband fluxes and their derivatives
We computed the fluxes, ${\cal F}_{x}$, for the BSTAR2006 NLTE models in 12
commonly used photometric passbands, i.e. in the Strömgren ($uvby$) and
Johnson-Cousins-Glass ($UBVRIJHK$) systems, according to the formula
${\cal F}_{x}=\frac{\int\limits^{\lambda_{1}}_{\lambda_{2}}{\cal
F}(\lambda)S(\lambda)\,d\lambda}{\int\limits^{\lambda_{1}}_{\lambda_{2}}S(\lambda)\,d\lambda},$
(7)
where $S(\lambda)$ is the response function of the passband $x$, adopted from
the Asiago Database on Photometric Systems (Moro & Munari moro (2000)). The
integral in Eq. (7) is computed in the wavelength range from $\lambda_{1}$ to
$\lambda_{2}$ where $S(\lambda)$ has non-zero values.
We considered the whole grid of effective temperature, $T_{\rm eff}$, gravity,
$\log g$, metallicity, $(Z/Z_{\odot})_{\rm atm}$, and microturbulent velocity,
$\xi_{t}$, for the BSTAR2006 models. Table 1 summarizes our results. The file
names are coded by values of $(Z/Z_{\odot})_{\rm atm}$ and $\xi_{t}$. We have
adopted the same designations as Lanz &Hubeny (lanzhubeny (2007)), i.e. BC,
BG, BL, BS, BT, BZ denote $(Z/Z_{\odot})_{\rm atm}$=2, 1, 0.5, 0.2, 0.1 and 0,
respectively, whereas v2 and v10 correspond to $\xi_{t}=2$ km/s and 10 km/s,
respectively. The index CN marks a model enriched in helium and nitrogen, and
depleted in carbon. Each file contains the following columns: line number,
effective temperature, $T_{\rm eff}$, logarithm of the surface gravity, $\log
g$, metallicity, $(Z/Z_{\odot})_{\rm atm}$, microturbulent velocity,
$\xi_{t}$, and the logarithmic flux, $\log{\cal F}_{x}$, in the $uvbyUBVRIJHK$
passbands.
Table 1: Tables of the passband fluxes, their derivatives over effective
temperature and gravity, and non-linear limb darkening coefficients, for the
BSTAR2006 NLTE models in the $uvbyUBVRIJHK$ passbands.
fluxes | derivatives | limb darkening | range of $T_{\rm eff}$ [K] | range of $\log g$ [dex] | $(Z/Z_{\odot})_{\rm atm}$ | $\xi_{t}$ [km/s] | mixture
---|---|---|---|---|---|---|---
| | coefficients | | | | |
flux_BCv2 | der_BCv2 | LDC_BCv2 | 15000-30000 | 1.75-4.75 | 2.0 | 2 | GS98
flux_BCv10 | der_BCv10 | LDC_BCv10 | 15000-30000 | 1.75-3.00 | 2.0 | 10 | GS98
flux_BCv10CN | der_BCv10CN | LDC_BCv10CN | 15000-30000 | 1.75-3.00 | 2.0 | 10 | CN
flux_BGv2 | der_BGv2 | LDC_BGv2 | 15000-30000 | 1.75-4.75 | 1.0 | 2 | GS98
flux_BGv10 | der_BGv10 | LDC_BGv10 | 15000-30000 | 1.75-3.00 | 1.0 | 10 | GS98
flux_BGv10CN | der_BGv10CN | LDC_BGv10CN | 15000-30000 | 1.75-3.00 | 1.0 | 10 | CN
flux_BLv2 | der_BLv2 | LDC_BLv2 | 15000-30000 | 1.75-4.75 | 0.5 | 2 | GS98
flux_BLv10 | der_BLv10 | LDC_BLv10 | 15000-30000 | 1.75-3.00 | 0.5 | 10 | GS98
flux_BLv10CN | der_BLv10CN | LDC_BLv10CN | 15000-30000 | 1.75-3.00 | 0.5 | 10 | CN
flux_BSv2 | der_BSv2 | LDC_BSv2 | 15000-30000 | 1.75-4.75 | 0.2 | 2 | GS98
flux_BSv10 | der_BSv10 | LDC_BSv10 | 15000-30000 | 1.75-3.00 | 0.2 | 10 | GS98
flux_BSv10CN | der_BSv10CN | LDC_BSv10CN | 15000-30000 | 1.75-3.00 | 0.2 | 10 | CN
flux_BTv2 | der_BTv2 | LDC_BTv2 | 15000-30000 | 1.75-4.75 | 0.1 | 2 | GS98
flux_BTv10 | der_BTv10 | LDC_BTv10 | 15000-30000 | 1.75-3.00 | 0.1 | 10 | GS98
flux_BTv10CN | der_BTv10CN | LDC_BTv10CN | 15000-30000 | 1.75-3.00 | 0.1 | 10 | CN
flux_BZv2 | der_BZv2 | LDC_BZv2 | 15000-30000 | 1.75-4.75 | 0.0 | 2 | GS98
flux_BZv10 | der_BZv10 | LDC_BZv10 | 15000-30000 | 1.75-3.00 | 0.0 | 10 | GS98
111 GS98 - chemical mixture by Grevesse & Sauval (grevese (1998)), CN -
chemical mixture enriched in helium and nitrogen, and depleted in carbon.
The most important atmospheric parameters in the expression for the brightness
variation of a pulsating star are the flux derivatives over effective
temperature, $T_{\rm eff}$, and gravity, $\log g$. In a given photometric
pasband $x$, they are defined as
$\alpha_{T}^{x}=\frac{\partial\log{\cal F}_{x}}{\partial\log T_{\rm
eff}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\alpha_{g}^{x}=\frac{\partial\log{\cal F}_{x}}{\partial\log g}.$
Values of these flux derivatives depend not only on effective temperature and
gravity but also on the metallicity, microturbulent velocity and the departure
from LTE in the star’s atmosphere.
Similarly to the passband fluxes, the derivatives were computed in the whole
range of parameters of the BSTAR2006 models. Names of the derivative tables
are given in Table 1. Each file contains the following columns: line number,
effective temperature, $T_{\rm eff}$, logarithm of the surface gravity, $\log
g$, metallicity, $(Z/Z_{\odot})_{\rm atm}$, microturbulent velocity,
$\xi_{t}$, and the flux derivatives over effective temperature, $\alpha_{T}$,
and gravity, $\alpha_{g}$, in the $uvbyUBVRIJHK$ passbands.
In Fig. 1, we show the NLTE flux derivatives as a function of temperature for
two Strömgren passbands $uy$ and three values of gravity, $\log
g=3.5,4.0,4.5$. In the left panel, we plot the temperature derivative,
$\alpha_{T}$, and in the right one the gravity derivative, $\alpha_{g}$. We
assumed the solar metallicity, $(Z/Z_{\odot})_{\rm atm}$=1 and the
microturbulent velocity of $\xi_{t}$=2 km/s. As we can see the lower the
gravity the larger values of $\alpha_{T}$ and the lower values of
$\alpha_{g}$.
The wavelength dependence of the flux derivatives is presented in Fig. 2. We
considered the central wavelengths, $\lambda_{c}$, of the $UBVRIJHK$ passbands
and a model with $T_{\rm eff}=20000$ K and $\log g$ =4.0. The most steep
derivatives are in the ultraviolet filter and then the absolute values of
$\alpha_{T}$ and $\alpha_{g}$ decrease quite rapidly with $\lambda_{c}$. This
is because the early B-type stars emit the most amount of their energy in the
UV wavelength range. In Fig. 2, we compare also the NLTE derivatives with the
LTE ones. The largest difference can be seen again in the $U$ passband.
Effect of NLTE on the flux derivatives as a function of effective temperature
for the Strömgren $uy$ passbands is presented in Fig. 3. In general,
differences between the LTE and NLTE derivatives are relatively small and they
increase with effective temperature, especially in the case of $\alpha_{g}$ in
the $u$ passband. In the case of both model atmospheres, the values of
$\alpha_{T}$ in the $u$ passband change only slightly with the temperature,
whereas in the $y$ passband they are larger for higher $T_{\rm eff}$. The
gravity derivatives get more steep with the higher temperature, except
$\alpha_{g}^{u}(T_{\rm eff})$ for NLTE models, which is a decreasing function
up to $T_{\rm eff}\approx 24000$ K and then it increases.
Figure 1: The NLTE flux derivatives over $\log T_{\rm eff}$ (the left panel)
and $\log g$ (the right panel) as a function of temperature for the Strömgren
$uy$ passbands. Three values of $\log g=3.5,4.0,4.5$ were considered. The
values of the atmospheric metallicity and microturbulent velocity were assumed
as $(Z/Z_{\odot})_{\rm atm}$=1 and $\xi_{t}$=2 km/s, respectively. Figure 2: A
comparison of NLTE and LTE flux derivatives as a function of the central
wavelengths of the $UBVRIJHK$ passbands, for a models with $T_{\rm eff}$=20000
K, $\log g=4.0$, $(Z/Z_{\odot})_{\rm atm}$=1, $\xi_{t}$=2 km/s. The left and
right panel correspond to the temperature ($\alpha_{T}^{x}$) and gravity
($\alpha_{g}^{x}$) derivatives, respectively. Figure 3: Values of
$\alpha_{T}^{x}$ (the left panel) and $\alpha_{g}^{x}$ (the right panel) as a
function of temperature for LTE and NLTE models and the Strömgren $uy$
passbands. The remaining parameters of the model are: $\log g=4.0$,
$(Z/Z_{\odot})_{\rm atm}$=1, $\xi_{t}$=2 km/s.
### 3.2 Limb darkening
Knowledge of a distribution of the specific intensity over the stellar disc is
crucial in many fields of astrophysics. In the case of pulsating stars, limb-
darkening function and its derivatives over effective temperature and gravity,
are the second atmospheric data needed to compute theoretical values of the
photometric amplitudes and phases.
A usage of analytical formulae for the specific intensity distribution saves
enormously the computation time. The first limb-darkening law was proposed by
Milne (milne (1921)) in a linear form. With the development of models of
stellar atmospheres, it turned out that this approximation has a poor
accuracy. Thereafter, Klinglesmith & Sobieski (kinglesmith (1970)) published a
logarithmic law for early type stars. A quadratic formula for the limb-
darkening law was proposed by Manduca at al. (manduca (1977)) and Wade &
Ruciński (1985). For stars hotter than 8500 K, Diáz-Cordovés & Giménez (diaz
(1992)) suggested a square root law.
However, all these above formulas were not adequate for all types of stars. A
more general law was proposed by Claret (claret2000 (2000)) in the following
non-linear form
$\frac{I(\lambda,\mu)}{I(\lambda,1)}=1-\sum_{k=1}^{4}a^{\lambda}_{k}(1-\mu^{\frac{k}{2}})$
(8)
where $I(\lambda,\mu)$ is a specific intensity at the wavelength $\lambda$,
$\mu$ is a cosine of the angle between a line of sight and the emergent
intensity, and $I(\lambda,1)$ is the value at the center of the stellar disc.
Limb darkening coefficients, $a^{\lambda}_{k}$, are determined to reproduce
the model intensity distribution and to conserve the flux with a high accuracy
in the whole range of effective temperatures and gravities.
These coefficients were calculated for LTE model atmospheres in many
photometric systems and a wide range of effective temperatures, gravities,
metallicities and for several values of the microturbulent velocities (Claret
2000, 2003, Claret & Hauschildt 2003, Claret 2008). A comparison of different
numerical methods of computations of limb darkening coefficients (LDC) were
discussed in detail by Díaz-Cordovés et al. (diaz1995 (1995)), Claret
(claret2000 (2000)), Heyrovský (heyrovsky (2007)) and Claret (claret2008
(2008)).
In this paper, we determine non-linear LDC for metal line-blanketed, NLTE,
plane-parallel, hydrostatic model atmospheres of Lanz & Hubeny (lanzhubeny
(2007)). In the first step, we computed monochromatic specific intensities for
20 equally separated points of $\mu$ in the range of $\left<0.001,1\right>$
and in the wavelength range of (2950,26500) $\AA$, for all parameters of the
BSTAR2006 atmospheres. All calculations were performed using the SYNSPEC
program (Hubeny at al. synspec2 (1985), Hubeny & Lanz synspec (2000)). This
code is intended to compute specific intensities and fluxes from the model
atmosphere input, which we took from NLTE models described above.
Subsequently, for each disc position, $\mu$, we integrated intensities over
the $uvby$ and $UBVRIJHK$ bands
$I_{x}(\mu)=\frac{\int\limits^{\lambda_{1}}_{\lambda_{2}}I(\lambda,\mu)S(\lambda)\,d\lambda}{\int\limits^{\lambda_{1}}_{\lambda_{2}}S(\lambda)\,d\lambda},$
(9)
where $I_{x}(\mu)$ is the specific intensity in the passband $x$ and
$I(\lambda,\mu)$ is the monochromatic specific intensity obtained by SYNSPEC.
The response function, $S(\lambda)$, was interpolated to the wavelengths of
computed monochromatic specific intensity by the cubic spline method.
The non-linear limb darkening coefficients, $a_{k}^{x}$, were determined using
the least squares method by minimizing
$\chi^{2}(a_{1},a_{2},a_{3},a_{4})=\sum_{i=1}^{20}\left(I_{LDC_{i}}-I_{i}\right)^{2},$
(10)
where $I_{LDC_{i}}$ is the specific intensity computed with the Claret
nonlinear law and $I_{i}$ is the corresponding model intensity, at the point
$\mu_{i}$.
Similarly to Heyrovský (heyrovsky (2007)), we evaluated the quality of our
fits by the relative residuals
$\sigma^{2}=\frac{\sum\limits_{i=1}^{20}\left(I_{i}-I_{LDC_{i}}\right)^{2}}{\sum\limits_{i=1}^{N}I_{i}^{2}},$
(11)
whereas, the conservation of the flux was controlled by computing the relative
flux excess
$\frac{\Delta{\cal F}_{x}}{{\cal
F}_{x}}=\frac{\int\limits_{0}^{1}I_{x}\mu\,d\mu-\int\limits_{0}^{1}I_{x,LDC}\mu\,d\mu}{\int\limits_{0}^{1}I_{x}\mu\,d\mu}.$
(12)
The limb darkening law, $h(\mu)$, given in Eq. 6, is defined as (Daszyńska-
Daszkiewicz et al. 2002)
$h_{x}(\mu)=2\pi\leavevmode\nobreak\ \frac{I_{x}(\mu)}{{\cal
F}_{x}}=2\pi\leavevmode\nobreak\
\frac{1-\sum\limits_{k=1}^{4}a_{k}^{x}(1-\mu^{k/2})}{1-\sum\limits_{k=1}^{4}\frac{k}{k+4}a_{k}^{x}}.$
(13)
Names of the LDC files are given Table 1 and they are coded in the same way as
the flux and flux derivatives tables (cf. Sect. 3.1). Each file contains the
following columns: names of LDC, $a_{k}$, effective temperature, $T_{\rm
eff}$, gravity, $\log g$, metallicity, $(Z/Z_{\odot})_{\rm atm}$,
mictroturbulent velocity, $\xi_{t}$, and values of LDC in the Strömgren and
Johnson-Cousins-Glass photometric passbands, in the order: $uvbyUBVRIJHK$.
In Fig. 4 we show distributions of the normalized specific intensity in the
$UBVRIJHK$ passbands for a model with the following parameters: $T_{\rm
eff}$=20000 K, log g=4.0, $\xi_{t}$=2 km/s, $(Z/Z_{\odot})_{\rm atm}$=1. In
this figure, we compare the specific intensity computed by means of SYNSPEC
for 20 equally separated points of $\mu$ (squares) and the fitted limb
darkening function defined by Eq. 8 (solid lines).
Figure 4: The angular distribution of the normalized specific intensity for
the $UBVRIJHK$ passbands and the NLTE model with $T_{\rm eff}$=20000 K, $\log
g$=4.0, $\xi_{t}$=2 km/s and $(Z/Z_{\odot})_{\rm atm}$=1. The actual model
intensities are marked as squares and intensities computed from LDC as solid
lines. For clarity, passbands from $B$ are shifted downward by 0.1.
Intensities for passbands $BVRIJHK$ were shifted downward by $n\cdot 0.1$,
where n=1,2,…,7. As we can see a quality of the limb darkening fit is very
accurate.
To compare values of the NLTE and LTE intensities, in Fig. 5 we plot $I_{x}$
[erg$\cdot$cm${}^{-2}\cdot$s${}^{-1}\cdot$ster-1] as a function of the angle
$\mu$ for the Strömgren $uvby$ passbands. The same model as in Fig. 4 was
considered. The NLTE intensities get lower values, in particular, for the $u$
passband. This is caused by the location of the $u$ filter on the Balmer
continuum where the difference in the amount of energy radiated in LTE and
NLTE models is more pronounced than in a region to the right from the Balmer
jump, where other filters ($vby$) are defined.
Figure 5: The intensity distribution for the $uvby$ passbands for the NLTE
(solid lines) and LTE (dashed lines) models with $T_{\rm eff}$=20000 K, $\log
g$=4.0, $\xi_{t}$=2 km/s and $(Z/Z_{\odot})_{\rm atm}$=1.
In Fig. 6 and 7, we plot the NLTE limb darkening coefficients,
$a_{1},a_{2},a_{3},a_{4}$, as a function of $T_{\rm eff}$ in the Strömgren $u$
passband, for different values of gravity, $\log g$, and metallicity,
$(Z/Z_{\odot})_{\rm atm}$, respectively. All panels have the same scale. As we
can see, there is a strong dependence of LDC on effective temperature, gravity
and metallicity. The sensitivity on $T_{\rm eff}$ is stronger for lower values
of $\log g=3.5$. The sensitivity to metallicity is similar for all LDC. The
effect of the microturbulent velocity, $\xi_{t}$, not shown here, is
comparable to the effect of metallicity.
Figure 6: The NLTE limb darkening coefficients in the $u$ passband as a
function of $T_{\rm eff}$ for three values of gravity: $\log=3.5$ (solid
line), $\log=4.0$ (dashed line) and $\log=4.5$ (dotted line). The solar
metallicity, $(Z/Z_{\odot})_{\rm atm}$=1, and microturbulent velocity of
$\xi_{t}$=2 km/s were assumed. Figure 7: The NLTE limb darkening coefficients
in the $u$ passband as a function of $T_{\rm eff}$ for six values of the
atmospheric metallicity: $(Z/Z_{\odot})_{\rm atm}$=2.0 (solid black line), 1.0
(dashed black line), 0.5 (dotted black line), 0.2 (solid grey line), 0.1
(dashed grey line) and 0.0 (dotted grey line). The gravity of $\log g$=4.0 and
microturbulent velocity of $\xi_{t}$=2 km/s were assumed.
Let us now discuss the accuracy of our determinations of the Claret limb-
darkening coefficients for the BSTAR2006 NLTE models. In Table. 2, we give the
average and maximum values of the merit functions, i.e. the relative
residuals, $\sigma$, and relative flux excess, $\left|\Delta F\over F\right|$.
These values were calculated for the whole range of $T_{\rm eff}$, i.e.
(15000,30000) K, and $\log g$, i.e. (1.75, 4.75) dex, assuming the solar
metallicity, $(Z/Z_{\odot})_{\rm atm}$=1, and the microturbulent velocity of
$\xi_{t}$=2 km/s. We used 20 equally spaced points of $\mu$ instead of 17
unequally spaced points as used in LTE Kurucz models. Moreover, our NLTE
values of $I(\mu)$ were computed closer to the stellar limb, i.e. up to
$\mu=0.001$, whereas the lowest angle of $\mu$ in LTE models is 0.01. We
considered two sets of passbands: $uvbyUBVRIJHK$ and $BVRI$. The second set
was used to compare our results with those of Heyrovský (heyrovsky (2007)) who
made computations for LTE Kurucz atmosphere models using 17 and 11 points of
$\mu$. As we can see the quality of our fit is very good. The average value of
$\sigma$ amounts to 0.119% and for the worst fitted profile we got
$\sigma$=0.432%. Our average value of $\sigma$ is smaller than the Heyrovský’s
one by a factor of two. When we limited our analysis to the $BVRI$ passbans,
as in Heyrovský (heyrovsky (2007)), the result is even better. In this case,
the average and maximum values of $\sigma$ amount to 0.089% and 0.251%,
respectively. This indicates that with the larger number of equally separated
points of $\mu$ in the fitting procedure one reproduces more accurately the
model intensities.
Table 2: Comparison of our limb-darkening fit quality with results of
Heyrovsky (2007) for the Claret non-linear law. In columns 2 and 3, we give
the average and maximum values of the relative rms residual, $\sigma$,
respectively. Columns 4 and 5 contain the average and maximum values of the
relative flux excess, $\left|\Delta F\over F\right|$. Passbands for which
these goodness-of-fit quantities were evaluated are given in notes.
Method | Average $\sigma$ | Max. $\sigma$ | Average $\left|\Delta F\over F\right|$ | Max. $\left|\Delta F\over F\right|$
---|---|---|---|---
| % | % | |
20-pointa | 0.119 | 0.432 | $14.5\times 10^{-5}$ | $9.10\times 10^{-4}$
20-pointb | 0.089 | 0.251 | $9.50\times 10^{-5}$ | $5.28\times 10^{-4}$
17-pointc | 0.190 | 0.625 | $8.27\times 10^{-5}$ | $5.05\times 10^{-4}$
11-pointc | 0.175 | 0.558 | $4.57\times 10^{-5}$ | $2.17\times 10^{-4}$
222 a values evaluated for all 12 passbands: $uvbyUBVRIJHK$. b values
evaluated for the $BVRI$ passbands. c values from Heyrovský (heyrovsky (2007))
for the $BVRI$ passbands.
Finally, let us check the conservation of the flux. The average value of
$\left|\Delta F\over F\right|$ from our fitting for NLTE models is slightly
worse than in Heyrovský (heyrovsky (2007)) for LTE models atmospheres, but of
the same order of magnitude. The maximum values of $\left|\Delta F\over
F\right|$ are $9.10\times 10^{-4}$ for all 12 passbnds and $5.28\times
10^{-4}$ for $BVRI$. The corresponding average values are $14.5\times 10^{-5}$
and $9.50\times 10^{-5}$, respectively. For larger values of gravities, the
fluxes computed with the Claret limb darkening law are overestimated whereas
for smaller values of $\log g$, fluxes are underestimated. This is caused by a
steeper slope of the intensity near the limb for lower values of $\log g$.
Consequently, the fitted limb darkening for lower gravities is slightly below
model intensities.
## 4 Uncertainties in photometric pulsational observables from model
atmospheres
Figure 8: A comparison of the NLTE and LTE values of the amplitude ratios,
$A_{u}/A_{y}$ (the left panel) and phase differences,
$\varphi_{u}-\varphi_{y}$ (the right panel) for 10 $M_{\odot}$ main sequence
models as a function of $T_{\rm eff}$ for the first three radial modes: p1,
p2, p3. The atmospheric metallicity of $(Z/Z_{\odot})_{\rm atm}=1$ and
microturbulent velocity of $\xi_{t}=2$ km/s were assumed. Linear nonadiabatic
pulsations were computed with hydrogen and metal abundance of $X=0.7$ and
$Z=0.02$, respectively, the OPAL opacities and the AGSS09 chemical mixture.
Figure 9: The same as in Fig. 8 but for two $\ell=1$ modes: p1 and g14. Figure
10: The same as in Fig. 8 for for two $\ell=2$ modes: p1 and g22. Figure 11:
Effect of the atmospheric metallicity on the amplitude ratios, $A_{u}/A_{y}$
(the left panel) and phase differences, $\varphi_{u}-\varphi_{y}$ (the right
panel) for 10 $M_{\odot}$ main sequence models as a function of $T_{\rm eff}$
for three radial modes: p1, p2, p3. NLTE model atmospheres with the
microturbulent velocity of $\xi_{t}=2$ km/s and two values of
$(Z/Z_{\odot})_{\rm atm}$ were used.
The atmospheric metallicity, $(Z/Z_{\odot})_{\rm atm}$, microturbulent
velocity, $\xi_{t}$, and effects of NLTE are the most important factors which
can affect the photometric amplitudes and phases of a pulsating star.
In all comparisons, we used a reference model computed with the mass of
$M=10M_{\odot}$, the OPAL opacity tables, the AGSS09 mixture, hydrogen
abundance of $X$=0.7, metal abundance of $Z$=0.02, without overshooting from a
convective core, $\alpha_{\rm ov}$=0.0, and NLTE-TLUSTY models of stellar
atmospheres with the metallicity of $(Z/Z_{\odot})_{\rm atm}$=1.0 and the
microturbulent velocity of $\xi_{t}$=2 km/s.
In Fig. 8, we compare the photometric observables computed with LTE and NLTE
model atmospheres at the same values of metallicity, $(Z/Z_{\odot})_{\rm
atm}=1$, and microturbulent velocity, $\xi_{t}=2$ km/s, for the $10M_{\odot}$
model in the course of its main sequence evolution. We considered the
Strömgren $uy$ passbands and the first three radial modes: p1, p2, p3. In the
left panel, we show the amplitude ratios, $A_{u}/A_{y}$, and in the right one
the corresponding phase differences, $\varphi_{u}-\varphi_{y}$, as a function
of $T_{\rm eff}$. As we can see, the fundamental mode is most sensitive to the
NLTE effects. In general, the NLTE values of the amplitude ratio,
$A_{u}/A_{y}$, are smaller than the LTE ones for the hotter models and larger
for more evolved models.
Subsequently we checked the NLTE effect for nonradial modes. In Fig. 9, we
plot the same photometric observables as in Fig. 8, but for the $\ell=1$ mode
considering one pressure mode, p1, and one high-order gravity mode, g14. Fig.
10 illustrates the same but for two $\ell=2$ modes: p1 and g22. In the case of
nonradial modes, the amplitude ratios, $A_{u}/A_{y}$, have larger values for
NLTE computations. We can see also a different behavior and values of
$A_{u}/A_{y}$ and $\varphi_{u}-\varphi_{y}$ for pressure and high-order
gravity modes, especially for the $\ell=2$ modes.
Effect of the atmospheric metallicity on the photometric observables for the
radial modes are shown in Fig. 11. Here, we considered $(Z/Z_{\odot})_{\rm
atm}=0.1$ in addition to our standard value of $(Z/Z_{\odot})_{\rm atm}=1.0$.
As we can see, the effect of the atmospheric metallicity is comparable to the
effect of the departure from LTE in stellar atmospheres. In the case of
nonradial modes the effect of $(Z/Z_{\odot})_{\rm atm}$ is negligible.
Influence of the microturbulent velocity, $\xi_{t}$, is of the same order as
$(Z/Z_{\odot})_{\rm atm}$, as has been shown by Szewczuk & Daszyńska-
Daszkiewicz (2010).
## 5 Uncertainties in pulsational photometric observables from pulsation
theory
Linear nonadiabatic theory of stellar pulsation provides eigenfrequencies,
corresponding eigenfunctions and information on mode excitation. The value of
the flux eigenfunction at the level of the photosphere enters the expression
for the light variation and it is called the $f$-parameter, as it has been
introduced in Sect. 2. For a given mode frequency, this parameter depends, in
general, on mean stellar parameters, chemical composition, opacity data,
subphotospheric convection etc. Moreover, in the case of high-order gravity
modes, the $f$-parameter depends also on the mode degree, $\ell$ (e.g.
Daszyńska-Daszkiewicz & Walczak 2010). There are also large differences in
values of $f$ between high and low frequency modes of a given degree, $\ell$.
This is why the amplitude ratios and phase differences for p- and high-order
g-modes behave so different, which is exemplified most pronounced in Fig. 10.
Figure 12: Effects of the internal abundances of hydrogen, $X$, and metals,
$Z$, on photometric observables for the first three radial modes as a function
of $T_{\rm eff}$ for the 10 $M_{\odot}$ main sequence models. The NLTE models
with the atmospheric metallicity of $(Z/Z_{\odot})_{\rm atm}=1$ and
microturbulent velocity of $\xi_{t}=2$ km/s were used. Linear nonadiabatic
pulsations were computed with OPAL opacity tables and AGSS09 chemical mixture.
Figure 13: The same as in Fig. 12 but for two $\ell=1$ modes: p1 and g14.
Figure 14: The same as in Fig. 12 but for two $\ell=2$ modes: p1 and g22.
Figure 15: Effects of the opacity tables and chemical mixture on photometric
observables for the first three radial modes as a function of $T_{\rm eff}$
for the 10 $M_{\odot}$ main sequence models. The NLTE models with the
metallicity of $(Z/Z_{\odot})_{\rm atm}=1$ and microturbulent velocity of
$\xi_{t}=2$ km/s were used. Linear nonadiabatic pulsations were computed with
$X=0.7$ and $Z=0.02$. Figure 16: The same as in Fig. 15 but for two $\ell=1$
modes: p1 and g14. Figure 17: The same as in Fig. 15 but for two $\ell=2$
modes: p1 and g22.
Pulsations of the B-type stars are driven by the $\kappa$ mechanism operating
in the ,,Z-bump” layer. The nonadiabatic complex parameter, $f$, is also
defined in this subphotospheric layer, therefore its value strongly depends on
metal abundances, chemical mixture and hence opacities (Daszyńska-Daszkiewicz,
Dziembowski, Pamyatnykh 2005).
In Fig. 12, we show effects of the internal abundances of hydrogen, $X$, and
metals, $Z$, on the photometric amplitude ratios, $A_{u}/A_{y}$, and phase
differences, $\varphi_{u}-\varphi_{y}$, for the first three radial modes: p1,
p2, p3. The same model as in Sect. 4 was selected. Here, we compare
computations obtained with $Z$=0.02 $vs.$ $Z$=0.03 and $X$=0.7 $vs.$ $X$=0.75.
As we can see, the amplitude ratios computed with $Z$=0.03 are larger than
those obtained with $Z$=0.02, whereas increasing hydrogen abundance, $X$,
decreases the amplitude ratios. The value of $\varphi_{u}-\varphi_{y}$ changes
by about 0.1 rad at most. These effects are particulary distinct for the
radial fundamental mode, p1. Comparing Fig. 12 with Fig. 8 and 11, we can see
also that the effects of $X$ and $Z$ on the photometric observables are much
more pronounced than the effects of NLTE and $(Z/Z_{\odot})_{\rm atm}$. Please
note that for a given mode degree, $\ell$, scales of the amplitude ratios and
phases differences are the same in all figures.
Effects of the hydrogen and metallicity abundance for the $\ell=1$, p1 and
$g_{14}$ modes is shown in Fig. 13, and for the $\ell=2$, p1 and g22 modes in
Fig. 14. As one can see, in the case of the nonradial modes these parameters
have smaller influence on the photometric observables than in the case of
radial modes. The amplitude ratio, $A_{u}/A_{y}$, for the $\ell=1$ mode
increases with increasing abundances of both $Z$ and $X$. This is true also
for the $\ell=2$, p1 mode, whereas the amplitude ratio of the $\ell=2$, g22
mode does not behave monotonically. The important result is that in the case
of nonradial modes the effects of $X$ and $Z$ are considerable smaller than
the NLTE effects (cf. Fig. 9 and 10), which is opposite to the radial mode
case.
Finally, we evaluated effects of the opacity data and chemical mixture. In
Fig. 15 we show the photometric observables for the radial modes computed with
two sources of the opacity tables and two chemical mixtures. Here, we compared
computations obtained with the OPAL $vs.$ OP tables and the GN93 $vs.$ AGSS09
mixture. Again, the largest effects are for the p1 mode in both the amplitude
ratios and phase differences. The amplitude ratios computed with GN93 are
smaller than those obtained with AGSS09, because of relatively higher
abundance of iron in AGSS09. Also computations with the OP data give smaller
values of $A_{u}/A_{y}$ comparing to those obtained with the OPAL data.
Moreover, the maximum of the OP amplitude ratio is shifted towards higher
effective temperature. This is connected with the location of the $Z-$bump
layer, which occurs at slightly higher temperature in the OP data. A usage of
different opacity data changes the value of $\varphi_{u}-\varphi_{y}$ by about
0.2 rad at most. The chemical mixture has rather a minor effect on phase
differences. We can see that also these input data affect the photometric
observables of radial modes far more than the atmospheric parameters (cf. Fig.
8 and 11).
Effects of the opacity tables and chemical mixture for nonradial modes with
$\ell=1$ and 2 are shown in Fig. 16 and Fig. 17, respectively. These data
affect the $\ell=1$ modes very subtle and are far less important than the NLTE
effects (cf. Fig. 9). In the case of the $\ell=2$ modes effects of opacities
and mixture are comparable to the NLTE effects (cf. Fig. 10).
As we could see, generally, in the case of radial modes effects of pulsational
parameters on photometric observables are much larger than effects of
atmospheric parameters discussed in previous section. The opposite is true for
nonradial modes which are more sensitive to the NLTE effects, in particular in
the case of dipole modes.
## 6 Conclusions
We have presented a comprehensive overview of possible sources of
uncertainties in theoretical values of the photometric amplitudes and phases
of early B-type pulsators. These data are of particular importance because
they serve as tools for mode identification in the case of main sequence
pulsators, as well as contain information on stellar physics.
The uncertainties are embedded in stellar model atmospheres and nonadiabatic
theory of stellar pulsation. The atmospheric input consists of the flux
derivatives over $T_{\rm eff}$ and $\log g$, and limb darkening and its
derivatives, in photometric passbands. These quantities are sensitive to the
atmospheric metallicity and microturbulent velocity, as well as to the
departure from the LTE approximation. From pulsation computations, we get the
nonadiabatic complex parameter, $f$, whose value depends on chemical
composition, opacities and subphotospheric convection if present. If the
$f$-parameter can be derived from observations then we get an extra seismic
probe by means of which we can test this input physics.
We begun with computations of tables with various quantities, needed to
evaluate the light variation in a given photometric band, for NLTE-TLUSTY
model atmospheres. These tables include data on the passband flux, flux
derivatives over effective temperature and gravity, and the non-linear limb
darkening coefficients in 12 most popular passbands: $uvbyUBVRIJHK$. All these
data are public available and can be retrieved from the Wrocław HELAS web
page.
Then, we studied effects of these input parameters on the photometric
observables, i.e. amplitude ratios and phase differences. We considered the 10
$M_{\odot}$ main sequence models and the low degree modes with $\ell=0,1,2$.
In particular, the effect of NLTE model atmospheres was studied for the first
time. In the case of radial modes, this effect is comparable to an influence
of the atmospheric metalicity and microturbulent velocity. The photometric
observables of nonradial modes appeared more sensitive to the departure from
LTE. Subsequently, we drew a parallel to effects of parameters related to
pulsation computations. These comparisons showed that the photometric
observables of the radial modes are by far more sensitive to the pulsation
input. In turn, in the case of nonradial modes the NLTE effect became more
important, especially for the $\ell=1$ modes.
A complete seismic model should reproduce not only pulsational frequencies but
also the observed values of the photometric amplitude and phases, which can be
translated into the empirical values of the $f$-parameter. Our studies showed
that such comprehensive approach should take into account all inaccuracies in
the photometric amplitudes and phases.
###### Acknowledgements.
This work was supported by the HELAS EU Network, FP6, No. 026138 and for JDD
by the Polish MNiSW grant N N203 379636.
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|
arxiv-papers
| 2010-09-02T11:49:25 |
2024-09-04T02:49:12.645536
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Daszy\\'nska-Daszkiewicz, W. Szewczuk",
"submitter": "Daszynska-Daszkiewicz Jadwiga",
"url": "https://arxiv.org/abs/1009.0398"
}
|
1009.0410
|
GENERALIZED NEWTON’S METHOD
BASED ON GRAPHICAL DERIVATIVES
T. HOHEISEL111Institute of Applied Mathematics and Statistics, University of
Würzburg, Am Hubland, 97074 Würzburg, Germany (hoheisel@mathematik.uni-
wuerzburg.de, kanzow@mathematik.uni-wuerzburg.de)., C. KANZOW11footnotemark:
1, B. S. MORDUKHOVICH222Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA (boris@math.wayne.edu, pmhung@wayne.edu). The research
of these authors was partially supported by the US National Science Foundation
under grants DMS-0603846 and DMS-1007132 and by the Australian Research
Council under grant DP-12092508. and H. PHAN22footnotemark: 2
Abstract. This paper concerns developing a numerical method of the Newton type
to solve systems of nonlinear equations described by nonsmooth continuous
functions. We propose and justify a new generalized Newton algorithm based on
graphical derivatives, which have never been used to derive a Newton-type
method for solving nonsmooth equations. Based on advanced techniques of
variational analysis and generalized differentiation, we establish the well-
posedness of the algorithm, its local superlinear convergence, and its global
convergence of the Kantorovich type. Our convergence results hold with no
semismoothness assumption, which is illustrated by examples. The algorithm and
main results obtained in the paper are compared with well-recognized
semismooth and $B$-differentiable versions of Newton’s method for nonsmooth
Lipschitzian equations.
Key words. nonsmooth equations, optimization and variational analysis,
Newton’s method, graphical derivatives and coderivatives, local and global
convergence
AMS subject classification. 49J53, 65K15, 90C30
## 1 Introduction
Newton’s method is one of the most powerful and useful methods in optimization
and in the related area of solving systems of nonlinear equations
(1.1) $H(x)=0$
defined by continuous vector-valued mappings $H\colon I\\!\\!R^{n}\to
I\\!\\!R^{n}$. In the classical setting when $H$ is a continuously
differentiable (smooth, $C^{1}$) mapping, Newton’s method builds the following
iteration procedure
(1.2) $x^{k+1}:=x^{k}+d^{k}\;\mbox{ for all }\;k=0,1,2,\ldots,$
where $x^{0}\in I\\!\\!R^{n}$ is a given starting point, and where $d^{k}\in
I\\!\\!R^{n}$ is a solution to the linear system of equations (often called
“Newton equation”)
(1.3) $H^{\prime}(x^{k})d=-H(x^{k}).$
A detailed analysis and numerous applications of the classical Newton’s method
(1.2), (1.3) and its modifications can be found, e.g., in the books [7, 14,
26] and the references therein.
However, in the vast majority of applications—including those to optimization,
variational inequalities, complementarity and equilibrium problems, etc.—the
underlying mapping $H$ in (1.1) is nonsmooth. Indeed, the aforementioned
optimization-related models and their extensions can be written via Robinson’s
formalism of “generalized equations,” which in turn can be reduced to standard
equations of the form above (using, e.g., the projection operator) while with
intrinsically nonsmooth mappings $H$; see [8, 19, 33, 29] for more details,
discussions, and references.
Robinson originally proposed (see [32] and also [34] based on his earlier
preprint) a point-based approximation approach to solve nonsmooth equations
(1.1), which then was developed by his student Josephy [11] to extend Newton’s
method for solving variational inequalities and complementarity problems.
Other approaches replace the classical derivative $H^{\prime}(x_{k})$ in the
Newton equation (1.3) by some generalized derivatives. In particular, the
$B$-differentiable Newton method developed by Pang [27, 28] uses the iteration
scheme (1.2) with $d^{k}$ being a solution to the subproblem
(1.4) $H^{\prime}(x^{k};d)=-H(x^{k}),$
where $H^{\prime}(x^{k};d)$ denotes the classical directional derivative of
$H$ at $x^{k}$ in the direction $d$. Besides the existence of the classical
directional derivative in (1.4), a number of strong assumptions are imposed in
[27, 28] to establish appropriate convergence results; see Section 5 below for
more discussions and comparisons.
In another approach developed by Kummer [16] and Qi and Sun [31], the
direction $d^{k}$ in (1.2) is taken as a solution to the linear system of
equations
(1.5) $A_{k}d=-H(x^{k}),$
where $A_{k}$ is an element of Clarke’s generalized Jacobian
$\partial_{C}H(x_{k})$ of a Lipschitz continuous mapping $H$. In [30], Qi
suggested to replace $A_{k}\in\partial_{C}H(x^{k})$ in (1.5) by the choice of
$A_{k}$ from the so-called $B$-subdifferential $\partial_{B}H(x^{k})$ of $H$
at $x^{k}$, which is a proper subset of $\partial_{C}H(x^{k})$; see Section 4
for more details. We also refer the reader to [8, 15, 34] and bibliographies
therein for wide overviews, historical remarks, and other developments on
Newton’s method for nonsmooth Lipschitz equations as in (1.1) and to [13] for
some recent applications.
It is proved in [31] and [30] that the Newton type method based on
implementing the generalized Jacobian and $B$-subdifferential in (1.5),
respectively, superlinearly converges to a solution of (1.1) for a class of
semismooth mappings $H$; see Section 4 for the definition and discussions.
This subclass of Lipschitz continuous and directionally differentiable
mappings is rather broad and useful in applications to optimization-related
problems. However, not every mapping arising in applications (from both
theoretical and practical viewpoints) is either directionally differentiable
or Lipschitz continuous. The reader can find valuable classes of functions and
mappings of this type in [24, 35] and overwhelmingly in spectral function
analysis, eigenvalue optimization, studying of roots of polynomials, stability
of control systems, etc.; see, e.g., [4] and the references therein.
The main goal and achievements of this paper are as follows. We propose a new
Newton-type algorithm to solve nonsmooth equations (1.1) described by general
continuous mappings $H$ that is based on graphical derivatives. It reduces to
the classical Newton method (1.3) when $H$ is smooth, being different from
previously known versions of Newton’s method in the case of Lipschitz
continuous mappings $H$. Based on advanced tools of variational analysis
involving metric regularity and coderivatives, we justify well-posedness of
the new algorithm and its superlinear local and global (of the Kantorovich
type) convergence under verifiable assumptions that hold for semismooth
mappings but are not restricted to them. Detailed comparisons of our algorithm
and results with the semismooth and $B$-differentiable Newton methods are
given and certain improvements of these methods are justified.
Note metric regularity and related concepts of variational analysis has been
employed in the analysis and justification of numerical algorithms starting
with Robinson’s seminal contribution; see, e.g., [1, 18, 25] and their
references for the recent account. However, we are not familiar with any usage
of graphical derivatives and coderivatives for these purposes.
The rest of the paper is organized as follows. In Section 2 we present basic
definitions and preliminaries from variational analysis and generalized
differentiation widely used for formulations and proofs of the main results.
Section 3 is devoted to the description of the new generalized Newton
algorithm with justifying its well-posedness/solvability and establishing its
superlinear local and global convergence under appropriate assumptions on the
underlying mapping $H$.
In Section 4 we compare our algorithm with the scheme of (1.5). We also
discuss in detail the major assumptions made in Section 3 deriving sufficient
conditions for their fulfillment and comparing them with those in the
semismooth Newton methods.
Section 5 contains applications of our algorithm to the $B$-differentiable
Newton method (1.4) with largely relaxed assumptions in comparison with known
ones. In Section 6 we give some concluding remarks and discussions on further
research.
Our notation is basically standard in variational analysis and numerical
optimization; cf. [8, 24, 35]. Recall that, given a set-valued mapping
$F\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$, the expression
(1.8) $\displaystyle\begin{array}[]{ll}\displaystyle\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}F(x):=\Big{\\{}y\in
I\\!\\!R^{m}\Big{|}&\exists\,x_{k}\to\bar{x}\;\mbox{ and }\;y_{k}\to y\;\mbox{
as }\;k\to\infty\;\mbox{ with}\\\ &y_{k}\in F(x_{k})\;\mbox{ for all }\;k\in
I\\!\\!N:=\\{1,2,\ldots\\}\Big{\\}}\end{array}$
defines the Painlevé-Kuratowski upper/outer limit of $F$ as $x\to\bar{x}$. Let
us also mention that the symbols ${\rm cone}\,\Omega$ and ${\rm co}\,\Omega$
stand, respectively, for the conic hull and convex hull of the set in
question, that ${\rm dist}(x;\Omega)$ denotes the Euclidean distance between a
point $x\in I\\!\\!R^{n}$ and a set $\Omega$, and that the notation $A^{T}$
signifies the matrix transposition. As usual, $B_{\varepsilon}(\bar{x})$
stands for the closed ball centered at $\bar{x}$ with radius $\varepsilon>0$.
## 2 Tools of Variational Analysis
In this section we briefly review some constructions and results from
variational analysis and generalized differentiation widely used in what
follows. The reader may consult the texts [3, 24, 35, 36] for more details and
additional material.
Given a nonempty set $\Omega\subset I\\!\\!R^{n}$ and a point
$\bar{x}\in\Omega$, the (Bouligand-Severi) tangent/contingent cone to $\Omega$
at $\bar{x}$ is defined by
(2.1) $T(\bar{x};\Omega):=\displaystyle\mathop{{\rm Lim}\,{\rm
sup}}_{t\downarrow 0}\frac{\Omega-\bar{x}}{t}$
via the outer limit (1.8). This cone is often nonconvex while its polar/dual
cone
(2.2) $\widehat{N}(\bar{x};\Omega):=\big{\\{}p\in I\\!\\!R^{n}\big{|}\;\langle
p,u\rangle\leq 0\;\mbox{ for all }\;u\in T(\bar{x};\Omega)\big{\\}}$
is always convex and can be intrinsically described by
$\widehat{N}(\bar{x};\Omega)=\Big{\\{}p\in
I\\!\\!R^{n}\Big{|}\;\limsup_{x\stackrel{{\scriptstyle\Omega}}{{\to}}\bar{x}}\frac{\langle
p,x-\bar{x}\rangle}{\|x-\bar{x}\|}\leq 0\Big{\\}},\quad\bar{x}\in\Omega,$
where the symbol $x\stackrel{{\scriptstyle\Omega}}{{\to}}\bar{x}$ signifies
that $x\to\bar{x}$ with $x\in\Omega$. The construction (2.2) is known as the
prenormal cone or the Fréchet/regular normal cone to $\Omega$ at
$\bar{x}\in\Omega$. For convenience we put
$\widehat{N}(\bar{x};\Omega)=\emptyset$ if $\bar{x}\notin\Omega$. Observe that
the prenormal cone (2.2) may not have natural properties of generalized
normals in the case of nonconvex sets $\Omega$; e.g., it often happens that
$\widehat{N}(\bar{x};\Omega)=\\{0\\}$ when $\bar{x}$ is a boundary point of
$\Omega$ and the cone (2.2) does not possesses required calculus rules. The
situation is dramatically improved when we consider a robust regularization of
(2.2) via the outer limit (1.8) and arrive at the construction
(2.3) $N(\bar{x};\Omega):=\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}\widehat{N}(x;\Omega)$
known as the (limiting, basic, Mordukhovich) normal cone to $\Omega$ at
$\bar{x}\in\Omega$. If $\Omega$ is locally closed around $\bar{x}$, the basic
normal cone (2.3) can be equivalently described as
$N(\bar{x};\Omega)=\mathop{{\rm Lim}\,{\rm sup}}_{x\to\bar{x}}\big{[}{\rm
cone}\big{(}x-\Pi(x;\Omega)\big{)}\big{]},\quad\bar{x}\in\Omega,$
via the Euclidean projector $\Pi(\cdot;\Omega)$ on $\Omega$; this was in fact
the original definition of the normal cone in [21]. Despite its nonconvexity,
the normal cone (2.3) and the corresponding subdifferential and coderivative
constructions for extended-real-valued functions and set-valued mappings enjoy
comprehensive calculus rules, which are particularly based on
variational/extremal principles of variational analysis.
Consider next a set-valued mapping $F\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$ with the graph
$\mbox{\rm gph}\,F:=\big{\\{}(x,y)\in I\\!\\!R^{n}\times
I\\!\\!R^{m}\big{|}\;y\in F(x)\big{\\}}$
and define the graphical derivative and coderivative constructions generated
by the tangent and normal cones, respectively. Given
$(\bar{x},\bar{y})\in\mbox{\rm gph}\,F$, the graphical/contingent derivative
of $F$ at $(\bar{x},\bar{y})$ is introduced in [2] as a mapping
$DF(\bar{x},\bar{y})\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$ with the values
(2.4) $DF(\bar{x},\bar{y})(z):=\big{\\{}w\in I\\!\\!R^{m}\big{|}\;(z,w)\in
T\big{(}(\bar{x},\bar{y});\mbox{\rm gph}\,F\big{)}\big{\\}},\quad z\in
I\\!\\!R^{n},$
defined via the contingent cone (2.1) to the graph of $F$ at the point
$(\bar{x},\bar{y})$; see [3, 35] for various properties, equivalent
representation, and applications. The coderivative of $F$ at
$(\bar{x},\bar{y})\in\mbox{\rm gph}\,F$ is introduced in [22] as a mapping
$D^{*}F(\bar{x},\bar{y})\colon I\\!\\!R^{m}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{n}$ with the values
(2.5) $D^{*}F(\bar{x},\bar{y})(v):=\big{\\{}u\in
I\\!\\!R^{n}\big{|}\;(u,-v)\in N\big{(}(\bar{x},\bar{y});\mbox{\rm
gph}\,F\big{)}\big{\\}},\quad v\in I\\!\\!R^{m},$
defined via the normal cone (2.3) to the graph of $F$ at $(\bar{x},\bar{y})$;
see [24, 35] for extended calculus and a variety of applications. We drop
$\bar{y}$ in the graphical derivative and coderivative notation when the
mapping in question is single-valued at $\bar{x}$. Note that the graphical
derivative and coderivative constructions in (2.4) and (2.5) are not dual to
each other, since the basic normal cone (2.3) is nonconvex and hence cannot be
tangentially generated.
In this paper we employ, together with (2.4) and (2.5), the following modified
derivative construction for mappings, which seems to be new in generality
although constructions of this (radial, Dini-like) type have been widely used
for extended-real-valued functions.
###### Definition 2.1
(restrictive graphical derivative of mappings). Let $F\colon
I\\!\\!R^{n}\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$, and let
$(\bar{x},\bar{y})\in\mbox{\rm gph}\,F$. Then a set-valued mapping
$\widetilde{D}F(\bar{x},\bar{y})\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$ given by
(2.6) $\widetilde{D}F(\bar{x},\bar{y})(z):=\mathop{{\rm Lim}\,{\rm
sup}}_{t\downarrow 0}\frac{F(\bar{x}+tz)-\bar{y}}{t},\quad z\in I\\!\\!R^{n},$
is called the restrictive graphical derivative of $F$ at $(\bar{x},\bar{y})$.
The next proposition collects some properties of the graphical derivative
(2.4) and its restrictive counterpart (2.6) needed in what follows.
###### Proposition 2.2
(properties of graphical derivatives). Let $F\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$, and let
$(\bar{x},\bar{y})\in\mbox{\rm gph}\,F$. Then the following assertions hold:
(a) We have $\widetilde{D}F(\bar{x},\bar{y})(z)\subset DF(\bar{x},\bar{y})(z)$
for all $z\in I\\!\\!R^{n}$.
(b) There are inverse derivative relationships
$DF(\bar{x},\bar{y})^{-1}=DF^{-1}(\bar{y},\bar{x})\;\mbox{ and
}\;\widetilde{D}F(\bar{x},\bar{y})^{-1}=\widetilde{D}F^{-1}(\bar{y},\bar{x}).$
(c) If $F$ is single-valued and locally Lipschitzian around $\bar{x}$, then
$\widetilde{D}F(\bar{x})(z)=DF(\bar{x})(z)\;\mbox{ for all }\;z\in
I\\!\\!R^{n}.$
(d) If $F$ is single-valued and directionally differentiable at $\bar{x}$,
then
$\widetilde{D}F(\bar{x})(z)=\big{\\{}F^{\prime}(\bar{x};z)\big{\\}}\;\mbox{
for all }\;z\in I\\!\\!R^{n}.$
(e) If $F$ is single-valued and Gâteaux differentiable at $\bar{x}$ with the
Gâteaux derivative $F^{\prime}_{G}(\bar{x})$, then we have
$\widetilde{D}F(\bar{x})(z)=\big{\\{}F^{\prime}_{G}(\bar{x})z\big{\\}}\;\mbox{
for all }\;z\in I\\!\\!R^{n}.$
(f) If $F$ is single-valued and $($Fréchet$)$ differentiable at $\bar{x}$ with
the derivative $F^{\prime}(\bar{x})$, then
$DF(\bar{x})(z)=\big{\\{}F^{\prime}(\bar{x})z\big{\\}}\;\mbox{ for all }\;z\in
I\\!\\!R^{n}.$
Proof. It is shown in [35, 8(14)] that the graphical derivative (2.4) admits
the representation
(2.7) $DF(\bar{x},\bar{y})(z)=\mathop{{\rm Lim}\,{\rm sup}}_{t\downarrow
0,\,h\to z}\frac{F(\bar{x}+th)-\bar{y}}{t},\quad z\in I\\!\\!R^{n}.$
The inclusion in (a) is an immediate consequence of Definition 2.1 and
representation (2.7).
The first equality in (b), observed from the very beginning [2], easily
follows from definition (2.4). We can similarly check the second one in (b).
To justify the equality in (c), it remains to verify by (a) the opposite
inclusion ‘$\supset$’ when $F$ is single-valued and locally Lipschitzian
around $\bar{x}$. In this case fix $z\in I\\!\\!R^{n}$, pick any $w\in
DF(\bar{x})(z)$, and find by representation (2.7) sequences $h_{k}\to z$ and
$t_{k}\downarrow 0$ such that
$\frac{F(\bar{x}+t_{k}h_{k})-F(\bar{x})}{t_{k}}\to w\;\mbox{ as
}\;k\to\infty.$
The local Lipschitz continuity of $F$ around $\bar{x}$ with constant $L\geq 0$
implies that
$\displaystyle\Big{\|}\frac{F(\bar{x}+t_{k}h_{k})-F(\bar{x})}{t_{k}}-\frac{F(\bar{x}+t_{k}z)-F(\bar{x})}{t_{k}}\Big{\|}$
$\displaystyle=$
$\displaystyle\Big{\|}\frac{F(\bar{x}+t_{k}h_{k})-F(\bar{x}+t_{k}z)}{t_{k}}\Big{\|}$
$\displaystyle\leq$ $\displaystyle L\|h_{k}-z\Big{\|}$
for all $k\in I\\!\\!N$ sufficiently large, and hence we have the convergence
$\frac{F(\bar{x}+t_{k}z)-F(\bar{x})}{t_{k}}\to w\;\mbox{ as }\;k\to\infty.$
Thus $w\in\widetilde{D}F(\bar{x})(z)$, which justifies (c). Assertions (d) and
(e) follow directly from the definitions. Finally, assertion (f) is implied by
(e) in the local Lipschitzian case (c) while it can be easily derived from the
(Fréchet) differentiability of $F$ at $\bar{x}$ with no Lipschitz assumption;
see, e.g., [35, Exercise 9.25(b)]. $\hfill\triangle$
Proposition 2.2 reveals important differences between the graphical derivative
(2.4) and the coderivative (2.5). Indeed, assertions (c) and (d) of this
proposition show that the graphical derivative of locally Lipschitzian and
directionally differentiable mappings $F\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$
is always single-valued. At the same time, the coderivative single-valuedness
for locally Lipschitzian mappings is equivalent to the strict/strong Fréchet
differentiability of $F$ at the point in question; see [24, Theorem 3.66]. It
follows from the well-known formula
(2.8) ${\rm
co}D^{*}F(\bar{x})(z)=\big{\\{}A^{T}z\big{|}\;A\in\partial_{C}F(\bar{x})\big{\\}}$
that the latter strict differentiability condition characterizes also the
single-valuedness of the generalized Jacobian of $F$ at $\bar{x}$.
In fact, in the case of $F=(f_{1},\ldots,f_{m})\colon I\\!\\!R^{n}\to
I\\!\\!R^{m}$ being locally Lipschitzian around $\bar{x}$ the coderivative
(2.5) admits the subdifferential description
(2.9)
$D^{*}F(\bar{x})(z)=\partial\Big{(}\sum_{i=1}^{m}z_{i}f_{i}\Big{)}(\bar{x})\;\mbox{
for any }\;z=(z_{1},\ldots,z_{m})\in I\\!\\!R^{m},$
where the (basic, limiting, Mordukhovich) subdifferential $\partial
f(\bar{x})$ of a general scalar function $f$ at $\bar{x}$ is defined
geometrically by
(2.10) $\partial f(\bar{x}):=\big{\\{}p\in I\\!\\!R^{n}\big{|}\;(p,-1)\in
N\big{(}(\bar{x},f(\bar{x}));\mbox{\rm epi}\,f\big{)}\big{\\}}$
via the normal cone (2.3) to the epigraph $\mbox{\rm epi}\,f:=\\{(x,\mu)\in
I\\!\\!R^{n+1}|\;\mu\geq f(x)\\}$ and admits analytical descriptions in terms
of the outer limit (1.8) of the Fréchet/regular and proximal subdifferentials
at points nearby; see [24, 35] with the references therein. Note also that the
basic subdifferential (2.10) of a continuous function $f$ can be also
described via the coderivative of $f$ by $\partial
f(\bar{x})=D^{*}f(\bar{x})(1)$; see [24, Theorem 1.80].
Finally in this section, we recall the notion of metric regularity and its
coderivative characterization that play a significant role in the paper. A
mapping $F\colon I\\!\\!R^{n}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;I\\!\\!R^{m}$ is metrically regular around
$(\bar{x},\bar{y})\in\mbox{\rm gph}\,F$ if there are neighborhoods $U$ of
$\bar{x}$ and $V$ of $\bar{y}$ as well as a number $\mu>0$ such that
(2.11) ${\rm dist}\big{(}x;F^{-1}(y)\big{)}\leq\mu\,{\rm
dist}\big{(}y;F(x)\big{)}\;\mbox{ for all }\;x\in U\;\mbox{ and }\;y\in V.$
Observe that it is sufficient to require the fulfillment of (2.11) just for
those $y\in V$ satisfying the estimate dist$(y;F(x))\leq\gamma$ for some
$\gamma>0$; see [24, Proposition 1.48].
We will see below that metric regularity is crucial for justifying the well-
posedness of our generalized Newton algorithm and establishing its local and
global convergence. It is also worth mentioning that, in the opposite
direction, a Newton-type method (known as the Lyusternik-Graves iterative
process) leads to verifiable conditions for metric regularity of smooth
mappings; see, e.g., the proof of [24, Theorem 1.57] and the commentaries
therein. The latter procedure is replaced by variational/extremal principles
of variational analysis in the case of nonsmooth and set-valued mappings under
consideration; cf. [10, 24, 35].
In this paper we broadly use the following coderivative characterization of
the metric regularity property for an arbitrary set-valued mapping $F$ with
closed graph, known also as the Mordukhovich criterion (see [23, Theorem 3.6],
[35, Theorem 9.45], and the references therein): $F$ is metrically regular
around $(\bar{x},\bar{y})$ if and only if the inclusion
(2.12) $0\in D^{*}F(\bar{x},\bar{y})(z)\;\mbox{ implies that }\;z=0,$
which amounts the kernel condition ${\rm ker}D^{*}F(\bar{x},\bar{y})=\\{0\\}$.
## 3 The Generalized Newton Algorithm
This section presents the main contribution of the paper: a new generalized
Newton method for nonsmooth equations, which is based on graphical
derivatives. The section consists of three parts. In Subsection 3.1 we
precisely describe the algorithm and justify its well-posedness/solvability.
Subsection 3.2 contains a local superlinear convergence result under
appropriate assumptions. Finally, in Subsection 3.3 we establish a global
convergence result of the Kantorovich type for our generalized Newton
algorithm.
### 3.1 Description and Justification of the Algorithm
Keeping in mind the classical scheme of the smooth Newton method in (1.2),
(1.3) and taking into account the graphical derivative representation of
Proposition 2.2(f), we propose an extension of the Newton equation (1.3) to
nonsmooth mappings given by:
(3.1) $-H(x^{k})\in DH(x^{k})(d^{k}),\quad k=0,1,2,\ldots.$
This leads us to the following generalized Newton algorithm to solve (1.1):
###### Algorithm 3.1
(generalized Newton’s method).
Step 0: Choose a starting point $x^{0}\in I\\!\\!R^{n}$.
Step 1: Check a suitable termination criterion.
Step 2: Compute $d^{k}\in I\\!\\!R^{n}$ such that (3.1) holds.
Step 3: Set $x^{k+1}:=x^{k}+d^{k}$, $k\leftarrow k+1$, and go to Step 1.
The proposed Algorithm 3.1 does not require a priori any assumptions on the
underlying mapping $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ in (1.1) besides its
continuity, which is the standing assumption in this paper. Other assumptions
are imposed below to justify the well-posedness and (local and global)
convergence of the algorithm. Observe that Proposition 2.2(c,d) ensures that
Algorithm 3.1 reduces to scheme (1.4) in the $B$-differentiable Newton method
provided that $H$ is directionally differentiable and locally Lipschitzian
around the solution point in question. In Section 5 we consider in detail
relationships with known results for the $B$-differentiable Newton method,
while Section 4 compares Algorithm 3.1 and the assumptions made with the
corresponding semismooth versions in the framework of (1.5).
To proceed further, we need to make sure that the generalized Newton equation
(3.1) is solvable, which is a major part of the well-posedness of Algorithm
3.1. The next proposition shows that an appropriate assumption to ensure the
solvability of (3.1) is metric regularity.
###### Proposition 3.2
(solvability of the generalized Newton equation). Assume that $H\colon
I\\!\\!R^{n}\to I\\!\\!R^{n}$ is metrically regular around $\bar{x}$ with
$\bar{y}=H(\bar{x})$ in (2.11), i.e., we have $\mbox{\rm
ker}\,D^{*}H(\bar{x})=\\{0\\}$. Then there is a constant $\varepsilon>0$ such
that for all $x\in B_{\varepsilon}(\bar{x})$ the equation
(3.2) $-H(x)\in DH(x)(d)$
admits a solution $d\in I\\!\\!R^{n}$. Furthermore, the set $S(x)$ of
solutions to (3.2) is computed by
(3.3) $S(x)=\mathop{{\rm Lim}\,{\rm sup}}_{t\downarrow
0,\,h\to-H(x)}\frac{H^{-1}\big{(}H(x)+th\big{)}-x}{t}\neq\emptyset.$
Proof. By the assumed metric regularity (2.11) of $H$ we find a number $\mu>0$
and neighborhoods $U$ of $\bar{x}$ and $V$ of $H(\bar{x})$ such that
${\rm dist}\big{(}x;H^{-1}(y)\big{)}\leq\mu\,{\rm dist}(y;H(x)\big{)}\;\mbox{
for all }\;x\in U\;\mbox{ and }\;y\in V.$
Pick now an arbitrary vector $x\in U$ and select sequences $h_{k}\to-H(x)$ and
$t_{k}\downarrow 0$ as $k\to\infty$. Suppose with no loss of generality that
$H(x)+t_{k}h_{k}\in V$ for all $k\in I\\!\\!N$. Then we have
${\rm dist}\big{(}x;H^{-1}(H(x)+t_{k}h_{k})\big{)}\leq\mu t_{k}\|h_{k}\|,\quad
k\in I\\!\\!N,$
and hence there is a vector $u_{k}\in H^{-1}(H(x)+t_{k}h_{k})$ such that
$\|u_{k}-x\|\leq\mu t_{k}\|h_{k}\|$ for all $k\in I\\!\\!N$. This shows that
the sequence $\\{\|u_{k}-x\|/t_{k}\\}$ is bounded, and thus it contains a
subsequence that converges to some element $d\in I\\!\\!R^{n}$. Passing to the
limit as $k\to\infty$ and recalling the definitions of the outer limit (1.8)
and of the tangent cone (2.1), we arrive at
$\big{(}d,-H(x)\big{)}\in\mathop{{\rm Lim}\,{\rm sup}}_{t\downarrow
0}\frac{\mbox{\rm gph}\,H-\big{(}x,H(x)\big{)}}{t}=T\big{(}(x,H(x));\mbox{\rm
gph}\,H\big{)},$
which justifies the desired inclusion (3.2). The solution representation (3.3)
follows from (2.7) and Proposition 2.2(b) in the case of single-valued
mappings, since
$S(x)=DH(x)^{-1}\big{(}-H(x)\big{)}$
due to (3.2). This completes the proof of the proposition. $\hfill\triangle$
### 3.2 Local Convergence
In this subsection we first formulate major assumptions of our generalized
Newton method and then show that they ensure the superlinear local convergence
of Algorithm 3.1.
* (H1)
There exist a constant $C>0$, a neighborhood $U$ of $\bar{x}$, and a
neighborhood $V$ of the origin in $I\\!\\!R^{n}$ such that the following
holds:
For all $x\in U$, $z\in V$, and for any $d\in I\\!\\!R^{n}$ with $-H(x)\in
DH(x)(d)$ there is a vector $w\in\widetilde{D}H(x)(z)$ such that
$C\|d-z\|\leq\|w+H(x)\|+o(\|x-\bar{x}\|).$
* (H2)
There exists a neighborhood $U$ of $\bar{x}$ such that for all
$v\in\widetilde{D}H(x)(\bar{x}-x)$ we have
$\|H(x)-H(\bar{x})+v\|=o(\|x-\bar{x}\|).$
A detailed discussion of these two assumptions and sufficient conditions for
their fulfillment are given in Section 4. Note that assumption (H2) means, in
the terminology of [8, Definition 7.2.2] focused on locally Lipschitzian
mappings $H$, that the family $\\{\widetilde{D}H(x)\\}$ provides a Newton
approximation scheme for $H$ at $\bar{x}$.
Now we establish our principal local convergence result that makes use of the
major assumptions (H1) and (H2) together with metric regularity.
###### Theorem 3.3
(superlinear local convergence of the generalized Newton method). Let
$\bar{x}\in I\\!\\!R^{n}$ be a solution to (1.1) for which the underlying
mapping $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ is metrically regular around
$\bar{x}$ and assumptions (H1) and (H2) are satisfied. Then there is a number
$\varepsilon>0$ such that for all $x^{0}\in B_{\varepsilon}(\bar{x})$ the
following assertions hold:
(i) Algorithm 3.1 is well defined and generates a sequence $\\{x^{k}\\}$
converging to $\bar{x}$.
(ii) The rate of convergence $x^{k}\to\bar{x}$ is at least superlinear.
Proof. To justify (i), pick $\varepsilon>0$ such that assumptions (H1) and
(H2) hold with $U:=B_{\varepsilon}(\bar{x})$ and $V:=B_{\varepsilon}(0)$ and
such that Proposition 3.2 can be applied. Then we choose a starting point
$x^{0}\in B_{\varepsilon}(\bar{x})$ and conclude by Proposition 3.2 that the
subproblem
$-H(x^{0})\in DH(x^{0})(d)$
has a solution $d^{0}$. Thus the next iterate $x^{1}:=x^{0}+d^{0}$ is well
defined. Let further $z^{0}:=\bar{x}-x^{0}$ and get $\|z^{0}\|\leq\varepsilon$
by the choice of the starting point $x^{0}$. By assumption (H1), find a vector
$w^{0}\in\widetilde{D}H(x^{0})(z^{0})$ such that
$C\|x^{1}-\bar{x}\|=C\|(x^{1}-x^{0})-(\bar{x}-x^{0})\|=C\|d^{0}-z^{0}\|\leq\|w^{0}+H(x^{0})\|+o(\|x^{0}-\bar{x}\|).$
Taking this into account and employing assumption (H2), we get the
relationships
$\displaystyle C\|x^{1}-\bar{x}\|$ $\displaystyle\leq$
$\displaystyle\|w^{0}+H(x^{0})\|+o(\|x^{0}-\bar{x}\|)$ $\displaystyle=$
$\displaystyle\|H(x^{0})-H(\bar{x})+w^{0}\|+o(\|x^{0}-\bar{x}\|)$
$\displaystyle=$ $\displaystyle o(\|x^{0}-\bar{x}\|)$ $\displaystyle\leq$
$\displaystyle\tfrac{C}{2}\|x^{0}-\bar{x}\|,$
which imply that $\|x^{1}-\bar{x}\|\leq\frac{1}{2}\|x^{0}-\bar{x}\|$. The
latter yields, in particular, that $x^{1}\in B_{\varepsilon}(\bar{x})$. Now
standard induction arguments allow us to conclude that the iterative sequence
$\\{x^{k}\\}$ generated by Algorithm 3.1 is well defined and converges to the
solution $\bar{x}$ of (1.1) with at least a linear rate. This justifies
assertion (i) of the theorem.
Next we prove assertion (ii) showing that the convergence $x^{k}\to\bar{x}$ is
in fact superlinear under the validity of assumption (H2). To proceed, we
basically follow the proof of assertion (i) and construct by induction
sequences $\\{d^{k}\\}$ satisfying
$-H(x^{k})\in DH(x^{k})(d^{k})\;\mbox{ for all }\;k\in I\\!\\!N,$
$\\{z^{k}\\}$ with $z^{k}:=\bar{x}-x^{k}$, and $\\{w^{k}\\}$ with
$w^{k}\in\widetilde{D}H(x^{k})(z^{k})$ such that
$C\|x^{k+1}-\bar{x}\|\leq\|w^{k}+H(x^{k})\|+o(\|x^{k}-\bar{x}\|),\quad k\in
I\\!\\!N.$
Applying then assumption (H2) gives us the relationships
$C\|x^{k+1}-\bar{x}\|\leq\|H(x^{k})-H(\bar{x})+w^{k}\|+o(\|x^{k}-\bar{x}\|)=o(\|x^{k}-\bar{x}\|),$
which ensure the superlinear convergence of the iterative sequence
$\\{x^{k}\\}$ to the solution $\bar{x}$ of (1.1) and thus complete the proof
of the theorem. $\hfill\triangle$
### 3.3 Global Convergence
Besides the local convergence in the classical Newton method based on suitable
assumptions imposed at the (unknown) solution of the underlying system of
equations, there are global (or semi-local) convergence results of the
Kantorovich type [12] for smooth systems of equations which show that, under
certain conditions at the starting point $x^{0}$ and a number of assumptions
to hold in a suitable region around $x^{0}$, Newton’s iterates are well
defined and converge to a solution belonging to this region; see [7, 12] for
more details and references. In the case of nonsmooth equations (1.1) results
of the Kantorovich type were obtained in [31, 34] for the corresponding
versions of Newton’s method. Global convergence results of different types can
be found in, e.g., [6, 8, 9, 28] and their references.
Here is a global convergence result for our generalized Newton method to solve
(1.1).
###### Theorem 3.4
(global convergence of the generalized Newton method). Let $x^{0}$ be a
starting point of Algorithm 3.1, and let
(3.4) $\Omega:=\big{\\{}x\in I\\!\\!R^{n}\big{|}\;\|x-x^{0}\|\leq r\big{\\}}$
with some $r>0$. Impose the following assumptions:
* (a)
The mapping $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ in (1.1) is metrically
regular on $\Omega$ with modulus $\mu>0$, i.e., it is metrically regular
around every point $x\in\Omega$ with the same modulus $\mu$.
* (b)
The set-valued map $DH(x)(z)$ uniformly on $\Omega$ converges to $\\{0\\}$ as
$z\to 0$ in the sense that: for all $\varepsilon>0$ there is $\delta>0$ such
that
$\|w\|\leq\varepsilon\;\mbox{ whenever }\;w\in
DH(x)(z),\;\|z\|\leq\delta,\;\mbox{ and }\;x\in\Omega.$
* (c)
There is $\alpha\in(0,1/\mu)$ such that
(3.5) $\mu\|H(x^{0})\|\leq r(1-\alpha\mu)$
and for all $x,y\in\Omega$ we have the estimate
(3.6) $\|H(x)-H(y)-v\|\leq\alpha\|x-y\|\;\mbox{ whenever }\;v\in DH(x)(y-x).$
Then Algorithm 3.1 is well defined, the sequence of iterates $\\{x^{k}\\}$
remains in $\Omega$ and converges to a solution $\bar{x}\in\Omega$ of (1.1).
Moreover, we have the error estimate
(3.7)
$\|x^{k}-\bar{x}\|\leq\frac{\alpha\mu}{1-\alpha\mu}\|x^{k}-x^{k-1}\|\;\mbox{
for all }\;k\in I\\!\\!N.$
Proof. The metric regularity assumption (a) allows us to employ Proposition
3.2 and, for any $x\in\Omega$ and $d\in I\\!\\!R^{n}$ satisfying the inclusion
$-H(x)\in DH(x)(d)$, to find sequences of $h_{k}\to-H(x)$ and $t_{k}\downarrow
0$ as $k\to\infty$ such that
$\|d\|=\lim_{k\to\infty}\Big{\|}\frac{H^{-1}\big{(}H(x)+t_{k}h_{k}\big{)}-x}{t_{k}}\Big{\|}\leq\lim_{k\to\infty}\mu\|h_{k}\|=\mu\|H(x)\|.$
In view of assumption (3.5) in (c) and the iteration procedure of the
algorithm, this implies
$\|x^{1}-x^{0}\|=\|d^{0}\|\leq\mu\|H(x^{0})\|\leq r(1-\alpha\mu),$
which ensures that $x^{1}\in\Omega$ due the form of $\Omega$ in (3.4) and the
choice of $\alpha$. Proceeding further by induction, suppose that
$x^{1},\ldots,x^{k}\in\Omega$ and get the relationships
$\displaystyle\|x^{k+1}-x^{k}\|$ $\displaystyle=\|d^{k}\|\leq\mu\|H(x^{k})\|$
$\displaystyle\leq\mu\|H(x^{k})-H(x^{k-1})+H(x^{k-1})\|$
$\displaystyle\leq\alpha\mu\|x^{k}-x^{k-1}\|\quad\Big{(}\text{using \eqref{g2}
and }-H(x^{k-1})\in DH(x^{k-1})(x^{k}-x^{k-1})\Big{)}$
$\displaystyle\leq(\alpha\mu)^{k}\|x^{1}-x^{0}\|\leq
r(\alpha\mu)^{k}(1-\alpha\mu),$
which imply the estimates
$\|x^{k+1}-x^{0}\|\leq\sum_{j=0}^{k}\|x^{j+1}-x^{j}\|\leq\sum_{j=0}^{k}r(\alpha\mu)^{j}(1-\alpha\mu)\leq
r$
and hence justify that $x^{k+1}\in\Omega$. Thus all the iterates generated by
Algorithm 3.1 remain in $\Omega$. Furthermore, for any natural numbers $k$ and
$m$, we have
$\|x^{k+m+1}-x^{k}\|\leq\sum_{j=k}^{k+m}\|x^{j+1}-x^{j}\|\leq\sum_{j=k}^{k+m}r(\alpha\mu)^{j}(1-\alpha\mu)\leq
r(\alpha\mu)^{k},$
which shows that the generated sequence $\\{x^{k}\\}$ is a Cauchy sequence.
Hence it converges to some point $\bar{x}$ that obviously belongs to the
underlying closed set (3.4).
To show next that $\bar{x}$ is a solution to the original equation (1.1), we
pass to the limit as $k\to\infty$ in the iterative inclusion
(3.8) $-H(x^{k})\in DH(x^{k})(x^{k+1}-x^{k}),\quad k\in I\\!\\!N.$
It follows from assumption (b) that $\lim_{k\to\infty}H(x^{k})=0$. The
continuity of $H$ then implies that $H(\bar{x})=0$, i.e., $\bar{x}$ is a
solution to (1.1).
It remains to justify the error estimate (3.7). To this end, first observe by
(3.5) that
$\|x^{k+m+1}-x^{k}\|\leq\sum_{j=k}^{k+m}\|x^{j+1}-x^{j}\|\leq\sum_{j=0}^{m}(\alpha\mu)^{j+1}\|x^{k}-x^{k-1}\|\leq\frac{\alpha\mu}{1-\alpha\mu}\|x^{k}-x^{k-1}\|$
for all $k,m\in I\\!\\!N$. Passing now to the limit as $m\to\infty$, we arrive
at (3.7) thus completes the proof of the theorem. $\hfill\triangle$
## 4 Discussion of Major Assumptions and Comparison with Semismooth Newton
Methods
In this section we pursue a twofold goal: to discuss the major assumptions
made in Section 3 and to compare our generalized Newton method based on
graphical derivatives with the semismooth versions of the generalized Newton
method developed in [30, 31]. As we will see from the discussions below, these
two aims are largely interrelated.
Let us begin with sufficient conditions for metric regularity in terms of the
constructions used in the semismooth versions of the generalized Newton
method. Given a locally Lipschitz continuous vector-valued mapping $H\colon
I\\!\\!R^{n}\to I\\!\\!R^{m}$, we have by the classical Rademacher theorem
that the set of points
(4.1) $S_{H}:=\\{x\in I\\!\\!R^{n}\big{|}\;H\;\mbox{ is differentiable at
}\;x\big{\\}}$
is of full Lebesgue measure in $I\\!\\!R^{n}$. Thus for any mapping $H\colon
I\\!\\!R^{n}\to I\\!\\!R^{m}$ locally Lipschitzian around $\bar{x}$ the set
(4.2)
$\partial_{B}H(\bar{x}):=\Big{\\{}\lim_{k\to\infty}H^{\prime}(x^{k})\Big{|}\;\exists\,\\{x^{k}\\}\subset
S_{H}\;\mbox{ with }\;x^{k}\to\bar{x}\Big{\\}}$
is nonempty and obviously compact in $I\\!\\!R^{m}$. It was introduced in [38]
for $m=1$ as the set of “almost-gradients” and then was called in [30] the
$B$-subdifferential of $H$ at $\bar{x}$. Clarke’s generalized Jacobian [5] of
$H$ at $\bar{x}$ is defined by the convex hull
(4.3)
$\partial_{C}H(\bar{x}):=\mbox{co}\big{\\{}\partial_{B}H(\bar{x})\big{\\}}.$
We also make use of the Thibault derivative/limit set [39] (called sometimes
the “strict graphical derivative” [35]) of $H$ at $\bar{x}$ defined by
(4.4) $D_{T}H(\bar{x})(z):=\mathop{{\rm Lim}\,{\rm sup}}_{x\to\bar{x}\atop
t\downarrow 0}\frac{H(x+tz)-H(x)}{t},\quad z\in I\\!\\!R^{n}.$
Observe the known relationships [15, 39] between the above derivative sets
(4.5) $\partial_{B}H(\bar{x})z\subset
D_{T}H(\bar{x})(z)\subset\partial_{C}H(\bar{x})z,\quad z\in I\\!\\!R^{n}.$
The next result gives a sufficient condition for metric regularity of
Lipschitzian mappings in terms of the Thibault derivative (4.4). It can be
derived from the coderivative characterization of metric regularity (2.12),
while we give here a direct independent proof.
###### Proposition 4.1
(sufficient condition for metric regularity in terms of Thibault’s
derivative). Let $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ be locally
Lipschitzian around $\bar{x}$, and let
(4.6) $0\notin D_{T}H(\bar{x})(z)\;\mbox{ whenever }\;z\neq 0.$
Then the mapping $H$ is metrically regular around $\bar{x}$.
Proof. Kummer’s inverse function theorem [17, Theorem 1.1] ensures that
condition (4.6) implies (actually is equivalent to) the fact that there are
neighborhoods $U$ of $\bar{x}$ and $V$ of $H(\bar{x})$ such that the mapping
$H\colon U\to V$ is one-to-one with a locally Lipschitzian inverse
$H^{-1}\colon V\to U$. Let $\mu>0$ be a Lipschitz constant of $H^{-1}$ on $V$.
Then for all $x\in U$ and $y\in V$ we have the relationships
$\displaystyle\text{dist}\big{(}x;H^{-1}(y)\big{)}$ $\displaystyle=$
$\displaystyle\|x-H^{-1}(y)\|$ $\displaystyle=$
$\displaystyle\|H^{-1}\big{(}H(x)\big{)}-H^{-1}(y)\|$ $\displaystyle\leq$
$\displaystyle\mu\|H(x)-y\|$ $\displaystyle=$
$\displaystyle\mu\,\text{dist}\big{(}y;H(x)\big{)},$
which thus justify the metric regularity of $H$ around $\bar{x}$.
$\hfill\triangle$
To proceed further with sufficient conditions for the validity of our
assumption (H1), we first introduce the notion of directional boundedness.
###### Definition 4.2
(directional boundedness). A mapping $H\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$ is
said to be directionally bounded around $\bar{x}$ if
(4.7) $\limsup_{t\downarrow 0}\left\|\frac{H(x+tz)-H(x)}{t}\right\|<\infty$
for all $x$ near $\bar{x}$ and for all $z\in I\\!\\!R^{n}$.
It is easy to see that if $H$ is either directionally differentiable around
$\bar{x}$ or locally Lipschitzian around this point, then it is directionally
bounded around $\bar{x}$. The following example shows that the converse does
not hold in general.
###### Example 4.3
(directional bounded mappings may not be directionally differentiable). Define
a real function $H\colon I\\!\\!R\to I\\!\\!R$ by
$H(x):=\left\\{\begin{array}[]{ll}x\sin\big{(}\frac{1}{x}\big{)}&\text{if
}\;x\neq 0,\\\ 0&\text{if }\;x=0.\end{array}\right.$
It is easy to see that this function is not directionally differentiable at
$\bar{x}=0$. However, it is directionally bounded around $\bar{x}$. Indeed,
for any $x\neq 0$ near $\bar{x}$ condition (4.7) holds because $H$ is simply
differentiable at $x\neq 0$. For $x=0$ we have
$\limsup_{t\downarrow
0}\Big{|}\frac{H(tz)-H(0)}{t}\Big{|}=\limsup_{t\downarrow
0}\frac{|H(tz)|}{t}=\limsup_{t\downarrow
0}\Big{|}z\sin\Big{(}\frac{1}{tz}\Big{)}\Big{|}=|z|<\infty.$
The next proposition and its corollary present verifiable sufficient
conditions for the fulfillment of assumption (H1).
###### Proposition 4.4
(assumption (H1) from metric regularity). Let $H\colon I\\!\\!R^{n}\to
I\\!\\!R^{n}$, and let $\bar{x}$ be a solution to (1.1). Suppose that $H$ is
metrically regular around $\bar{x}$ $($i.e., $\mbox{\rm
ker}\,D^{*}H(\bar{x})=0)$, that it is directionally bounded and one-to-one
around this point. Then assumption (H1) is satisfied.
Proof. Recall that the metric regularity of $H$ around $\bar{x}$ is equivalent
to the condition $\mbox{\rm ker}\,D^{*}H(\bar{x})=\\{0\\}$ by the coderivative
criterion (2.12). Let $U\subset I\\!\\!R^{n}$ be a neighborhood of $\bar{x}$
such that $H$ is metrically regular and one-to-one on $U$. Choose further a
neighborhood $V\subset I\\!\\!R^{n}$ of $H(\bar{x})=0$ from the definition of
metric regularity of $H$ around $\bar{x}$. Then pick $x\in U$, $z\in V$ and an
arbitrary direction $d\in I\\!\\!R^{n}$ satisfying $-H(x)\in DH(x)(d)$.
Employing now Proposition 3.2, we get
$d\in\mathop{{\rm Lim}\,{\rm sup}}_{h\to-H(x),\;t\downarrow
0}\frac{H^{-1}\big{(}H(x)+th\big{)}-x}{t}.$
By the local single-valuedness of $H^{-1}$ and the metric regularity of $H$
around $\bar{x}$ there exists a number $\mu>0$ such that
$\left\|\frac{H^{-1}(H(x)+th)-x}{t}-z\right\|\leq\mu\left\|\frac{H(x)+th-H(x+tz)}{t}\right\|=\mu\left\|\frac{H(x+tz)-H(x)}{t}-h\right\|$
for all $t>0$ sufficiently small. It follows that
$\|d-z\|\leq\limsup_{t\downarrow 0\atop
h\to-H(x)}\left\|\frac{H^{-1}\big{(}H(x)+th\big{)}-x}{t}-z\right\|\leq\mu\limsup_{t\downarrow
0\atop h\to-H(x)}\left\|\frac{H(x+tz)-H(x)}{t}-h\right\|<\infty$
by the directional boundedness of $H$ around $\bar{x}$. The boundedness of the
family
$\Big{\\{}v(t):=\frac{H(x+tz)-H(x)}{t}\Big{\\}},\quad t\downarrow 0,$
allows us to select a sequence $t_{k}\downarrow 0$ such that $v(t_{k})\to w$
for some $w\in I\\!\\!R^{n}$. By passing to the limit above as $k\to\infty$
and employing Definition 2.1 we get that
$w\in\widetilde{D}H(x)(z)\quad{\rm
and}\quad\frac{1}{\mu}\|d-z\|\leq\|w+H(x)\|,$
which completes the proof of the proposition. $\hfill\triangle$
###### Corollary 4.5
(sufficient conditions for (H1) via Thibault’s derivative). Let $\bar{x}$ be a
solution to (1.1), where $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ is locally
Lipschitzian around $\bar{x}$ and such that condition (4.6) holds, which is
automatic when ${\rm det}\,A\neq 0$ for all $A\in\partial_{C}H(\bar{x})$. Then
(H1) is satisfied with $H$ being both metrically regular and one-to-one around
$\bar{x}$.
Proof. Indeed, both metric regularity and bijectivity of $H$ around $\bar{x}$
assumed in Proposition 4.4 follow from Proposition 4.1 and its proof.
Nonsingularity of all $A\in\partial_{C}H(\bar{x})$ clearly implies (4.6) by
the second inclusion in (4.5). $\hfill\triangle$
Note that other conditions ensuring the fulfillment of assumption (H1) for
Lipschitzian and non-Lipschitzian mappings $H\colon I\\!\\!R^{n}\to
I\\!\\!R^{n}$ can be formulated in terms of Warga’s derivate containers by
[40, Theorems 1 and 2] on “fat homeomorphisms” that also imply the metric
regularity and one-to-one properties of $H$.
Next we proceed with the discussion of assumption (H2) and present, in
particular, sufficient conditions for their fulfillment via semismoothness.
First observe the following.
###### Proposition 4.6
(relationship between graphical derivative and generalized Jacobian). Let
$H\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$ be locally Lipschitzian around
$\bar{x}$. Then we have
(4.8) $DH(\bar{x})(z)\subset\partial_{C}H(\bar{x})z\;\mbox{ for all }\;z\in
I\\!\\!R^{n}.$
Proof. Pick $w\in DH(\bar{x})(z)$ and get by Proposition 2.2(c) and Definition
2.1 a sequence of $t_{k}\downarrow 0$ as $k\to\infty$ such that
(4.9) $w=\lim_{k\to\infty}\frac{H(\bar{x}+t_{k}z)-H(\bar{x})}{t_{k}}.$
It follows from [5, Proposition 2.6.5] that
$\frac{H(\bar{x}+t_{k}z)-H(\bar{x})}{t_{k}}\in{\rm
co}\big{\\{}\partial_{C}H[\bar{x},\bar{x}+t_{k}z]\big{\\}}z\;\mbox{ for all
}\;k\in I\\!\\!N.$
Applying to the latter the classical Carathéodory theorem, we find scalars
$\gamma^{k}_{i}\in[0,t_{k}]$, $\lambda^{k}_{i}\in[0,1]$ and matrices
$A^{k}_{i}\in\partial_{C}H(\bar{x}+\gamma^{k}_{i}z)$ for $i=1,\ldots,m+1$ such
that
$\frac{H(\bar{x}+t_{k}z)-H(\bar{x})}{t_{k}}=\Big{[}\sum_{i=1}^{m+1}\lambda_{i}^{k}A_{i}^{k}\Big{]}z\quad\text{and}\quad\sum_{i=1}^{m+1}\lambda_{i}^{k}=1\;\mbox{
for all }\;k\in I\\!\\!N.$
Due to the boundedness of the sequences $\\{\lambda^{k}_{i}\\}_{k\in
I\\!\\!N}$, the convergence $\bar{x}+\gamma^{k}_{i}z\to\bar{x}$ as
$k\to\infty$ for all $i=1,\ldots,m+1$, and the outer/upper semicontinuity of
the mapping $x\mapsto\partial_{C}H(x)$ proved in [5, Proposition 2.6.2] we
have that the sequences $\\{A^{k}_{i}\\}$ are bounded as well. Hence there are
subsequences of these sequences (without relabelling), scalars
$\lambda_{i}\in[0,1]$, and matrices $A_{i}$ as $i=1,\ldots,m+1$ such that
$\lambda^{k}_{i}\to\lambda_{i},\quad\sum_{i=1}^{m+1}\lambda_{i}=1,\;\mbox{ and
}\;A^{k}_{i}\to A_{i}\in\partial_{C}H(\bar{x})\;\mbox{ as }\;k\to\infty.$
By (4.9) and the subsequent relationships therein, we get
$w=\lim_{k\to\infty}\Big{[}\sum_{i=1}^{m+1}\lambda_{i}^{k}A_{i}^{k}\Big{]}z=\Big{[}\sum_{i=1}^{m+1}\lambda_{i}A_{i}\Big{]}z\in{\rm
co}\big{\\{}\partial_{C}H(\bar{x})\big{\\}}z=\partial_{C}H(\bar{x})z$
and thus complete the proof of the proposition. $\hfill\triangle$
Inclusion (4.8)—which may be strict as illustrated by Example 4.7 below—shows
that our generalized Newton Algorithm 3.1 based on the graphical derivative
provides in the case of Lipschitz equations (1.1) a more accurate choice of
the iterative direction $d^{k}$ via (3.1) in comparison with the iterative
relationship
(4.10) $-H(x^{k})\in\partial_{C}H(x^{k})d^{k},\quad k=0,1,2,\ldots,$
used in the semismooth Newton method [31] and related developments [15, 16]
based on the generalized Jacobian. If in addition to the assumptions of
Proposition 4.6 the mapping $H$ is directionally differentiable at $\bar{x}$,
then $DH(\bar{x})(z)=\\{H^{\prime}(\bar{x};z)\\}$ by Proposition 2.2(c,d).
Thus in this case we have from Proposition 4.6 that for any $z\in
I\\!\\!R^{n}$ there is $A\in\partial_{C}H(\bar{x})$ such that
$H^{\prime}(\bar{x};z)=Az$, which recovers a well-known result from [31, Lemma
2.2].
The following example shows that the converse inclusion in Proposition 4.6 is
not satisfied in general even with the replacement of the set $DH(\bar{x})(z)$
in (4.8) by its convex hull co$DH(\bar{x})(z)$ in the case of real functions.
Furthermore, the same holds if we replace the generalized Jacobian in (4.8) by
the smaller $B$-subdifferential $\partial_{B}H(\bar{x})$ from (4.2).
###### Example 4.7
(graphical derivative is strictly smaller than $B$-subdifferential and
generalized Jacobian). Consider the simplest nonsmooth convex function
$H(x)=|x|$ on $I\\!\\!R$. In this case $\partial_{B}H(0)=\\{-1,1\\}$ and
$\partial_{C}H(0)=[-1,1]$. Thus
$\partial_{B}H(0)z=\\{-1,1\\}\;\mbox{ and }\;\partial_{C}H(0)z=[-1,1]\;\mbox{
for }\;z=1.$
Since $H(x)=|x|$ is locally Lipschitzian and directionally differentiable, we
have
$DH(0)(z)=\big{\\{}H^{\prime}(0;z)\big{\\}}=|z|=\\{1\\}\;\mbox{ for }\;z=1.$
Hence it gives the relationships
$DH(0)(z)={\rm
co}\big{\\{}DH(0)(z)\big{\\}}\subset\partial_{B}H(0)z\subset\partial_{C}H(0)z,$
where both inclusions are strict. Observe also the difference between the
convexification of the graphical derivative and of the coderivative; in the
latter case we have equality (2.8).
As mentioned in Section 1, there is an improvement [30] of the iterative
procedure (4.10) with the replacement the generalized Jacobian therein by the
$B$-subdifferential
(4.11) $-H(x^{k})\in\partial_{B}H(x^{k})d^{k},\quad k=0,1,2,\ldots.$
Note that, along with obvious advantages of version (4.11) over the one in
(4.10), in some settings it is easier to deal with the generalized Jacobian
than with its $B$-subdifferential counterpart due to much better calculus and
convenient representations for $\partial_{C}H(\bar{x})$ in comparison with the
case of $\partial_{B}H(\bar{x})$, which does not even reduce to the classical
subdifferential of convex analysis for simple convex functions as, e.g.,
$H(x)=|x|$. A remarkable common feature for both versions in (4.10) and (4.11)
is the efficient semismoothness assumption imposed on the underlying mapping
$H$ to ensure its local superlinear convergence. This assumption, which
unifies and labels versions (4.10) and (4.11) as the “semismooth Newton
method”, is replaced in our generalized Newton method by assumption (H2). Let
us now recall the notion of semismoothness and compare it with (H2).
A mapping $H\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$, locally Lipschitzian and
directionally differentiable around $\bar{x}$, is semismooth at this point if
the limit
(4.12) $\lim_{h\to z,\;t\downarrow 0\atop
A\in\partial_{C}H(\bar{x}+th)}\big{\\{}Ah\big{\\}}$
exists for all $z\in I\\!\\!R^{n}$; see [8, Definition 7.4.2]. This notion was
introduced in [20] for real-valued functions and then extended in [31] to the
vector mappings for the purpose of applications to a nonsmooth Newton’s
method. It is not hard to check [31, Proposition 2.1] that the existence of
the limit in (4.12) implies the directional differentiability of $H$ at
$\bar{x}$ (but may not around this point) with
$H^{\prime}(\bar{x};z)=\lim_{h\to z,\;t\downarrow 0\atop
A\in\partial_{C}H(\bar{x}+th)}\big{\\{}Ah\big{\\}}\;\mbox{ for all }\;z\in
I\\!\\!R^{n}.$
One of the most useful properties of semismooth mappings is the following
representation for them obtained in [29, Proposition 1]:
(4.13) $\|H(\bar{x}+z)-H(\bar{x})-Az\|=o(\|z\|)\;\mbox{ for all }\;z\to
0\;\mbox{ and }\;A\in\partial_{C}H(\bar{x}+z),$
which we exploit now to relate semismoothness to our assumption (H2).
###### Proposition 4.8
(semismoothness implies assumption (H2)). Let $H\colon I\\!\\!R^{n}\to
I\\!\\!R^{m}$ be semismooth at $\bar{x}$. Then assumption (H2) is satisfied.
Proof. Since any semismooth mapping is Lipschitz continuous on a neighborhood
$U$ of $\bar{x}$, we have by Proposition 2.2(c) that
$\widetilde{D}H(x)(\bar{x}-x)=DH(x)(\bar{x}-x)\;\mbox{ for all }\;x\in U.$
Proposition 4.6 yields therefore that
$\widetilde{D}H(x)(\bar{x}-x)\subset\partial_{C}H(x)(\bar{x}-x)\;\mbox{
whenever }\;x\in U.$
Given any $v\in\widetilde{D}H(x)(\bar{x}-x)$ and using the latter inclusion,
find a matrix $A\in\partial_{C}H(x)$ such that $v=A(\bar{x}-x)$. Applying
finally property (4.13) of semismooth mappings, we get
$\|H(x)-H(\bar{x})+v\|=\|H(x)-H(\bar{x})-A(x-\bar{x})\|=o(\|x-\bar{x}\|)\;\mbox{
for all }\;x\in U,$
which thus verifies (H2) and completes the proof of the proposition.
$\hfill\triangle$
Note that the previous proposition actually shows that condition (4.13)
implies (H2). The next result states that the converse is also true, i.e., we
have that assumption (H2) is completely equivalent to (4.13) for locally
Lipschitzian mappings.
###### Proposition 4.9
(equivalent description of (H2)). Let $H\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$
be locally Lipschitzian around $\bar{x}$, and let assumption (H2) hold with
some neighborhood $U$ therein. Then
(4.14) $\|H(x)-H(\bar{x})-A(x-\bar{x})\|=o(\|\bar{x}-x\|)\;\mbox{ for all
}\;x\in U\;\mbox{ and }\;A\in\partial_{B}H(x).$
Therefore assumption (H2) is equivalent to (4.13).
Proof. Arguing by contradiction, suppose that (4.14) is violated and find
sequences $x_{k}\to\bar{x}$, $A_{k}\in\partial_{B}H(x_{k})$ and a constant
$\gamma>0$ such that
$\|H(x_{k})-H(\bar{x})-A_{k}(x_{k}-\bar{x})\|\geq\gamma\|\bar{x}-x_{k}\|,\quad
k\in I\\!\\!N.$
By the Lipschitz property of $H$ and by construction (4.2) of the
$B$-subdifferential there are points of differentiability $u_{k}\in S_{H}$
close to $x_{k}$ with $H^{\prime}(u_{k})$ sufficiently close to $A_{k}$
satisfying
$\|H(u_{k})-H(\bar{x})-H^{\prime}(u_{k})(u_{k}-\bar{x})\|\geq\tfrac{\gamma}{2}\|\bar{x}-u_{k}\|,\quad
k\in I\\!\\!N.$
Then Proposition 2.2(c,f) gives us the representations
$\widetilde{D}H(u_{k})(\bar{x}-u_{k})=DH(u_{k})(\bar{x}-u_{k})=-H^{\prime}(u_{k})(u_{k}-\bar{x})$
for all $k\in I\\!\\!N$, which imply therefore that
$\|H(u_{k})-H(\bar{x})+v\|\geq\tfrac{\gamma}{2}\|\bar{x}-u_{k}\|\;\mbox{
whenever }\;v\in\widetilde{D}H(u_{k})(\bar{x}-u_{k}),\quad k\in I\\!\\!N.$
This clearly contradicts assumption (H2) for $k$ sufficiently large and thus
ensures property (4.14). The equivalence between (H2) and (4.13) follows now
from the implication (H2)$\Longrightarrow$(4.14) and the proof of Proposition
4.8. $\hfill\triangle$
It is well known that, for the class of locally Lipschitzian and directionally
differentiable mappings, condition (4.13) is equivalent to the original
definition of semismoothness; see, e.g., [8, Theorem 7.4.3]. Proposition 4.9
above establishes the equivalence of (4.13) to our major assumption (H2)
provided that $H$ is locally Lipschitzian around the reference point while it
may not be directionally differentiable therein. In fact, it follows from
Example 4.11 that assumption (H2) may hold for locally Lipschitzian functions,
which are not directionally differentiable and hence not semismooth. Let us
now illustrate that (H2) may also be satisfied for non-Lipschitzian mappings,
in which case it is not equivalent to property (4.13).
###### Example 4.10
(assumption (H2) holds for non-Lipschitzian one-to-one mappings). Consider the
mapping $H\colon I\\!\\!R^{2}\to I\\!\\!R^{2}$ defined by
(4.15)
$H(x_{1},x_{2}):=\Big{(}x_{2}\sqrt{|x_{1}|+|x_{2}|^{3}},x_{1}\Big{)}\;\mbox{
for }\;x_{1},x_{2}\in I\\!\\!R.$
It is easy to check that this mapping is one-to-one around $(0,0)$. Focusing
for definiteness on the nonnegative branch of the mapping $H$, observe that at
any point $(x_{1},x_{2})\in I\\!\\!R^{2}$ with either $x_{1},x_{2}>0$, the
classical Jacobian $JH(x_{1},x_{2})$ is computed by
$\displaystyle
JH(x_{1},x_{2})=\left[\begin{array}[]{c}\begin{array}[]{ll}\displaystyle\frac{x_{2}}{2\sqrt{x_{1}+x^{3}_{2}}}\qquad\sqrt{x_{1}+x^{3}_{2}}+\displaystyle\frac{3x^{3}_{2}}{2\sqrt{x_{1}+x^{3}_{2}}}\\\
\qquad 1\qquad\qquad\qquad\qquad 0\end{array}\end{array}\right].$
Setting $x_{1}=x_{2}^{3}$, we see that the first component
$\frac{x_{2}}{2\sqrt{x_{1}+x_{2}^{3}}}=\frac{x_{2}}{2\sqrt{x_{2}^{3}+x_{2}^{3}}}$
is unbounded when $x_{1},x_{2}\downarrow 0$. This implies that the Jacobian
$JH(x_{1},x_{2})$ is unbounded around $(\bar{x}_{1},\bar{x}_{2})=(0,0)$, and
hence $H$ is not locally Lipschitzian around the origin.
Let us finally verify that the underlying assumption (H2) is satisfied for the
mapping $H$ in (4.15). First assume that $x_{1},x_{2}>0$. Then we need to
check that
$\displaystyle\|H(x_{1},x_{2})-H(\bar{x}_{1},\bar{x}_{2})+JH(x_{1},x_{2})(-x_{1},-x_{2})\|$
$\displaystyle=\left|x_{2}\sqrt{x_{1}+x_{2}^{3}}-\frac{x_{1}x_{2}}{2\sqrt{x_{1}+x_{2}^{3}}}-x_{2}\sqrt{x_{1}+x_{2}^{3}}-\frac{3x_{2}^{4}}{2\sqrt{x_{1}+x_{2}^{3}}}\right|$
$\displaystyle=\left|\frac{x_{1}x_{2}}{2\sqrt{x_{1}+x_{2}^{3}}}+\frac{3x_{2}^{4}}{2\sqrt{x_{1}+x_{2}^{3}}}\right|=o\big{(}\sqrt{x_{1}^{2}+x_{2}^{2}}\big{)}.$
The latter surely holds as $(x_{1},x_{2})\to(0,0)$ due to the estimates
$\frac{x_{1}x_{2}}{2\sqrt{x_{1}+x_{2}^{3}}\sqrt{x_{1}^{2}+x_{2}^{2}}}\leq\frac{x_{1}}{\sqrt{x_{1}+x_{2}^{3}}}\leq\sqrt{x_{1}},$
$\frac{3x_{2}^{4}}{2\sqrt{x_{1}+x_{2}^{3}}\sqrt{x_{1}^{2}+x_{2}^{2}}}\leq\frac{3x_{2}^{3}}{2\sqrt{x_{1}+x_{2}^{3}}}\leq
3x_{2},$
which thus justify the fulfillment of assumption (H2) in this case. The other
cases where $x_{1}>0,x_{2}\leq 0$ or $x_{1}<0,x_{2}>0$ or $x_{1}<0,x_{2}\leq
0$ or, finally, $x_{1}=0,x_{2}$ arbitrary (here $H$ is not differentiable) can
be treated in a similar way.
To complete our discussion on the major assumptions in this section, let us
present an example of a locally Lipschitzian function, which satisfies
assumptions (H1) and (H2) being locally one-to-one and metrically regular
around the point in question while not being directionally differentiable and
hence not semismooth at this point.
###### Example 4.11
(non-semismooth but metrically regular, Lipschitzian, and one-to-one functions
satisfying (H1) and (H2)). We construct a function $H\colon[-1,1]\to I\\!\\!R$
in the following way. First set $H(\bar{x}):=0$ at $\bar{x}=0$. Then define
$H$ on the interval $(1/2,1]$ staying between two lines
$\Big{(}1-\frac{1}{2}\Big{)}x+\frac{1}{4}\leq H(x)\leq x$
in the following way: start from $(1,1)$ and let $H$ be continuous piecewise
linear when $x$ goes from 1 to 1/2 with the slope 1+1/4 and then with the
slope $1/2-1/4$ alternatively until $x$ reaches 1/2. Consider further each
interval $(2^{-k},2^{-(k-1)}]$ for $k=2,3,\ldots$ and, starting from the point
$\big{(}2^{-(k-1)},2^{-(k-1)}\big{)}$, define $H$ to be continuous piecewise
linear with the corresponding slopes of either $1+2^{-2k}$ or
$1-2^{-k}-2^{-2k}$ staying between the two lines
(4.18) $\Big{(}1-\frac{1}{2^{k}}\Big{)}x+\frac{1}{2^{2k}}\leq H(x)\leq x.$
Thus we have constructed $H$ on the whole interval $[0,1]$; see Figure 1 for
illustration. On the interval $[-1,0]$, define the function $H$ symmetrically
with respect to the origin. Then it is easy to see that $H$ in continuous on
$[-1,1]$ and satisfies the following properties:
* •
$H$ is clearly Lipschitz continuous around $\bar{x}=0$.
* •
Since $H$ is continuous and monotone with a positive uniform slope, it is one-
to-one and metrically regular around $\bar{x}$, which directly follows, e.g.,
from the coderivative criterion (2.12). This ensures the fulfillment of
assumption (H1) by Proposition 4.4.
* •
To verify assumption (H2), fix $k\in I\\!\\!N$ and $x\in(2^{-k},2^{-(k-1)}]$
and then pick any
$v\in
DH(x)(\bar{x}-x)\subset\Big{[}1-\frac{1}{2^{k}}-\frac{1}{2^{2k}},1+\frac{1}{2^{2k}}\Big{]}(\bar{x}-x).$
Since $\bar{x}=0$, the latter implies that
$-\Big{(}1+\frac{1}{2^{2k}}\Big{)}x\leq
v\leq\Big{(}1-\frac{1}{2^{k}}-\frac{1}{2^{2k}}\Big{)}x$
Thus we have by (4.18) and simple computations that
$|H(x)-H(\bar{x})+v|\leq\frac{1}{2^{k}}|x|+\frac{1}{2^{2k}}+\frac{1}{2^{2k}}=o\Big{(}\frac{1}{2^{k}}\Big{)}=o(|x-\bar{x}|),$
which shows that assumption (H2) is satisfied. In fact, it follows from above
that the latter value is $O(2^{-2k})=O(\|x-\bar{x}\|^{2})$.
* •
Let us finally check that $H$ is not directionally differentiable at
$x_{k}=2^{-k}$ for any $k\in I\\!\\!N$; therefore it is not directionally
differentiable around the reference point $\bar{x}=0$ and hence not semismooth
at $\bar{x}$. Indeed, this follows directly from computing the graphical
derivative by
$DH(x_{k})(1)=\Big{[}1-\frac{1}{2^{k}},1\Big{]},\quad k\in I\\!\\!N,$
which is not single-valued at $x_{k}$, and thus $H$ is not directionally
differentiable at $x_{k}$ due to Proposition 2.2(c,d).
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aa555555 55aaaaaa aa555555 55aaaaaa aa555555 55aaaaaa aa555555 55aaaaaa
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aa555555 55aaaaaa aa555555 55aaaaaa aa555555 55aaaaaa aa555555 55aaaaaa
128128128128128128128128128128128128(12,912)(8562,912)
(8442.000,882.000)(8562.000,912.000)(8442.000,942.000)(8478.000,912.000)(8442.000,882.000)
(912,12)(912,8562)
(942.000,8442.000)(912.000,8562.000)(882.000,8442.000)(912.000,8478.000)(942.000,8442.000)
(8112,912)(8112,732) (4512,912)(4512,732) (2712,912)(2712,732)
60.000(912,912)(8112,8112) (732,2712)(912,2712) (732,4512)(912,4512)
(732,8112)(912,8112) 60.000(8112,6762)(507,1092) 60.000(8112,6312)(417,2487)
60.000(2712,912)(2712,2712) 60.000(4512,912)(4512,4512)
60.000(8112,912)(8112,8112) (8112,8112)(5772,5142)(5052,5052)
(4782,4647)(4647,4647) (4512,4512)(3477,3297)(3207,3207)
(3117,3072)(3072,3027)
$1/4$$1/2$$1$$1$$1/2$$1/4$$H$$x$$\frac{3}{4}x+\frac{1}{16}$$\frac{1}{2}x+\frac{1}{4}$
Figure 1: Construction of the mapping from Example 4.11: Illustration
## 5 Application to the $B$-differentiable Newton Method
In this section we present applications of the graphical derivate-based
generalized Newton method developed above to the $B$-differentiable Newton
method for nonsmooth equations (1.1) originated by Pang [27].
Throughout this section, suppose that $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$
is locally Lipschitzian and directionally differentiable around the reference
solution $\bar{x}$ to (1.1). Proposition 2.2(c,d) yields in this setting that
the generalized Newton equation (3.1) in our Algorithm 3.1 reduces to
(5.1) $-H(x^{k})=H^{\prime}(x^{k};d^{k})$
with respect to the new search direction $d^{k}$ and that the new iterate
$x^{k+1}$ is computed by
(5.2) $x^{k+1}:=x^{k}+d^{k},\quad k=0,1,2,\ldots.$
Note that Pang’s $B$-differentiable Newton method and its further developments
(see, e.g., [8, 9, 28, 30, 31]) are based on Robinson’s notion of the
$B$(ouligand)-derivative [32] for nonsmooth mappings; hence the name. As was
then shown in [37], the $B$-derivative of a locally Lipschitzian mapping
agrees with the classical directional derivative. Thus the iteration scheme in
Pang’s $B$-differentiable method reduces to (5.1) and (5.2) in the
Lipschitzian and directionally differentiable case, and so we keep the
original name of [27].
The next theorem shows what we get from applying our local convergence result
from Theorem 3.3 and the subsequent analysis developed in Sections 3 and 4 to
the $B$-differentiable Newton method. This theorem employs an equivalent
description of assumption (H2) held in the setting under consideration and the
coderivative criterion (2.12) for metric regularity of the underlying
Lipschitzian mapping $H$ ensuring the validity of assumption (H1).
###### Theorem 5.1
(solvability and local convergence of the $B$-differentiable Newton method via
metric regularity). Let $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ be semismooth,
one-to-one, and metrically regular around a reference solution $\bar{x}$ to
(1.1), i.e.,
(5.3) $0\in\partial\langle z,H\rangle(\bar{x})\Longrightarrow z=0.$
Then the $B$-differentiable Newton method (5.1), (5.2) is well defined
$($meaning that equation (5.1) is solvable for $d^{k}$ as $k\in I\\!\\!N)$ and
converges at least superlinearly to the solution $\bar{x}$.
Proof. Since $H$ is locally Lipschitzian around $\bar{x}$, the coderivative
criterion (2.12) is equivalently written in form (5.3) via the limiting
subdifferential (2.10) due to the scalarization formula (2.9). Applying
Theorem 3.3 to the $B$-differentiable Newton method, we need to check that
assumptions (H1) and (H2) are satisfied in the setting under consideration.
Indeed, it follows from Proposition 4.9 and the discussion right after it that
(H2) is equivalent to the semismoothness for locally Lipschitzian and
directionally differentiable mappings. The fulfillment of assumption (H1) is
guaranteed by Proposition 4.4. $\hfill\triangle$
More specific sufficient conditions for the well-posedness and superlinear
convergence of the $B$-differentiable Newton method are formulated via of the
Thibault derivative (4.4).
###### Corollary 5.2
($B$-differentiable Newton method via Thibault’s derivative). Let $H\colon
I\\!\\!R^{n}\to I\\!\\!R^{n}$ be semismooth at the reference solution point
$\bar{x}$ of equation (1.1), and let condition (4.6) be satisfied. Then the
$B$-subdifferential Newton method (5.1), (5.2) is well defined and converges
superlinearly to the solution $\bar{x}$.
Proof. Follows from Theorem 5.1 and Proposition 4.5. $\hfill\triangle$
Observe by the second inclusion in (4.5) that the assumptions of Corollary 5.2
are satisfied when all the matrices from the generalized Jacobian
$\partial_{C}H(\bar{x})$ are nonsingular. In the latter case the solvability
of subproblem (5.1) and the superlinear convergence of the $B$-differentiable
Newton method follow from the results of [31] that in turn improve the
original ones in [27], where $H$ is assumed to be strongly Fréchet
differentiable at the solution point.
Further, it is shown in [30] that the $B$-differentiable method for semismooth
equations (1.1) converges superlinearly to the solution $\bar{x}$ if just
matrices $A\in\partial_{B}H(\bar{x})$ are nonsingular while assuming in
addition that subproblem (5.1) is solvable. As illustrated by the example
presented on pp. 243–244 of [30], without the latter assumption the
$B$-differentiable Newton method may not be well defined for semismooth
mappings $H$ on the plane with all the nonsingular matrices from
$\partial_{B}H(\bar{x})$. We want to emphasize that the solvability assumption
for (5.1) is not imposed in Theorem 5.1—it is ensured by metric regularity.
Let us now discuss interconnections between the metric regularity property of
locally Lipschitzian mappings $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ via its
coderivative characterization (5.3) and the nonsingularity of the generalized
Jacobian and $B$-subdifferential of $H$ at the reference point. To this end,
observe the following relationships between the corresponding constructions.
###### Proposition 5.3
(relationships between the $B$-subdifferential, generalized Jacobian, and
coderivative of Lipschitzian mappings). Let $H\colon I\\!\\!R^{n}\to
I\\!\\!R^{m}$ be locally Lipschitzian around $\bar{x}$. Then we have
(5.4) $\partial_{B}H(\bar{x})^{T}z\subset\partial\langle
z,H\rangle(\bar{x})\subset\partial_{C}H(\bar{x})^{T}z\;\mbox{ for all }\;z\in
I\\!\\!R^{m},$
where both inclusions in (5.4) are generally strict.
Proof. Recall that the middle term in (5.4) expressed via the limiting
subdifferential (2.10) is exactly the coderivative $D^{*}H(\bar{x})(z)$ due to
the scalarization formula (2.9) for locally Lipschitzian mappings. Thus the
second inclusion in (5.4) follows immediately from the well-known equality
(2.8) involving convexification, and it is strict as a rule due to the usual
nonconvexity of the limiting subdifferential; see [24, 35].
To justify the first inclusion in (5.4), observe that the limiting
subdifferential $\partial f(\bar{x})$ of every function $f\colon
I\\!\\!R^{n}\to I\\!\\!R$ continuous around $\bar{x}$ admits the
representation
(5.5) $\partial f(\bar{x})=\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}\widehat{\partial}f(x)$
via the outer limit (1.8) of the Fréchet/regular subdifferentials
(5.6) $\widehat{\partial}f(x):=\Big{\\{}p\in
I\\!\\!R^{n}\Big{|}\;\liminf_{u\to x}\frac{f(u)-f(x)-\langle
p,u-x\rangle}{\|u-x\|}\geq 0\Big{\\}}$
of $f$ at $x$; see, e.g., [24, Theorem 1.89]. We obviously have from (5.6)
that $\widehat{\partial}f(\bar{x})=\\{f^{\prime}(\bar{x})\\}$ if $f$ is
(Fréchet) differentiable at $\bar{x}$ with its derivative/gradient
$f^{\prime}(\bar{x})$.
Having the mapping $H=(h_{1},\ldots,h_{m})\colon I\\!\\!R^{n}\to I\\!\\!R^{m}$
in the proposition and fixing an arbitrary vector
$\bar{z}=(\bar{z}_{1},\ldots,\bar{z}_{m})\in I\\!\\!R^{m}$, form now a scalar
function $f_{\bar{z}}\colon I\\!\\!R^{n}\to I\\!\\!R$ by
(5.7) $f_{\bar{z}}(x):=\sum_{i=1}^{m}\bar{z}_{i}h_{i}(x),\quad x\in
I\\!\\!R^{n}.$
Then the first inclusion in (5.4) amounts to say that
(5.8) $\partial_{B}H(\bar{x})^{T}\bar{z}\subset\partial f_{\bar{z}}(\bar{x}).$
To proceed with proving (5.8), pick any matrix
$A\in\partial_{B}H(\bar{x})^{T}\bar{z}$ and denote by $a_{i}\in I\\!\\!R^{n}$,
$i=1,\ldots,n$, its vector rows. By definition (4.2) of the
$B$-subdifferential $\partial_{B}H(\bar{x})$ there is a sequence
$\\{x^{k}\\}\subset S_{H}$ from the set of differentiability (4.1) such that
$x^{k}\to\bar{x}$ and $H^{\prime}(x^{k})\to A$ as $k\to\infty$. It is clear
from (5.7) that the function $f_{\bar{z}}$ is differentiable at each $x^{k}$
with
$f^{\prime}_{\bar{z}}(x^{k})=\sum_{i=1}^{m}\bar{z}_{i}h^{\prime}_{i}(x^{k})\to\sum_{i=1}^{m}\bar{z}_{i}a_{i}=A^{T}\bar{z}\;\mbox{
as }\;k\to\infty.$
Since $\widehat{\partial}f_{\bar{z}}(x^{k})=\\{f^{\prime}_{\bar{z}}(x^{k})\\}$
at all the points of differentiability, we arrive at (5.8) by representation
(5.5) of the limiting subdifferential and thus justify the first inclusion in
(5.4).
To illustrate that the latter inclusion may be strict, consider the function
$H(x):=|x|$ on $I\\!\\!R$. Then $\partial_{B}H(0)z=\\{-z,z\\}$ for all $z\in
I\\!\\!R$, while
$\displaystyle\partial(zH)(0)=D^{*}H(0)(z)=\left\\{\begin{array}[]{ll}[-z,z]&\mbox{for
}\;z\geq 0,\\\ \\{-z,z\\}&\mbox{for }\;z<0.\end{array}\right.$
This completes the proof of the proposition. $\hfill\triangle$
It follows from Proposition 5.3 in the case of Lipschitzian transformations
$H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$ that the nonsingularity of all the
matrices $A\in\partial_{C}H(\bar{x})$ is a sufficient condition for the metric
regularity of $H$ around $\bar{x}$ due to the coderivative criterion (5.3)
while the nonsingularity of all $A\in\partial_{B}H(\bar{x})$ is a necessary
condition for this property. Note however, as it has been discussed above,
that the nonsingularity condition for $\partial_{B}H(\bar{x})$ alone does not
ensure the solvability of subproblem (5.1) in the $B$-differentiable Newton
method, and thus it cannot be used alone for the justification of algorithm
(5.1), (5.2) in the $B$-differentiable semismooth case. Furthermore, we are
not familiar with any verifiable condition to support the nonsingularity of
$\partial_{B}H(\bar{x})$ in the full justification of the $B$-differentiable
Newton method.
In contrast to this, the metric regularity itself—via its verifiable pointwise
characterization (5.3)—ensures the solvability of (5.1) and fully justifies
the B-differentiable Newton method with its superlinear convergence provided
that the mapping $H$ is semismooth and locally invertible around the reference
solution point. Note that the nonsingularity of the generalized Jacobian
$\partial_{C}H(\bar{x})$ implies not only the metric regularity but
simultaneously the semismoothness and local invertibility of a Lipschitzian
transformation $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$. However, the latter
condition fails to spot a number of important situations when all the
assumptions of Theorem 5.1 are satisfied; see, in particular, Corollary 5.2
and the corresponding conditions in terms of Warga’s derivate containers
discussed right after Corollary 4.5. We refer the reader to the specific
mappings $H\colon I\\!\\!R^{2}\to I\\!\\!R^{2}$ from [17, Example 2.2] and
[40, Example 3.3] that can be used to illustrate the above statement.
## 6 Concluding Remarks
In this paper we develop a new generalized Newton method for solving systems
of nonsmooth equations $H(x)=0$ with $H\colon I\\!\\!R^{n}\to I\\!\\!R^{n}$.
Local superlinear convergence and global (of the Kantorovich type) convergence
results are derived under relatively mild conditions. In particular, the local
Lipschitz continuity and directional differentiability of $H$ are not
necessarily required. We show that the new method and its specifications have
some advantages in comparison with previously known results on the semismooth
and $B$-differentiable versions of the generalized Newton method for nonsmooth
Lipschitz equations.
Our approach is heavily based on advanced tools of variational analysis and
generalized differentiation. The algorithm itself is built by using the
graphical/contingent derivative of $H$, while other graphical derivatives and
coderivatives are employed in formulating appropriate assumptions and proving
solvability and convergence results. The fundamental property of metric
regularity and its pointwise coderivative characterization play a crucial role
in the justification of the algorithm and its satisfactory performance.
In the other lines of developments, it seems appealing to develop an
alternative Newton-type algorithm, which is constructed by using the basic
coderivative instead of the graphical derivative. This requires certain
symmetry assumptions for the given problem, since the coderivative is an
extension of the adjoint derivative operator. Major advantages of a
coderivative-based Newton method would be comprehensive calculus rules held
for the coderivative in contrast to the contingent derivative, complete
coderivative characterizations of Lipschitzian stability, and explicit
calculations of the coderivative in a number of settings important for
applications. The details of these ideas are part of our future research.
## References
* [1] F. J. Aragón Artacho and M. H. Goeffroy, Uniformity and inexact version of a proximal method for metrically regular mappings, J. Math. Anal. Appl., 135 (2007), pp. 168–183.
* [2] J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, in Mathematical Analysis and Applications, L. Nachbin, ed., Academic Press, New York, 1981, pp. 159–229.
* [3] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
* [4] J. V. Burke, A. S. Lewis and M. L. Overton, Variational analysis of functions of the roots of polynomials, Math. Program., 104 (2005), pp. 263–292.
* [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
* [6] T. De Luca, F. Facchinei, and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Program., 75 (1996), pp. 407–439.
* [7] J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, NJ, 1983.
* [8] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalitites and Complementarity Problems, Volumes I and II, Springer, New York, 2003.
* [9] S. P. Han, J.-S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations, Math. Oper. Res., 17 (1992), pp. 586–607.
* [10] A. D. Ioffe, Metric regularity and subdifferential calculus, Russian Math. Surveys, 55 (2000), pp. 501–558.
* [11] N. H. Josephy, Newton’s method for generalized equations and the PIES energy model, Ph.D. Dissertation, Department of Industrial Engineering, Univesity of Wisconsin–Madison, 1979.
* [12] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan, New York, NY, 1964.
* [13] C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity, SIAM J. Optim., 20 (2009), pp. 297–320.
* [14] C. T. Kelley, Solving Nonlinear Equations with Newton’s Method, SIAM, Philadelphia, 2003.
* [15] D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications, Kluwer, Dordrecht, The Netherlands, 2002.
* [16] B. Kummer, Newton’s method fort non-differentiable functions, in Advances in Mathematical Optimization, J. Guddat et al., eds., Akademi-Verlag, Berlin, 1988, pp. 114–125.
* [17] B. Kummer, Lipschitzian inverse functions, directional derivatives, and applications in $C^{1,1}$ optimization, J. Optim. Theory Appl., 70 (1991), 561–582.
* [18] A. S. Lewis, D. R. Luke and J. Malick, Local linear convergence for alternative and averaged nonconvex projections, Found. Comp. Math., 9 (2009), pp. 485–513.
* [19] Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
* [20] R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), pp. 957–972.
* [21] B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), pp. 960–960.
* [22] B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of extremal problems, Soviet Math. Dokl., 22 (1980), pp. 526–530.
* [23] B. S. Mordukhovich, Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 340 (1993), pp. 1–35.
* [24] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II; Applications, Springer, Berlin, 2006.
* [25] B. S. Mordukhovich, J. Peña and V. Roshchina, Applying metric regularity to compute a condition measure of smoothing algorithm for matrix games, http://arxiv.org/abs/1007.4458, to appear in SIAM J. Optim.
* [26] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
* [27] J.-S. Pang, Newton’s method for B-differentiable equations, Math. Oper. Res., 15 (1990), pp. 311–341.
* [28] J.-S. Pang, A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Program., 51 (1991), pp. 101–131.
* [29] J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM J. Optim., 3 (1993), pp. 443–465.
* [30] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227–244.
* [31] L. Qi and J. Sun, A nonsmooth version of Newton’s method, Math. Program., 58 (1993), pp. 353–367.
* [32] S. M. Robinson, Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity, Math. Program. Studies, 30 (1987), pp. 45–66.
* [33] S. M. Robinson, An implicit function theorem for a class of nonsmooth functions, Math. Oper. Res., 16 (1991), pp. 292–309.
* [34] S. M. Robinson, Newton’s method for a class of nonsmooth functions, Set-Valued Anal., 2 (1994), pp. 291–305.
* [35] R. T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer, Berlin, 1998.
* [36] W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
* [37] A. Shapiro, On concepts of directional differentiability, J. Optim. Theory Appl., 66 (1990), pp. 477–487.
* [38] N. Shor, On a class of almost-differentiable functions and on a minimization method for functions from this class, Kibernetika, No. 4 (1972), pp. 65–70.
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|
arxiv-papers
| 2010-09-02T12:34:17 |
2024-09-04T02:49:12.653809
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. Hoheisel, C. Kanzow, B. S. Mordukhovich, H. Phan",
"submitter": "Hung Phan",
"url": "https://arxiv.org/abs/1009.0410"
}
|
1009.0600
|
# Young Supernova Remnants and the Knee in the Cosmic Ray Sectrum
Anatoly Erlykin Physics Department, Durham University, Durham DH1 3LE, UK
Lebedev Physical Institute, Moscow 117924, Russia Tadeusz Wibig Physics
Dept., University of Łódź;
The Andrzej Sołtan Institute for Nuclear Studies, Uniwersytecka 5, 90-950
Łódź, Poland. Arnold W. Wolfendale Physics Department, Durham University,
Durham DH1 3LE, UK wibig@zpk.u.lodz.pl
###### Abstract
It has recently been suggested that neutron stars inside the shells of young
supernova remnants (SNR) are the sources of PeV cosmic rays and that the
interaction of the particles with the radiation field in the SNR causes
electron pair production, which has relevance to recent observations of ’high’
positron fluxes. Furthermore, the character of the interaction is such that
the well-known knee in the cosmic ray energy spectrum can be explained. Our
examination of the mechanism leads us to believe that the required parameters
of SN and pulses are so uncommon that the knee and positron fraction can only
be explained if a single, local and recent SN – and associated pulsar – are
concerned.
Cosmic rays – spectrum, composition, origin
Although it is over 50 years since the ’Knee’ in the cosmic ray (CR) spectrum,
at about 3 PeV, was discovered, its origin is still the subject of
controversy. Many have considered that it is simply due to Galactic Diffusion,
the entrapment of CR by Galactic Magnetic fields becoming increasingly
inefficient above this energy. We ourselves, however, favour a ’single source
model’ in which a single, recent local supernova (SN) is responsible (e.g.
Erlykin and Wolfendale, 1997, 2001). We have argued that the knee is too sharp
to allow Galactic Diffusion to work. This view has relevance to the recent
work by Hu et al (2009), (to be referred to as I) which includes the
observation of a sharp knee in the all-particle spectrum measured by the
impressive Tibet AS-$\gamma$ array (Amenomori et al, 2008).
The advance made by Hu et al. (2009) is their proposal that pulsars (P) close
to their parent young supernovae (SN) can give a spectrum of the appropriate
shape, viz a sharp knee. The mechanism involved is nucleus-SN optical
radiation interactions. A bonus is an explanation of recent excesses of
electron/positron fluxes.
Hu et al. choose appropriate parameters and conclude that there are 3
possibilities: (1) all the sources (to be designated P, SN) are ’standard’,
(2) the average effect is equivalent to using one set of parameters or (3) one
single nearby source dominates the observed fluxes of CR. In what follows we
endeavour to determine which, if any, of these possibilities is valid. Insofar
as they, and we, incline to the view that helium nuclei predominate at the
knee, and, indeed the knee is in the helium component itself, we concentrate
on this component. In I the parameters used (needed) are: an effective black
body radiation temperature of 7000K, a period for acceleration of $\tau$ =
0.19 y and $nc\tau=12.9\times 10^{29}$ cm-2 (where $n$ is the number of
photons per cm3 in the interaction region). We will look at various
parameters, in turn:
* The ’sharpness’ of the predicted helium spectrum.
* The characteristics of pulsars.
* The characteristics of young SN.
In a number of papers we have examined the sharpness of the knee from the
published spectra in many experiments, most recently in Erlykin and Wolfendale
(2001); to be referred to as II. We have defined ’sharpness’, $S$, as the
second differential of the logarithm of the intensity with respect to the
logarithm of the energy. In II we pointed out that GM prediction was $S=0.3$;
anything above this is regarded as needing a contribution from a sharply
peaked (in log intensity versus log energy) spectrum, delivered from a single
source. In II the adopted model gave $S\approx-1.8$ for the all particle
spectrum, using $\Delta\log E=0.2$ in the derivation of $S$. The adopted model
spectrum for helium (the preferred nucleus at the knee) had a very sharp cut-
off in intensity, in fact, an unnecessary feature. In I the helium component
itself has $S=-3.1$ and the all – particle spectrum has $S=-1.7$, (the ’3.1’
is diluted by the smoothly varying intensities for the other mass components).
The spectral knee shown in I for the Tibet-III spectrum is similarly sharp.
The near equality of the S-values from I and II means to us that the single
source model (II) gives an all particle spectrum in good agreement with that
predicted in I.
It remains to examine the reasonableness of the new model involving PSN (I)
and, in particular to see whether ’standard’ sources of the type specified are
likely to occur.
We start with the pulsars. Millisecond pulsars are involved, following the
work of Gaisser et al. (1999). Such pulsars (period $\lesssim$10 ms) and
having high magnetic fields ($\gtrsim$1012 G) are required,. Here we encounter
the first snag, however: it is apparent that such pulsars are very rare.
Millisecond pulsars themselves account for about 3% of all pulsars but most of
these are low magnetic field ’spun-up’ objects and the number of potentially
useful millisecond pulsars is less than 1%. This is too few to give the whole
of the PeV CR spectrum. The recently discovered ’magnetastars’ (Kasen and
Bildsten, 2010), may provide the answer, however. These objects appear to have
periods in the range 2-20ms, magnetic fields $\sim 5\times 10^{14}$ G, and
luminosities above about 1048 erg s-1. The bulk of their energy loss occurs
over days to weeks. Their frequency may be as high as a few %.
Turning to the associated young SN, we agree that of order 1 eV photons are
common and that $nc\tau\sim 10^{30}$ cm${-2}$ is both needed and available.
However, it appears that the condition of a near constant photon energy during
the interaction period ($y\sim 0.2\gamma$) is not (often?) met. The
brightness, too, is, hardly surprisingly, variable, depending on such
parameters as progenitor radius, the ejected mass and the explosion energy
(e.g. Young, 2004). The spectra of different SN differ somewhat from one to
another, too, and are not black body, some also emit copious fluxes of soft
X-rays, eg $3\times 10^{46}$ erg over 1000s for red supergiants (Nakar and
Sari, 2010) which would interact with lower energy nuclei and cause
complications.
Of the above, the time dependence of the mean photon energy is the most
serious and this needs further examination. 3D radioactive transfer
calculations have been made by Kasen et al. (2006) and can be taken as an
example. These workers quote mean temperatures as a function of time from the
explosion as follows.
$6\times 10^{4}$ K (5 days), $3\times 10^{4}$ K (10 days) and $2\times 10^{4}$
K (19 days). Multiplying by the radiation intensity as a function of time
yields intensity times temperature for the times listed as 6, 9 and 4 units.
Thus, there is near constancy of effective intensity over a range of a factor
3 in temperature, with consequent smoothing of the knee in the derived CR
spectrum.
We have examined in more detail the behaviour of the sharpness of different CR
mass components effected by the e+e- pair production mechanism for the model
of expanding SNR when the temperature varies by a factor of a few within days.
For the background photon energies of a fraction of an eV and initial energy
density of 1015 eV/cm3 we found a significant effect above 1015 eV. The
sharpness parameter values obtained form upper limits which could be reduced
in a particular source by the flux of CR produced and propagated undisturbed
later on, when the opaqueness of the SNR diminishes. We have observed for
constant SNR with photon temperatures of 500 K and 1000 K peak values of $S$
equal to about 4 and 2.5, respectively. The temperature change from 1000 K to
500 K was chosen to obtain the peak position for the proton component at
$3\times 10^{15}$ eV and the peak value is $\sim 2$ there (these temperatures
are,in fact,unusually low for a conventional SNR). The maximum of the
sharpness scales with the CR particle gamma factor, is proportional to the CR
particle mass number. The sharpness maximum is related to the initial part of
the steepening of the spectrum: when the flux is reduced by only 20%. The
significant reduction, about a factor 5-10 appears at an energy about 5 times
higher than the one for maximum $S$, independently of the CR particle mass.
The sensitivity of the sharpness, $S$, to the spread of the other parameters
involved needs further examination.
Other problems of the model, which are beyond computation, include the
following:
* i
The effect of the shock wave of the SN, propagating at $\sim 3\times 10^{4}$
km/s, on the emerging optical radiation and on the rotation characteristics of
the pulsar,
* ii
Non-isotropic interactions between SN photons, which are strongly collimated,
and the Pulsar-accelerated nuclei.
* iii
The likelihood of the required helium nuclei dominating in the PeV region, as
distinct from other nuclei.
In II an analysis was made of $S$ as a function of the standard deviation
$\sigma$ of the error in the logarithm of the energy. As a check, the form of
the helium energy spectrum given in I has been taken, and uncertainties in
energy of magnitude $\sigma$ (in $\log E$) have been applied. Similar results
to those in II were derived; these have the following $S(\sigma)$ values with
respect to unity at $\sigma=0$: 0.6 (0.1), 0.33 (0.2) and, extrapolating, 0.1
at $\sigma=0.42$.
Returning to the $S$-value derived for Helium in I (3.0), this resulted in a
fit to the data for the all-particle spectrum with adequate sharpness, as
already mentioned. We consider that a Helium S-value less than 1.0 would be
quite unacceptable. From the above this indicates $\sigma\lesssim$ 0.2, or, in
linear scale, $\pm 60$%.
Applying these argument to the model advanced in I we have the following. To
be acceptable there can be no deviation from the adopted model parameters
overall by more than about $\pm 60$%. For a single source, already there are
difficulties in that the likely SN temperature variation gives a range of
about 3, ie $\approx\pm 50$% (ie $(1+0.5)/(1-0.5)$) and the temperatures
needed are low. Nevertheless, this model is ’in with a chance’.
As applied to the origin of the knee in terms of many standard sources of the
type specified in I, however, there appears to be no possibility at all. SN
vary too much from one to the next and pulsars of the required characteristics
are far too infrequent.
## References
* (1) Amenomori, M., et al 2008, J. Phys. Conf. Ser. 20. 062024.
* (2) Erlykin, A. D., and Wolfendale, A.W., 1997, Astropart. Phys. 7,1; 2001, J.Phys.6, 27,1005.
* (3) Gaisser, T.K., Stanev, T. and Harding, A.K., 1989, ApJ, 345, 423.
* (4) Hu, H-B., et al, 2009, ApJ, 700, L170.
* (5) Kasen, D. and Bildsten, 2010, ApJ, 717, 215.
* (6) Kasen, D., Thomas, R.C. and Nugent, P., 2006, ApJ, 651,366.
* (7) Nakar, E, and Sari R, 2010, arXiv: 1004.2496v1 [astro-ph. HE].
* (8) Young, T.R., 2004, ApJ, 617, 1233.
|
arxiv-papers
| 2010-09-03T07:47:24 |
2024-09-04T02:49:12.667283
|
{
"license": "Public Domain",
"authors": "Anatoly Erlykin, Tadeusz Wibig, Arnold W. Wolfendale",
"submitter": "Tadeusz Wibig",
"url": "https://arxiv.org/abs/1009.0600"
}
|
1009.0956
|
# Axial anomaly contribution to the parity nonconservation effects in atoms
and ions.
Gavriil Shchedrin1 and Leonti Labzowsky1,2 1V. A. Fock Institute of Physics,
St. Petersburg State University, Uljanovskaya 1, Petrodvorets, St. Petersburg
198904, Russia 2Petersburg Nuclear Physics Institute, Gatchina, St.
Petersburg 188350, Russia
###### Abstract
The contribution of the axial triangle anomalous graph to the parity non-
conservation effect in atoms is evaluated. The final answer looks like the
emission of the electric photon by the magnetic dipole. The relative
contribution to the parity non-conservation effect in neutral atoms appears to
be negligible but is essentially larger in case of multicharged ions.
###### pacs:
31.30 Jv, 12.20 Ds, 31.15.-p
The problem of testing the standard model (SM) in the low-energy physics is
one of the interesting topics in physics during the last few decades. The SM
in the low energy limit is tested in particular by observing the parity
nonconservation (PNC) effects in atoms. The most accurate of these experiments
is the experiment with the neutral Cs atom, first proposed in B and performed
with the utmost precision in W .
The basic transition, employed in the Cs experiment was the strongly forbidden
$6s-7s$ transition with the absorption of $M1$ photon. In the real experiment
this very weak transition was opened by the external electric field but it
does not matter for our further derivations. The Feynman graphs illustrating
the PNC effect in Cs are given in Figs. 1(a) and 1(b).
The atomic experiments are indirect and require very accurate calculations of
the PNC effects in Cs to extract the value of the free parameter of the SM,
the Weinberg angle which can be compared with the corresponding high-energy
value. The main difficulty with the PNC calculations in neutral atoms is the
necessity to take into account the electron correlation within the system of
all electrons. Therefore the experiments with much simpler systems, such as
the few-electron highly charge ions (HCIs) would be highly desirable. Several
proposals on the subject were considered in Sof1 ; D ; Lab1 ; Bud .
The radiative corrections to the PNC effect appeared to be important in Cs
calculations to reach the agreement with the high energy SM predictions. These
radiative corrections include electron self-energy, vertex and vacuum
polarization corrections. They are even more important in the case of the HCI.
The entire set of these corrections for neutral Cs atom was calculated in Fl ;
Mil ; Sh ; flamb . The electron self-energy and vertex corrections for HCI
were obtained in Sap ; the vacuum polarization correction was given in Lab2 .
However, the full set of radiative corrections including $Z$-boson loops is
not yet calculated, neither for neutral Cs nor for the HCI. Therefore the
problem cannot be considered as fully solved.
In the present work we will consider a very special radiative correction to
the PNC effect, presented by a triangle Feynman graph, or axial anomaly (AA).
We understand the triangle AA as a fermion loop with at least one weak vertex
Weinberg . Our conclusion will be that in neutral Cs the contribution of the
axial anomaly is negligible, but in HCI it is comparable with the electron
self-energy, vertex, and vacuum polarization corrections.
The leading contribution of the AA to the atomic PNC effect is depicted in
Fig. 1.c. This contribution corresponds to the Adler-Bell-Jackiw anomaly ABJ .
In this work we will concentrate exclusively on this term.
We employ the standard expression for the effective parity nonconserving
interaction of the atomic electron with the nucleus X in the form
$H_{W}=A_{PNC}\rho_{N}(\vec{r})\gamma_{5}$, with
$A_{PNC}=-G_{F}Q_{W}/2\sqrt{2}$, where $G_{F}$ is the Fermi constant and
$Q_{W}$ is the weak charge of the nucleus:
$Q_{W}=-N+Z(1-4\sin^{2}\theta_{w})$ where Z and N are the numbers of protons
and neutrons in the nucleus, and $\theta_{w}$ is the Weinberg angle. The
recently accepted value for this parameter deduced from all available
experiments in the high and low energy physics is
$\sin^{2}\theta_{w}\approx{0.23}$. The function $\rho_{N}(\vec{r})$ represents
the nucleon density distribution within the nucleus, and the $\gamma_{5}$ is
the Dirac pseudoscalar matrix.
Figure 1: The Feynman graphs that describe PNC effect in Cs. The double solid
line denotes the electron in the field of the nucleus. The wavy line denotes
the photon (real or virtual) and the dashed horizontal line with the short fat
solid line at the end denotes the effective weak potential, i.e. the exchange
by $Z$-boson between the atomic electron and the nucleus. Graph (a)
corresponds to the basic $M1$ transition amplitude, the graph (b) corresponds
to the $E1$ transition amplitude, induced by the effective weak potential. The
latter violates the spatial parity and allows for the arrival of np-states in
the electron propagator in Fig 1.b, of which the contribution of $6p,7p$
states states dominates. The standard PNC effect arises due to the
interference between amplitude graphs (a) and (b). Graph (c) corresponds to
the axial anomaly. The ordinary solid line represents the free electron. To
graph (c) the graph with interchanged external photon and $Z$-boson lines
should be added.
We write down the $S$-matrix element corresponding to the amplitude Fig. 1(c)
in the momentum representation:
$\displaystyle
S=-4\pi{e}^{3}\int{\frac{d^{4}p_{1}^{\prime}}{(2\pi)^{4}}}\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p_{2}}{(2\pi)^{4}}\hskip
2.84526pt\overline{\Psi}_{n^{\prime}s}(p_{1})\gamma^{\rho}\Psi_{ns}(p_{1}^{\prime})\frac{g_{\rho{\nu}}}{q^{2}+i\epsilon}$
$\displaystyle\times
Tr\left[\gamma^{\mu}\frac{\not{p}+m_{e}}{p^{2}-m_{e}^{2}}\gamma^{\nu}\frac{\not{p}+\not{q}+m_{e}}{(p+q)^{2}-m_{e}^{2}}\gamma^{\lambda}\gamma^{5}\frac{\not{p}+\not{k}+m_{e}}{(p+k)^{2}-m_{e}^{2}}\right]V^{PNC}_{\lambda}(q-k)A_{\mu}(p_{2}-k).$
(1)
Here $e$ and $m_{e}$ are the electron charge and mass,
$\Psi_{ns}(p)=\Psi_{ns}(\vec{p})\delta(p_{0}-\epsilon_{ns})$ are the wave
functions of the atomic bound electron in the state $ns$ with $\epsilon_{ns}$
being the energy of this state;
$A_{\mu}(p_{2}-k)=(2\pi)^{4}\sqrt{2\pi/k_{0}}e_{\mu}\delta(p_{2}-k)$ is the
wave function of the emitted photon, where $k_{0}\equiv{\omega}$ is the
frequency and $e_{\mu}$ are 4-vector of the polarization for this photon; and
$g_{\mu{\nu}}$ is the pseudo-Euclidean metric tensor.
The potential $V^{PNC}_{\lambda}$ for the parity nonconserving interaction of
the electron with the nucleus looks like
$V^{PNC}_{\lambda}(q-k)=A_{PNC}(q-k)\delta_{\lambda 0}$, where
$\rho(q-k)=(2\pi)\delta(q_{0}-k_{0})$, $q=p_{1}-p_{1}^{\prime}$ is the
transferred momentum and $k$ is the momentum of emitted photon. In Eq(Axial
anomaly contribution to the parity nonconservation effects in atoms and ions.)
we use the relativistic units: $\hbar=c=1$.
To begin with we consider $Z$-boson [with spin $J(Z)=1$] decay into two
photons Anomaly . The Landau theorem forbids this decay because two-photon
system can not exist with full momentum $J=1$ Landau1 ; Ax in contrast to the
allowed decay $\pi_{0}\rightarrow{\gamma\gamma}$ since $J(\pi_{0})=0$ Zuber .
We shall see this directly from the $S$-matrix element and see also nonzero
contribution in the $S$-matrix element corresponding to the virtual photon as
in our case.
The $S$-matrix element is proportional to
$\displaystyle S_{\mu\nu\lambda}(k_{1},k_{2})=\int{d^{4}p}\hskip
5.69054ptTr\left[\gamma^{\mu}\frac{\not{p}\hskip
2.84526pt+m_{e}}{p^{2}-m_{e}^{2}}\gamma^{\nu}\frac{\not{p}-\not{k}_{2}+m_{e}}{(p-k_{2})^{2}-m_{e}^{2}}\gamma^{\lambda}\gamma^{5}\frac{\not{p}+\not{k}_{1}+m_{e}}{(p+k_{1})^{2}-m_{e}^{2}}\right]$
(2)
One has to note that Eq(2) turns to the integral over the the loop in Eq(Axial
anomaly contribution to the parity nonconservation effects in atoms and ions.)
under the change of variables
$k_{1}\rightarrow{k};k_{2}\rightarrow{-q}$. Due to the identity
$Tr\left[\gamma^{5}\gamma^{\tau}\gamma^{\mu}\gamma^{\nu}\gamma^{\lambda}\right]=4i\varepsilon_{\tau\mu\nu\lambda},$
where $\varepsilon_{\tau\mu\nu\lambda}$ is the unit antisymmetric tensor of
the IV rank with definition $\varepsilon_{0123}=-1$, we will have nonzero
contribution in Eq(2) if and only if we retain one momentum (with one
$\gamma$-matrix) or three momenta (with three $\gamma$-matrices) in the square
bracket in Eq(2). All other combinations will give zero result. The integrals
over loop with expressions containing three momenta are convergent and could
be calculated using the standard Feynman parametrization technique Ax . But
the integrals with expressions containing one momentum are divergent and
additional conditions are necessary to get rid of these divergences. These
conditions consist of demanding a gauge-invariance of the $S$-matrix element
and look like
$k_{1\mu}S_{\mu\nu\lambda}=0$ (3) $k_{2\nu}S_{\mu\nu\lambda}=0$ (4)
After imposing Eqs(3) and (4) on the $S$-matrix element (2) it becomes gauge-
invariant and is presented by the finite expression (it is depicted in the
Fig. 1.c with additional graph with interchanged external photon and $Z$-boson
lines):
$\displaystyle
S_{\mu\nu\lambda}(k_{1},k_{2})=J_{110}(k_{1},k_{2}){\varepsilon_{\mu\nu\alpha\beta}}k_{1\alpha}k_{2\beta}(k_{1}+k_{2})_{\lambda}+$
(5) $\displaystyle
J_{101}(k_{1},k_{2})({\varepsilon_{\lambda\nu\alpha\beta}}k_{1\alpha}k_{2\beta}k_{1\mu}+k^{2}_{1}{\varepsilon_{\lambda\mu\nu\alpha}}k_{2\alpha})$
$\displaystyle-
J_{011}(k_{1},k_{2})({\varepsilon_{\lambda\mu\alpha\beta}}k_{1\alpha}k_{2\beta}k_{2\nu}+k^{2}_{2}{\varepsilon_{\lambda\mu\nu\alpha}}k_{1\alpha})$
where
$\displaystyle
J_{rst}(k_{1},k_{2})=-\frac{1}{\pi^{2}}\int^{1}_{0}d{\xi_{1}}\int^{1}_{0}d{\xi_{2}}\int^{1}_{0}d{\xi_{3}}$
(6)
$\displaystyle\frac{({\xi_{1}}^{r}{\xi_{2}}^{s}{\xi_{3}}^{t})\delta(1-\xi_{1}-\xi_{2}-\xi_{3})}{({\xi_{1}}{\xi_{2}}(k_{1}+k_{2})^{2}+{\xi_{1}}{\xi_{3}}k^{2}_{1}+{\xi_{2}}{\xi_{3}}k^{2}_{2}-m^{2})}$
Then due to the transversality conditions for $Z$-boson and real photons
$(k_{1}+k_{2})_{\lambda}\epsilon_{\lambda}=0$, $\epsilon_{1\mu}k_{1\mu}=0$,
$\epsilon_{2\nu}k_{2\nu}=0$ and to the conditions for the real photons
$k^{2}_{1}=0$, $k^{2}_{2}=0$ we get the Landau theorem result
$S_{Z\gamma\gamma}=0$. But in our case one of the photons (e.g. with index 2)
is virtual, as well as $Z$-boson. Therefore the initial $S$-matrix Eq(Axial
anomaly contribution to the parity nonconservation effects in atoms and ions.)
will give nonzero result.
Returning to our former variables $k,q$ we see that the first term in Eq(5) is
proportional to
$V_{0}(q-k)(q-k)_{0}\sim{(q_{0}-k_{0})\delta(q_{0}-k_{0})=0}$ (7)
and is therefore absent.
Thus $S_{\mu\nu\lambda}(k_{1},k_{2})$ in our case reduces to:
$S_{\mu\nu\lambda}(k,q)=-J_{011}(k,q)({\varepsilon_{\lambda\mu\alpha\beta}}k_{\alpha}q_{\beta}q_{\nu}+q^{2}{\varepsilon_{\lambda\mu\nu\alpha}}k_{\alpha})$
(8)
Integrating over the time variables in Eq(Axial anomaly contribution to the
parity nonconservation effects in atoms and ions.), reducing to the three-
dimensional vectors and using the three-dimensional notations
$\gamma_{0}\vec{\gamma}=\vec{\alpha}$,
$\varepsilon_{0\mu\nu\tau}=-\varepsilon_{\mu\nu\tau}$ ($\mu,\nu,\tau=1,2,3$)
results in
$\displaystyle
S=-4\pi{e}^{3}A_{PNC}\delta(E_{f}-E_{in}-\omega_{0})\sqrt{\frac{4\pi}{2\omega_{0}}}\int{\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}}}\frac{d^{3}p_{1}}{(2\pi)^{3}}\hskip
2.84526pt\Psi^{+}_{n^{\prime}s}(\vec{p}_{1})J_{011}(\vec{k},\vec{q})\left[\frac{(\vec{\epsilon}\cdot[\vec{k}\times{\vec{q}}])(\vec{\alpha}\cdot\vec{q})}{q^{2}}+(\vec{\epsilon}\cdot[\vec{\alpha}{\times{\vec{k}}}])\right]\Psi_{ns}(\vec{p}_{1}^{\prime})$
(9)
In the following we represent the $S$-matrix element in the nonrelativistic
limit which is obviously justified in case of Cs atom. Recalling that the
lower component $\chi$ of the Dirac wave function could be expressed via the
upper one as $\chi=\frac{(\vec{\sigma}\cdot\vec{p})}{2m}\varphi$ and using
properties of Pauli-matrices
$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b})=(\vec{a}\cdot\vec{b})+i(\vec{\sigma}\cdot{[\vec{a}\times{\vec{b}}]})$
we obtain the following expression for the square bracket in Eq.(9) (without
the factor $1/2m_{e}$)
$(\vec{q}\cdot\vec{P})\frac{(\vec{q}\cdot[\vec{\epsilon}\times{\vec{k}}])}{\vec{q}^{2}}+(\vec{P}\cdot[\vec{k}\times{\vec{\epsilon}}])+i(\vec{\sigma}\cdot\vec{k})(\vec{q}\cdot\vec{\epsilon})-i(\vec{\sigma}\cdot\vec{\epsilon})(\vec{q}\cdot\vec{k})$
(10)
where $\vec{P}\equiv{\vec{p}_{1}+\vec{p}^{\prime}_{1}}$ and
$\vec{q}=\vec{p}_{1}-\vec{p}^{\prime}_{1}$. Expression (10) changes sign under
the inversion
$\vec{p}_{1}\rightarrow{-\vec{p}_{1}};\vec{p}^{\prime}_{1}\rightarrow{-\vec{p}^{\prime}_{1}}$.
Then, remembering that the wave functions $\Psi_{n^{\prime}s},\Psi_{ns}$ are
of the same parity, the only reason for the whole expression (9) not to be
zero is the presence of the scalar product
$(\vec{k}_{1}\cdot\vec{k}_{2})=-(\vec{k}\cdot\vec{q})$ in the denominator of
Eq(6). In the three-dimensional notations we have to analyze the integral:
$\displaystyle
I(\vec{k},\vec{q})=-\frac{1}{\pi^{2}}\int^{1}_{0}d\xi_{1}\int^{1-\xi_{1}}_{0}d\xi_{2}$
(11)
$\displaystyle\times\frac{\xi_{1}(\xi_{1}+\xi_{2}-1)}{-m_{e}^{2}-2\xi_{1}\xi_{2}(\vec{k}\cdot\vec{q})+\xi_{1}(1-\xi_{1})\vec{q}^{2}}$
The order of magnitude for the atomic electron momenta is
$|\vec{q}|\sim{m_{e}\alpha{Z}}$ and the order of magnitude for the emitted
photon momenta is $|k|=\omega\sim{m_{e}(\alpha{Z})^{2}}$. Therefore, in the
nonrelativistic limit we have to expand the denominator in Eq. (11) and keep
only the leading nonvanishing term.
This yields
$\displaystyle
S=-4\pi{e}^{3}A_{PNC}\delta(E_{f}-E_{in}-\omega_{0})\sqrt{\frac{4\pi}{2\omega_{0}}}\frac{I}{m_{e}^{5}}\int{\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}}}\frac{d^{3}p_{1}}{(2\pi)^{3}}\hskip
2.84526pt\Psi^{+}_{n^{\prime}s}(\vec{p}_{1})[-i(\vec{\sigma}\cdot\vec{\epsilon})(\vec{q}\cdot\vec{k})^{2}]\Psi_{ns}(\vec{p}_{1}^{\prime})$
(12)
where
$I=-1/\pi^{2}\int^{1}_{0}d\xi_{1}\int^{1-\xi_{1}}_{0}d\xi_{2}(\xi^{2}_{1}\xi_{2})(\xi_{1}+\xi_{2}-1)=1/{360\pi^{2}}$
After the Fourier-transform to the coordinate representation in Eq(12) and
using the standard relation between $S$-matrix and the PNC-amplitude $E_{PNC}$
$S=-2\pi{i}E_{PNC}\delta(E_{n^{\prime}s}-E_{ns}-\omega)$, we get following
expression for $E_{PNC}$:
$\displaystyle
E_{PNC}=-2i{e}^{3}\frac{(G_{F}m_{p}^{2})Q_{W}}{2\sqrt{2}}(m_{e}/m_{p})^{2}\frac{\sqrt{2\pi}}{360\pi^{2}}\frac{\omega^{3/2}_{0}}{m_{e}}(\vec{\sigma}\cdot\vec{\epsilon})\varphi^{*}_{6s}(0)\varphi^{{}^{\prime\prime}}_{7s}(0)$
(13)
We would like to note that diagram in Fig. 1.c corresponds to the transition
of the bound electron between states of the same parity $n^{\prime}s,ns$, in
particular between $7s$ and $6s$ states in Cs atom, like in experiments B ; W
. But the resulting $S$-matrix element in Eq. (13) is proportional to
$(\vec{\sigma}\cdot\vec{\epsilon})$, i.e. the magnetic dipole moment of the
electron $\hat{\mu}=\frac{1}{2}\mu_{0}\hat{\sigma}$ emits an electric type of
photon $\vec{\epsilon}$. This could be considered as a unique effect which
occurs due to the parity violation in atoms. Note, that no other radiative
correction to the atomic PNC effect, evaluated up to now, can be interpreted
in such a way. The $T$-invariance of $E_{PNC}$ is satisfied due to the
presence of the imaginary unit in Eq. (13).
For the probability of the process, combining two amplitudes [Figs 1.a and 1.c
correspondingly], we get
$W_{7s\rightarrow{6s}}=W_{M1}+\frac{1}{2j_{0}+1}\sum_{m_{0}m_{1}}2Re\left[E_{M1}E_{PNC}\right]+O\left(E^{2}_{PNC}\right)$
(14)
where $m_{0},m_{1}$ are the angular momentum projections for the initial and
the final electron states.
Performing the summation over the electron angular momentum projections
$m_{0},m_{1}$ and applying the Wigner-Eckart theorem to the product
$E_{M1}E_{PNC}\sim{<n^{\prime}s|\vec{\mu}(\vec{\nu}\times{\vec{\epsilon}})|ns>^{*}<n^{\prime}s|(\vec{\sigma}\cdot{\vec{\epsilon}})|ns>}$
we get the final answer in the form
$W_{7s\rightarrow{6s}}=W_{M1}(1+R(\vec{\nu}\cdot{\vec{s}_{ph}}))$ (15)
where $\nu=\vec{k}/|\vec{k}|$,
$\vec{s}_{ph}=i[\vec{\epsilon}{\times{\vec{\epsilon}^{*}}}]$ is the spin of
the photon and $R$ is so called ”degree of the parity violation”. In our case
$R$ is equal to the ratio $E_{PNC}/E_{M1}$, where the amplitudes are expressed
via the angular reduced matrix elements.
Using the estimate $\varphi(0)\varphi^{\prime\prime}(0)\sim{\alpha}^{5}Z^{3}$
for neutral atoms and
$\varphi(0)\varphi^{\prime\prime}(0)\sim{\alpha}^{5}Z^{5}$ for HCI Landau2 we
get following result for the anomaly contribution to the PNC-amplitudes in the
neutral atoms (Fig. 1.c):
$E^{A_{atoms}}_{PNC}\sim{\frac{1}{360\pi^{2}}\left(\frac{m_{e}}{m_{p}}\right)^{2}\alpha^{3/2}(G_{F}m_{p}^{2})Q_{W}\alpha^{5}Z^{3}}$
(16)
and the estimate
$E^{A_{HCI}}_{PNC}\sim{\frac{1}{360\pi^{2}}\left(\frac{m_{e}}{m_{p}}\right)^{2}\alpha^{3/2}(G_{F}m_{p}^{2})Q_{W}\alpha^{5}Z^{5}}$
(17)
for the anomaly contribution to the PNC effects in HCI.
Using a well-known estimate for the PNC-amplitude X (Fig. 1.b) in neutral
atoms
$E^{B_{atoms}}_{PNC}\sim{\left(\frac{m_{e}}{m_{p}}\right)^{2}\alpha^{3/2}Z^{2}(G_{F}m_{p}^{2})Q_{W}}$
(18)
we get for the relative AA contribution a negligible value
$\sim(10)^{-3}\alpha^{5}Z$. In the $H$-like HCI this relative contribution
will be on the order of $\sim(10)^{-3}\alpha$.
In conclusion we should stress that the observation of the AA contribution to
the PNC effects would be of a special interest since it would be an
observation of AA in atomic physics.
The authors are grateful to the participants of the Petersburg Nuclear Physics
Institute Theoretical Seminar Ya. Asimov, D. Diakonov, E. Drukarev, I.
Dyatlov, L. Lipatov and N. Ural’tsev for many helpful remarks; they are also
indebted to G. Plünien for the valuable discussions and for the hospitality
during their stay in TU Dresden.
This work was supported by RFBR Grant No. 08-02-00026. G. S. was supported
also by non-profit foundation ”Dynasty” and L. L. was supported by the PDSPHS-
MES of RF Grant No. 2.1.1/1136.
## References
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* (3) A. Schafer, G. Soff, P. Indelicato, B. Muller, and W. Greiner, Phys. Rev. A 40, 7362 (1989)
* (4) R. W. Dunford, Phys. Rev. A 54, 3820 (1996)
* (5) L. N. Labzowsky, A. V. Nefiodov, G. Plunien, G. Soff, R. Marrus and D. Liesen, Phys. Rev. A 63, 054105 (2001)
* (6) M. Zolotorev and D. Budker, Phys. Rev. Lett. 78, 4717 (1997)
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* (8) A. I. Milstein, O. P. Sushkov and I. S. Terekhov, Phys. Rev. Lett. 89, 283003 (2002)
* (9) V. M. Shabaev, K. Pachucki, I. I. Tupitsyn and V. A. Yerokhin, Phys. Rev. Lett. 94, 213002 (2005)
* (10) V. V. Flambaum and J. S. M. Ginges, Phys. Rev. A 72 , 052115 (2005)
* (11) J. Sapirstein, K. Pachucki, A. Vietta and K. T. Cheng, Phys. Rev. A 67, 052110 (2003)
* (12) I. Bednyakov, L. Labzowsky, G. Plunien, G. Soff and V. Karasiev, Phys. Rev. A 61, 012103 (1999)
* (13) S. Weinberg, The Quantum Theory of Fields, Vol. 2 Modern Applications, Cambridge University Press, Cambridge, UK, (2000).
* (14) S. Adler, Phys. Rev. 177, 2426 (1969); J. S.Bell and R. Jackiw, Nuovo Cimento A 60, 47, (1969)
* (15) I. B. Khriplovich, Parity Noncoservation in Atomic Phenomena (Gordon and Breach, London 1991).
* (16) L. Rosenberg, Phys. Rev. 129, 2786 - 2788 (1963); M.L. Laursen and M.A. Samuel, Z. Phys. C 14, 325-344 (1982); A. Barroso et al., Z. Phys. C 28, 149-154 (1985)
* (17) L. D. Landau, Dokl. Akad. Nauk. S.S.S.R. 60, 207-209 (1948); C.N.Yang Phys. Rev. 77, 242 - 245 (1950)
* (18) A. I. Akhiezer and V. B. Beresteckij, Quantum Electrodynamics, Interscience Publishers, New York, (1965)
* (19) C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill (1980)
* (20) L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Butterworth-Heinemann, Oxford, (1991)
|
arxiv-papers
| 2010-09-05T23:30:37 |
2024-09-04T02:49:12.686000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gavriil Shchedrin and Leonti Labzowsky",
"submitter": "Gavriil Shchedrin",
"url": "https://arxiv.org/abs/1009.0956"
}
|
1009.1031
|
††thanks: Current address: ICFO–Institut de Ciències Fotòniques, Av. Carl
Friedrich Gauss, num. 3, 08860 Castelldefels (Barcelona), Spain
# A mathematical model of the Mafia game
Piotr Migdał pmigdal@gmail.com http://migdal.wikidot.com/en Institute of
Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
Banacha 2, 02-097 Warsaw, Poland
###### Abstract
Mafia (also called Werewolf) is a party game. The participants are divided
into two competing groups: citizens and a mafia. The objective is to eliminate
the opponent group. The game consists of two consecutive phases (day and
night) and a certain set of actions (e.g. lynching during day). The mafia
members have additional powers (knowing each other, killing during night)
whereas the citizens are more numerous.
We propose a simple mathematical model of the game, which is essentially a
pure death process with discrete time. We find the closed-form solutions for
the mafia winning-chance $w(n,m)$ as well as for the evolution of the game.
Moreover, we investigate the discrete properties of results, as well as their
continuous-time approximations.
It turns out that a relatively small number of the mafia members, i.e.
proportional to the square root of the total number of players, gives equal
winning-chance for both groups. Furthermore, the game strongly depends on the
parity of the total number of players.
## I Introduction
Mafia (also called Werewolf) is a popular party game [1]. The participants
collect in a circle and a game coordinator assigns each player to one of two
groups: a mafia or citizens. Citizens know only their own identity, whereas
mafia members know identity of their fellows. The objective is to eliminate
the opponent group. The game consists of two alternating phases (day and
night). During the day, all players can discuss and vote who they want to
lynch. During the night the mafia kills a citizen of their choice.
There are many variants of the mafia game. The most common modification of the
gameplay is the addition of characters with special abilities. Usually there
are two special citizens: Detective (who checks every night if a chosen person
is in the mafia) and Nurse (who may protect a victim from being killed, if she
chooses correctly).
In this paper we analyze the simplest version of the Mafia game. That is, with
only mafia and citizens, and without any special players or additional rules.
Even if it may not be the most popular variant, it is the best one for
mathematical modelling. Despite the game having a complex psychological
component it is possible to create a stochastic model of the Mafia game. We
consider a game beginning with $n$ players, out of which there are $m$ mafia
members. The main questions we address are:
* •
What is the probability that the mafia wins $w(n,m)$?
* •
How does the dynamics of the game look like? That is, what is the chance that
after a given time there is exactly a certain number of mafia members?
Besides the direct answers (i.e. closed form expressions), we study
approximations, qualitative behavior and some special cases.
There are only a few previous papers on the Mafia game. Works [2] and [3]
basically propose the same model as the one presented in this paper. They
calculate a simple asymptotic formula for the mafia-winning chance,
$w(n,m)\propto m/\sqrt{n}$.
Other research projects concentrate on the psychological aspect of the Mafia
game, in particular — deceiving. Topics analyzed include: patterns of a
deceiver’s interruptions and voice parameters [4], movement of face and hands
[5] and usage of language [6]. Also, the Mafia game is applied as a test for
some videoconferencing setups [7].
The research is motivated by two goals, besides the sheer fun of calculation
and writing. First, a good theoretical model may be useful for the
investigation of the psychological aspect of the Mafia game. Experimental
deviations from the idealized behavior may give a valuable insight into
psychology of strategy choosing, manipulation, deceiving, following others and
hiding identity. Second, when a small, but well-informed and powerful group
fights against the majority, the nature of the process may be similar to that
of the Mafia game. Such real-life phenomena include actions of secret
societies as in high-stake corruption, terrorism and illegal oppositions.
The paper is organized as follows. In Section II we introduce the rules of the
Mafia game and propose a simple mathematical model. Section III contains a
special case of play with only a single mafia member. The results are not only
simple and didactic, but also useful for a more general case. Section IV gives
simple results on the qualitative behavior of the mafia-winning chance.
Section V presents the dynamical aspect of the Mafia game and gives closed-
form expressions for dynamics of the game. We consider both the discrete-time
model and its continuous time approximation. In Section VI we provide the
exact result for the mafia-winning chance, together with its asymptotic
approximation. Section VII concludes the work and gives insight into possible
extensions and applications. Appendices A and B contain derivations of
formulas found in Sec. V.
## II Model
Before proceeding to mathematics, we need to write down our arbitrarily chosen
rules of the Mafia game. Some assumptions are made to simplify the model,
others purely for convenience. Note that this Section is only a hand-waving
transition from the real-life game to its mathematical model.
* •
The game needs $n$ players and one more person to coordinate it.
* •
At the beginning players are randomly divided into ($n-m$) citizens and $m$
mafia members.
* •
Mafia members know the identity of each other, citizens — only their own.
* •
There are two alternating phases, day and night, which together comprise a
turn.
* •
During day, there are two consecutive subphases:
* –
Debate. Everyone still alive can say anything related to accusing or
defending.
* –
Voting. Everyone has exactly one vote, who he wants to lynch. The player who
gets the highest number of votes is eliminated (in case of a tie, a random
’winner’ is eliminated).
* •
During the night:
* –
Mafia jointly decides who they want to kill (eliminate).
* •
The game continues until there is only one group (either the citizens or the
mafia) left. That group wins.
Stated briefly, during the day one player is eliminated (either a citizen or a
mafia member) whereas during the night the mafia kills one citizen.
Consequently, during a single turn the possible transitions are:
$(n,m)\rightarrow(n-2,m)$ and $(n,m)\rightarrow(n-2,m-1)$. Their
probabilities, in principle, depend on $n$, $m$ and the course of the play
(i.e. previous discussions and votings). To start with, we need to get rid of
the psychological aspect, restricting ourselves only to strategic and
probabilistic parts of the Mafia game. It is a coarse approximation, as
psychology plays an essential role in the Mafia game. Nevertheless, even such
a bare model has interesting properties. Moreover, we want to neglect the
discussion phase as it is extremely difficult to formalize in a meaningful and
useful way.
Now, let’s make a hand-waving argument why the transition probability depends
only on the current state $(n,m)$. Consider the scenario, in which players are
lynched at random, that is
$\displaystyle P\left[(n-2,m)|(n,m)\right]=\frac{n-m}{n}\quad\hbox{and}\quad
P\left[(n-2,m-1)|(n,m)\right]=\frac{m}{n}.$ (1)
Let’s denote by $w(n,m)$ the mafia-winning chance, induced by the above
transition probabilities.
Suppose that there is a better strategy (in the common sense of the word) for
citizens, giving it $w^{\prime}(n,m)<w(n,m)$. Then the mafia can force a
random lynch, just by pretending they are citizens. It is possible as:
* •
Each mafia member knows more than any citizen.
* •
In such case there is no way to detect a mafia member.
Suppose that there is a better strategy for the mafia,
$w^{\prime\prime}(n,m)>w(n,m)$. If the citizens are in majority, they can
force a random lynching by the following procedure:
* •
One citizen says "let’s every of us give a number $k_{i}$ from $1$ to $n$, we
will kill the $\sum k_{i}\bmod n$-th".
* •
The number $\sum k_{i}\bmod n$ is random, so is the chosen player.
* •
As there are more citizens than mafia members, mafia votes do not play any
role.
If the citizens are in the minority (i.e. $m>n-m$) no matter what the moves
are, the mafia will win. For $m=n-m$ let’s assume a random player is lynched.
Consequently, random lynch is a well-justified strategy. A more detailed
argument is shown in [2]. As a direct consequence of (1), the mafia winning-
chance can be expressed as a recurrence equation
$\displaystyle w(n,m)=\left\\{\begin{array}[]{ll}0&\hbox{if }m=0,\\\
1&\hbox{if }m>n-m,\\\ \frac{n-m}{n}w(n-2,m)+\frac{m}{n}w(n-2,m-1)&\hbox{in all
other cases}.\\\ \end{array}\right.$ (5)
The above equation is the core of our paper. We investigate it for one mafia
member (Section III), its qualitative behavior (Section IV) and closed-form
solution along with its asymptotic formula (Section VI).
It is important to point out that we may set different boundary conditions,
i.e. in which if there is the same number of citizens and mafia members, mafia
wins. That is, in (5) we may write $(1\hbox{ if }m\geq n-m)$ instead of
$(1\hbox{ if }m>n-m)$. In that case all results can be reproduced, perhaps in
a slightly changed form (i.e. different constants and
even$\leftrightarrow$odd).
Figure 1: A numerical plot of the probability that the mafia wins $w(n,m)$.
## III Game with one mafia member
To start with, we want to consider a very simple case — the game with only a
single mafia member. That is, in this Section we analyze $w(n,1)$. Besides the
simplicity, there are two more motivations: this case gives insight into some
general properties and the results will be useful in the further part of the
paper.
In a game with one mafia member, their winning probability is the chance that
during every lynch a citizen is killed. Bearing in mind that we start the game
with the day phase, we get
$\displaystyle
w(n,1)=\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{1+(n\bmod
2)}{2+(n\bmod 2)}=\frac{(n-1)!!}{n!!}.$ (6)
The above formula has an explicit dependence on the parity of $n$. Even though
it is obvious that addition of $2$ citizens reduces mafia-winning chance
$w(n+2,1)<w(n,1)$, it is not the case for addition of a single citizen. In
fact, addition of an odd player (holding $m=1$) increases the mafia-winning
chance. To understand this let’s use an example. In a play with one citizen
and one mafia member, there is tie in voting, so lynching relies on tossing a
coin and $w(2,1)=1/2$. In a game with two citizen, the mafia also wins after
lynching a citizen (the second one is to be killed during the night). This
time it is more difficult to hunt mafia, and $w(3,1)=2/3$. One may think that
we encountered an issue of boundary conditions (i.e. killing a random player
in a tie) or that we considered too small $n$. Neither is the case. A careful
reader may easily check that changing $(1$ if $m>n-m)$ to $(1$ if $m\geq n-m)$
in (5) we get an analogous phenomena.
We may simply check that adding an odd player always increases the mafia
winning-chance,
$\displaystyle w(2k+1,1)-w(2k,1)$
$\displaystyle=\frac{2k}{2k+1}\cdot\frac{2k-2}{2k-1}\cdot\ldots\cdot\frac{4}{5}\cdot\frac{2}{3}+$
(7)
$\displaystyle-\frac{2k-1}{2k}\cdot\frac{2k-3}{2k-2}\cdot\ldots\cdot\frac{3}{4}\cdot\frac{1}{2}$
$\displaystyle>\frac{\sqrt{(2k+1)(2k-1)}}{2k+1}\cdot\frac{\sqrt{(2k-1)(2k-3)}}{2k-1}\cdot\ldots\cdot\frac{\sqrt{5\cdot
3}}{5}\cdot\frac{\sqrt{3\cdot 1}}{3}+$
$\displaystyle-\frac{2k-1}{\sqrt{(2k+1)(2k-1)}}\cdot\frac{2k-3}{\sqrt{(2k-1)(2k-3)}}\cdot\ldots\cdot\frac{3}{\sqrt{5\cdot
3}}\cdot\frac{1}{\sqrt{3\cdot 1}}=0,$
that is, $w(2k+1,1)>w(2k,1)$.
Let’s consider $w(n,1)$ averaged over neighbouring numbers. We use the
geometric mean, as it simplifies the result
$\displaystyle\sqrt{w(n-1,1)w(n,1)}=\sqrt{\frac{(n-2)!!}{(n-1)!!}\frac{(n-1)!!}{n!!}}=\frac{1}{\sqrt{n}}.$
(8)
The averaging may be considered an approximation in which we get a monotonic
function (without the sawtooth pattern). To get some information on the
dependence of $w(n,1)$ on the parity of $n$ we consider the following ratio
$\displaystyle\frac{w(2k+1,1)}{w(2k,1)}$
$\displaystyle=\frac{(2k)!!}{(2k+1)!!}/\frac{(2k-1)!!}{(2k)!!}=\left(\frac{(2k)!!}{(2k-1)!!}\right)^{2}\frac{1}{2k+1}.$
(9)
The above expression has a limit when $k$ goes to infinity, which can be found
with the aid of the Wallis formula:
$\displaystyle\frac{\pi}{2}=\lim_{k\rightarrow\infty}\left(\frac{(2k)!!}{(2k-1)!!}\right)^{2}\frac{1}{2k+1}.$
(10)
We may develop approximate formulas for the single mafia member winning-
chance, which take into account parity of the number of players
$\displaystyle w(2k,1)$
$\displaystyle=\left(\frac{w(2k+1,1)}{w(2k,1)}\right)^{-\frac{1}{2}}\sqrt{w(2k,1)w(2k+1,1)}$
(11) $\displaystyle\approx\sqrt{\frac{2}{\pi}}\frac{1}{\sqrt{2k+1}},$
$\displaystyle w(2k+1,1)$
$\displaystyle=\left(\frac{w(2k+1,1)}{w(2k,1)}\right)^{\frac{1}{2}}\sqrt{w(2k,1)w(2k+1,1)}$
$\displaystyle\approx\sqrt{\frac{\pi}{2}}\frac{1}{\sqrt{2k+1}}.$
Or, bearing in mind $(n+1)/n\rightarrow 1$,
$\displaystyle w(n,1)$
$\displaystyle\approx\left(\frac{\pi}{2}\right)^{(n\bmod
2)-1/2}\frac{1}{\sqrt{n}}.$ (12)
As we see, the recurrence equation produces results that might be counter-
intuitive. Imagine there are four of us and we are playing with a single mafia
member. His winning-chance is $w(4,1)=\frac{3}{8}=0.375$. Then we invite five
more friends to play as additional citizens. Consequently, the mafia winning-
chance rise to $w(9,1)=\frac{128}{315}\approx 0.406$ (contrary to a naive
expectation).
Figure 2: Plot of the mafia winning-chance for a game with a single mafia
member, that is $w(n,1)$. Dots illustrate the exact result (6), whereas the
lines represent the approximations (12) — solid and dashed line for even and
odd number of players, respectively.
## IV Qualitative properties of $w(n,m)$
It is worth investigating the qualitative behavior of $w(n,m)$. Some
questions, which naturally arise, are:
* •
Does the addition of two citizens increase their winning-chance, i.e. is
$w(n+2,m)<w(n,m)$?
* •
Does the addition of one citizen and one mafia member increase the mafia
winning-chance, i.e. is $w(n+1,m+1)>w(n,m)$?
* •
Does the change of one player from a mafia member to a citizen decrease the
mafia winning-chance, i.e. is $w(n,m+1)>w(n,m)$?
In fact, the above questions are equivalent, as we can see using using the
recurrence relation (5)
$w(n,m)=\frac{n-m}{n}w(n-2,m)+\frac{m}{n}w(n-2,m-1)$
for $n-m\geq m>0$. After some cosmetic arithmetical operations we get
$\displaystyle(n-m)\left[w(n-2,m)-w(n,m)\right]$
$\displaystyle=m\left[w(n,m)-w(n,m-1)\right],$ (13) $\displaystyle
n\left[w(n,m)-w(n-2,m-1)\right]$
$\displaystyle=(n-m)\left[w(n-2,m)-w(n-2,m-1)\right].$
The above relations can be directly translated into
$\displaystyle w(n-2,m)>w(n,m)\Leftrightarrow w(n,m)>w(n-2,m-1)\Leftrightarrow
w(n-2,m)>w(n-2,m-1).$ (14)
In addition to the equivalence of the ’obvious’ inequalities, we are
interested whether they are fulfilled. Let’s utilize the third part of (14) in
the following form:
If a mafia member is changed into a citizen, the mafia winning-chance does not
increase, that is $w(n+1,m+1)\geq w(n+1,m)$. If additionally $n-m\geq m$, the
inequality is strong, $w(n+1,m+1)>w(n+1,m)$.
Let’s introduce an agent to a game with $n$ players among which there are $m$
mafia members. The agent is an additional player who simulates either a
citizen or a mafia member. We are going to show that the agent does not have
to reveal its identity until the very late part of the game.
As citizens are indistinguishable, let’s assume that mafia kills them during
night in a fixed sequence. Let the agent have the last position. During the
lynching there are three possibilities
* •
A citizen dies.
* •
A mafia member dies.
* •
The agent dies.
In the case of the agent’s death, his affiliation has no meaning. In the two
remaining situations, the game continues with the agent. The agent’s
affiliation plays a decisive role in two situations:
* •
When all mafia members are dead.
* •
When $0$–$1$ citizens and $1$–$2$ mafia members remain.
In both cases when the agent is a mafia member, the mafia winning-chance
increase:
Game state $(n,m)$ | $w$ if the agent is of mafia | $w$ if the agent is of citizens
---|---|---
$(n,0)$ | $0$ | $w(n,1)$
$(1,1)$ | $\frac{1}{2}$ | $1$
$(2,1)$ | $\frac{1}{3}$ | $1$
$(3,2)$ | $\frac{1}{4}$ | $1$
The total mafia winning-chance is equal to the mean winning-chance, averaged
over disjoint games. The weight is the probability of achieving a such game.
The change of the agent’s affiliation does not affect the weights, but in the
ending always increases the mafia winning-chance, $w(n+1,m+1)\geq w(n+1,m)$.
Additionally, for $n-m\geq m\geq 1$ there is a game in which all mafia members
are killed and the agent’s affiliation plays a role, hence
$w(n+1,m+1)>w(n+1,m)$.
Let’s show one more property — when $n-m\geq m$ adding an odd player always
increases the mafia winning-chance, not only in games with one mafia member
(7). We will use mathematical induction, using the boundary conditions (5) and
the already shown properties of $w(n,1)$ as the basis. The inductive
assumption is that the property holds for the total number of players $n\leq
2k+1$, and the inductive step is the following:
$\displaystyle w(2(k+1)+1,m)$
$\displaystyle=\frac{2k+3-m}{2k+3}w(2k+1,m)+\frac{m}{2k+3}w(2k+1,m-1)$ (15)
$\displaystyle>\frac{2k+3-m}{2k+3}w(2k,m)+\frac{m}{2k+3}w(2k,m-1)$
$\displaystyle>\frac{2k+2-m}{2k+2}w(2k,m)+\frac{m}{2k+2}w(2k,m-1)$
$\displaystyle=w(2(k+1),m),$
where we subsequently used the inductive assumption and made use of the fact
$w(2k,m)>w(2k,m-1)$.
## V Evolution
In the two previous Sections we considered the mafia winning-chance with
respect to the initial number of players ($N$) and mafia members ($M$). It may
be interesting to explore dynamics of the play, that is, analyze the
probability $p_{m}(t)$ that after the $t$-th turn there will be exactly $m$
mafia members. Even though this goes beyond the (5), it is a direct
consequence of the random lynching model (1). Throughout this section we use
capital letters $N$ and $M$ to denote the initial conditions.
Note that every turn, or day and night, two players are killed, so the total
number of player decreases $n(t)=N-2t$.
### V.1 Discrete time
The evolution of each $p_{m}(t)$ is governed by a set of recurrent equations
$\displaystyle
p_{m}(t+1)=\frac{(N-2t)-m}{N-2t}p_{m}(t)+\frac{m+1}{N-2t}p_{m+1}(t),$ (16)
where $m\in\mathbbm{N}$ and with the initial condition $p_{m}(0)=\delta_{mM}$.
Note that the above is a stochastic process with $M+1$ distinct states and
$(m)\rightarrow(m-1)$. In general, such a Markov process is called a pure
death process (a subclass of birth and death processes). In our case the
transition probabilities change with time, which is a slight complication.
It is essential to point that the equations in (16) are correct only when
$m\leq n(t)$, that is, when there is at least one citizen alive and the game
can be still played. Otherwise they have no meaning. But why does an erroneous
equation for $m>n(t)$ not spoil states with $m\leq n(t)$? For a given time $t$
let $p_{m}(t)$ be the erroneous probability with the smallest index, i.e.
$m=N-2t+1$. In the next turn it affects $p_{m-1}(t+1)$. But, now the smallest
wrong index is $N-2(t+1)+1=m-2<m-1$. So the error propagates more slowly than
the exclusion of the states.
The closed form solution for $p_{m}(t)$ reads
$\displaystyle p_{m}(t)=\sum_{i=m}^{M}{M\choose i}{i\choose
m}(-1)^{i-m}\frac{(N-2t)!!}{N!!}\frac{(N-i)!!}{(N-2t-i)!!}$ (17)
and its derivation is presented in Appendix A. An example is plotted in Fig. 3
(a).
However, the expression above, although exact, is fairly complicated. Let’s
analyze the mean number of mafia players after $t$ days and nights, that is
$\displaystyle\langle m\rangle(t)=\sum_{m=0}^{M}mp_{m}(t).$ (18)
In birth and death processes with transition coefficients proportional to
state’s label (as radioactive decay process or Yule process) such mean may be
obtained by a straightforward calculation with the use of (16)
$\displaystyle\langle m\rangle(t)$ $\displaystyle=\frac{N-2t+1}{N-2t+2}\langle
m\rangle(t-1)=\left(\prod_{i=0}^{t-1}\frac{N-2i-1}{N-2i}\right)M.$ (19)
The result is correct only when $N-2t-M\geq 0$, that is, the average is taken
over probabilities of the correct states.
### V.2 Continuous time approximation
Usually differential equations are simpler than difference equations. In this
Section we crudely approximate the discrete-time evolution (16) by its
continuous-time version. We know that
$\displaystyle
p_{m}(t+1)-p_{m}(t)\xleftarrow[\Delta=1]{}\frac{p_{m}(t+\Delta)-p_{m}(t)}{\Delta}\xrightarrow[\Delta\rightarrow
0]{}\frac{d}{dt}p_{m}(t).$ (20)
Let’s change in (16) the difference $p_{m}(t+1)-p_{m}(t)$ into the
differential $\frac{d}{dt}\tilde{p}_{m}(t)$
$\displaystyle\frac{d}{dt}\tilde{p}_{m}(t)=-\frac{m}{N-2t}\tilde{p}_{m}(t)+\frac{m+1}{N-2t}\tilde{p}_{m+1}(t)$
(21)
where $m\in\mathbbm{N}$ and with the initial condition
$\tilde{p}_{m}(0)=\delta_{mM}$. When a relative change of the function
derivative is over unit length interval $[t,t+1]$, the approximation should be
good $\tilde{p}_{m}(t)\approx p_{m}(t)$. Intuitively speaking, when the chance
of lynching a mafia member during one day is small, such an approximation is
justified. However, in the course of this paper we do not estimate the
approximation error.
The solution of the differential equation (21) reads
$\displaystyle\tilde{p}_{m}(t)={M\choose
m}\left(1-\sqrt{1-\frac{2t}{N}}\right)^{M-m}\left(\sqrt{1-\frac{2t}{N}}\right)^{m},$
(22)
see Appendix B for detailed derivation. This is more convenient to work with
than the exact expression (17). An example is plotted in Fig. 3 (a).
For example, maxima of $\tilde{p}_{m}(t)$ are easily found
$\displaystyle t_{m}=\frac{N}{2}\left(1-\left(\frac{m}{M}\right)^{2}\right).$
(23)
The mean number of mafia players, defined as in (18), is
$\displaystyle\langle\tilde{m}\rangle(t)=M\sqrt{1-\frac{2t}{N}},$ (24)
which is a simpler expression than its analogue (19). The comparison of the
discrete-time formula with its continuous-time approximation is in Fig. 3 (b).
|
---|---
(a) | (b)
Figure 3: Dynamics of the Mafia game for the initial conditions $N=32$ and
$M=4$. We compare results for the discrete-time evolution with its continuous-
time counterparts. (a) The probability that after $t$ turns there are exactly
$m$ mafia members alive — the exact results $p_{m}(t)$ (dots) and their
approximations $\tilde{p}_{m}(t)$ (lines). (b) The mean number mafia members —
the discrete result $\langle m\rangle(t)$ (dots) and its continuous time
variant $\langle\tilde{m}\rangle(t)$ (lines).
It is tempting to estimate the mafia winning-chance. As $\tilde{p}_{0}(t)$ is
the probability that the citizens won, setting time for which there is only
one player left gives the total citizen winning-chance:
$\displaystyle\tilde{w}(n,m)$
$\displaystyle=1-\tilde{p}_{0}\left({\scriptstyle\frac{n-1}{2}}\right)=1-\left(1-\sqrt{1-\frac{n-1}{n}}\right)^{m}$
(25) $\displaystyle\approx\frac{m}{\sqrt{n}}.$
Not surprisingly, in the continuous-time approximation, there is no explicit
dependence on parity (the discrete properties are lost).
## VI General solution
The result for $p_{m}(t)$ (17) is not only interesting by itself — it also
gives us the solution for the mafia winning-chance (5). Note that $p_{0}(t)$
is the probability that after $t$ turns citizens win. When total number of
players is odd, game lasts for $t_{odd}=(N-1)/2$ days and nights, leaving 1
player alive. Thus $p_{0}(t_{odd})$ is the probability the citizen win the
game. For an even number of players, after $t_{even}=N/2$ turns there are no
players alive. But now $p_{0}(t_{even})$ is the probability that there are no
mafia members, so in fact — no-one was killed during the last night and
citizen win. So $w(n,m)=1-p_{0}({\scriptstyle\frac{N-N\bmod 2}{2}})$.
Consequently, the closed form formula for the mafia winning-chance reads
$\displaystyle w(n,m)$ $\displaystyle=1-\sum_{i=0}^{m}{m\choose
i}(-1)^{i}\frac{(n-i)!!}{n!!((n\bmod 2)-i)!!}.$ (26)
Let’s show what we have proven for a game with only one mafia member — the
asymptotic behavior (12).
To get the asymptotic formula for $w(n,m)$ it suffices to notice that in the
limit $n\rightarrow\infty$ (taken over a selected parity) only the first two
terms of the sum contribute. For $i=0$ the term is equal to $1$. The terms
with $i>1$ do not matter as $(n-i)!!/(n-1)!!\rightarrow 0$. Consequently, we
may write
$\displaystyle w(n,m)$ $\displaystyle\approx m\frac{(n-1)!!}{n!!}$ (27)
$\displaystyle\approx\left(\frac{\pi}{2}\right)^{(n\bmod
2)-1/2}\frac{m}{\sqrt{n}},$
where we used the result from game with one mafia member, that is,
approximating (6) with (12). The approximate formula (27) can be formally
expressed as $\frac{w(n,m)}{m/\sqrt{m}}$
It is tempting to ask for the optimal number of mafia members $m_{opt}$, for a
given number of players. An interesting game is one in which both groups have
the same chance of winning
$\displaystyle m_{opt}$
$\displaystyle\approx\frac{1}{2}\left(\frac{\pi}{2}\right)^{-(n\bmod
2)+1/2}\sqrt{n},$ (28)
which are plotted in Fig. 4.
Figure 4: Optimal number of mafia members for a given number of players.
Points show numerical results, that is, number of mafia members $m_{opt}$ for
which $w(n,m_{opt})$ is the closest to $1/2$. Lines show the approximation
(28). Dots and solid line are for even number of players, whereas squares and
dashed line — odd.
## VII Conclusion
In this paper we make, and solve, a mathematical model of the Mafia game. The
starting point of our calculations is random lynching process (1).
The main interest of this research is analysis of the mafia winning-chance
$w(n,m)$ (5). We find the closed-form solution (26), as well as its asymptotic
approximation (27). We prove some discrete properties of $w(n,m)$, namely that
adding two citizens decreases the mafia winning-chance, whereas adding a
citizen (which is an odd player) — increases.
The dynamics of the game is also of our interest. We calculate the probability
that after $t$ days and nights there is exactly a certain number of mafia
members alive (16). We obtain closed-form expressions (17) and their
convenient approximations (21).
In every realistic Mafia game, there is a psychological part. A dry
mathematical model may be a backbone of a more complex one, but certainly, is
not enough to describe a real game. During course of play, citizens gain some
information — either by discovering another’s identity by themselves, or
trying to catch messages. Furthermore, voting may be subject to some kind of
witch-hunt mentality. Moreover, rarely are all players the same — usually
there are ones with higher and lower influence on the others.
All these processes may be investigated mathematically, or numerically. There
is still a lot of challenge in finding an appropriate model and obtaining the
results. Comparison of the theory with the experiment may be crucial.
Out of author’s personal experience, in games when players don’t know each
other, usually a random person is killed (e.g. a person who is always nervous,
not just because of being in the mafia). When they know each other very well,
very dim signals can be used to reveal one’s identity.
###### Acknowledgements.
The paper is roughly a translation of my bachelor’s thesis Matematyczny model
gry w mafię (in Polish), written under supervision of prof. Jacek Miękisz at
the Faculty of Mathematics, Informatics and Mechanics, University of Warsaw. I
would like to thank G. John Lapeyre and Marcin Kotowski for valuable comments
and language corrections. Additionally, I am grateful to all with whom I have
played the Mafia game, with a special emphasis on friends from the Polish
Children’s Fund. Personally, I am a fan of Ktulu, a rather complex variant of
the Mafia game. In Ktulu there are 4 distinct factions, every player has a
different special ability and the winning conditions rely on a special item.
## Appendix A Solution of difference equation for $p_{m}(t)$
We want to solve the following set of difference equations (16),
$\displaystyle
p_{m}(t+1)=\frac{(N-2t)-m}{N-2t}p_{m}(t)+\frac{m+1}{N-2t}p_{m+1}(t)\quad:\quad
m\in\mathbbm{Z}$
with the initial condition $p_{m}(0)=\delta_{mM}$. Let’s introduce a
generating function
$\displaystyle F(t,z)=\sum_{m=0}^{\infty}p_{m}(t)z^{m},$ (29)
with the respective initial condition $F(0,z)=z^{M}$. After side-by-side
multiplication of (16) by $z^{m}$ and summation we get
$\displaystyle F(t+1,z)$ $\displaystyle=F(t,z)-\frac{z}{N-2t}\frac{\partial
F(t,z)}{\partial z}+\frac{1}{N-2t}\frac{\partial F(t,z)}{\partial z}$ (30)
$\displaystyle=\underbrace{\left(1-\frac{z-1}{N-2t}\frac{\partial}{\partial
z}\right)}_{\mathbf{B}(t)}F(t,z).$
The linear operator $\mathbf{B}(t)$ has the eigenvectors of the form
$(z-1)^{k}$ with the corresponding eigenvalues $(1-k/(N-2t))$. As the
eigenvectors do not depend on $t$, the computation of $F(t,z)$ is
straightforward:
$\displaystyle F(t,z)$
$\displaystyle=\mathbf{B}(t-1)\mathbf{B}(t-2)\cdots\mathbf{B}(0)z^{M}$ (31)
$\displaystyle=\sum_{i=0}^{M}{M\choose
i}(z-1)^{i}\left(1-\frac{i}{N-2(t-1)}\right)\left(1-\frac{i}{N-2(t-2)}\right)\cdots\left(1-\frac{k}{N}\right)$
$\displaystyle=\sum_{k=0}^{M}{M\choose
i}(z-1)^{i}\frac{(N-2t)!!}{N!!}\frac{(N-i)!!}{(N-2t-i)!!}.$
Applying once more the binomial theorem, and comparing the result (31) to the
definition of the generating function (29) we get
$\displaystyle p_{m}(t)=\sum_{i=m}^{M}{M\choose i}{i\choose
m}(-1)^{i-m}\frac{(N-2t)!!}{N!!}\frac{(N-i)!!}{(N-2t-i)!!},$
or (17).
## Appendix B Solution of differential equation for $\tilde{p}_{m}(t)$
We want to solve the following set of differential equations (21),
$\displaystyle\frac{d}{dt}\tilde{p}_{m}(t)=-\frac{m}{N-2t}\tilde{p}_{m}(t)+\frac{m+1}{N-2t}\tilde{p}_{m+1}(t)\quad:\quad
m\in\mathbbm{Z},$
with the initial condition $\tilde{p}(0)=\delta_{mM}$. Once again, we employ a
generating function
$\displaystyle G(t,z)=\sum_{m=0}^{\infty}\tilde{p}_{m}(t)z^{m},$ (32)
with the respective initial condition $G(0,z)=z^{M}$. We get the partial
differential equation
$\displaystyle\frac{\partial G(t,z)}{\partial
t}=\frac{(-z+1)}{N-2t}\frac{\partial G(t,z)}{\partial z},$ (33)
which we want to solve by the method of characteristics. Let’s have integral
curves of $G(t,z)$ in the form of $(t(\varphi),z(\varphi))$,
$\displaystyle\frac{\partial G(t,z)}{\partial
t}\frac{dt}{d\varphi}+\frac{\partial G(t,z)}{\partial
z}\frac{dz}{d\varphi}=0.$ (34)
Comparison of (33) with (34) yields in
$\displaystyle\frac{dt}{N-2t}=d\varphi=\frac{dz}{z-1},$ (35)
or
$\displaystyle G(t,z)$
$\displaystyle=f\left(\ln\left|\sqrt{N-2t}(1-z)\right|\right).$ (36)
All we need is to find a function $f(x)$ that satisfies the initial condition.
The explicit form of the generating function reads
$\displaystyle
G(t,z)=\left(1-\sqrt{1-{\scriptstyle\frac{2t}{N}}}(1-z)\right)^{M},$ (37)
which gives the result
$\displaystyle\tilde{p}_{m}(t)={M\choose
m}\left(1-\sqrt{1-\frac{2t}{N}}\right)^{M-m}\left(\sqrt{1-\frac{2t}{N}}\right)^{m}$
or (22).
## References
* [1] Wikipedia. Mafia (party game). Wikipedia, The Free Encyclopedia [Online; accessed 28-July-2011].
* [2] M. Braverman, O. Etesami, E. Mossel. Mafia: A theoretical study of players and coalitions in a partial information environment. The Annals of Applied Probability, 18, 825, (2008), doi:10.1214/07-AAP456, arXiv:math/0609534v4 [math.PR].
* [3] E. Yao. A Theoretical Study of Mafia Games (2008), arXiv:0804.0071v1 [math.PR].
* [4] G. Chittaranjan, H. Hung. Are you a werewolf? Detecting deceptive roles and outcomes in a conversational role-playing game. International Conference on Acoustics, Speech and Signal Processing (ICASSP), 5334, (2010), doi:10.1109/ICASSP.2010.5494961.
* [5] F. Xia, H. Wang, J. Huang. Deception Detection via Behavioral State Analysis. International Conference on Affective Computing and Intelligent Interaction, (2007).
* [6] L. Zhou, Y. Sung. Cues to Deception in Online Chinese Groups. Proceedings of the 41st Hawaii International Conference on System Sciences, 146, (2008), doi:10.1109/HICSS.2008.109.
* [7] A. L. Batcheller, B. Hilligoss, K. Nam, E. Rader, M. Rey-Babarro, X. Zhou. Testing the technology: Playing Games with Video Conferencing. Proceedings of the SIGCHI conference on Human factors in computing systems, 849, (2007). doi:10.1145/1240624.1240751.
|
arxiv-papers
| 2010-09-06T11:30:24 |
2024-09-04T02:49:12.692500
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Piotr Migda{\\l}",
"submitter": "Piotr Migda{\\l}",
"url": "https://arxiv.org/abs/1009.1031"
}
|
1009.1178
|
Calibrations in hyperkähler geometry
Gueo Grantcharov, Misha Verbitsky111Misha Verbitsky is partially supported by
RFBR grant 10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, Simons-IUM
fellowship, Science Foundation of the SU-HSE award No. 10-09-0015 and AG
Laboratory HSE, RF government grant, ag. 11.G34.31.0023.
Abstract
We describe a family of calibrations arising naturally on a hyperkähler
manifold $M$. These calibrations calibrate the holomorphic Lagrangian,
holomorphic isotropic and holomorphic coisotropic subvarieties. When $M$ is an
HKT (hyperkähler with torsion) manifold with holonomy $SL(n,{\mathbb{H}})$, we
construct another family of calibrations $\Phi_{i}$, which calibrates
holomorphic Lagrangian and holomorphic coisotropic subvarieties. The
calibrations $\Phi_{i}$ are (generally speaking) not parallel with respect to
any torsion-free connection on $M$.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Calibrations in Riemannian geometry
2. 2.2 Hyperkähler manifolds and calibrations
3. 2.3 Calibrations in HKT-geometry
3. 3 $SL(n,{\mathbb{H}})$-manifolds
1. 3.1 An introduction to $SL(n,{\mathbb{H}})$-geometry
2. 3.2 Balanced HKT-manifolds
4. 4 Differential forms on hypercomplex manifolds
1. 4.1 The quaternionic Dolbeault complex
2. 4.2 Positive $(2,0)$-forms on hypercomplex manifolds
3. 4.3 The map ${\cal V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$ on $SL(n,{\mathbb{H}})$-manifolds
4. 4.4 Algebra generated by $\omega_{I}$, $\omega_{J}$, $\omega_{K}$
5. 5 Calibrations on hyperkähler manifolds
1. 5.1 Hodge decomposition and $U(1)$-action
2. 5.2 An $SU(2)$-invariant calibration
3. 5.3 A holomorphic Lagrangian calibration
4. 5.4 Isotropic and coisotropic calibrations
5. 5.5 Holomorphic Lagrangian calibrations of degree two
6. 5.6 Examples
6. 6 Calibrations on $SL(n,{\mathbb{H}})$-manifolds
## 1 Introduction
The theory of calibrations was developed by R. Harvey and B. Lawson in [HL],
and proved to be very useful in describing the geometric structures associated
with special holonomies. Since then calibrations have become a central notion
in many geometric developments in string physics and M-theory. Up to dimension
8, the calibrations are thoroughly studied and pretty much understood ([DHM]),
but in the higher dimensions, the classification problem seems to be immense.
Even in more special situations, such as in hyperkähler geometry, the problem
of classification of natural111For hyperkähler geometry, “natural” would mean
“$Sp(n)$-invariant”. calibrations is unsolved.
On a Kähler manifold, the normalized power of the Kähler form
$\frac{\omega^{p}}{p!}$ is a calibration. A subvariety is complex analytic if
and only if it is calibrated. This is actually very easy to see, because a
subspace $V\subset TM$ is a face of $\frac{\omega^{p}}{p!}$ if and only if $V$
is complex linear (this follows from the so-called “Wirtinger inequalities”,
see e.g. [HL]).
In this paper we study a family of calibrations which appear naturally in
quaternionic geometry, and describe the corresponding calibrated subvarieties.
These calibrations are in many ways analogous to the powers of the Kähler
form. We define several new calibrations, for hyperkähler, hypercomplex and
HKT-geometry. From the calibration-theoretic point of view, the last of these
is most interesting, because it is (generally speaking) not preserved by any
torsionless connection on $M$. Some of these forms were considered previously
in [V6, AV2, V7].
In hyperkähler geometry, the role of a Kähler form is played by a 4-form
$\Theta:=\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2}$. In Section 5.2 we show
that the normalized powers $\Theta^{p}$ are calibrations. It is easy to see
that $V\subset TM$ is a face of $\Theta$ if and only if $V$ is a quaternionic
subspace (5.2).
The corresponding calibrated subvarieties are those which are complex analytic
with respect to $I$, $J$ and $K$. Such subvarieties are called trianalytic. In
[V1, V2], the theory of trianalytic subvarieties was developed to some extent.
It was shown that the trianalytic subvarieties admit a canonical
desingularizaton, which is hyperkähler. Also it was shown that any complex
analytic subvariety of $(M,I)$ is trianalytic, if the complex structure $I$ is
generic in its twistor family.
Any homogeneous polynomial $P(x,y,z)$ of degree $p$ gives a closed $2p$-form
$P(\omega_{I},\omega_{J},\omega_{K})$ on $M$, and (when the holonomy of $M$ is
maximal) all parallel differential forms on $M$ are obtained this way. When
$P(x,y,z)=\frac{x^{p}}{p!}$, it is a Kähler calibration, when
$P(x,y,z)=c_{p}(x^{2}+y^{2}+z^{2})^{p}$, where
$c_{p}=\sum_{k=0}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$, it is the
trianalytic calibration defined above( Theorem 5.3). It would be interesting
to classify all calibrations obtained this way.
The calibrations $\Psi_{k}$ and $\Phi_{n+k}$ we study in this paper are also
polynomials on $\omega_{I},\omega_{J},\omega_{K}$. These calibrations are
called holomorphic Lagrangian, holomorphic isotropic and holomorphic
coisotropic calibrations. The form $\Psi_{k}$ is obtained as a
$(k,k)$-component of
$\operatorname{Re}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{k}$, normalized in
appropriate way, where $\omega_{I}-\sqrt{-1}\>\omega_{K}$ is a holomorphic
symplectic form on $(M,J)$, and the $(k,k)$-part is taken with respect to the
complex structure $I$. In [V6, AV2] it was proven that this form is closed and
weakly positive.
We show in Section 5.4 that a subvariety $Z\subset M$ is calibrated by
$\Psi_{k}$ if and only if $Z$ is holomorphic Lagrangian in $(M,I)$ (for
$k=\frac{1}{2}\dim_{\mathbb{C}}M$) and isotropic (for
$k<\frac{1}{2}\dim_{\mathbb{C}}M$) (5.3, 5.4). Note that holomorphic
Lagrangian calibrations have been found previously in [BrH] in dimension
eight.
In [F] a different holomorphic Lagrangian calibration in any dimension was
constructed as part of an investigation relating the faces of some
calibrations to intersecting supersymmetric branes in M-theory. In String
Theory the holomorphic Lagrangian submanifolds were related to 3-dimensional
topological field theory with target hyperkähler manifold [KRS]. In Section
5.6 we provide some examples of holomorphic Lagrangian subvarieties of
hypercomplex manifolds which are not hyperkähler.
The proof of this result relies on a particular partial order defined on the
set of precalibrations. We say that $\eta\preceq\eta_{1}$ if all faces of
$\eta$ are also faces of $\eta_{1}$. For instance, the calibrations
$c_{p}\Theta^{p}$,
$c_{p}=\sum_{k=0}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$, and
$\frac{\omega_{I}^{k}}{k!}$ defined above can be compared:
$c_{p}\Theta^{p}\preceq\frac{\omega_{I}^{2p}}{(2p)!}$
because the faces of $c_{p}\Theta^{p}$ are quaternionic subspaces in $TM$, and
the faces of $\frac{\omega_{I}^{2p}}{(2p)!}$ are complex subspaces (5.2).
Let $\rho$ be a precalibration on a complex manifold (2.1), and $\rho^{p,p}$
be its $(p,p)$-part. We show that a plane $V\subset TM$ is a face of
$\rho^{p,p}$ if and only for $\zeta(V)$ is a face of $\rho$ for all $\zeta\in
U(1)$, for the standard $U(1)$-action on $TM$ (5.1).
Applying this result to the special Lagrangian calibration on $(M,J)$ defined
in [HL] (see also [McL]), we obtain the form $\Psi_{n}$,
$n=\dim_{\mathbb{H}}M$, which calibrates complex analytic Lagrangian
subvarieties on $(M,I)$ (these subvarieties are known to be special Lagrangian
on $(M,J)$; see e.g. [Hit]). This argument is not hard to generalize to
arbitrary dimension.
In most cases listed in [HL] and elsewhere, a calibration form is parallel
with respect to the Levi-Civita connection. An interesting side effect of our
construction of holomorphic Lagrangian calibrations is an appearance of a
family of calibrations which are not parallel, under any torsionless
connection (6). These calibrations are associated with the so-called HKT
structures in hypercomplex geometry. In physics the HKT manifolds appear as
target manifolds with $N=(4,0)$ supersymmetric $\sigma$-models with Wess-
Zumino term [HP].
We construct calibrations on a special class of hypercomplex manifolds with
holonomy of its Obata connection in $SL(n,{\mathbb{H}})$, the commutator
subgroup of $GL(n,{\mathbb{H}})$. Such manifolds are called
$SL(n,{\mathbb{H}})$-manifolds. For more examples and an introduction to
$SL(n,{\mathbb{H}})$-geometry, see Section 3. For any
$SL(n,{\mathbb{H}})$-manifold $M$, and an induced complex structure $I$, there
is a holomorphic volume form $\Phi\in\Lambda^{2n,0}(M,J)$, which is parallel
with respect to the Obata connection ([V5], [BDV]). The space $V$ of parallel
holomorphic volume forms is 1-dimensional. A choice of an auxiliary induced
complex structure such that $I\circ J=-J\circ I$ endows $V$ with a real
structure and a positive direction (Subsection 4.2). We choose $\Phi$ to be
real and positive. Denote by $\Pi^{n,n}_{I}$ the projection to
$(n,n)$-component with respect to the complex structure $I$, such that $I\circ
J=-J\circ I$.
In Section 6 we show that $\operatorname{Re}(\Pi^{n,n}_{I}\Phi)$ is a
calibration for any quaternionic Hermitian metric $g$ for which $|\Phi|=2^{n}$
(6). This calibration calibrates complex subvarieties of $Z\subset(M,I)$ which
are Lagrangian with respect to the $(2,0)$-form
$\Omega=\omega_{J}+\sqrt{-1}\>\omega_{K}$, defined as in (2.2).
This calibration is defined for any quaternionic Hermitian metric, subject to
the condition $|\Phi|=1$ (and there are always many). When $(M,I,J,K,\Phi,g)$
is an HKT manifold with $\operatorname{Hol}(M)\subset SL(n,{\mathbb{H}})$,
more calibrations can be defined.
We choose $\Phi$ to be positive, real $(2n,0)$-form on $(M,J)$, and let
$\Phi_{n}:=\operatorname{Re}\Pi^{n,n}_{I}(\Phi)$. In [V7] it was shown that
the form $\Phi_{n+k}:=\frac{1}{2^{k}k!}\Phi_{n}\wedge\omega_{I}^{k}$ is always
closed and positive (4.3). In 6, we prove that this form is a calibration, for
a metric
$g^{\prime}:=g\cdot\left|\frac{\Phi_{n+k}}{2^{n}}\right|^{(2n+2k)^{-1}}$,
conformally equivalent to $g$. When $g$ is also balanced, $|\Phi|=const$, the
conformal weight $\left|\frac{\Phi_{n+k}}{2^{n}}\right|^{(2n+2k)^{-1}}$ is
constant (6), and $g^{\prime}$ is also HKT, but otherwise $g^{\prime}$ is not
an HKT metric. In either case, the calibration $\Phi_{n+k}$ is (generally
speaking) not parallel with respect to any connection on $M$ (6).
We show that $\Phi_{n+k}$ calibrates complex subvarieties of $(M,I)$ which are
coisotropic with respect to the (2,0)-form
$\Omega=\omega_{J}+\sqrt{-1}\>\omega_{K}$ (6). The situation with isotropic
subvarieties is completely different. Using the examples from Section 5.6, we
notice in Remark 6.5 that complex isotropic submanifolds in this case do not
have to be calibrated by any form, since they could be homologous to zero.
## 2 Preliminaries
### 2.1 Calibrations in Riemannian geometry
We provide here the basic definitions of the theory of calibrations which we
use in the paper. The standard reference for this material is [HL] and the
reader may also consult [J2] for recent progress and developments related to
manifolds with restricted holonomy.
Definition 2.1: Let $W\subset V$ be a $p$-dimensional subspace in a Euclidean
space, and $\operatorname{Vol}(W)$ denote the Riemannian volume form of
$W\subset V$, defined up to a sign. For any $p$-form $\eta\in\Lambda^{p}V$,
let comass $\operatorname{\sf comass}(\eta)$ be the maximum of
$\frac{\eta(v_{1},v_{2},...,v_{p})}{|v_{1}||v_{2}|...|v_{p}|}$, for all
$p$-tuples $(v_{1},...,v_{p})$ of vectors in $V$ and face be the set of planes
$W\subset V$ where $\frac{\eta}{\operatorname{Vol}(W)}=\operatorname{\sf
comass}(\eta)$.
Definition 2.2: A precalibration on a Riemannian manifold is a differential
form with comass $\leqslant 1$ everywhere.
Definition 2.3: A calibration is a precalibration which is closed.
Definition 2.4: Let $\eta$ be a $k$-dimensional precalibration on a Riemannian
manifold, and $Z\subset M$ a $k$-dimensional subvariety (we usually assume
that the Hausdorff dimension of the set of singular points of $Z$ is
$\leqslant k-2$, because in this case a compactly supported differential form
can be integrated over $Z$). We say that $Z$ is calibrated by $\eta$ if at any
smooth point $z\in Z$, the space $T_{z}Z$ is a face of the precalibration
$\eta$.
Remark 2.5: Clearly, for any precalibration $\eta$,
$\operatorname{Vol}(Z)\geqslant\int_{Z}\eta,$ (2.1)
where $\operatorname{Vol}(Z)$ denotes the Riemannian volume of a compact $Z$,
and the equality happens iff $Z$ is calibrated by $\eta$. If, in addition,
$\eta$ is closed, $\int_{Z}\eta$ is a cohomological invariant, and the
inequality (2.1) implies that $Z$ minimizes the Riemannian volume in its
homology class.
### 2.2 Hyperkähler manifolds and calibrations
The following definitions are standard.
Definition 2.6: A manifold $M$ is called hypercomplex if $M$ is equipped with
a triple of complex structures $I,J,K$, satisfying the quaternionic relations
$I\circ J=-J\circ I=K$. If, in addition, $M$ is equipped with a Riemannian
metric $g$ which is Kähler with respect to $I,J,K$, $(M,I,J,K,g)$ is called
hyperkähler. This is equivalent to $\nabla I=\nabla J=\nabla K=0$, where
$\nabla$ is the Levi-Civita connection of $g$; see [Bes].
Remark 2.7: It has been known since 1955 that any hypercomplex manifold
admits a torsion-free connection preserving $I,J$ and $K$, which is
necessarily unique. This connection is called the Obata connection, after M.
Obata, who discovered it in [Ob]. Any almost complex structure which is
preserved by a torsion-free connection is necessarily integrable (this is an
easy consequence of Newlander-Nirenberg theorem). Therefore, for any
$a,b,c\in{\mathbb{R}}$, with $a^{2}+b^{2}+c^{2}=1$, the almost complex
structure $aI+bJ+cK$ is in fact integrable. We denote by $(M,L)$ the manifold
$M$ considered as a complex manifold with the complex structure induced by
$L=aI+bJ+cK$.
Definition 2.8: Such complex structures are called induced by quaternions, and
the corresponding family, parametrized by $S^{2}$ – the twistor family, or the
hypercomplex family. This family is holomorphic, and its total space (fibered
over ${\mathbb{C}}P^{1}$) is called the twistor space of $M$. It is a complex
analytic space, non-Kähler even in simplest cases (for $M$ a torus or a K3
surface).
Hyperkähler geometry has a long history and is already well established. For
more details and background definitions, please see [Bes, J2]. In algebraic
geometry, the word hyperkähler is essentialy synonymous with “holomorphic
symplectic”. The reason is that any hyperkähler manifold is equipped with a
complex-valued form $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$.111We always
write $\omega_{I},\omega_{J},\omega_{K}$ for the corresponding Kähler forms.
This form has Hodge type (2,0) on $(M,I)$ and is closed, hence holomorphically
symplectic.
The converse follows from the Yau’s proof of Calabi’s conjecture: a
holomorphically symplectic, Kähler manifold admits a unique hyperkähler metric
in a given Kähler class ([Bes]). For survey of recent advances in hyperkähler
geometry see [H1, H2].
Some of the main objects of this paper are holomorphic Lagrangian, isotropic
and coisotropic subvarieties of $(M,I)$, where $(M,I,J,K,g)$ is hyperkähler.
Definition 2.9: A complex analytic subvariety $Z$ of a holomorphically
symplectic manifold $(M,\Omega)$ is called holomorphic Lagrangian if
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=0$, and
$\dim_{\mathbb{C}}Z=\frac{1}{2}\dim_{\mathbb{C}}M$, and isotropic if
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=0$, and
$\dim_{\mathbb{C}}Z<\frac{1}{2}\dim_{\mathbb{C}}M$. It is called coisotropic
if $\Omega$ has rank
$\frac{1}{2}\dim_{\mathbb{C}}M-\operatorname{codim}_{\mathbb{C}}Z$ on $TZ$ in
all smooth points of $Z$, which is the minimal possible rank for a
$2n-p$-dimensional subspace in a $2n$-dimensional symplectic space.
### 2.3 Calibrations in HKT-geometry
Let $(M,I,J,K)$ be a hypercomplex manifold. Then the tangent bundle $TM$ is
equipped with a natural quaternionic action. In particular, the group $SU(2)$
of unitary quaternions acts on $TM$, in a canonical way. A Riemannian metric
on $M$ is called quaternionic Hermitian if it is $SU(2)$-invariant. A
hyperkähler metric is obviously quaternionic Hermitian, but the converse is
manifestly false, as we shall explain presently.
With every quaternionic Hermitian metric $g$ we associate 2-forms
$\omega_{I}:=g(I\cdot,\cdot),\omega_{J}:=g(J\cdot,\cdot)$ and
$\omega_{K}:=g(K\cdot,\cdot)$ which are clearly antisymmetric, because $g$ is
$SU(2)$-invariant. It is easy to check that
$\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ (2.2)
is a (2,0)-form on $(M,I)$. This form is closed if and only if $(M,I,J,K,g)$
is hyperkähler ([Bes]).
For a weaker form of this condition, consider the $(1,0)$-part of the de Rham
differential,
$\partial:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p+1,q}(M).$
A quaternionic Hermitian hypercomplex manifold is called HKT (short for
“hyperkähler with torsion”) if $\partial\Omega=0$.
The theory of HKT-manifolds is a rapidly developing subfield of quaternionic
geometry. Originally this notion appeared in physics ([HP]), but
mathematicians found it very useful. For an early survey of HKT-geometry,
please see [GP].
Another ingredient of an HKT calibration theory is the notion of Obata
connection (2.2). Since this connection preserves the quaternionic structure,
its holonomy $\operatorname{Hol}(M)$ lies in $GL(n,{\mathbb{H}})$. The
holonomy of the Obata connection is one of the most important invariants of a
hypercomplex manifold. Many properties of $M$ can be related directly to its
holonomy group. In particular, the group $\operatorname{Hol}(M)$ is compact if
and only if $(M,I,J,K)$ admits a hyperkähler metric.
There seems to be no holonomy characterization of HKT structures. In fact the
holonomy of Obata connection is rarely known explicitly, except on hyperkähler
manifolds, where it is equal to the Levi-Civita connection. However the
knowledge of holonomy is still quite useful for the study of HKT geometry. For
many examples of compact hypercomplex manifolds, the group
$\operatorname{Hol}(M)\subset GL(n,{\mathbb{H}})$ is strictly smaller than
$GL(n,{\mathbb{H}})$. Only recently it was found that the group $SU(3)$ with
the left-invariant hypercomplex structure has $GL(n,{\mathbb{H}})$ as its
holonomy group ([Sol]).
An important subgroup inside $GL(n,{\mathbb{H}})$ is its commutator
$SL(n,{\mathbb{H}})$. This group can be defined as a group of quaternionic
matrices $A\subset\operatorname{End}({\mathbb{H}}^{n})$ preserving a non-zero
complex-valued form
$\Phi\in\Lambda^{2n,0}_{\mathbb{C}}({\mathbb{H}}^{n}_{I})$, where
${\mathbb{H}}^{n}_{I}$ is ${\mathbb{H}}^{n}$ considered as a $2n$-dimensional
complex space, with the complex structure $I$ induced by quaternions. The
coefficient $\lambda:=\frac{A(\Phi)}{\Phi}$ is called the Moore determinant of
the matrix $A$ ([A], [AV1]); it is always a positive real number, with
$\lambda^{4}$ equal to the determinant of $A$, considered as an element of
$GL(4n,{\mathbb{R}})$. The group $SL(n,{\mathbb{H}})$ is a group of
quaternionic matrices with Moore determinant 1.
## 3 $SL(n,{\mathbb{H}})$-manifolds
### 3.1 An introduction to $SL(n,{\mathbb{H}})$-geometry
As Obata has shown ([Ob]), a hypercomplex manifold $(M,I,J,K)$ admits a
necessarily unique torsion-free connection, preserving $I,J,K$. The converse
is also true: if a manifold $M$ equipped with an action of ${\mathbb{H}}$
admits a torsion-free connection preserving the quaternionic action, it is
hypercomplex. This implies that a hypercomplex structure on a manifold can be
defined as a torsion-free connection with holonomy in $GL(n,{\mathbb{H}})$.
This connection is called the Obata connection on a hypercomplex manifold.
Connections with restricted holonomy are one of the central notions in
Riemannian geometry, due to Berger’s classification of irreducible holonomy of
Riemannian manifolds. However, a similar classification exists for general
torsion-free connections ([MS]). In the Merkulov-Schwachhöfer list, only three
subroups of $GL(n,{\mathbb{H}})$ occur. In addition to the compact group
$Sp(n)$ (which defines hyperkähler geometry), also $GL(n,{\mathbb{H}})$ and
its commutator $SL(n,{\mathbb{H}})$ appear, corresponding to hypercomplex
manifolds and hypercomplex manifolds with trivial determinant bundle,
respectively. Both of these geometries are interesting, rich in structure and
examples, and deserve detailed study.
It is easy to see that $(M,I)$ has holomorphically trivial canonical bundle,
for any $SL(n,{\mathbb{H}})$-manifold $(M,I,J,K)$ ([V5]). For a hypercomplex
manifold with trivial canonical bundle admitting an HKT metric, a version of
Hodge theory was constructed ([V3]). Using this result, it was shown that a
compact hypercomplex manifold with trivial canonical bundle has holonomy in
$SL(n,{\mathbb{H}})$, if it admits an HKT-structure ([V5]).
In [BDV], it was shown that holonomy of all hypercomplex nilmanifolds lies in
$SL(n,{\mathbb{H}})$. Many working examples of hypercomplex manifolds are in
fact nilmanifolds, and by this result they all belong to the class of
$SL(n,{\mathbb{H}})$-manifolds.
The $SL(n,{\mathbb{H}})$-manifolds were studied in [AV2] and [V6], because on
such manifolds the quaternionic Dolbeault complex is identified with a part of
de Rham complex (4.3). Under this identification, ${\mathbb{H}}$-positive
forms become positive in the usual sense, and $\partial$,
$\partial_{J}$-closed or exact forms become
$\partial,\overline{\partial}$-closed or exact (see Section 3.1). This linear-
algebraic identification is especially useful in the study of the quaternionic
Monge-Ampère equation ([AV2]).
### 3.2 Balanced HKT-manifolds
The following lemma is contained in [BDV] (Theorem 3.2; see also [V7], Lemma
4.3). Recall that the map $\eta{\>\longrightarrow\>}J(\overline{\eta})$
defines a real structure on $\Lambda^{2p,0}(M,I)$. A $(p,0)$-form $\eta$ is
called ${\mathbb{H}}$-real if $J(\eta)=\overline{\eta}$.
Lemma 3.1: Let $(M,I,J,K)$ be a hypercomplex manifold, and $\eta$ a top
degree $(2n,0)$-form, which is ${\mathbb{H}}$-real and holomorphic. Then
$\eta$ is Obata-parallel.
Definition 3.2: Let $(M,I,g)$ be a complex Hermitian manifold,
$\dim_{\mathbb{C}}M=n$, and $\omega\in\Lambda^{1,1}(M)$ its Hermitian form.
One says that $M$ is balanced if $d(\omega^{n-1})=0$.
Remark 3.3: It is easy to see that $d(\omega^{m})=0$ for $1\leqslant
m\leqslant n-2$ implies that $\omega$ is Kähler; the balancedness makes sense
as the only non-trivial condition of form $d(\omega^{m})=0$ which is not
equivalent to the Kähler property.
Theorem 3.4: Let $(M,I,J,K,\Omega)$ be an HKT-manifold as in Section 2.3,
$\dim_{\mathbb{H}}M=n$. If $\overline{\partial}$ is the standard Dolbeault
operator on $(M,I)$, then the following conditions are equivalent.
(i)
$\overline{\partial}(\Omega^{n})=0$
(ii)
$\nabla(\Omega^{n})=0$, where $\nabla$ is the Obata connection
(iii)
The manifold $(M,I)$ with the induced quaternionic Hermitian metric is
balanced as a Hermitian manifold:
$d(\omega_{I}^{2n-1})=0.$
Proof: [V7], Theorem 4.8.
Remark 3.5: A balanced HKT-manifold has holonomy in $SL(n,{\mathbb{H}})$. This
statement follows immediately from the implication (iii) $\Rightarrow$ (ii) of
3.2. However the balanced HKT condition is a little stronger. It is shown in
[IP] that an HKT manifold has (restricted) holonomy of the Obata connection in
$SL(n,{\mathbb{H}})$ if and only if it is (locally) conformally balanced.
Remark 3.6: The condition $\nabla(\Omega^{n})=0$ is independent from the
choice of a basis $I,J,K$, $IJ=-JI=K$ of ${\mathbb{H}}$. Indeed, suppose that
$g\in SU(n)$, and $(I_{1},J_{1},K_{1})=(g(I),g(J),g(K))$ is a new basis in
${\mathbb{H}}$. The corresponding HKT-form
$\Omega_{1}=\omega_{J_{1}}+\sqrt{-1}\>\omega_{K_{1}}$ can be expressed as
$\Omega_{1}=g(\Omega)$, hence
$\nabla(\Omega_{1}^{n})=\nabla(g(\Omega_{1}^{n}))=g(\nabla(\Omega^{n}))=0.$
Therefore, 3.2 leads to the following corollary.
Corollary 3.7: Let $(M,I,J,K,\Omega)$ be an HKT-manifold, such that the
corresponding complex Hermitian manifold $(M,I)$ is balanced. Then $(M,I_{1})$
is balanced for any complex structrure $I_{1}$ induced by the quaternions.
Moreover, $(M,I,J,K,\Omega)$ is an $SL({\mathbb{H}},n)$-manifold.
## 4 Differential forms on hypercomplex manifolds
In this section, we give an introduction to the linear algebraic structures on
the de Rham algebra of a hypercomplex manifold. We follow [V6] and [V7].
### 4.1 The quaternionic Dolbeault complex
It is well-known that any irreducible representation of $SU(2)$ over
${\mathbb{C}}$ can be obtained as a symmetric power $S^{i}(V_{1})$, where
$V_{1}$ is a fundamental 2-dimensional representation. We say that a
representation $W$ has weight $i$ if it is isomorphic to $S^{i}(V_{1})$. A
representation is said to be pure of weight $i$ if all its irreducible
components have weight $i$.
Remark 4.1: The Clebsch-Gordan formula (see [Hu]) claims that the weight is
multiplicative, in the following sense: if $i\leqslant j$, then
$V_{i}\otimes V_{j}=\bigoplus_{k=0}^{i}V_{i+j-2k},$
where $V_{i}=S^{i}(V_{1})$ denotes the irreducible representation of weight
$i$.
Let $M$ be a hypercomplex manifold, $\dim_{\mathbb{H}}M=n$. There is a natural
multiplicative action of $SU(2)\subset{\mathbb{H}}^{*}$ on $\Lambda^{*}(M)$,
associated with the hypercomplex structure.
Let $V^{i}\subset\Lambda^{i}(M)$ be a maximal $SU(2)$-invariant subspace of
weight $<i$. The space $V^{i}$ is well defined, because it is a sum of all
irreducible representations $W\subset\Lambda^{i}(M)$ of weight $<i$. Since the
weight is multiplicative (4.1), $V^{*}=\bigoplus_{i}V^{i}$ is an ideal in
$\Lambda^{*}(M)$.
It is easy to see that the de Rham differential $d$ increases the weight by 1
at most. Therefore, $dV^{i}\subset V^{i+1}$, and $V^{*}\subset\Lambda^{*}(M)$
is a differential ideal in the de Rham DG-algebra $(\Lambda^{*}(M),d)$.
Definition 4.2: Denote by $(\Lambda^{*}_{+}(M),d_{+})$ the quotient algebra
$\Lambda^{*}(M)/V^{*}$. It is called the quaternionic Dolbeault algebra of
$M$, or the quaternionic Dolbeault complex (qD-algebra or qD-complex for
short).
Remark 4.3: The complex $(\Lambda^{*}_{+}(M),d_{+})$ was constructed earlier
by Capria and Salamon ([CS]) in a different (and more general) situation, and
much studied since then.
The Hodge bigrading is compatible with the weight decomposition of
$\Lambda^{*}(M)$, and gives a Hodge decomposition of $\Lambda^{*}_{+}(M)$
([V3]):
$\Lambda^{i}_{+}(M)=\bigoplus_{p+q=i}\Lambda^{p,q}_{+,I}(M).$
The spaces $\Lambda^{p,q}_{+,I}(M)$ are the weight spaces for a particular
choice of a Cartan subalgebra in $\mathfrak{su}(2)$. The
$\mathfrak{su}(2)$-action induces an isomorphism of the weight spaces within
an irreducible representation. This gives the following result.
Proposition 4.4: Let $(M,I,J,K)$ be a hypercomplex manifold and
$\Lambda^{i}_{+}(M)=\bigoplus_{p+q=i}\Lambda^{p,q}_{+,I}(M)$
the Hodge decomposition of qD-complex defined above. Then there is a natural
isomorphism
$\Lambda^{p,q}_{+,I}(M)\cong\Lambda^{p+q,0}(M,I).$ (4.1)
Proof: See [V3].
This isomorphism is compatible with a natural algebraic structure on
$\bigoplus_{p+q=i}\Lambda^{p+q,0}(M,I),$
and with the Dolbeault differentials, in the following way.
Let $(M,I,J,K)$ be a hypercomplex manifold. We extend
$J:\;\Lambda^{1}(M){\>\longrightarrow\>}\Lambda^{1}(M)$
to $\Lambda^{*}(M)$ by multiplicativity. Recall that
$J(\Lambda^{p,q}(M,I))=\Lambda^{q,p}(M,I),$
because $I$ and $J$ anticommute on $\Lambda^{1}(M)$. Denote by
$\partial_{J}:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p+1,q}(M,I)$
the operator $J\circ\overline{\partial}\circ J$, where
$\overline{\partial}:\;\Lambda^{p,q}(M,I){\>\longrightarrow\>}\Lambda^{p,q+1}(M,I)$
is the standard Dolbeault operator on $(M,I)$, that is, the $(0,1)$-part of
the de Rham differential. Since $\overline{\partial}^{2}=0$, we have
$\partial_{J}^{2}=0$. In [V3] it was shown that $\partial$ and $\partial_{J}$
anticommute:
$\\{\partial_{J},\partial\\}=0.$ (4.2)
Consider the quaternionic Dolbeault complex $(\Lambda^{*}_{+}(M),d_{+})$
constructed in 4.1. Using the Hodge bigrading, we can decompose this complex,
obtaining a bicomplex
$\Lambda^{*,*}_{+,I}(M)\ext@arrow
0099\arrowfill@\relbar\relbar\longrightarrow{}{d^{1,0}_{+,I},d^{0,1}_{+,I}}\Lambda^{*,*}_{+,I}(M)$
where $d^{1,0}_{+,I}$, $d^{0,1}_{+,I}$ are the Hodge components of the
quaternionic Dolbeault differential $d_{+}$, taken with respect to $I$.
Theorem 4.5: Under the multiplicative isomorphism
$\Lambda^{p,q}_{+,I}(M)\cong\Lambda^{p+q,0}(M,I)$
constructed in 4.1, $d^{1,0}_{+}$ corresponds to $\partial$ and $d^{0,1}_{+}$
to $\partial_{J}$:
$\textstyle{\Lambda^{0}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\textstyle{\Lambda^{0,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{1,0}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\textstyle{\Lambda^{0,1}_{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0,1}_{+}}$$\scriptstyle{d^{1,0}_{+}}$$\cong$$\textstyle{\Lambda^{1,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{1,0}_{I}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{J}}$$\textstyle{\Lambda^{2,0}_{+}(M)}$$\textstyle{\Lambda^{1,1}_{+}(M)}$$\textstyle{\Lambda^{0,2}_{+}(M)}$
$\textstyle{\Lambda^{2,0}_{I}(M)}$$\textstyle{\Lambda^{2,0}_{I}(M)}$$\textstyle{\Lambda^{2,0}_{I}(M)}$
(4.3)
Moreover, under this isomorphism, the form
$\omega_{I}\in\Lambda^{1,1}_{+,I}(M)$ corresponds to
$\Omega\in\Lambda^{2,0}_{I}(M)$.
Proof: See [V3] or [V4].
### 4.2 Positive $(2,0)$-forms on hypercomplex manifolds
The notion of positive $(2p,0)$-forms on hypercomplex manifolds (sometimes
called q-positive, or ${\mathbb{H}}$-positive) was developed in [AV1] (see
also [AV2] and [V6]).
Let $\eta\in\Lambda^{p,q}_{I}(M)$ be a differential form. Since $I$ and $J$
anticommute, $J(\eta)$ lies in $\Lambda^{q,p}_{I}(M)$. Clearly,
$J^{2}{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}_{I}(M)}}\right.}=(-1)^{p+q}$.
For $p+q$ even,
$J{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}_{I}(M)}}\right.}$ is an
anticomplex involution, that is, a real structure on $\Lambda^{p,q}_{I}(M)$. A
form $\eta\in\Lambda^{2p,0}_{I}(M)$ is called real if
$J(\overline{\eta})=\eta$.
For a real $(2,0)$-form $\eta$,
$\eta\left(x,J(\overline{x}))\right)=\overline{\eta}\left(J(x),J^{2}(\overline{x})\right)=\overline{\eta}\left(\overline{x},J(x)\right),$
for any $x\in T^{1,0}_{I}(M)$. From the definition of a real form, we obtain
that the scalar $\eta\left(x,J(\overline{x})\right)$ is always real.
Definition 4.6: A real $(2,0)$-form $\eta$ on a hypercomplex manifold is
called positive if $\eta\left(x,J(\overline{x})\right)\geqslant 0$ for any
$x\in T^{1,0}_{I}(M)$, and strictly positive if this inequality is strict, for
all $x\neq 0$.
An HKT-form $\Omega\in\Lambda^{2,0}_{I}(M)$ of any HKT-structure is strictly
positive. Moreover, HKT-structures on a hypercomplex manifold are in one-to-
one correspondence with $\partial$-closed, strictly positive $(2,0)$-forms.
The analogy between Kähler forms and HKT-forms can be pushed further; it turns
out that any HKT-form $\Omega\in\Lambda^{2,0}_{I}(M)$ has a local potential
$\varphi\in C^{\infty}(M)$, in such a way that
$\partial\partial_{J}\varphi=\Omega$ ([AV1]). Here $\partial\partial_{J}$ is a
composition of $\partial$ and $\partial_{J}$ defined on quaternionic Dolbeault
complex as above (these operators anticommute).
### 4.3 The map ${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$
on $SL(n,{\mathbb{H}})$-manifolds
Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold, $\dim_{\mathbb{R}}M=4n$,
and
${\cal
R}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M)$
the isomorphism induced by $\mathfrak{su}(2)$-action as in 4.1. Consider the
projection
$\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M),$ (4.4)
and let $R:\;\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p+q,0}_{I}(M)$
denote the composition of (4.4) and ${\cal R}_{p,q}^{-1}$.
Let $\Phi_{I}$ be a nowhere degenerate holomorphic section of
$\Lambda^{2n,0}_{I}(M)$. Assume that $\Phi_{I}$ is real, that is,
$J(\Phi_{I})=\overline{\Phi}_{I}$, and positive. Existence of such a form is
equivalent to $\operatorname{Hol}(M)\subset SL(n,{\mathbb{H}})$ (3.2). It is
often convenient to define $SL(n,{\mathbb{H}})$-structure by fixing the
quaternionic action and the holomorphic form $\Phi_{I}$.
Define the map
${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$
by the relation
${\cal V}_{p,q}(\eta)\wedge\alpha=\eta\wedge
R(\alpha)\wedge\overline{\Phi}_{I},$ (4.5)
for any test form $\alpha\in\Lambda^{n-p,n-q}_{I}(M)$.
The map ${\cal V}_{p,p}$ is especially remarkable, because it maps closed,
positive $(2p,0)$-forms to closed, positive $(n+p,n+p)$-forms, as the
following proposition implies.
Proposition 4.7: Let $(M,I,J,K,\Phi_{I})$ be an
$SL(n,{\mathbb{H}})$-manifold, and
${\cal
V}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{n+p,n+q}_{I}(M)$
the map defined above. Then
(i)
${\cal V}_{p,q}(\eta)={\cal R}_{p,q}(\eta)\wedge{\cal V}_{0,0}(1)$.
(ii)
The map ${\cal V}_{p,q}$ is injective, for all $p$, $q$.
(iii)
$(\sqrt{-1}\>)^{(n-p)^{2}}{\cal V}_{p,p}(\eta)$ is real if and only
$\eta\in\Lambda^{2p,0}_{I}(M)$ is real, and positive if and only if $\eta$ is
positive.
(iv)
${\cal V}_{p,q}(\partial\eta)=\partial{\cal V}_{p-1,q}(\eta)$, and ${\cal
V}_{p,q}(\partial_{J}\eta)=\overline{\partial}{\cal V}_{p,q-1}(\eta)$.
(v)
${\cal V}_{0,0}(1)=\lambda{\cal R}_{n,n}(\Phi_{I})$, where $\lambda$ is a
positive rational number, depending only on the dimension $n$.
Proof: See [V6], Proposition 4.2, or [AV2], Theorem 3.6.
Remark 4.8: For the purposes of the present paper, we are interested in 4.3
for the case $\eta=\Omega^{k}$, where $\Omega$ is an HKT-form. In this case,
${\cal R}_{p,p}(\Omega^{k})$ is a projection of $\omega_{I}^{k}$ to the
component of maximal weight (see 4.4 below). Now, ${\cal
V}_{p,q}(\Omega^{k})={\cal R}_{p,q}(\Omega^{k})\wedge{\cal V}_{0,0}(1)$, as
follows from 4.3 (i). However, ${\cal V}_{0,0}(1)$ has weight $2n$, by 4.3
(v), and $\omega_{I}^{k}$ has weight $\leqslant 2k$, hence their product is of
weight $\geqslant 2n-2k$. Since this product is $(2n-2k)$-form, it is pure of
weight $(2n-2k)$, and components of $\omega_{I}^{k}$ of weight $<2k$ do not
contribute to the product $\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$. We obtain
that the closed, positive form ${\cal V}_{k,k}(\Omega^{k})$ is proportional to
$\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$, with positive coefficient.
### 4.4 Algebra generated by $\omega_{I}$, $\omega_{J}$, $\omega_{K}$
Let $(M,I,J,K,g)$ be a quaternionic Hermitian manifold. Consider the algebra
$A^{*}=\oplus A^{2i}$ generated by $\omega_{I}$, $\omega_{J}$, and
$\omega_{K}$. In [V1], this algebra was computed explicitly. It was shown
that, up to the middle degree, $A^{*}$ is a symmetric algebra with generators
$\omega_{I}$, $\omega_{J}$, $\omega_{K}$. The algebra $A^{*}$ has Hodge
bigrading $A^{k}=\bigoplus\limits_{p+q=k}A^{p,q}$. From the Clebsch-Gordan
formula, we obtain that $A^{2i}_{+}:=\Lambda^{2i}_{+}(M)\cap A^{2i}$, for
$i\leqslant n$, is an orthogonal complement to $Q(A^{2i-4})$, where
$Q(\eta)=\eta\wedge(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})$. Moreover,
$A^{2i}_{+}$ is irreducible as a representation of $SU(2)$. Therefore, the
space $A^{p,p}_{+}=\ker
Q^{*}{\left|{}_{{\phantom{|}\\!\\!}_{A^{p,p}}}\right.}$ is 1-dimensional. This
argument also implies that the form ${\cal V}_{0,0}(1)$ is proportional to
$\Phi_{J}|^{n,n}_{I}$, where $\Phi_{J}$ is a holomorphic volume form on
$(M,J)$, obtained as a top power of the appropriate holomorphic symplectic
form, and $\Phi_{J}|^{n,n}_{I}$ its $(n,n)$-part, taken with respect to $I$.
Proposition 4.9: Let $(M,I,J,K,\Phi_{I})$ be an
$SL(n,{\mathbb{H}})$-manifold, equipped with an HKT-structure $\Omega$. Assume
that $\Omega^{n}=\Phi_{I}$. Let
$\Pi_{+}:\;\Lambda^{n+k,n+k}_{I}(M){\>\longrightarrow\>}\Lambda^{n+k,n+k}_{I,+}(M)$
be the projection to the component of maximal weight with respect to the
$SU(2)$-action. Then $\Xi_{k}:=\Pi_{+}(\omega_{I}^{n+k,n+k})$ is a closed,
weakly positive $(n+k,n+k)$-form, which is proportional to
$\omega^{k}_{I}\wedge\Phi_{J}|^{n,n}_{I}$ and to $\omega^{k}_{I}\wedge{\cal
V}_{0,0}(1)$.
Proof: The form $\omega^{k}_{I}\wedge\Phi_{J}|^{n,n}_{I}$ is proportional to
$\omega^{k}_{I}\wedge{\cal V}_{0,0}(1)$ as indicated above. Consider the
algebra $A^{*}=\oplus A^{2i}$ generated by $\omega_{I}$, $\omega_{J}$, and
$\omega_{K}$. The map $R^{p,q}$ is induced by the $SU(2)$-action, hence it
maps $A^{*,*}$ to itself. Since ${\cal V}_{p,q}(\eta)={\cal
R}_{p,q}(\eta)\wedge{\cal V}_{0,0}(1)$, and ${\cal V}_{0,0}(1)$ is
proportional to ${\cal R}_{n,n}(\Phi_{I})\in A^{*}$, we obtain
${\cal V}_{p,q}(A^{p+q,0})\subset A^{n+p,n+q}.$
Since ${\cal V}_{p,p}(\Omega^{p})\subset A^{n+p,n+p}_{+}$, the 1-dimensional
space $A^{n+p,n+p}_{+}$ is generated by ${\cal V}_{p,p}(\Omega^{p})$. This
form is closed and positive by 4.3. Therefore, the projection of
$\omega_{I}^{n+p}$ to $A^{n+p,n+p}_{+}$ is closed and positive (see 4.3).
## 5 Calibrations on hyperkähler manifolds
### 5.1 Hodge decomposition and $U(1)$-action
Let $I$ be a complex structure on a vector space $V$ and
$\rho:\;U(1){\>\longrightarrow\>}\operatorname{End}(V)$ a real
$U(1)$-representation given by $\rho(t)(X)=(\cos t+\sin tI)X$. This is
extended by multiplicativity to a representation in the tensor powers of $V$
with $\rho(t)(\alpha)(X)=\alpha(\rho(t)X)$ for a 1-form $\alpha$. In the usual
fashion, we define the weight decomposition associated with this
$U(1)$-action: the tensor $z$ has weight $p$ if $\rho(t)z=(\cos
pt)z+\sqrt{-1}\>(\sin pt)z$. We need also the definition of average over
$U(1)$ of $Y$:
$\operatorname{\sf Av}_{\rho}Y=\frac{1}{2\pi}\int_{0}^{2\pi}\rho(t)Ydt.$
Note that $\rho(t)Y=Y$ for every $t$ implies $IY=Y$ for any tensor $Y$ and
that $I\operatorname{\sf Av}_{\rho}Y=\operatorname{\sf Av}_{\rho}Y$.
Lemma 5.1: Let $\rho$ be a U(1)-action on $W$, and $W=\bigoplus W^{i}$ the
corresponding weight decomposition. Then the projection to $W^{0}$ along the
sum of other $W^{i}$, $i\neq 0$, coincides with taking the average over
$U(1)$.
Proof: For each $\eta\in W^{i}$, $i\neq 0$, one has $\int_{U(1)}\rho(t)\eta
dt=0$, because $\int_{0}^{2\pi}\cos(t)dt=0$.
Theorem 5.2: Let $\eta$ be a $2p$-form on a complex vector space $W$, with
$\operatorname{\sf comass}(\eta)\leqslant 1$, and
$\eta^{p,p}=\operatorname{\sf Av}_{\rho}\eta$ be the $(p,p)$-part of $\eta$.
Then $\operatorname{\sf comass}(\eta^{p,p})\leqslant 1$. Moreover, a
$2p$-dimensional plane $V$ is a face of $\eta^{p,p}$ if and only if
$\rho(t)(V)$ is a face of $\eta$ for all $t\in{\mathbb{R}}$.
Proof: For any decomposable $2p$-vector $\xi$, its image $\rho(t)(\xi)$ is
again decomposable for any $t$ and $|\rho(t)(\xi)|=|\xi|$. Then
$\eta^{(p,p)}(\xi)=(\operatorname{\sf
Av}_{\rho}(\eta))(\xi)=\frac{1}{2\pi}\int_{0}^{2\pi}\eta(\rho(t)(\xi))\leqslant
1$
since $\eta(\rho(t)\xi)\leqslant 1$ for every $t$. The equality holds iff
$\eta(\rho(t)\xi)=1$ for every $t$.
### 5.2 An $SU(2)$-invariant calibration
The most obvious example of a calibration on a hyperkähler manifold is
provided by the following theorem (see [Ber] for similar statement about a
quaternionic Wirtinger’s inequality).
Theorem 5.3: Let $(M,I,J,K,g)$ be a hyperkähler manifold,
$\omega_{I},\omega_{J},\omega_{K}$ the corresponding symplectic forms, and
$\Theta_{p}:=\frac{(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{p}}{c_{p}}$
the standard $SU(2)$-invariant $4p$-form normalized by
$c_{p}=\sum_{k=1}^{p}\frac{(p!)^{2}}{(k!)^{2}}(2k)!4^{p-k}$. Then $\Theta_{p}$
is a calibration, and its faces are $p$-dimensional quaternionic subspaces of
$TM$. Moreover, the form
$\Xi_{p}:=\frac{(\omega_{J}^{2}+\omega_{K}^{2})^{p}}{(p!)^{2}4^{p}}$ is also a
calibration, with the same faces.
Proof: Consider the form $\widetilde{\Xi}_{p}:=\frac{\omega_{J}^{2p}}{(2p)!}$.
By 5.1, ${(\tilde{\Xi}_{p})}^{2p,2p}_{I}=\Xi_{p}$, where $(\cdot)^{2p,2p}_{I}$
is an operation of taking $(2p,2p)$-part under the complex structure $I$.
Indeed, $\omega_{J}^{2p}=\frac{(\Omega+\overline{\Omega})^{2p}}{4^{p}}$, where
$\Omega$ is the standard $(2,0)$ form on $(M,I)$. Then the $(2p,2p)$-part of
$\omega_{J}^{2p}$ is equal to
$\frac{(2p)!}{(p!)^{2}}\frac{\Omega^{p}\wedge\overline{\Omega}^{p}}{4^{p}}=\frac{(2p)!(\omega_{J}^{2}+\omega_{K}^{2})^{p}}{(p!)^{2}4^{p}}.$
By 5.1, a subspace $V\subset TM$ is a face of $\Xi_{p}$ if and only if
$\rho_{I}(t)(V)$ is a face of ${\widetilde{\Xi}_{p}}$ for all $t$, with
$\rho_{I}(t)$ the $U(1)$-action associated with $I$. The form
${\tilde{\Xi}_{p}}$ is a standard Kähler calibration associated with $J$; it
follows from [HL] that $V\subset TM$ is a face of ${\widetilde{\Xi}_{p}}$ if
and only if it is $J$-linear, that is, ${\mathbb{C}}$-linear with respect to
the action of ${\mathbb{C}}$ induced by $J$. Since $\rho(t)(V)$ is $J$-linear
for all $t$, it remains $J$-linear if we act on $V$ by a group $G$ generated
by $\rho_{I}$ and $\rho_{J}$, with $\rho_{J}$ a $U(1)$-action associated with
$J$. Clearly, $G\cong SU(2)$ is the group of unitary quaternions acting on
$\Lambda^{*}M$. Therefore, $V$ is a face of $\Xi_{p}$ if and only if $V$ is
$g(J)$-linear, for all $g\in SU(2)$. This is equivalent to $V$ being a
quaternionic subspace. Taking the average of $\Xi_{p}$ with respect to $SU(2)$
will not change its faces, because they are already $SU(2)$-invariant.
Therefore, $\operatorname{\sf Av}_{SU(2)}(\Xi_{p})$ is a calibration with its
faces quaternionic subspaces. Moreover it is $Sp(n)Sp(1)$-invariant $4p$-form,
so it is proportional to $(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{p}$.
Then, using 5.4 below, we obtain that that $\operatorname{\sf
Av}_{SU(2)}(\Xi_{p})=\Theta_{p}$ by evaluating both forms on a fixed
quaternionic subspace.
Remark 5.4: Subvarieties calibrated by $\Theta_{p}$ are called trianalytic
subvarieties. They were studied, at some length, in [V1] and [V2].
### 5.3 A holomorphic Lagrangian calibration
Proposition 5.5: Let $(V^{4p},I,J,K,g)$ be a quaternionic Hermitian vector
space with fundamental forms $\omega_{I},\omega_{J},\omega_{K}$, and
$\Psi\in\Lambda^{2p}(V)$ a $2p$-form which is the real part of
$\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$ (it is a $(2p,0)$-form
with respect to $J$). Denote by $\Psi^{p,p}_{I}$ the $(p,p)$-part of $\Psi$
with respect to $I$. Then $\Psi^{p,p}_{I}$ has comass 1. Moreover, a
$2p$-dimensional subspace $W\subset V$ is calibrated by $\Psi^{p,p}_{I}$ if
and only if $W$ is complex $I$-linear and calibrated by $\Psi$.
Proof: The real part of $\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$
calibrates special Lagrangian subspaces taken with respect to the symplectic
form $\omega_{J}$ (see [HL]). Therefore, any face of
$\frac{1}{p!}(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$ is
$\omega_{J}$-Lagrangian. By 5.1, a $2p$-dimensional plane $W$ is a face of
$\Psi^{p,p}_{I}$ if and only if $\rho(t)(W)$ is a face of $\Psi$ for all
$t\in{\mathbb{R}}$. It follows by taking $t=0$ that $W$ is
$\omega_{J}$-Lagrangian and by taking $t=\pi/2$ that $I(W)$ is
$\omega_{J}$-Lagrangian too. But $I(W)$ is $\omega_{J}$-Lagrangian iff $W$ is
$\omega_{K}$-Lagrangian. By [Hit] (see also 5.3 below) $W$ determines an
$I$-complex subspace.
Remark 5.6: Let $V$ be a quaternionic Hermitian space,
$\dim_{\mathbb{H}}V=p$, and $\xi\in\Lambda^{2p}V$ a decomposable $2p$-vector
which is associated with a $2p$-dimensional subspace $W\subset V$. Clearly,
$W$ is Lagrangian with respect to $\omega_{J}$ if and only if
$L_{\omega_{J}}\xi=0$ and $\Lambda_{\omega_{J}}\xi=0$, where $L_{\omega_{J}}$,
$\Lambda_{\omega_{J}}$ are the corresponding Hodge operators,
$L_{\omega_{J}}(\eta):=\eta\wedge\omega_{J}$, and
$\Lambda_{\omega_{J}}=*L_{\omega_{J}}*$ its Hermitian adjoint. If $W$ is
Lagrangian with respect to $J$ and $K$, one has
$[L_{\omega_{J}},\Lambda_{\omega_{K}}]\xi=0.$ (5.1)
However, the commutator $[L_{\omega_{J}},\Lambda_{\omega_{K}}]$ acts on forms
of type $(p,q)$ with respect to $I$ as a multiplication by $(p-q)\sqrt{-1}\>$
(see [V0]). Then (5.1) implies that $\xi$ is of type $(p,p)$ with respect to
$I$.
Claim 5.7: Let $V$ be an $n$-dimensional quaternionic Hermitian space, and
${\cal V}^{0,0}:\;{\mathbb{R}}{\>\longrightarrow\>}\Lambda^{n,n}_{I}(V)$ be a
map defined in Subsection 4.3 (in Subsection 4.3 it was defined for
$SL(n,{\mathbb{H}})$-manifolds, but the definition can be repeated for
quaternionic spaces word by word). Then ${\cal V}^{0,0}(1)=\Psi^{n,n}_{I}$,
where $\Psi^{n,n}_{I}$ is a form defined as in 5.3.
Proof: From 4.3 (v), we know that ${\cal V}^{0,0}(1)$ and $\Psi^{n,n}_{I}$ are
proportional and we only have to calculate the coefficient of proportionality.
For this we use ${\cal
V}^{0,0}(1)\wedge\alpha=R(\alpha)\wedge\overline{\Phi}_{I}$ for a particular
choice of $\alpha$ as
$\alpha=\xi_{1}\wedge...\wedge\xi_{n}\wedge\overline{\xi_{n+1}}\wedge...\overline{\xi_{2n}},$
where $\xi_{i}$ are orthogonal and of unit norm. Then
$R(\alpha)=\xi_{1}\wedge...\wedge
J\overline{\xi_{n+1}}\wedge...J\overline{\xi_{2n}}.$
From here if ${\cal V}^{0,0}(1)=\lambda\Psi^{n,n}_{I}$, then $\lambda=1$.
Comparing 4.3 and 5.3, we find that the form $\Psi^{n,n}_{I}$ is positive.
### 5.4 Isotropic and coisotropic calibrations
A similar argument can be applied to other powers of $\Omega_{J}$.
Proposition 5.8: Consider an $n$-dimensional quaternionic Hermitian space
$V$, and let $\Omega_{J}:=\omega_{I}-\sqrt{-1}\>\omega_{K}$ be the usual
$(2,0)$-form on the complex space $(V,J)$. When $p\leqslant n$ denote by
$\Psi_{p}:=\frac{1}{p!}\operatorname{Re}(\Omega_{J}^{p})$, and let
$\Psi^{p,p}_{I}$ be its $(p,p)$-part taken with respect to $I$. Then
$\Psi^{p,p}_{I}$ has comass 1, and its faces are complex isotropic subspaces
of $(V,I)$
Proof: Let $W\subset V$ be a real $2p$-dimensional subspace, and $W_{1}$ be
the smallest complex subspace of $(V,J)$ containing $W$. Adding more vectors
if necessary, we can always assume that $\dim_{\mathbb{C}}W_{1}=2p$. Denote by
$\xi$ the decomposable $4p$-vector associated with $W_{1}$, and $I(\xi)$ its
image under the action of a quaternion $I$. Then $\frac{1}{p!}\Omega_{J}^{p}$
is a $(2p,0)$-form on $W_{1}$, proportional to the unit holomorphic volume
form $\operatorname{Vol}^{2p,0}(W_{1})$ with a coefficient $\kappa$ which
satisfies
$|\kappa|=\frac{(\xi,I(\xi))}{|\xi|^{2}}$
where $(,)$ is the induced scalar product. By Cauchy-Schwarz inequality
$|\xi|\leqslant 1$, where the equality holds iff $I\xi=\xi$ or, equivalently,
$W_{1}$ is quaternionic. Since $\operatorname{Vol}^{2p,0}(W_{1})$ has comass
1,
$\operatorname{\sf comass}\left(\frac{1}{p!}\Omega_{J}^{p}\right)\leqslant 1$
with equality if and only if $W_{1}$ is quaternionic. In the latter case, $W$
is a face of $\frac{1}{p!}\Omega_{J}^{p}$ if and only if $W$ is complex
Lagrangian in $W_{1}$, as follows from 5.3.
We provide also an expression of $\Psi^{p,p}$ as a polynomial of
$\omega_{I},\omega_{J}$ and $\omega_{K}$ for even $p$.
Proposition 5.9: Let $\Psi^{p,p}$ be the $(p,p)$ part with respect to $I$ of
$Re(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}$. Then
$\Psi^{p,p}=\sum_{k=0}^{q}\frac{(-1)^{k}}{4^{k}}\binom{p}{2k}\binom{2k}{k}\omega_{I}^{p-2k}\wedge(\omega_{K}^{2}+\omega_{J}^{2})^{k}$
where $q=\llcorner\frac{p}{2}\lrcorner$ is the greatest integer not exceeding
$\frac{p}{2}$.
Proof: First we notice that
$Re(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}=\sum_{k=0}^{\llcorner\frac{p}{2}\lrcorner}(-1)^{k}\binom{p}{2k}\omega_{I}^{p-2k}\wedge\omega_{K}^{2k}.$
Since $\omega_{I}^{p-2k}$ is of type $(p-2k,p-2k)$ with respect to $I$ we need
to determine the type of $\omega_{K}^{2k}$. To do this we use the fact that
$\omega_{K}=\frac{1}{2}\Omega+\frac{1}{2}\overline{\Omega}$ is the
decomposition of $\omega_{K}$ in $(2,0)+(0,2)$ parts with respect to $I$ where
$\Omega=\omega_{K}+\sqrt{-1}\>\omega_{J}$. Then
$\omega_{K}^{2k}=\frac{1}{4^{k}}\sum_{s=0}^{p-2k}\binom{2k}{s}\Omega^{s}\wedge\overline{\Omega}^{2k-s}$
and each term in the sum has degree $(2s,4k-s)$ with respect to $I$. So the
only term which will contribute to $\Psi^{p,p}$ above will be when $s=k$.
Obviously the term is
$\frac{1}{4^{k}}\binom{2k}{k}\Omega^{k}\wedge\overline{\Omega}^{k}$. Then the
proposition follows from the fact that
$\Omega\wedge\overline{\Omega}=\omega_{K}^{2}+\omega_{J}^{2}$.
Notice that one can take the imaginary part of $\Omega_{J}^{p}$ instead of the
real part. The resulting calibrated subspaces are again complex isotropic. To
identify the complex coisotropic subspaces, however, one has to be more
careful.
Proposition 5.10: Consider an $n$-dimensional quaternionic Hermitian space
$V$, and let $\Omega_{J}:=\omega_{I}-\sqrt{-1}\>\omega_{K}$ be the usual
$(2,0)$-form on the complex space $(V,J)$. Let
$\Phi_{p}+\sqrt{-1}\Phi^{\prime}_{p}:=\frac{1}{2^{p}p!n!}(\Omega_{J})^{n}\wedge\omega_{I}^{p}$,
and $\Phi^{p,p}_{I}$ (resp. $\Phi^{\prime p,p}_{I}$) be the corresponding
$(n+p,n+p)$-parts taken with respect to $I$. Then $\Phi^{p,p}_{I}$ (resp.
$\Phi^{\prime p,p}_{I}$) have comass 1 and their faces are complex coisotropic
subspaces of $(V,I)$
Proof: First we notice that if a form $\alpha$ is calibration, then its Hodge
dual $*\alpha$ is again calibration and its faces are orthogonal complements
to the faces of $\alpha$. Then the form $*\Psi^{p,p}$ is a calibration with
faces $I$-complex $\Omega_{J}$-coisotropic subspaces. The same is true also if
we consider the imaginary part of $\Omega_{J}^{p}$ instead of $\Psi^{p}$. Then
it remains to check that the complex form in the proposition is Hodge dual to
$\Omega_{J}^{p}$ up to a real constant. To this end we first notice that
$*\Omega_{J}^{n-p}=c_{1}\overline{\Omega_{J}^{n}}\wedge\Omega_{J}^{p}$ for a
real positive constant $c_{1}$. Then $\Phi^{p,p}+\sqrt{-1}\Phi^{\prime p,p}$
and $\overline{\Omega_{J}^{n}}\wedge\Omega_{J}^{p}$ are both highest vectors
in an irreducible representation $A^{2n+2p}$ of $SU(2)$ (see Subsection 4.4),
hence they are proportional up to a complex constant. More explicitly we have:
$(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge(\omega_{I}-\sqrt{-1}\>\omega_{K})^{p}=\Omega_{J}^{n}\wedge(2\omega_{I}-\Omega_{J})^{p}$
$=(\Omega_{J})^{n}\wedge\sum_{s=0}^{p}\binom{p}{s}(-\Omega_{J})^{s}\wedge
2^{p-s}\omega_{I}^{p-s}$
Since $\Omega_{J}^{n+s}=0$ for $s>0$ all terms in the sum above vanish except
the first one. Then
$(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge(\omega_{I}+\sqrt{-1}\>\omega_{K})^{p}=(\omega_{I}-\sqrt{-1}\>\omega_{K})^{n}\wedge
2^{p}\omega_{I}^{p}$
From here and 5.4 $ii)$ the proposition follows.
To calculate the comass of the forms above we need the following well-known
preliminary Lemma:
Lemma 5.11: If $(V^{2n},I,g)$ is an Hermitian vector space and $\omega$ is
the fundamental 2-form, then for any subset $X_{1},...,X_{2k}$ of a given
unitary basis $(e_{1},Ie_{1},...,e_{n},Ie_{n})$ we have:
i) $\omega^{k}(X_{1},....,X_{2k})=\pm k!$ if $span\\{X_{1},...,X_{2k}\\}$ is
complex and
ii) $\omega^{k}(X_{1},....,X_{2k})=0$ otherwise.
The proof of $i)$ is standard, while $ii)$ follows from the definition of
wedge product and the fact that $\omega(X_{i},X_{j})\neq 0$ only if
$IX_{i}=\pm X_{j}$.
Lemma 5.12: Let $(V^{4n},I,J,K,g)$ be a real vector space with anti-commuting
complex structures $I,J,K$ compatible with the positive scalar product $g$.
Denote by $\omega_{I},\omega_{J},\omega_{k}$ the fundamental 2-forms
corresponding to $I,J$ and $K$ respectively and
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$ be the standard $I$-complex
symplectic 2-form. Consider the form
$\Psi_{I}^{n}=Re(\omega_{I}+\sqrt{-1}\omega_{J})^{n}|_{I}^{(n,n)}$, where
$|_{I}^{(n,n)}$ denotes the $(n,n)$ component with respect to $I$. Then:
i)
$\Omega_{I}^{n}\wedge\overline{\Omega_{I}}^{n}=4^{n}(n!)^{2}\operatorname{Vol}$
for the volume form $\operatorname{Vol}$ on $V$.
ii)
$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=c_{n}\operatorname{Vol}$
where $c_{n}=\sum_{k=0}^{n}\frac{(n!)^{2}}{(k!)^{2}}(2k)!4^{n-k}$
iii)
$\omega_{I}^{k}\wedge\Psi^{n}=2^{k}k!n!\operatorname{Vol}_{E_{n+k}}$, where
$E_{n+k}$ is an $(n+k)$-dimensional $I$-complex and $\omega_{J}$-coisotropic
subspace.
Proof: Fix a quaternionic-Hermitian co-basis
$(e^{1},Ie^{1},Je^{1},Ke^{1},e^{2},Ie^{2},...,Ke^{n})$
of $V^{*}$ so that $\operatorname{Vol}=e^{1}\wedge...\wedge Ke^{n}$ and let
$e_{1},Ie_{1},...,Ke_{n}$ be the dual basis of $V$. From the fact that
$\Omega_{I}=\sum_{i}dz^{i}\wedge dw^{i}$ for coordinates
$dz_{i}=e^{i}+\sqrt{-1}Ie^{i}$ and $dw_{i}=Je^{i}+\sqrt{-1}Ke^{i}$, follows
that $\Omega_{I}^{n}=n!dz^{1}\wedge dw^{1}...dz^{n}\wedge dw^{n}$. Then to
obtain $i)$ we notice that $dz_{i}\wedge
d\overline{z_{i}}=-2\sqrt{-1}e^{i}\wedge Ie^{i}$ and $dw_{i}\wedge
d\overline{w_{i}}=-2\sqrt{-1}Je^{i}\wedge Ke^{i}$.
To prove $ii)$ we write
$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=(\omega_{I}^{2}+\Omega_{I}\wedge\overline{\Omega_{I}})^{n}=\sum_{k=0}^{n}\binom{n}{k}\omega_{I}^{2k}\wedge\Omega_{I}^{n-k}\wedge\overline{\Omega_{I}}^{n-k}.$
Then we consider the term
$\omega_{I}^{2k}\wedge\Omega_{I}^{n-k}\wedge\overline{\Omega_{I}}^{n-k}$. Let
$s_{i}=e^{i}\wedge Ie^{i}+Je^{i}\wedge Ke^{i}$ and $t_{j}=dz^{j}\wedge
dw^{j}$, so $\omega_{I}=\sum s_{i}$ and $\Omega_{I}=\sum t_{j}$. Then
$s_{i}^{3}=s_{i}t_{i}=t_{i}^{2}=0$ $s_{i},t_{j}$ commute and
$s_{i}^{2}=2\operatorname{Vol}_{i},t_{i}\overline{t_{i}}=4\operatorname{Vol}_{i}$,
where $\operatorname{Vol}_{i}=e^{i}\wedge Ie^{i}\wedge Je^{i}\wedge Ke^{i}$.
Fix $n-k$ indexes $(i_{k+1},i_{k+2},...,i_{n})$. Then notice that in the
product $\omega_{I}^{2k}\wedge
t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}\wedge\overline{\Omega_{I}}^{n-k}$ the only
non-vanishing terms are of the form
$s_{i_{1}}^{2}s_{i_{2}}^{2}...s_{i_{k}}^{2}t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}\overline{t_{i_{k+1}}t_{i_{k+2}}...t_{i_{n}}}$
for the complementary indexes $(i_{1},...,i_{k})$, such that
$(i_{1},...,i_{n})$ is a permutation of $(1,2...,n)$. Every such product is
equal to $2^{k}4^{n-k}\operatorname{Vol}$. Then we may select
$i_{1}=1,..,i_{k}=k,i_{k+1}=k+1,...,i_{n}=n$ and count the number of terms
corresponding to it; clearly, this number does not depend on the choice of the
permutation. The number is the product of the coefficients in front of
$s_{1}^{2}...s_{k}^{2}$ $t_{k+1}...t_{n}$ and
$\overline{t_{k+1}}...\overline{t_{n}}$ in the expansions of
$(s_{1}+...s_{k})^{2k}$ $(t_{k+1}+...+t_{n})^{n-k}$ and
$(\overline{t_{k+1}}+...+\overline{t_{n}})^{n-k}$ respectively, which is
$\frac{(2k)!}{2^{k}}((n-k)!)^{2}$. Since there are $\frac{n!}{k!(n-k)!}$
different choices for $n-k$ indexes, we obtain
$(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2})^{n}=\sum_{k=0}^{n}\frac{(n!)^{2}}{(k!)^{2}}(2k)!4^{n-k}\operatorname{Vol}$
and $ii)$ follows.
To prove $iii)$ we notice that $Sp(n)$ acts transitively on complex
coisotropic subspaces of fixed dimension. Then we choose the coisotropic
subspace $L$ spanned by
$e_{1},Ie_{1},....,e_{n},Ie_{n},Je_{1},Ke_{1},....,Je_{k},Ke_{k}$. Let
$\Omega_{K}=\omega_{I}+\sqrt{-1}\omega_{J}$, $\alpha\in L$ a subspace spanned
by $2n$ vectors and $\beta$ be a subspace generated by $2k$ vectors among
$e_{1},Ie_{1},....,e_{n},Ie_{n},Je_{1},Ke_{1},....,Je_{k},Ke_{k}$. Since
$\Psi^{n}=Re(\Omega_{K})|^{n,n}_{I}$, then $\Psi^{n}|_{\alpha}=0$ if $\alpha$
contains a quaternionic subspace or is not $I$-invariant. Similarly,
$\omega_{I}^{k}\bigg{|}_{\beta}=0$ if $\beta$ is not $I$-invariant as follows
from Lemma 5.4.
From the calculations in [HL] p. 88, we have
$\Psi^{n}(e_{1},Ie_{1},...,e_{n},Ie_{n})=n!Re(dz_{1}\wedge....\wedge
dw_{n})(e_{1},Ie_{1},...e_{n},Ie_{n})=n!,$
and from 5.4 above, $\omega_{I}^{K}(Je_{1},Ke_{1},...,Je_{k},Ke_{k})=k!$. Then
in the expression for
$\Psi^{n}\wedge\omega_{I}^{k}(e_{1},Ie_{1},....Je_{k},Ke_{k})$ the only non-
vanishing summands are $\omega_{I}^{K}(Je_{1},Ke_{1},...,Je_{k},Ke_{k})$ and
the terms where one or more pairs $e_{i},Ie_{i}$ are interchanged with
$Je_{i},Ke_{i}$. If we have exactly $s$ pairs interchanged, then there will be
$\binom{l}{s}$ terms each with value $n!k!$. So
$\Psi^{n}\wedge\omega_{I}^{k}(e_{1},Ie_{1},....Je_{k},Ke_{k})=n!k!\left(1+k+\binom{k}{2}+...+\binom{k}{k}\right)=2^{n}n!k!,$
which proves the Lemma. Note that for $k=n$ the result fits with the case $i)$
and the calculations in 5.4.
### 5.5 Holomorphic Lagrangian calibrations of degree two
The calibration 4-forms with constant coefficients in ${\mathbb{R}}^{8}$ were
studied systematically in [DHM]. Also various 4-forms which are calibrations
in ${\mathbb{H}}^{n}$ or any hyperkähler manifold are considered in [BrH]. We
want to relate our results to these works.
If $p=2$, from 5.4 we obtain
$\displaystyle\Psi^{2,2}_{I}=$
$\displaystyle\left.\frac{1}{2}\operatorname{Re}\left(\omega_{K}+\sqrt{-1}\omega_{I}\right)^{2}\right|_{I}^{2,2}$
$\displaystyle=$
$\displaystyle\left.\left(-\frac{1}{2}\omega_{I}^{2}+\frac{1}{2}\omega^{2}_{K}\right)\right|_{I}^{2,2}=-\frac{1}{2}\omega^{2}_{I}+\frac{1}{4}(\omega_{J}^{2}+\omega_{K}^{2}).$
In [BrH] R.Bryant and R. Harvey considered the forms
$\Psi_{\lambda,\mu,\nu}=\frac{\lambda}{2}\omega_{I}^{2}+\frac{\mu}{2}\omega_{J}^{2}+\frac{\nu}{2}\omega_{K}^{2}$
and showed that they are calibrations iff $-1\leqslant\nu,\lambda,\mu\leqslant
1$ and $-1\leqslant\nu+\lambda+\mu\leqslant 1$. We show here that the
”generic” form of this type calibrates either quaternionic or complex
isotropic subspaces.
Proposition 5.13: For the forms $\Psi_{\lambda,\mu,\nu}$ the following is
valid:
i) If $\lambda,\mu,\nu\geqslant 0$ and $\lambda+\mu+\nu=1$ with at least two
of $\lambda,\mu,\nu$ non-zero, the form $\Psi_{\lambda,\mu,\nu}$ has comass 1
and the faces are the quaternionic ones.
ii) If $\mu,\nu\leqslant 0$ and $\mu+\nu\geqslant-1$ with at least two of the
inequalities being strict, then $\Psi_{1,\mu,\nu}$ has comass 1 and the faces
are the I-complex $\Omega_{I}$-isotropic subspaces of
${\mathbb{H}}^{n}={\mathbb{C}}^{2n}$.
Proof: First we note that a convex hull of calibrations is a calibration. In
case $i)$, for any unit 4-vector $\psi$,
$\Psi_{\lambda,\mu,\nu}(\psi)=\frac{\lambda}{2}\omega_{I}^{2}(\psi)+\frac{\mu}{2}\omega_{J}^{2}(\psi)+\frac{\nu}{2}\omega_{K}^{2}(\psi)\leqslant(\lambda+\mu+\nu)|\psi|=|\psi|,$
and the equality is achieved only when $\psi$ spans a subspace which is
invariant with respect to at least two of $I,J$ and $K$, hence quaternionic.
For $ii)$ we note that
$\frac{1}{2}\omega_{I}^{2}+\frac{\mu}{2}\omega_{J}^{2}+\frac{\nu}{2}\omega_{K}^{2}=\frac{1+\mu+\nu}{2}\omega_{I}^{2}-\frac{\mu}{2}(\omega_{I}^{2}-\omega_{J}^{2})-\frac{\nu}{2}(\omega_{I}^{2}-\omega_{K}^{2})$
Then according to [BrH], Theorem 2.38,
$\frac{1}{2}(\omega_{I}^{2}-\omega_{J}^{2})$ and
$\frac{1}{2}(\omega_{I}^{2}-\omega_{K}^{2})$ are calibrations with comass 1
and faces which are $\omega_{K}$ or $\omega_{J}$ isotropic and contained in
2-dimensional quaternionic subspaces. So as in $i)$ if $\psi$ is a unit
4-vector, then $\Psi_{1,\mu,\nu}(\psi)\leqslant|\psi|$ with equality if and
only if $\psi$ is a face for all terms with nonvanishing coefficients on the
right-hand-side above. If the span of $\psi$ satisfies at least two of the
following:
i) $\psi$ is I-complex
ii) $\psi$ is $\omega_{J}$ isotropic and
iii) $\psi$ is $\omega_{K}$ isotropic
then $\psi$ satisfies also the third one and the Proposition follows.
In [BrH, Theorem 6.4], 5.5 is implicit. We note also that in String Theory,
the holomorphic Lagrangian submanifolds in 8-dimensional manifolds were
related to the notion of intersecting branes [F].
### 5.6 Examples
Examples of complex Lagrangian submanifolds in hyper-Kähler manifolds are
given by many authors. In [Vo], C. Voisin has proven a result about the
stability of such submanifolds under small deformation of the complex
structure of the ambient space; she gave also several classes of examples. N.
Hitchin noticed the fact that such subspaces are coming in complete families
([Hit]). In [M], D. Matsushita has shown that the families of holomorphic
Lagrangian fibrations on a hyperkaehler manifold always deform with a
deformation of a manifold, if the cohomology class of a fiber remains of Hodge
type $(n,n)$. Existence of such families is postulated by a conjecture called
“SYZ conjecture”, or, sometimes, the “Huybrechts-Sawon conjecture”. It is also
known as a hyperkähler version of abundance conjecture, related to the minimal
model program. For a survey of related questions, please see [Saw]. Recently
in String Theory the holomorphic Lagrangian submanifolds were related to
3-dimensional topological field theory with target hyperkähler manifold [KRS].
In this section we provide examples of complex Lagrangian submanifolds of
hypercomplex manifolds with holonomy $SL(n,{\mathbb{H}})$.
The known examples of manifolds with holonomy $SL(n,{\mathbb{H}})$ are either
nilmanifolds ([BDV]) or obtained via the twist construction of A. Swann [S],
which is based on previous examples by D. Joyce. The later construction
provides also simply-connected examples. We describe briefly a simplified
version of it.
Let $(X,I,J,K,g)$ be a compact hyper-Kähler manifold. By definition, an anti-
self-dual 2-form on it is a form which is of type (1,1) with respect to $I$
and $J$ and hence with respect to all complex structures of the hypercomplex
family. Let $\alpha_{1},...,\alpha_{4k}$ be anti-self-dual closed 2-forms
representing integral cohomology classes on $X$ (instatons). Consider the
principal $T^{4k}$-bundle $\pi:M\rightarrow X$ with characteristic classes
determined by $\alpha_{1},...,\alpha_{4k}$. It admits an instanton connection
$A$ given by $4k$ 1-forms $\theta_{i}$ s.t. $d\theta_{i}=\pi^{*}(\alpha_{i})$.
Then $M$ carries a hypercomplex structure determined in the following way: on
the horizontal spaces of $A$ we have the pull-backs of $I,J,K$ and on the
vertical spaces we fix a linear hypercomplex structure of the $4k$-torus. The
structures $\cal{I},\cal{J},\cal{K}$ on $M$ are extended to act on the
cotangent bundle $T^{*}M$ using the following relations:
$\displaystyle{\cal I}(\theta_{4i+1})=\theta_{4i+2},$ $\displaystyle{\cal
I}(\theta_{4i+3})=\theta_{4i+4},$ $\displaystyle{\cal
J}(\theta_{4i+1})=\theta_{4i+3},$ $\displaystyle{\cal
J}(\theta_{4i+2})=-\theta_{4i+4},$ $\displaystyle{\cal
I}(\pi^{*}\alpha)=\pi^{*}(I\alpha),$ $\displaystyle{\cal
J}(\pi^{*}\alpha)=\pi^{*}(J\alpha)$
for any 1-form $\alpha$ on $X$ and $i=0,1,...k-1$. Similarly one can define a
hyperhermitian (or quaternion-Hermitian) metric on $M$ from $g$ and a fixed
hyper-Kähler metric on $T^{4k}$ using the splitting of $TM$ in horizontal and
vertical subspaces. As A. Swann [S] has shown the structure is HKT and has a
holonomy $SL(n,{\mathbb{H}})$.
Suppose now that $Y$ is a complex Lagrangian subspace in $X$ with respect to
$I$. Consider the $T^{2k}$-bundle over $X$ determined by
$\alpha_{4i+1},\alpha_{4i+3}$. Suppose that $N$ is its restriction to $Y$ i.e
$N$ is a principal $T^{2k}$-subbundle of $M$ over $Y$ determined by
$\alpha_{4i+1}|_{Y},\alpha_{4i+3}|_{Y}$. Then $N$ is naturally embedded in $M$
and by the definiton above $N$ is $\cal{J}$-invariant and Lagrangian with
respect to the fundamental 2-form of $\cal{I}$. Notice that in general the
complex Lagrangian subspace could be Kähler or non-Kähler depending on whether
$\alpha_{1}|Y$ and $\alpha_{3}|Y$ define zero classes or not.
As a particular case assume $X$ to be a $K3$ surface with large enough Picard
group such that there are 4 independent anti-self-dual integral classes
defining a principal $T^{4}$-bundle $M$ over $X=K3$ with finite fundamental
group. After passing to a finite cover we may assume that $M$ is simply-
connected. Now if $vol$ denotes the volume form on $X$, then we can choose
representatives $\alpha_{1},...,\alpha_{4}$ in the characteristic classes of
$M$ such that $\alpha_{i}^{2}=-F\operatorname{Vol}$ where $F$ is a function
and $F>0$ almost everywhere. We want to see what is the structure of an
arbitrary complex Lagrangian subspace $N$ of $M$. Since $N$ is 4-dimensional
and ${\cal J}$-complex, we claim that its intersection with a generic fiber of
$\pi:M{\>\longrightarrow\>}X$ is at least complex 1-dimensional. Indeed,
otherwise $N$ would be a multisection of $M$ and will intersect a generic
fiber transversally. However then $\int_{N}\pi^{*}(\alpha_{1}^{2})<0$ since
$\alpha_{1}^{2}=-vol$ on one hand, and $\int_{N}\pi^{*}(\alpha_{1}^{2})=0$
since $\pi^{*}(\alpha_{1})=d\theta_{1}$ for some connection form $\theta_{1}$
on the other. The contradiction proves the claim and we have:
Proposition 5.14: If $M$ is a principal instanton $T^{4}$-bundle over a $K3$
surface then any complex Lagrangian subspace is fibered by complex Lagrangian
curves of the fibers of $M$ over a Lagrangian curve of the base $K3$.
Remark 5.15: Notice that any complex curve is a priori Lagrangian in a K3
surface.
In general one can use a similar construction to obtain complex isotropic and
coisotropic subspaces of the instanton bundle $M$.
## 6 Calibrations on $SL(n,{\mathbb{H}})$-manifolds
Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold, that is, a
hypercomplex manifold with $\Phi_{I}$ a holomorphic volume form on $(M,I)$
preserved by the Obata connection. Clearly, $\overline{\Phi}_{I}$ is
proportional to $J(\Phi_{I})$. After a rescaling to $e^{\sqrt{-1}t}\Phi_{I}$
if necessary, we can assume that $\Phi_{I}$ is ${\mathbb{H}}$-real, i.e.
$J(\Phi_{I})=\overline{\Phi}_{I}$, and ${\mathbb{H}}$-positive (Subsection
4.2). A number of interesting calibrations can be constructed in this
situation.
Theorem 6.1: Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$. Pick a quaternionic Hermitian metric on $M$. Using a conformal change,
we may assume that $|\Phi_{I}|_{g}=2^{n}$. Then $Re((\Phi_{I})_{J}^{n,n})$ is
a calibration, and it calibrates complex subvarieties of $(M,J)$ which are
Lagrangian with respect to the $(2,0)$-form
$\omega_{K}+\sqrt{-1}\>\omega_{I}$.
Proof: It follows from the assumptions of 6 that
$\Phi_{I}=\lambda\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}.$
Since both forms are real and ${\mathbb{H}}$-positive, $\lambda$ is real and
positive. It is easy to check that in local quaternionic Hermitian frame
$(dz_{1},dw_{1},...,dz_{n},dw_{n})$ the norm is calculated as
$\left|\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}\right|^{2}=|dz_{1}|^{2}|dw_{1}|^{2}...|dz_{n}|^{2}|dw_{n}|^{2}=4^{n}.$
Then
$\left|\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}\right|=2^{n}=|\Phi_{I}|$
and $\lambda=1$. Now the proof follows from the fact that $Re(\Phi_{I})$ and
$Re(\Phi_{I})_{J}^{n,n}$ are both closed,111The form $(\Phi_{I})_{J}^{n,n}$ is
parallel with respect to the Obata connection, which is torsion-free. and 5.3.
Theorem 6.2: Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$. Assume that $(M,I,J,K)$ is equipped with an HKT metric $g$ which is
balanced and $|\Phi_{I}|=2^{n}$. Then
$V_{n+i,n+i}:=\frac{1}{2^{i}i!}Re((\Phi_{I})_{J}^{n,n}\wedge\omega_{J}^{i})$
is a calibration, which calibrates complex subvarieties of $(M,J)$ which are
coisotropic with respect to the $(2,0)$-form
$\omega_{K}+\sqrt{-1}\>\omega_{I}$.
Proof: As in the previous proof,
$\Phi_{I}=\frac{(\omega_{J}+\sqrt{-1}\omega_{K})^{n}}{n!}$, so the form
$V_{n+i,n+i}$ is a pre-calibration by 5.4. It is closed, as follows from 4.4.
Remark 6.3: Notice that the form $V_{n+i,n+i}$ is not parallel with respect to
any torsion-free connection on $M$ (6), unless $M$ is hyperkähler.
Existence of a balanced HKT metric is a hard problem, which is equivalent to a
quaternionic version of a Calabi-Yau theorem ([V7]). However, even if $g$ is
not balanced, an analogue of the calibration $V_{n+i,n+i}$ is possible to
construct.
Theorem 6.4: Let $(M,I,J,K,\Phi_{I})$ be an $SL(n,{\mathbb{H}})$-manifold,
and $(\Phi_{I})_{J}^{n,n}$ the $(n,n)$-part of $\Phi_{I}$ taken with respect
to $J$, and $g$ an HKT metric. Then there exists a function $c_{i}(m)$ on $M$,
such that $V_{n+i,n+i}:=(\Phi_{I})_{J}^{n,n}\wedge\omega_{J}^{i}$ is a
calibration with respect to the conformal metric $\widetilde{g}=c_{i}g$,
calibrating complex subvarieties of $(M,J)$ which are coisotropic with respect
to the $(2,0)$-form
$\widetilde{\omega}_{K}+\sqrt{-1}\>\widetilde{\omega}_{I}$.
Proof: Since $\Phi_{I}$ is ${\mathbb{H}}$-positive and Obata parallel, the
form $(\Phi_{I})^{n,n}_{J}$ is closed. Then 4.4 implies that $V_{n+i,n+i}$ is
also closed. If we denote by $\widetilde{\omega_{J}}$ and
$\widetilde{\Omega_{I}}$ the corresponding forms after the conformal change
$\widetilde{g}=c_{i}(m)g$, then we can find the function $c_{i}(m)$ such that
$V_{n+i,n+i}=\frac{1}{2^{i}n!i!}(\widetilde{\Omega}_{I}^{n})^{n,n}_{J}\wedge\widetilde{\omega}_{J}^{i}.$
6 then follows from 5.4.
Remark 6.5: Similarly to the hyperkähler case, it is a natural question to ask
whether the complex isotropic submanifolds are also calibrated in
$SL(n,{\mathbb{H}})$-manifolds with an HKT structure. However we can see in
the examples from Section 4.6 that this is not the case. Consider again a
toric bundle $M$ over $K3$-surface which has 4-dimensional fiber and is
simply-connected. Such fiber contains a 2-torus which will be a complex
isotropic curve with respect to some of the structures. By a spectral sequence
argument as in Lemma 4.7 of [S], one can see that all second cohomology
classes of $M$ are pull-backs from classes on the base $K3$-surface. Then such
a torus is homologous to zero, since the integral of any closed 2-form on it
vanishes. Therefore, it can not be calibrated by any form.
Claim 6.6: Let $M$ be an $SL(n,{\mathbb{H}})$-manifold, $\Omega$ an HKT-form,
and $V_{n+i,n+i}$ the corresponding calibration, constructed above. Assume
that $\Omega$ is not hyperkähler. Then, the form $V_{n+i,n+i}$ is not
preserved by any torsion-free connection, for any $0<i<n$.
Proof: It is easy to check that the stabilizer
$St_{GL(4n,{\mathbb{R}})}(V_{n+i,n+i})$ is equal to the group $Sp(n)$ of
quaternionic Hermitian matrices. Therefore, any connection preserving
$V_{n+i,n+i}$ would also preserve an $Sp(n)$-structure. However, a torsion-
free connection preserving $Sp(n)$-structure is hyperkähler.
Acknowledgements: We are grateful to the referee for the careful reading and
many suggestions which improved the presentation of the paper.
## References
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Misha Verbitsky
Laboratory of Algebraic Geometry, SU-HSE,
7 Vavilova Str. Moscow, Russia, 117312
verbit@maths.gla.ac.uk, verbit@mccme.ru
Gueo Grantcharov
Department of Mathematics and Statistics
Florida International University
Miami Florida, 33199, USA
grantchg@fiu.edu
|
arxiv-papers
| 2010-09-06T23:25:07 |
2024-09-04T02:49:12.705484
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gueo Grantcharov, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1009.1178"
}
|
1009.1269
|
# Optimal Dividend and reinsurance strategy of a Property Insurance Company
under Catastrophe Risk
Zongxia Liang
Department of Mathematical Sciences, Tsinghua University, Beijing, China
Email: zliang@math.tsinghua.edu.cn
Lin He
The School of Finance, Renmin University of China, Beijing, China
Email: helin@ruc.edu.cn
Jiaoling Wu
Department of Mathematical Sciences, Tsinghua University, Beijing, China
Email: maths2005ling@hotmail.com
###### Abstract.
We consider an optimal control problem of a property insurance company with
proportional reinsurance strategy. The insurance business brings in
catastrophe risk, such as earthquake and flood. The catastrophe risk could be
partly reduced by reinsurance. The management of the company controls the
reinsurance rate and dividend payments process to maximize the expected
present value of the dividends before bankruptcy. This is the first time to
consider the catastrophe risk in property insurance model, which is more
realistic. We establish the solution of the problem by the mixed singular-
regular control of jump diffusions. We first derive the optimal retention
ratio, the optimal dividend payments level, the optimal return function and
the optimal control strategy of the property insurance company, then the
impacts of the catastrophe risk and key model parameters on the optimal return
function and the optimal control strategy of the company are discussed.
MSC(2000): Primary 91B30, 91B70, 91B28; Secondary 60H10, 60H30. Keywords:
Optimal dividend and reinsurance strategy; Optimal return function;
Catastrophe risk; Jump diffusions; Regular-singular control.
## 1\. Introduction
In this paper we consider a property insurance company in which the dividend
payments process and risk exposure are controlled by the management. The
property insurance business brings in catastrophe risk, such as earthquake and
flood. We assume that the company can only reduce its risk exposure by
proportional insurance strategy for simplicity. The catastrophe risk could
also be partly reduced by reinsurance. The regulation of the catastrophe risk
determines to what extent the catastrophe risk could be eliminated, here we
use reinsurance rate and adjusted risk rate in the regulation. We equate the
value of the company to the expected present value of the dividend payments
before bankruptcy. This is a mixed singular-regular control on diffusion
models with jumps. These optimization problems of diffusion models for
property insurance companies that control their risk exposure by means of
dividend payments have attracted significant interests recently. We refer
readers to Radner and Shepp [22], Paulsen and Gjessing [21], Højgaard and
Taksar[17, 19] and Asmussen[2]. Optimizing dividend payments is a classical
problem starting from the early work of Borch[6, 7], Gerber[10]. For some
applications of control theory in insurance mathematics, see Højgaard and
Taksar[16, 18], Martin-löf[20], Asmussen and Taksar[4, 9] and He and Liang[13,
14, 15], Basse, Reddemann, Riegler and Schulenburg[8], Guo, Liu and Zhou [11]
and other author’s work. Recent surveys can be found in Taksar[23], Avanzi
[3], Albrecher and Thonhauser[1]. Unfortunately, there is little work
concerned with the catastrophe risk of the property insurance company in the
problem of optimal risk control/dividend distribution via the reinsurance
rate. In the real financial market, the property insurance business generally
brings in catastrophe risk, such as earthquake and flood. The asset of the
company evolves as a lévy process with jump diffusions. Harrison and Taksar
[12] provides a good idea to solve this kind of problems. Bernt Øksendal and
Agnès Sulem[5] study the stochastic control problem of jump diffusions.
Enlightened by these innovative ideas, we can solve effectively the optimal
control problem of the company under catastrophe risk. Firstly, we establish
the control problem of the Lévy processes with jump diffusions which is a
realistic model of the property insurance company facing catastrophe risk.
Then we work out the solution of singular-regular control of the jump
diffusions, that is, we establish the optimal return function, the optimal
reinsurance rate and the optimal dividend strategy of the insurance company.
Finally we study the impacts of some key model parameters on the optimal
return function and the optimal dividend strategy. The paper is organized as
follows: In next section, we establish the mathematical control model of the
insurance company facing catastrophe risk. In section 3, we work out a
solution of HJB equations associated with the singular-regular control on Lévy
processes with jump diffusions. In section 4, we establish the solution of the
optimal control problem, i.e., we derive the optimal return function, the
optimal reinsurance rate and the optimal dividend strategy of the property
insurance company. In section 5, we use numerical calculations to discuss the
influences of the key model parameters on the optimal retention ratio, the
optimal dividend payments level, the optimal return function and optimal
control strategy of the company. In section 6, we summarize main results of
this paper.
## 2\. Mathematical model with proportional reinsurance strategy under
catastrophe risk
In this paper, we consider a property insurance company with proportional
reinsurance strategy. The property insurance business brings in catastrophe
risk, such as earthquake and flood. The catastrophe risk could only be partly
reduced by reinsurance. The company’s management can accommodate the profit
and the risk by choosing dividend payments process and reinsurance rate. The
asset of the company evolves as the Lévy processes with jump diffusions. In
this model, if there is no dividend payments and only the proportional
reinsurance strategy is used to control the risk, then the asset of the
property insurance company is approximated by the following processes(see
Øksendal and Sulem[5]),
$\displaystyle dR_{t}=\mu a(t)dt+\sigma
a(t)d{W}_{t}+ka(t)\int_{\Re}z\widetilde{N}(dt,dz),$
where ${W}_{t}$ is a standard Brownian motion, $\mu$ is the premium rate, and
$\sigma^{2}$ is the volatility rate, it is a normal description of the
property insurance company. $1-a(t)\in[0,1]$ is the proportional reinsurance
rate. $\widetilde{N}(dt,dz)=N(dt,dz)-{I}_{\\{|z|<R\\}}\nu(dz)dt$ is the
compensated Poisson random measure of Lévy process $\\{N_{t}\\}$ with finite
Lévy measure $\nu$. The jump diffusions stand for the catastrophe risk
produced by earthquake and flood in the property insurance business. The
catastrophe risk could be partly reduced by reinsurance strategy. Since the
catastrophe risk is huge, the reinsurance strategy is not the same as the
normal reinsurance. Denote $k$ as the adjusted risk rate according to the
reinsurance regulation of the catastrophe risk. $k$ is a constant. Throughout
this paper we assume that $k\in(0,\frac{\mu}{2\int_{\Re}z\nu(dz)}]$, which
ensures that the company does not go into bankruptcy as soon as the
catastrophe risk appears. To give a mathematical foundation of the
optimization problem, we fixed a filtered probability space
$(\Omega,\mathcal{F},\mathcal{F}_{t},P)$, $\\{{W}_{t}\\}$ is a standard
Brownian motion, $\widetilde{N}(dt,dz)=N(dt,dz)-I_{\\{|z|<R\\}}\nu(dz)dt$ is
also the compensated Poisson random measure of Lévy process $\\{N_{t}\\}$ with
finite Lévy measure $\nu$ on this probability space. $\mathcal{F}_{t}$
represents the information available at time $t$ and any decision is made
based on this information. In our model, we denote $L_{t}$ as the cumulative
amount of dividend payments from time $0$ to time $t$. We assume that the
dividend payments process $L_{t}$ is an $\mathcal{F}_{t}$ -adapted, non-
decreasing and right-continuous with left limits. A control strategy $\pi$ is
described by a pair of $\mathcal{F}_{t}$ -adapted stochastic processes
$\\{a_{\pi},L^{\pi}\\}$. A strategy $\pi=\\{a_{\pi}(t),L_{t}^{\pi}\\}$ is
called admissible if $0\leq a_{\pi}(t)\leq 1$ and $L_{t}^{\pi}$ is a
nonnegative, non-decreasing and right-continuous function. We denote $\Pi$ the
set of all admissible policies. When a admissible strategy $\pi$ is applied,
we can rewrite the asset of the insurance company by the following processes,
$\displaystyle dR^{\pi}_{t}=\mu a_{\pi}(t)dt+\sigma
a_{\pi}(t)d{W}_{t}+ka_{\pi}(t)\int_{\Re}z\widetilde{N}(dt,dz)-dL^{\pi}_{t},\quad
R^{\pi}_{0}=x.$
In this case, we consider transaction cost in the dividend procedures. To
simplify the problem, we consider the proportional transaction cost, that is,
if the company pays $l$, as dividend payments, then the shareholders can get
$\beta l,\beta<1$. The company is considered bankruptcy as soon as its asset
falls below $0$. We define the bankrupt time as $\tau_{\pi}=\inf\\{t\geq
0:R^{\pi}_{t}\leq 0\\}$. $\tau_{\pi}$ is clearly an $\mathcal{F}_{t}$
-stopping time. The performance function associated with each $\pi$ is defined
by
$\displaystyle J(s,x,\pi)$ $\displaystyle=$
$\displaystyle\mathbf{E}\big{[}\int_{0}^{\tau_{\pi}}e^{-c(s+t)}\beta
dL^{\pi}_{t}\big{]},$ (2.1)
and the optimal return function is
$\displaystyle V(s,x)$ $\displaystyle=$
$\displaystyle\sup\limits_{\pi\in\Pi}\big{\\{}J(s,x,\pi)\big{\\}},$ (2.2)
where $c$ denotes the discount rate. If a strategy $\pi^{*}$ is such that
$J(s,x,\pi^{*})=V(s,x)$, then we call $\pi^{*}$, $a_{\pi^{*}}(t)$ and
$L^{\pi^{*}}_{t}$ the optimal dividend strategy, the optimal retention ratio
and the optimal dividend payments process, respectively. This paper aims at
working out the optimal strategy as well as the optimal return function, and
then discussing impacts of key model parameters(e.g. $k$, $\nu$, $\mu$ and
$\sigma^{2}$) on $V(s,x)$, $a_{\pi^{*}}(t)$ and $L^{\pi^{*}}_{t}$.
## 3\. The solution of HJB equations for(2.1) and (2.2)
In order to solve the optimal stochastic control problem (2.1) and (2.2) of
jump diffusions in next section, we establish a solution of HJB equation
associated with the control problem in this section. The main result of this
section is the following.
###### Theorem 3.1.
Assume that the Lévy measure $\nu$ and the adjusted risk rate $k$ satisfy
$0<\nu(\Re)<+\infty$, $0<\int_{\Re}z\nu(dz)<+\infty$ and
$0<k\leq\frac{\mu}{2\int_{\Re}z\nu(dz)}$. Let $\phi(s,x)$ be the function
defined by
$\phi(s,x)=e^{-cs}\psi(x)\ \mbox{and}$
$\displaystyle\psi(x)=\left\\{\begin{array}[]{l l
l}\psi_{1}(x)=C_{1}x^{\gamma},\ 0\leq x\leq x_{0},\\\
\psi_{2}(x)=C_{3}e^{d_{-}x}+C_{4}e^{d_{+}x},\ \ x_{0}\leq x\leq x^{*},\\\
\psi_{3}(x)=\beta(x-x^{*})+\psi_{2}(x^{*}),\ \ x\geq x^{*},\end{array}\right.$
(3.4)
where $x_{0}=\frac{(1-\gamma)\sigma^{2}}{\mu}$, $\gamma$, $d_{-}$ and $d_{+}$
are solutions of (3) and (3.16) below with
$\displaystyle\frac{x_{0}}{\gamma}+\frac{1}{|d_{-}|}-\frac{1}{d_{+}}<0.$ (3.5)
$x^{*}$, $C_{1}$, $C_{2}$ and $C_{3}$ are determined by (3.24), (3),( 3.19)
and (3.20) below, respectively. Then $\phi(s,x)\in C^{2}$ and is a solution of
the following HJB equation
$\displaystyle\max\big{\\{}-\frac{\partial\phi}{\partial x}(s,x)+\beta
e^{-cs},\max\limits_{a\in[0,1]}\\{\mathcal{A}\phi\\}\big{\\}}=0,$ (3.6)
where
$\displaystyle\mathcal{A}\phi$ $\displaystyle=$
$\displaystyle\frac{\partial\phi}{\partial s}+\frac{\partial\phi}{\partial
x}a\mu+\frac{1}{2}a^{2}\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\int_{\Re}\big{\\{}\phi(s,x+akz)-\phi(s,x)$ $\displaystyle-$
$\displaystyle akz\frac{\partial\phi}{\partial x}(s,x)\big{\\}}\nu(dz).$
###### Proof.
Define $D$ as
$\displaystyle D=\\{(s,x):-\frac{\partial\phi}{\partial x}(s,x)+\beta
e^{-cs}<0\\}.$
We guess that
$\displaystyle D=\\{(s,x):s\geq 0,\ \ 0<x<x^{*}\\}$
for some unidentified $x^{*}$. Inside $D$, the $\phi$ satisfies
$\displaystyle\max\limits_{a\in[0,1]}\\{\mathcal{A}\phi\\}=0,$ (3.7)
i.e.,
$\displaystyle\max\limits_{a\in[0,1]}\big{\\{}\frac{\partial\phi}{\partial s}$
$\displaystyle+$ $\displaystyle\frac{\partial\phi}{\partial
x}a\mu+\frac{1}{2}a^{2}\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\int_{\Re}\\{\phi(s,x+akz)-\phi(s,x)$ $\displaystyle-$ $\displaystyle
akz\frac{\partial\phi}{\partial x}(s,x)\\}\nu(dz)\big{\\}}=0.$
Differentiating $\mathcal{A}\phi=0$ w.r.t. $a$, we get
$\displaystyle\frac{\partial\phi}{\partial
x}\mu+a\sigma^{2}\frac{\partial^{2}\phi}{\partial x^{2}}=0.$ (3.9)
The equation (3.9) implies that the maximizer of the right-hand side of the
equation (3), $a(x)$, is the following
$\displaystyle a(x)=-\frac{\mu\frac{\partial\phi}{\partial
x}}{\sigma^{2}\frac{\partial^{2}\phi}{\partial x^{2}}}.$ (3.10)
Putting the expression (3.10) into the equation (3), we derive
$\displaystyle\frac{\partial\phi}{\partial s}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\frac{\mu^{2}(\frac{\partial\phi}{\partial
x})^{2}}{\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}}+\int_{\Re}\\{\phi(s,x-\frac{\mu\frac{\partial\phi}{\partial
x}}{\sigma^{2}\frac{\partial^{2}\phi}{\partial x^{2}}}kz)-\phi(s,x)$
$\displaystyle+$ $\displaystyle\frac{\mu\frac{\partial\phi}{\partial
x}}{\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}}kz\frac{\partial\phi}{\partial x}(s,x)\\}\nu(dz)=0.$
Define $\phi=e^{-cs}\psi(x)$, then it is easy to see from (3) that the
function $\psi(x)$ satisfies
$\displaystyle-c\psi-\frac{1}{2}\frac{\mu^{2}(\psi^{{}^{\prime}})^{2}}{\sigma^{2}\psi^{{}^{\prime\prime}}}+\int_{\Re}\\{\psi(x-\frac{\mu\psi^{{}^{\prime}}}{\sigma^{2}\psi^{\prime\prime}}kz)-\psi+\frac{\mu(\psi^{{}^{\prime}})^{2}}{\sigma^{2}\psi^{\prime\prime}}kz\\}\nu(dz)=0.$
Because $a(x)\in[0,1)$, $0\leq x\leq x_{0}$ and $a(x)=1$, $x\geq x_{0}$ for
some $x_{0}\geq 0$, we guess that
$\psi(x)=\psi_{1}(x):=C_{1}x^{\gamma}+C_{2}$, $0\leq x\leq x_{0}$. Using
$\psi(0)=0$, we have $\psi(x)=C_{1}x^{\gamma}$. Putting it into (3), we derive
the following equation
$\displaystyle-c-\frac{1}{2}\frac{\mu^{2}}{\sigma^{2}}\frac{\gamma}{\gamma-1}+\int_{\Re}\\{(1-\frac{\mu}{\sigma^{2}}\frac{1}{\gamma-1}kz)^{\gamma}-1+\frac{\mu}{\sigma^{2}}\frac{\gamma}{\gamma-1}kz\\}\nu(dz)=0.$
By the assumption of Lévy measure $\nu$ every term in the (3) is well-defined.
Let $h(\gamma)$ denote the left hand side of the (3). Then by the assumption
of $k$ we have $h(1_{-}):=\lim\limits_{\gamma<1,\gamma\rightarrow
1}\\{h(\gamma)\\}=+\infty$ and $h(0)=-c<0$. So there is at least a $\gamma$ to
solve the equation (3). Thus $\psi_{1}(x)=C_{1}x^{\gamma}$ and $a(x)=\frac{\mu
x}{\sigma^{2}(1-\gamma)}$ for $0\leq x\leq
x_{0}=\frac{(1-\gamma)\sigma^{2}}{\mu}$ because of $a(x)\in[0,1]$. If
$x_{0}\leq x\leq x^{*}$, then $a(x)=1$ and the (3 ) becomes
$\displaystyle\frac{\partial\phi}{\partial s}+\frac{\partial\phi}{\partial
x}\mu+\frac{1}{2}\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\int_{\Re}\\{\phi(s,x+kz)-\phi(s,x)-kz\frac{\partial\phi}{\partial
x}(s,x)\\}\nu(dz)=0.$
Define $\phi(x)=\phi_{2}(x):=e^{-cs}\psi_{2}(x)$ for $x_{0}\leq x\leq x^{*}$,
then we derive from the (3) that
$\displaystyle\frac{1}{2}\sigma^{2}\psi_{2}^{{}^{\prime\prime}}(x)+\mu\psi_{2}^{{}^{\prime}}(x)-c\psi_{2}(x)+\int_{\Re}\\{\psi_{2}(x+kz)-\psi_{2}(x)-kz\psi_{2}^{{}^{\prime}}(x)\\}\nu(dz)=0.$
We guess that
$\psi_{2}(x)=e^{dx}\ \mbox{ for some constant $d\in\Re$ }$
and further get the equation
$\displaystyle l(d):=\frac{1}{2}\sigma^{2}d^{2}+\mu
d-c+\int_{\Re}\big{\\{}e^{kdz}-1-kdz\big{\\}}\nu(dz)=0.$ (3.16)
Since $l(0)<0$ and
$\lim\limits_{d\rightarrow+\infty}l(d)=\lim\limits_{d\rightarrow-\infty}l(d)=+\infty$,
the equation(3.16) has two solutions $d_{-}$ and $d_{+}$ with $d_{-}<0<d_{+}$,
and so the $\psi_{2}(x)$ should have the following form
$\displaystyle\psi_{2}(x)=C_{3}e^{d_{-}x}+C_{4}e^{d_{+}x}\ \mbox{ for
$x_{0}\leq x\leq x^{*}$}$
where $C_{3}$ and $C_{4}$ are constants. For $x\geq x^{*}$, the solution
$\phi=e^{-cs}\psi_{3}(x)$ and
$\displaystyle\psi_{3}(x)=\beta(x-x^{*})+\psi_{2}(x^{*})\ \mbox{ for $x\geq
x^{*}$}.$
Since $\psi^{\prime}$ and $\psi^{\prime\prime}$ are continuous at $x^{*}$,
$\displaystyle\psi_{2}^{{}^{\prime}}(x^{*})=\psi_{3}^{{}^{\prime}}(x^{*}),$
(3.17)
$\displaystyle\psi_{2}^{{}^{\prime\prime}}(x^{*})=\psi_{3}^{{}^{\prime\prime}}(x^{*}).$
(3.18)
So
$\displaystyle
C_{3}(x^{*})d_{-}e^{d_{-}x^{*}}+C_{4}(x^{*})d_{+}e^{d_{+}x^{*}}=\beta,$
$\displaystyle
C_{3}(x^{*})d^{2}_{-}e^{d_{-}x^{*}}+C_{4}(x^{*})d^{2}_{+}e^{d_{+}x^{*}}=0.$
Solving the last two equations, we have
$\displaystyle C_{3}(x^{*})$ $\displaystyle=$ $\displaystyle\frac{\beta
d_{+}}{e^{d_{-}x^{*}}d_{-}(d_{+}-d_{-})}<0,$ (3.19) $\displaystyle
C_{4}(x^{*})$ $\displaystyle=$ $\displaystyle\frac{\beta
d_{-}}{e^{d_{+}x^{*}}d_{+}(d_{-}-d_{+})}>0.$ (3.20)
Also, since $\psi$ and $\psi^{{}^{\prime}}$ are continuous at $x_{0}$,
$\displaystyle\psi_{1}(x_{0})=\psi_{2}(x_{0}),$
$\displaystyle\psi_{1}^{{}^{\prime}}(x_{0})=\psi_{2}^{{}^{\prime}}(x_{0}),$
that is,
$\displaystyle
C_{1}x_{0}^{\gamma}=C_{3}(x^{*})e^{d_{-}x_{0}}+C_{4}(x^{*})e^{d_{+}x_{0}},$
(3.21) $\displaystyle C_{1}\gamma
x_{0}^{\gamma-1}=C_{3}(x^{*})d_{-}e^{d_{-}x_{0}}+C_{4}(x^{*})d_{+}e^{d_{+}x_{0}}.$
(3.22)
We deduce from the equations (3.21) and (3.22) that
$\displaystyle q(x^{*})$ $\displaystyle:=$
$\displaystyle(\frac{x_{0}}{\gamma}-\frac{1}{d_{-}})\frac{\beta
d_{+}}{(d_{+}-d_{-})}e^{d_{-}(x_{0}-x^{*})}$
$\displaystyle-(\frac{x_{0}}{\gamma}-\frac{1}{d_{+}})\frac{\beta
d_{-}}{(d_{+}-d_{-})}e^{d_{+}(x_{0}-x^{*})}=0.$
We claim that the $x^{*}$ satisfying the last equation does exist. In fact,
differentiating $q(x)$, we have
$\displaystyle q^{{}^{\prime}}(x)=-(\frac{x_{0}}{\gamma}d_{-}-1)\frac{\beta
d_{+}}{(d_{+}-d_{-})}e^{d_{-}(x_{0}-x)}-(\frac{x_{0}}{\gamma}d_{+}-1)\frac{\beta
d_{-}}{(d_{-}-d_{+})}e^{d_{+}(x_{0}-x)}$
$\displaystyle=-\beta\\{\frac{x_{0}d_{+}d_{-}}{\gamma(d_{+}-d_{-})}(e^{d_{-}(x_{0}-x)}-e^{d_{+}(x_{0}-x)})+\frac{d_{-}e^{d_{+}(x_{0}-x)}-d_{+}e^{d_{-}(x_{0}-x)}}{d_{+}-d_{-}}\\}>0$
for $x>x_{0}$. So $q(x)$ is an increasing function of $x$ and reaches its
minimum at $x_{0}$. Furthermore, by (3.5) we have
$\displaystyle q(x_{0})=(\frac{x_{0}}{\gamma}-\frac{1}{d_{-}})\frac{\beta
d_{+}}{(d_{+}-d_{-})}-(\frac{x_{0}}{\gamma}-\frac{1}{d_{+}})\frac{\beta
d_{-}}{(d_{+}-d_{-})}<0.$ (3.23)
Also $\lim\limits_{x\rightarrow+\infty}q(x)=+\infty$. Thus there exists an
$x^{*}(>x_{0})$ satisfying $q(x^{*})=0$. Solving the equation $q(x^{*})=0$, we
get
$\displaystyle
x^{*}=x_{0}-\frac{1}{d_{+}-d_{-}}\ln\big{\\{}\frac{d_{+}^{2}(d_{-}x_{0}-\gamma)}{d^{2}_{-}(d_{+}x_{0}-\gamma)}\big{\\}}.$
(3.24)
Clearly, (3.5) implies that
$0<\frac{d_{+}^{2}(d_{-}x_{0}-\gamma)}{d^{2}_{-}(d_{+}x_{0}-\gamma)}<1,$ so
$x^{*}>x_{0}$. Moreover,
$\displaystyle C_{1}(x^{*})=\frac{\beta
d_{+}}{x_{0}^{\gamma}e^{d_{-}x^{*}}d_{-}(d_{+}-d_{-})}e^{d_{-}x_{0}}+\frac{\beta
d_{-}}{x_{0}^{\gamma}e^{d_{+}x^{*}}d_{+}(d_{-}-d_{+})}e^{d_{+}x_{0}}>0.$
Therefore the function $\phi(s,x)$ defined by the (3.6) should be the
following form
$\phi(s,x)=e^{-cs}\psi(x)\ \mbox{ and}$
$\displaystyle\psi(x)=\left\\{\begin{array}[]{l l
l}\psi_{1}(x)=C_{1}(x^{*})x^{\gamma},\ 0\leq x\leq x_{0},\\\
\psi_{2}(x)=C_{3}(x^{*})e^{d_{-}x}+C_{4}(x^{*})e^{d_{+}x},\ \ x_{0}\leq x\leq
x^{*},\\\ \psi_{3}(x)=\beta(x-x^{*})+\psi_{2}(x^{*}),\ \ x\geq
x^{*},\end{array}\right.$ (3.29)
where $x_{0}=\frac{(1-\gamma)\sigma^{2}}{\mu}$. $x^{*}$, $\gamma$, $d_{-}$ and
$d_{+}$ are solutions of (3.24), (3) and (3.16), and $C_{1}$, $C_{2}$ and
$C_{3}$ are determined by (3),(3.19) and (3.20), respectively. The problem
remained is to approve the following inequalities.
For $0\leq x\leq x^{*}$,
$\displaystyle-\frac{\partial\phi}{\partial x}(s,x)+\beta e^{-cs}<0,$ (3.30)
$\displaystyle\max\limits_{a\in[0,1]}\big{\\{}\frac{\partial\phi}{\partial s}$
$\displaystyle+$ $\displaystyle\frac{\partial\phi}{\partial
x}a\mu+\frac{1}{2}a^{2}\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\int_{\Re}\\{\phi(s,x+akz)-\phi(s,x)$ (3.31) $\displaystyle-$
$\displaystyle akz\frac{\partial\phi}{\partial x}(s,x)\\}\nu(dz)\big{\\}}\leq
0.$
For $x\geq x^{*}$,
$\displaystyle-$ $\displaystyle\frac{\partial\phi}{\partial x}(s,x)+\beta
e^{-cs}=0,$ (3.32)
$\displaystyle\max\limits_{a\in[0,1]}\big{\\{}\frac{\partial\phi}{\partial s}$
$\displaystyle+$ $\displaystyle\frac{\partial\phi}{\partial
x}a\mu+\frac{1}{2}a^{2}\sigma^{2}\frac{\partial^{2}\phi}{\partial
x^{2}}+\int_{\Re}\\{\phi(s,x+akz)-\phi(s,x)$ (3.33) $\displaystyle-$
$\displaystyle akz\frac{\partial\phi}{\partial x}(s,x)\\}\nu(dz)\big{\\}}\leq
0.$
Since
$\displaystyle\phi_{1}^{{}^{\prime\prime}}(x)=e^{-cs}C_{1}\gamma(\gamma-1)x^{\gamma-2}<0,$
$\displaystyle\phi_{2}^{{}^{\prime\prime}}(x)=e^{-cs}\frac{\beta
d_{+}d_{-}}{d_{+}-d_{-}}(e^{d_{-}(x-x^{*})}-e^{d_{+}(x-x^{*})})<0$
for $\gamma<1$ and $x\leq x^{*}$, the inequality (3.30) is trivial due to
$\phi(x)\in C^{2}$ is a convex function, and the inequality (3.32) is a direct
consequence of $\phi_{3}^{\prime\prime}(x)=\beta$ for $x\geq x^{*}$ . For
$0\leq x\leq x_{0}$, by the expression of $\phi$,
$\max\limits_{a\in[0,1]}\\{\mathcal{A}\phi\\}=0$ is obvious. For $x_{0}\leq
x\leq x^{*}$, the inequality (3.31) is equal to
$\displaystyle\max\limits_{a\in[0,1]}\\{\frac{1}{2}a^{2}\sigma^{2}\psi_{2}^{{}^{\prime\prime}}(x)$
$\displaystyle+$ $\displaystyle
a\mu\psi_{2}^{{}^{\prime}}(x)-c\psi_{2}(x)+\int_{\Re}\\{\psi_{2}(x+akz)-\psi_{2}(x)$
(3.34) $\displaystyle-$ $\displaystyle
akz\psi_{2}^{{}^{\prime}}(x)\\}\nu(dz)\\}\leq 0.$
Denote the function in bracket $\\{\cdot\\}$ at the left side of the
inequality (3.34) as $p(a)$, we will prove that $p(a)$ is an increasing
function of $a$.
$\displaystyle p^{{}^{\prime}}(a)$ $\displaystyle=$ $\displaystyle
a\sigma^{2}\psi^{{}^{\prime\prime}}_{2}(x)+\mu\psi^{{}^{\prime}}_{2}(x)=a\sigma^{2}[C_{3}(d_{-})^{2}e^{d_{-}x}+C_{4}(d_{+})^{2}e^{d_{+}x}]$
$\displaystyle+$
$\displaystyle\mu[C_{3}d_{-}e^{d_{-}x}+C_{4}d_{+}e^{d_{+}x}]=\frac{\beta
d_{+}d_{-}}{d_{+}-d_{-}}(e^{d_{-}(x-x^{*})}-e^{d_{+}(x-x^{*})})$
$\displaystyle+$
$\displaystyle\mu[C_{3}d_{-}e^{d_{-}x}+C_{4}d_{+}e^{d_{+}x}]\geq 0.$
as $d_{-}<0$, $d_{+}>0$, $x\leq x^{*}$, $C_{3}<0$, and $C_{4}>0$. Then
$p(a)\leq p(1)=0$ for $0\leq a\leq 1$. For $x\geq x^{*}$, the inequality(3.33)
is equal to
$\displaystyle\max\limits_{a\in[0,1]}\frac{1}{2}a^{2}\sigma^{2}\psi_{3}^{{}^{\prime\prime}}(x)$
$\displaystyle+$ $\displaystyle
a\mu\psi_{3}^{{}^{\prime}}(x)-c\psi_{3}(x)+\int_{\Re}\\{\psi_{3}(x+akz)-\psi_{3}(x)$
$\displaystyle-$ $\displaystyle
akz\psi_{3}^{{}^{\prime}}(x)\\}\nu(dz)=a\mu\beta-c\beta(x-x^{*})-c\psi_{2}(x^{*})$
$\displaystyle\leq$
$\displaystyle\mu\beta-c\psi_{2}(x^{*})-c\beta(x-x^{*})\leq 0$
due to $x\geq x^{*}$ and $\mu\beta-c\psi_{2}(x^{*})=0$. So we end the proof. ∎
## 4\. The solution of the optimal control problem with jump diffusions
We now give a verification theorem for singular -regular control problem(2.1)
and (2.2). We first prove the following.
###### Theorem 4.1.
Let W(s, x) satisfy the following HJB equation,
$\displaystyle\max\\{-\frac{\partial W}{\partial x}(t,x)+\beta
e^{-ct},\max\limits_{a\in[0,1]}\\{\mathcal{A}W(t,x)\\}\\}=0$ (4.1) for $t\geq
0$ and $x\geq 0$, $\displaystyle W(t,0)=0\ \mbox{ for any $t\geq 0$}.$ (4.2)
Then $W(s,x)\geq J(s,x,\pi)$ for any admissible strategy $\pi$ and
$(s,x)\in\Re^{2}_{+}$.
###### Proof.
For any fixed strategy $\pi$, let $\Lambda=\\{s:L_{s-}^{\pi}\neq
L_{s}^{\pi}\\}$, $\hat{L}=\sum_{s\in\Lambda,s\leq
t}(L_{s}^{\pi}-L_{s-}^{\pi})$ be the discontinuous part of $L_{s}^{\pi}$ and
$\tilde{L}_{t}^{\pi}=L_{t}^{\pi}-\hat{L}_{t}^{\pi}$ be the continuous part of
$L_{s}^{\pi}$. Let $\tau_{\pi}$ be the first time that the corresponding cash
flow $R^{\pi}_{t}$ defined by (2.2) hit $(-\infty,0)$. Then, by applying the
generalized Itô formula to the stochastic process
$Y^{\pi}(t):=(s+t,R^{\pi}_{t})^{T}$ and the function $W(s,x)$, we have
$\displaystyle\mathbf{E}[W(s+t\wedge\tau_{\pi},R_{t\wedge\tau_{\pi}}^{\pi})]$
$\displaystyle=W(s,x)+\mathbf{E}[\int_{0}^{t\wedge\tau_{\pi}}\mathcal{A}W(s+u,R_{u}^{\pi})du$
$\displaystyle-\int_{0}^{t\wedge\tau_{\pi}}\frac{\partial
W(s+u,R^{\pi}_{u})}{\partial x}dL^{(c)}_{u}+\sum\limits_{0<t_{n}\leq
t\wedge\tau_{\pi}}\Delta_{L}W(s+t_{n},R^{\pi}_{t_{n}})],$
where
$\displaystyle\mathcal{A}W(s,x)$ $\displaystyle=$ $\displaystyle\frac{\partial
W}{\partial s}+a\mu\frac{\partial W}{\partial
x}+\frac{1}{2}a^{2}\sigma^{2}\frac{\partial^{2}W}{\partial
x^{2}}+\int_{\Re}\\{W(s,x+akz)-W(s,x)$ $\displaystyle-$ $\displaystyle
akz\frac{\partial W}{\partial x}(s,x)\\}\nu(dz),$
$\displaystyle\Delta_{L}W(s+t_{n},R^{\pi}_{t_{n}}):=W(Y^{\pi}(t_{n}))-W(Y^{\pi}(t^{-}_{n})+\Delta_{N}Y^{\pi}(t_{n})),$
$\displaystyle\Delta_{N}Y^{\pi}(t_{n})):=\big{(}0,ka_{\pi}(t_{n})\int_{\Re}z\widetilde{N}(\\{t_{n}\\},dz)\big{)},$
$\displaystyle\\{t_{k}\\}\mbox{ is the set of jumping times of $L$ }.$
Using $\mathcal{A}W\leq 0$ in the equation (4), we see that
$\displaystyle\mathbf{E}[W(s+t\wedge\tau_{\pi},R_{t\wedge\tau_{\pi}}^{\pi})]\leq
W(s,x)$ $\displaystyle-\mathbf{E}[\int_{0}^{t\wedge\tau_{\pi}}\frac{\partial
W(s+u,R^{\pi}_{u})}{\partial x}dL^{(c)}_{u}-\sum\limits_{s<t_{n}\leq
t\wedge\tau_{\pi}}\Delta_{L}W(Y^{\pi}_{t_{n}})].$ (4.4)
By the mean value theorem we have
$\displaystyle\Delta_{L}W(Y^{\pi}_{t_{n}})=-\frac{\partial W}{\partial
x}(\hat{Y}^{(n)}_{t_{n}})\Delta L(t_{n}),$
where $\hat{Y}^{(n)}_{t_{n}}$ is some point on the straight line between
$Y^{\pi}_{t_{n}}$ and $Y^{\pi}_{t_{n}^{-}}+\Delta_{N}(Y^{\pi}_{t_{n}})$. Since
$W^{{}^{\prime}}(Y_{u}^{\pi})\geq\beta e^{-c(s+u)}$,
$\Delta_{L}W(Y^{\pi}_{t_{n}})\leq-\beta
e^{-c(s+t_{n})}(L_{t_{n}}^{\pi}-L_{t_{n}-}^{\pi}),$
which, together with the inequality (4), implies
$\displaystyle\mathbf{E}[W(s+t\wedge\tau_{\pi},R_{t\wedge\tau_{\pi}}^{\pi})]$
$\displaystyle+$
$\displaystyle\mathbf{E}\big{\\{}\int_{0}^{t\wedge\tau_{\pi}}\beta
e^{-c(s+u)}dL_{u}^{\pi}\big{\\}}\leq W(s,x).$
By the definition of $\tau_{\pi}$, the boundary condition (4.2) and
$W^{{}^{\prime}}(Y_{u}^{\pi})\geq\beta e^{-c(s+u)}$, it is easy to prove that
$\liminf\limits_{t\rightarrow\infty}W(Y_{t})I_{\\{\tau_{\pi}=\infty\\}}=0$ and
$\displaystyle\liminf\limits_{t\rightarrow\infty}W(s+t\wedge\tau_{\pi},R_{t\wedge\tau_{\pi}}^{\pi})$
$\displaystyle=$ $\displaystyle
W(s+\tau_{\pi},0)I_{\\{\tau_{\pi}<\infty\\}}+\liminf\limits_{t\rightarrow\infty}W(Y_{t})I_{\\{\tau_{\pi}=\infty\\}}$
(4.6) $\displaystyle\geq$ $\displaystyle
W(s+\tau_{\pi},0)I_{\\{\tau_{\pi}<\infty\\}}=0.$
So, we deduce from the inequalities (4) and (4.6) that
$\displaystyle
J(s,x,\pi)=\mathbf{E}\big{\\{}\int_{0}^{\tau_{\pi}}e^{-c(s+t)}\beta
dL_{t}^{\pi}\big{\\}}\leq W(s,x),$
thus we complete the proof. ∎
Let
$\displaystyle a(x)=\left\\{\begin{array}[]{l l l}\frac{\mu
x}{\sigma^{2}(1-\gamma)},\quad x<x_{0},\\\ 1,\qquad\quad x\geq
x_{0}\end{array}\right.$
where $x_{0}=\frac{(1-\gamma)\sigma^{2}}{\mu}$. We call $a(x)$ the feedback
control function of the control problem (2.1) and (2.2). We can now state the
main result of this paper.
###### Theorem 4.2.
Assume that (3.5)holds, the Lévy measure $\nu$ and the adjusted risk rate $k$
satisfy $0<\nu(\Re)<+\infty$, $0<\int_{\Re}z\nu(dz)<+\infty$ and
$0<k\leq\frac{\mu}{2\int_{\Re}z\nu(dz)}$. Then the optimal return function and
the optimal dividend strategy of the control problem (2.1) and (2.2) are
$V(s,x)=\phi(s,x)=e^{-cs}\psi(x)$ and
$\pi^{*}=(a(R_{t}^{\pi^{*}}),L^{\pi^{*}}_{t})$, respectively, where
$(R_{t}^{\pi^{*}},L^{\pi^{*}}_{t})$ is uniquely determined by the following
stochastic differential equations with reflection,
$\displaystyle\left\\{\begin{array}[]{l l l}R_{t}^{\pi^{*}}=x+\int_{0}^{t}\mu
a(R_{s}^{\pi^{*}})ds+\int_{0}^{t}\sigma
a(R_{s}^{\pi^{*}})d{W}_{s}+k\int_{0}^{t}\int_{\Re}a(R_{s}^{\pi^{*}})z\widetilde{N}(ds,dz)\\\
-L_{t}^{\pi^{*}},\\\ R_{t}^{\pi^{*}}\leq x^{*},\\\
\int_{0}^{\infty}I_{\\{t:R_{t}^{\pi^{*}}<x^{*}\\}}(t)dL_{t}^{\pi^{*}}=0,\end{array}\right.$
(4.12)
$\psi(x)$ is the function defined by (3.4) and the optimal dividend payments
level $x^{*}$ is given by (3.24).
###### Proof.
Since the function $\phi(s,x)$ satisfies the HJB equations (3.6), it is not
hard to see that $\phi(s,x)$ also satisfies conditions in Theorem 4.1. So
$\phi(s,x)\geq J(s,x,\pi)$ for any $\pi$, i.e.,
$\displaystyle\phi(s,x)\geq V(s,x).$ (4.13)
Next, we will prove $V(s,x)=\phi(s,x)=J(s,x,\pi^{*})$ corresponding to
$\pi^{*}$. By applying the generalized Itô formula, noting that the
construction of $\phi(s,x)$ and the last two equations in (4.12), we deduce
from the inequality(3.30)and the equations (4.1) that
$\mathcal{A}\phi(s+t,R_{t}^{\pi^{*}})=0$ for any $t\geq 0$,
$\int_{0}^{t\wedge\tau^{*}}\frac{\partial\phi(Y^{\pi^{*}}_{u})}{\partial
x}dL^{(c)}_{u}=\int_{0}^{t\wedge\tau^{*}}\beta e^{-c(s+u)}dL_{u}^{(c)}$ and
$\sum\limits_{s<t_{n}\leq\tau^{*}}\Delta_{L}\phi(Y^{\pi^{*}}_{t_{n}})=-\sum\limits_{s<t_{n}\leq\tau^{*}}\frac{\partial\phi}{\partial
x}(s+t_{n},x^{*})\Delta L(t_{n})=-\sum\limits_{s<t_{n}\leq\tau^{*}}\beta
e^{-c(s+t_{n})}\Delta L(t_{n})$, where $\tau^{*}=\inf\\{t\geq
0:R^{\pi^{*}}_{t}<0\\}$. So
$\displaystyle\mathbf{E}[\phi(s+t\wedge\tau^{*},R_{t\wedge\tau^{*}}^{\pi^{*}})]$
$\displaystyle=$
$\displaystyle\phi(s,x)+\mathbf{E}[\int_{0}^{t\wedge\tau^{*}}\mathcal{A}\phi(Y_{u}^{\pi^{*}})du$
(4.14) $\displaystyle-$
$\displaystyle\int_{0}^{t\wedge\tau^{*}}\frac{\partial\phi(Y^{\pi^{*}}_{u})}{\partial
x}dL^{(c)}_{u}+\sum\limits_{s<t_{n}\leq\tau^{*}}\Delta_{L}\phi(Y^{\pi^{*}}_{t_{n}})]$
$\displaystyle=$
$\displaystyle\phi(s,x)-\mathbf{E}[\int_{0}^{t\wedge\tau^{*}}\beta
e^{-c(s+u)}dL_{u}^{(c)}$ $\displaystyle+\sum\limits_{s<t_{n}\leq\tau^{*}}\beta
e^{-c(s+t_{n})}\Delta L(t_{n})]$ $\displaystyle=$
$\displaystyle\phi(s,x)-\mathbf{E}[\int_{0}^{t\wedge\tau^{*}}\beta
e^{-c(s+u)}dL_{u}^{\pi^{*}}].$
Since
$\lim\limits_{t\rightarrow\infty}\phi(s+t\wedge\tau^{*},R_{t\wedge\tau^{*}}^{\pi^{*}})=\lim\limits_{t\rightarrow\infty}e^{-c(s+t\wedge\tau^{*})}\psi(R_{t\wedge\tau^{*}}^{\pi^{*}})=e^{-c(s+\tau^{*})}\psi(R_{\tau^{*}}^{\pi^{*}})=e^{-c(s+\tau^{*})}\psi(0)=0$,
we see from the inequality (4.13) and the equation(4.14) that
$V(s,x)\leq\phi(s,x)=\lim\limits_{t\rightarrow\infty}\mathbf{E}\big{[}\int_{0}^{t\wedge\tau^{*}}e^{-cs}\beta_{1}dL^{\pi^{*}}_{s}\big{]}=J(s,x,\pi^{*})\leq
V(s,x).$
So $V(s,x)=\phi(s,x)=J(s,x,\pi^{*})$, that is, $\phi(s,x)$ is the optimal
return function, $\pi^{*}$ is the optimal dividend strategy and $x^{*}$ is the
optimal dividend payments level. Thus the proof has been done. ∎
## 5\. Numerical examples
In this section, based on Theorem 4.2, we present some numerical examples,
together with the feedback control function $a(x)$ and the comparison theorem
for SDE, to portray how the key model parameters(e.g. $k$, $\mu$, $\sigma^{2}$
and $\nu$) impact on $V(s,x)$ and the optimal control strategy $\pi^{*}$, that
is, $a_{\pi^{*}}(t)$ and $L^{\pi^{*}}_{t}$, respectively.
###### Example 5.1.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)dz$. Figure 1 below explains that the
adjusted risk rate will increase the optimal dividend payments level
$x^{*}(k)$, so to avoid bankruptcy the company should decrease the times of
dividend or increase $k$ if possible, that is, the company needs to maintain
the cash inside the company to cover the catastrophe risk ,so it pays dividend
at a higher level. On the other hand, $L^{\pi^{*}}_{t}$ decreases with $k$ by
(4.12), $R_{t}^{\pi^{*}}$ increases with $k$, so we see that $a_{\pi^{*}}(t)$
also increases with $k$. In fact, the catastrophe risk business brings in more
risk as well as more income, and the higher asset level raises the risk
sustainment of the company. It could reduce its reinsurance level
$(i.e.,1-a_{\pi^{*}}(t))$ according with the optimal control strategy
$\pi^{*}$.
Figure 1. The optimal dividend payments level $x^{*}(k)$ as a function of
$k$.The parameter values are $\sigma^{2}=5$, $c=0.05$, $\mu=2$, $s=0$.
###### Example 5.2.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)dz$. Figure 2 below states that the
property insurance company’s profit increases with the initial capital $x$ and
the adjusted risk rate $k$. So the property insurance company can get some
return from its catastrophe risk insurance business, but the return’s
increment is small by adjusting $k$. However, the company can receive a good
public reputation by constant $k$, and interest from the catastrophe insurance
business.
Figure 2. The optimal return function $V(x,k)$ as a function of $x$ and $k$.
The parameter values are $\sigma^{2}=5$, $c=0.05$, $\beta=0.8$, $s=0$,
$\mu=2$.
###### Example 5.3.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)$. Figure 3 below portrays that the
optimal dividend payments level $x^{*}(\mu)$ decreases with the premium rate
$\mu$, so $L^{\pi^{*}}_{t}$ increases with $\mu$, but $a_{\pi^{*}}(t)$
decreases with the premium rate. These facts mean that the higher growth rate
of the insurance company’s asset raise the company’s risk tolerance level and
the company could pay dividend at a lower level. Meanwhile, the company should
adopt a higher reinsurance rate to avoid bankruptcy due to the lower dividend
payments level.
Figure 3. The optimal dividend payments level $x^{*}(\mu)$ as a function of
$\mu$. The parameter values are $\sigma^{2}=5$, $c=0.05$, $s=0$, $k=0.5$.
###### Example 5.4.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)dz$. Figure 4 states that the optimal
return function $V(x,\mu)$ is an increasing function of $\mu$, and high
premium rate can notably increase the company’s return, that is, a higher
growth rate of the insurance company’s asset results in a higher return.
Figure 4. The optimal return function $V(x,\mu)$ as a function of $x$ and
$\mu$. The parameter values are $\sigma^{2}=5$, $c=0.05$, $\beta=0.8$, $s=0$,
$k=0.5$.
###### Example 5.5.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)dz$. Figure 5 below portrays that the
optimal dividend payments level $x^{*}(\sigma^{2})$ increases with the risk
volatility rate $\sigma^{2}$ of normal insurance business, so
$L^{\pi^{*}}_{t}$ decreases with $\sigma^{2}$, but $a_{\pi^{*}}(t)$ increases
it. These mean that the higher volatility make the insurance company’s asset
reduce the company’s risk tolerance level and the company prefer to maintain
the cash inside the company to cover the risk. Meanwhile, the company should
adopt a lower reinsurance rate to get lower optimal dividend payments level.
Figure 5. The optimal dividend payments level $x^{*}(\sigma^{2})$ as a
function of $\sigma^{2}$. The parameter values are $c=0.05$, $s=0$, $k=0.5$,
$\mu=2$.
###### Example 5.6.
Let $\nu(dz)=e^{-z}I_{\\{z\geq 0\\}}(z)dz$. Figure 6 below states that the
increment of the optimal return function $V(x,\sigma^{2})$ due to $\sigma^{2}$
is very large, so higher risk can also notably increase the company’s return.
Figure 6. The optimal return function $V(x,\sigma^{2})$ as a function of $x$
and $\sigma^{2}$. The parameter values are $c=0.05$, $\beta=0.8$, $s=0$,
$k=0.5$, $\mu=2$.
###### Example 5.7.
Let $\nu_{t}(dz)=e^{-tz}I_{\\{z\geq 0\\}}(z)dz(t\geq 1)$. Figure 7 below
portrays that the optimal dividend payments level $x^{*}(t)$ has obvious
decrements on $[1,4]$, but on $[4,+\infty)$ the optimal dividend payments
level has no visibly changes, so $a_{\pi^{*}}(\cdot)$ and
$L^{\pi^{*}}_{\cdot}$ change greatly for different Lévy measures
$\nu_{t}(dz)=e^{-tz}I_{\\{z\geq 0\\}}(z)dz$, $t\in[1,4]$. However, they are
almost same for different Lévy measures $\nu_{t}(dz)=e^{-tz}I_{\\{z\geq
0\\}}(z)dz$, $t\geq 4$.
Figure 7. The optimal dividend payments level $x^{*}(t)$ as a function of
$t$. The parameter values are $\sigma^{2}=5$, $c=0.05$, $s=0$, $k=0.5$,
$\mu=2$.
###### Example 5.8.
Let $\nu_{t}(dz)=e^{-tz}I_{\\{z\geq 0\\}}(z)dz$. Figure 8 below states that
the change of the optimal return function $V(x,t)$ for different $t$ is not
distinct. So the optimal return function $V(x,t)$ is nearly stable for
different Lévy measures $\nu_{t}(dz)=e^{-tz}I_{\\{z\geq 0\\}}(z)dz$ ($t\geq
0$).
Figure 8. The optimal return function $V(x,t)$ as a function of $x$ and $t$.
The parameter values are $\sigma^{2}=5$, $c=0.05$, $\beta=0.8$, $s=0$,
$k=0.5$, $\mu=2$. )
## 6\. Conclusion
We consider the optimal dividend and the reinsurance strategy of a property
insurance company. The property insurance business brings in catastrophe risk,
such as earthquake and flood. The catastrophe risk could be partly reduced by
reinsurance. Due to the huge risk, the company needs to add a adjusted risk
rate in the regulation. The management of the company controls the reinsurance
rate and dividend payments to maximize the expected present value of the
dividends before bankruptcy. This is the first time to consider the
catastrophe risk in an insurance model, which is more realistic. The
catastrophe risk is modeled as the jump process in the stochastic control
problem. In order to find the solution of the problem, we implore the mixed
singular-regular control methods of jump diffusions. We establish the optimal
reinsurance rate, the optimal dividend strategies and explicit the optimal
return function of the company. The influences of the catastrophe risk and the
reinsurance regulation of the catastrophe risk on the optimal control strategy
of the insurance company are also discussed. Based on the main results we have
just established, we present some numerical examples to analyze in detail how
the key model parameters impact on the optimal retention ratio, the optimal
dividend payments strategies and the optimal return of the company.
Acknowledgements. This work is supported by Projects 10771114 and 11071136 of
NSFC, Project 20060003001 of SRFDP, the SRF for ROCS, SEM and the Korea
Foundation for Advanced Studies. We would like to thank the institutions for
the generous financial support. Special thanks also go to the participants of
the seminar stochastic analysis and finance at Tsinghua University for their
feedbacks and useful conversations. Li He also thanks the generous financial
support of 10XNF057 from Renmin University of China. We are also grateful to
Dr. Huan Fan and Yifeng Yin for useful discussions.
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* [15] Lin He, Zongxia Liang, 2008. Optimal Dividend Control of the Insurance Company with Proportional Reinsurance Policy under solvency constraints. Insurance: Mathematics and Economics, Vol.43, 474-479.
* [16] Højgaard, B., Taksar, M.: Optimal Proportional Reinsurance Policies for Diffusion Models, Scandinavian Actuarial Journal, Vol. 2, 166-180, 1998\.
* [17] Højgaard, B., Taksar, M.: Controlling Risk Exposure and Dividends Payments Schemes: Insurance company Example, Mathematical Finance, Vol. 9, No. 2, 153-182, April 1999.
* [18] Højgaard, B., Taksar, M.: Optimal Proportional Reinsurance Policies with Transaction Costs, Insurance: Mathematics and Economics, Vol. 22, 41-51, 1998.
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|
arxiv-papers
| 2010-09-07T12:32:08 |
2024-09-04T02:49:12.716364
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zongxia Liang, Lin He, Jiaoling Wu",
"submitter": "Zongxia Liang",
"url": "https://arxiv.org/abs/1009.1269"
}
|
1009.1274
|
# Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-
Zygmund operators on non-homogeneous spaces
The Anh Bui The Anh Bui was supported by a Macquarie University scholarship
Xuan Thinh Duong Xuan Thinh Duong was supported by a research grant from
Macquarie University
2010 Mathematics Subject Classification: 42B20, 42B35.
Key words: non-homogeneous spaces, Hardy spaces, BMO, Calderón - Zygmund
operator.
###### Abstract
One defines a non-homogeneous space $(X,\mu)$ as a metric space equipped with
a non-doubling measure $\mu$ so that the volume of the ball with center $x$,
radius $r$ has an upper bound of the form $r^{n}$ for some $n>0$. The aim of
this paper is to study the boundedness of Calderón-Zygmund singular integral
operators $T$ on various function spaces on $(X,\mu)$ such as the Hardy
spaces, the $L^{p}$ spaces and the regularized BMO spaces. This article thus
extends the work of X. Tolsa [T1] on the non-homogeneous space
$(\mathbb{R}^{n},\mu)$ to the setting of a general non-homogeneous space
$(X,\mu)$. Our framework of the non-homogeneous space $(X,\mu)$ is similar to
that of [Hy] and we are able to obtain quite a few properties similar to those
of Calderón-Zygmund operators on doubling spaces such as the weak type $(1,1)$
estimate, boundedness from Hardy space into $L^{1}$, boundedness from
$L^{\infty}$ into the regularized BMO and an interpolation theorem.
Furthermore, we prove that the dual space of the Hardy space is the
regularized BMO space, obtain a Calderón-Zygmund decomposition on the non-
homogeneous space $(X,\mu)$ and use this decomposition to show the boundedness
of the maximal operators in the form of Cotlar inequality as well as the
boundedness of commutators of Calderón-Zygmund operators and BMO functions.
###### Contents
1. 1 Introduction
2. 2 Non-homogeneous spaces, families of doubling balls and singular integrals
1. 2.1 Non-homogeneous spaces and families of doubling balls
2. 2.2 Calderón-Zygmund operators
3. 3 The RBMO spaces
1. 3.1 Definition of RBMO$(\mu)$
2. 3.2 Some characterizations of RBMO$(\mu)$
4. 4 Interpolation results
1. 4.1 The sharp maximal operator
2. 4.2 An Interpolation Theorem for linear operators
5. 5 Atomic Hardy spaces and their dual spaces
1. 5.1 The space $H^{1,\infty}_{at}(\mu)$
2. 5.2 The space $H^{1,p}_{at}(\mu)$
6. 6 Calderón-Zygmund decomposition
1. 6.1 Calderón-Zygmund decomposition
2. 6.2 The weak $(1,1)$ boundedness of Calderón-Zygmund operators
3. 6.3 Cotlar inequality
7. 7 The boundedness of Calderón-Zygmund operators
1. 7.1 The boundedness of Calderón-Zygmund operators from $L^{\infty}$ to RBMO space
2. 7.2 The boundedness of Calderón-Zygmund operators on Hardy spaces
3. 7.3 Commutators of Calderón-Zygmund operators with RBMO functions
## 1 Introduction
In the last few decades, Calderón-Zygmund theory of singular integrals has
played a central part of modern harmonic analysis with lots of extensive
applications to other fields of mathematics. This theory has established
criteria for singular integral operators to be bounded on various function
spaces including $L^{p}$ spaces, $1<p<\infty$, Hardy spaces, BMO spaces and
Besov spaces.
One of the main features of the standard Calderón-Zygmund singular integral
theory is the requirement that the underlying spaces or domains to possess the
doubling (volume) property. Recall that a space $X$ equipped with a distance
$d$ and a measure $\mu$ is said to have the doubling property if there exists
a constant $C$ such that for all $x\in X$ and all $r>0$,
$\mu(B(x,2r))\leq C\mu(B(x,r))$
where $B(x,r)$ denotes the ball with center $x$ and radius $r>0$.
In the last ten years or so, there has been substantial progress in obtaining
boundedness of singular integrals acting on spaces without the doubling
property. Many features of the standard Calderón-Zygmund singular integral
theory was extended to spaces with a mild volume growth condition in place of
doubling property through the works of Nazarov, Treil, Volberg, Tolsa and
others. See, for example [NTV1], [NTV2], [NTV3], [T1] and [T2]. These
breakthroughs disproved the long held belief of the decades of 70’s and 80’s
that the doubling property is indispensable in the theory of Calderón-Zygmund
singular integrals and lead to more powerful techniques and estimates in
harmonic analysis.
Let $X$ be a metric space equipped with a measure $\mu$, possibly non-
doubling, satisfying
$\mu(B(x,r))\leq Cr^{n}$
for some positive constants $C$, $n$ and all $r>0$. We will call such a space
$(X,\mu)$ a non-homogeneous space. Let $T$ be a Calderón-Zygmund operator
acting on a non-homogeneous space $X$, i.e. the associated kernel of $T$
satisfies appropriate bounds and has Hölder continuity (for the precise
definition, see Section 2.1). Assume that $T$ is bounded on $L^{2}(X)$, then
it is shown in [NTV2] that the Calderón-Zygmund operator $T$ is of weak type
$(1,1)$, hence by interpolation, is bounded on $L^{p}(X),1<p<\infty$. See also
[T2].
Hardy spaces and BMO spaces on a non-homogeneous space $X$ were studied by a
number of authors, for example [T1], [MMNO], [Hy]. In [MMNO], the authors
studied the spaces BMO$(\mu)$ and $H^{1}_{at}(\mu)$ on $\mathbb{R}^{n}$ (with
BMO$(\mu)$ space being defined via the standard bounded oscillations and the
Hardy space $H^{1}_{at}(\mu)$ being defined by an atomic decomposition) for a
non doubling measure $\mu$ and showed some standard properties of these spaces
such as the John-Nirenberg inequality, an interpolation theorem between
BMO$(\mu)$ and $H^{1}_{at}(\mu)$, and BMO$(\mu)$ being the dual space of
$H^{1}_{at}(\mu)$. However, it is shown by Verdera [V] that an $L^{2}$ bounded
Calderón-Zygmund operator may be unbounded from $L^{\infty}(\mu)$ into
BMO$(\mu)$ as well as from $H^{1}_{at}(\mu)$ into $L^{1}(\mu)$. This shows the
need to introduce variants of the BMO spaces characterized by bounded
oscillation estimates so that the Calderón-Zygmund operators are bounded from
$L^{\infty}(\mu)$ into these variants of BMO spaces.
In [T1], the author introduced the RBMO space, a variant of the space BMO, on
the non-homogeneous space $(\mathbb{R}^{n},\mu)$ which retains some of the
properties of the standard BMO such as the John-Nirenberg inequality. See
Section 3 for the definition of RBMO spaces. While Calderón-Zygmund operators
might not be bounded from $L^{\infty}(\mathbb{R}^{n},\mu)$ into
BMO$(\mathbb{R}^{n},\mu)$, they are bounded from
$L^{\infty}(\mathbb{R}^{n},\mu)$ into RBMO$(\mathbb{R}^{n},\mu)$, [T1].
Recently, Hytönen studied the RBMO spaces on non-homogeneous spaces $(X,\mu)$
(instead of $(\mathbb{R}^{n},\mu)$) [Hy]. He proved that the space RBMO$(\mu)$
on $X$ still satisfies John-Nirenberg inequality. However, the boundedness of
Calderón-Zygmund operators from $L^{\infty}(\mu)$ into RBMO$(\mu)$ and a
number of other properties are still open questions for the setting of general
non-homogeneous spaces $(X,\mu)$.
In this article, our aim is to conduct an extensive study on the RBMO spaces
on general non-homogeneous spaces. More specifically, for a non-homogeneous
space $(X,\mu)$ equipped with a measure $\mu$ which is dominated by some
doubling measure (the same setting as in [Hy]), we are able to prove the
following new results:
1. (i)
An $L^{2}$ bounded Calderón-Zygmund operator is bounded from $L^{\infty}(\mu)$
into the RBMO space, see Theorem 7.1.
2. (ii)
The dual space of the atomic Hardy spaces is shown to be the RBMO space. We
also show that an $L^{2}$ bounded Calderón-Zygmund operator is bounded from
the atomic Hardy space $H^{1}_{at}(\mu)$ into $L^{1}(\mu)$, see Theorems 5.6
and 7.3.
3. (iii)
An interpolation theorem between the RBMO space and the Hardy space
$H^{1}_{at}(\mu)$: if an operator is bounded from $H^{1}_{at}(\mu)$ into
$L^{1}(\mu)$ and from $L^{\infty}(\mu)$ into the RBMO space, then the operator
is bounded on $L^{p}(\mu)$ for all $1<p<\infty$, see Theorem 6.4.
4. (iv)
A Cotlar type inequality for Calderón-Zygmund operators which gives the
boundedness of several maximal operators associated with $T$, see Theorem 6.6.
5. (v)
The boundedness of commutators of Calderón-Zygmund operators and RBMO
functions on $L^{p}$ spaces, see Theorem 7.6.
6. (vi)
A Calderón-Zygmund decomposition using a variant of Vitali covering lemma, see
Theorem 6.3, and the weak type $(1,1)$ of an $L^{2}$ bounded Calderón-Zygmund
operator, see Theorem 6.5.
We remark that, while this manuscript was in finishing touch, we learned that
similar results concerning the Hardy spaces as in (ii) have been obtained
independently in [HyYY].
We now give a brief comment about some techniques used in this paper. In
addition to using some ideas and techniques in [T1], we obtain certain key
estimates through careful investigation of the family of doubling balls in a
non-homogeneous space $(X,\mu)$. Let us recall that the main techniques used
in [T1] rely on the Besicovitch covering lemma and the construction of the
$(\alpha,\beta)$-doubling balls in $\mathbb{R}^{n}$. However, the Besicovitch
covering lemma is only applicable to $\mathbb{R}^{n}$ and it is not applicable
in the setting of general non-homogeneous spaces. In the general setting, one
can construction the $(\alpha,\beta)$-doubling balls by using a covering lemma
in [He] in place of the Besicovitch covering lemma. In [Hy], the author used
this substitution to obtain the John-Nirenberg inequality for the RBMO spaces.
However, it seems that to obtain further results similar to the standard
theory as in the case of doubling spaces, more refined techniques are needed.
An important technical detail in this paper is our construction of the three
consecutive $(\alpha,\beta)$-doubling balls (see, Proposition 2.4) which we
employ successfully to obtain the important characterizations (9) and (10),
similar to those in [T1, Lemma 2.10]. By using these three consecutive
$(\alpha,\beta)$-doubling balls, we show the boundedness of Calderón-Zygmund
operators from $L^{\infty}(X,\mu)$ into the space RBMO (see Theorem 7.1) as
well as an interpolation theorem of RBMO spaces (see Theorem 4.3).
Acknowledgement: The second named author would like to thank El Maati Ouhabaz
for helpful discussion.
## 2 Non-homogeneous spaces, families of doubling balls and singular
integrals
### 2.1 Non-homogeneous spaces and families of doubling balls
In this paper, for the sake of simplicity we always assume that $(X,d)$ is a
metric space. With minor modifications, similar results hold when $X$ is a
quasi-metric space.
Geometrically doubling regular metric spaces. We adopt the definition that the
space $(X,d)$ is geometrically doubling if there exists a number
$N\in\mathbb{N}$ such that every open ball $B(x,r)=\\{y\in X:d(y,x)<r\\}$ can
be covered by at most $N$ balls of radius $r/2$. Our using of this somewhat
non-standard name is to differentiate this property from other types of
doubling properties. If there is no specification, the ball $B$ means the ball
center $x_{B}$ with radius $r_{B}$. Also, we set $n=\log_{2}N$, which can be
viewed as (an upper bound for) a geometric dimension of the space. Let us
recall the following well-known lemma. See, for example [Hy].
###### Lemma 2.1
In a geometrically doubling regular metric space, a ball $B(x,r)$ can contain
the centers $x_{i}$ of at most $N\alpha^{-n}$ disjoint balls $B(x_{i},\alpha
r)$ for any $\alpha\in(0,1]$.
Upper doubling measures. A measure $\mu$ in the metric space $(X,\mu)$ is said
to be an upper doubling measure if there exists a dominating function
$\lambda$ with the following properties:
1. (i)
$\lambda:X\times(0,\infty)\mapsto(0,\infty)$;
2. (ii)
for any fixed $x\in X$, $r\mapsto\lambda(x,r)$ is increasing;
3. (iii)
there exists a constant $C_{\lambda}>0$ such that $\lambda(x,2r)\leq
C_{\lambda}\lambda(x,r)$ for all $x\in X$, $r>0$;
4. (iv)
the inequality $\mu(x,r):=\mu(B(x,r))\leq\lambda(x,r)$ holds for all $x\in X$,
$r>0$;
5. (v)
and $\lambda(x,r)\approx\lambda(y,r)$ for all $r>0,\ x,y\in X$ and $d(x,y)\leq
r$.
We note that in [Hy], the condition (v) is not assumed.
###### Lemma 2.2
Every family of balls $\\{B_{i}\\}_{i\in F}$ of uniformly bounded diameter in
a metric space $X$ contains a disjoint sub-family $\\{B_{i}\\}_{i\in E}$ with
$E\subset F$ such that
$\cup_{i\in F}B_{i}\subset\cup_{i\in E}5B_{i}.$
For a proof of Lemma 2.2, see [He].
Assumptions: Throughout the paper, we always assume that $(X,\mu)$ is a
geometrically doubling regular metric space and the measure $\mu$ is an upper
doubling measure.
We adopt the following definition as in [T1]. For $\alpha,\beta>1$, a ball
$B\subset X$ is called $(\alpha,\beta)$-doubling if
$\mu(\alpha)\leq\beta\mu(B)$. The following result states the existence of
plenty of doubling balls with small radii and with large radii.
###### Lemma 2.3 ([Hy])
The following statements hold:
1. (i)
If $\beta>C_{\lambda}^{\log_{2}\alpha}$, then for any ball $B\subset X$ there
exists $j\in\mathbb{N}$ such that $\alpha^{j}B$ is $(\alpha,\beta)$-doubling.
2. (ii)
If $\beta>\alpha^{n}$ where $n$ is the doubling order of $\lambda$, then for
any ball $B\subset X$ there exists $j\in\mathbb{N}$ such that $\alpha^{-j}B$
is $(\alpha,\beta)$-doubling.
Our following result which shows the existence of three consecutive
$(\alpha,\beta)$ doubling balls will play an important role in this paper.
###### Proposition 2.4
If $B$ is a $(\alpha^{3},\beta)$ doubling ball ($\alpha>1$), then $B,\alpha B$
and $\alpha^{2}B$ are three consecutive $(\alpha,\beta)$ doubling balls.
_Proof:_ The proof of Proposition 2.4 is simple, hence we omit the details
here.
For any two balls $B\subset Q$, we defined
$K_{B,Q}=1+\int_{r_{B}\leq d(x,x_{B})\leq
r_{Q}}\frac{1}{\lambda(x_{B},d(x,x_{B}))}d\mu(x).$ (1)
This definition is a variant of the definition in [T1, pp.94-95]. Similarly to
the results [T1, Lemma 2.1] we have the following properties:
###### Lemma 2.5
1. (i)
If $Q\subset R\subset S$ are balls in $X$, then
$\max\\{K_{Q,R},K_{R,S}\\}\leq K_{Q,S}\leq C(K_{Q,R}+K_{R,S}).$
2. (ii)
If $Q\subset R$ are compatible size, then $K_{Q,R}\leq C.$
3. (iii)
If $\alpha Q,\ldots\alpha^{N-1}Q$ are non $(\alpha,\beta)$-doubling balls
$(\beta>C_{\lambda}^{\log_{2}\alpha})$ then $K_{Q,\alpha^{N}Q}\leq C.$
The proof of Lemma 2.5 is not difficult, hence we omit the details here.
As in [T1], for two balls $B\subset Q$ we can define the coefficient
$K^{\prime}_{B,Q}$ as follows: let $N_{B,Q}$ be the smallest integer
satisfying $6^{N_{B,Q}}r_{B}\geq r_{Q}$, then we set
$K^{\prime}_{B,Q}:=1+\sum_{k=1}^{N_{B,Q}}\frac{\mu(6^{k}B)}{\lambda(x_{B},6^{k}r_{B})}.$
In the case that $\lambda(x,ar)=a^{m}\lambda(x,r)$ for all $x\in X$ and
$a,r>0$, it is not difficult to show that $K_{B,Q}\approx K^{\prime}_{B,Q}$.
However, in general, we only have $K_{B,Q}\leq CK^{\prime}_{B,Q}$.
### 2.2 Calderón-Zygmund operators
A kernel $K(\cdot,\cdot)\in L^{1}_{{\rm loc}}(X\times
X\backslash\\{(x,y):x=y\\})$ is called a Calderón-Zygmund kernel if
1. (i)
$|K(x,y)|\leq
C\min\Big{\\{}\frac{1}{\lambda(x,d(x,y))},\frac{1}{\lambda(y,d(x,y))}\Big{\\}}.$
(2)
2. (ii)
There exists $0<\delta\leq 1$ such that
$|K(x,y)-K(x^{\prime},y)|+|K(y,x)-K(y,x^{\prime})|\leq
C\frac{d(x,x^{\prime})^{\delta}}{d(x,y)^{\delta}\lambda(x,d(x,y))}$ (3)
if $d(x,x^{\prime})\leq Cd(x,y).$
A linear operator $T$ is called a Calderón-Zygmund operator with kernel
$K(\cdot,\cdot)$ satisfying (2) and (3) if for all $f\in L^{\infty}(\mu)$ with
bounded support and $x\notin{supp}f$,
$Tf(x)=\int_{X}K(x,y)f(y)d\mu(y).$
The maximal operator $T_{*}$ associated with the Calderón-Zygmund operator $T$
is defined by
$T_{*}f(x)=\sup_{\epsilon>0}|T_{\epsilon}f(x)|,$
where $T_{\epsilon}f(x)=\int_{d(x,y)\geq\epsilon}K(x,y)f(y)d\mu(y)$.
We would like to give an example for the operator whose the associated kernel
satisfies the conditions (2) and (3). As in [Hy], we consider Bergman-type
operators which are studied by Volberg and Wick. In [VW], the authors obtained
a characterization of measures $\mu$ in the unit ball $\mathbb{B}_{2n}$ of
$\mathbb{C}^{n}$ for which the analytic Besov-Sobolev space
$B^{\sigma}_{2}(\mathbb{B}^{2n})$ embeds continuously into $L^{2}(\mu)$. Their
proof goes through a new $T1$ theorem for what they call Bergman-type
operators. Let us describe the situation of this application. The measures
$\mu$ in [VW] satisfy the upper power bound $\mu(B(x,r))\leq r^{m}$, except
possibly when $B(x,r)\subset H$, where $H$ is a fixed open set. However, in
the exceptional case there holds $r\leq\delta(x):=d(x,H^{c})$, and hence
$\mu(B(x,r))\leq\lim_{\epsilon\rightarrow
0}B(x,\delta(x)+\epsilon)\leq\lim_{\epsilon\rightarrow
0}(\delta(x)+\epsilon)^{m}=\delta^{m}.$
Thus we find that their measures are actually upper doubling with
$\mu(B(x,r))\leq\max\\{\delta(x)^{m},r^{m})=:\lambda(x,r).$
It is not difficult to show that $\lambda(\cdot,\cdot)$ satisfies the
conditions (i)-(v) in definition of upper doubling measures.
In [VW], as a main application concerning the Besov-Sobolev spaces, the
authors introduced the operator associated to the kernel
$K(x,y)=(1-\overline{x}\cdot y)^{-m},$ (4)
for $x,y\in\overline{\mathbb{B}}_{2n}\subset\mathbb{C}^{n}$. Here
$\overline{x}$ stands for the componentwise complex conjugation, and
$\overline{x}\cdot y$ designates the usual dot product of $n$-vectors
$\overline{x}$ and $y$. Moreover, one equips $\overline{\mathbb{B}}_{2n}$ with
the regular quasi-distance, see [Tch, Lemma 2.6],
$d(x,y):=\Big{|}|x|-|y|\Big{|}+\Big{|}1-\frac{\overline{x}\cdot
y}{|x||y|}\Big{|}.$
Finally, the set $H$ related to the exceptional balls is now the open unit
ball $\overline{\mathbb{B}}_{2n}$. It was proved in [HyM] that the kernel
$K(x,y)$ defined by (4) satisfies (2) and (3).
## 3 The RBMO spaces
### 3.1 Definition of RBMO$(\mu)$
The RBMO (Regularized BMO) space was introduced by Tolsa for
$(\mathbb{R}^{n},\mu)$ in [T1] and it was adopted by T. Hytönen for general
non-homogeneous space $(X,\mu)$ in [Hy].
###### Definition 3.1
Fix a parameter $\rho>1$. A function $f\in L^{1}_{{\rm loc}}(\mu)$ is said to
be in the space RBMO$(\mu)$ if there exists a number $C$, and for every ball
$B$, a number $f_{B}$ such that
$\frac{1}{\mu(\rho B)}\int_{B}|f(x)-f_{B}|d\mu(x)\leq A$ (5)
and, for any two balls $B$ and $B_{1}$ such that $B\subset B_{1}$,
$|f_{B}-f_{B_{1}}|\leq CK_{B,B_{1}}.$ (6)
The infimum of the values $C$ in (6) is taken to be the RBMO norm of $f$ and
denoted by $\|f\|_{{\rm RBMO}(\mu)}$.
The RBMO norm $\|\cdot\|_{{\rm RBMO}(\mu)}$ is independent of $\rho>1$.
Moreover the John-Nirenberg inequality holds for RBMO$(X)$. More precisely, we
have the following result (see Corollary 6.3 in [Hy]).
###### Proposition 3.2
For any $\rho>1$ and $p\in[1,\infty)$, there exists a constant $C$ so that for
every $f\in{\rm RBMO}(\mu)$ and every ball $B_{0}$,
$\Big{(}\frac{1}{\mu(\rho
B_{0})}\int_{B_{0}}|f(x)-f_{B_{0}}|^{p}d\mu(x)\Big{)}^{1/p}\leq C\|f\|_{{\rm
RBMO}(\mu)}.$
### 3.2 Some characterizations of RBMO$(\mu)$
In the rest of paper, unless $\alpha$ and $\beta$ are specified otherwise, by
an $(\alpha,\beta)$ doubling ball we mean a $(6,\beta_{0})$-doubling with a
fixed number $\beta_{0}>\max\\{C_{\lambda}^{3\log_{2}6},6^{3n}\\}$.
Given a ball $B\subset X$, let $N$ be the smallest non-negative integer such
that $\widetilde{B}=6^{N}B$ is doubling. Such a ball $\widetilde{B}$ exists
due to Lemma 2.5.
Let $\rho>1$ be some fixed constant. We say that $f\in L^{1}_{{\rm loc}}(\mu)$
is in RBMO$(\mu)$ if there exists some constant $C>0$ such that for any ball
$Q$
$\frac{1}{\mu(\rho B)}\int_{B}|f(x)-m_{\widetilde{B}}f|d\mu(x)\leq C$ (7)
and
$|m_{Q}f-m_{R}f|\leq CK_{Q,R},\ \ \text{for any two doubling balls $Q\subset
R$},$ (8)
here $m_{B}f$ is the mean value of $f$ over the ball $B$. Then we take
$\|f\|_{*}:=\inf\\{C:\text{(\ref{cond1-RBMOdefn2}) and (\ref{cond2-RBMOdefn2})
hold}\\}.$
By the same proof as in Lemma 2.8 of [T1], we have the following result.
###### Proposition 3.3
For a fixed $\rho>1$, the norms $\|\cdot\|_{*}$ and $\|\cdot\|_{{\rm
RBMO}(\mu)}$ are equivalent.
We now extend certain characterizations of ${\rm RBMO}(\mu)$ in [T1] in the
case of $(\mathbb{R}^{n},\mu)$ to the case of non-homogeneous spaces
$(X,\mu)$. In the case of $\mathbb{R}^{n}$, Besicovitch covering lemma was
used but this lemma is not applicable in our setting. We overcome this problem
by using the three consecutive doubling balls in Proposition 2.4.
###### Proposition 3.4
For $f\in L^{1}_{{\rm loc}}(\mu)$, the following are equivalent:
1. (a)
$f\in{\rm RBMO}(\mu)$.
2. (b)
There exists some constant $C_{b}$ such that for any ball $B$
$\frac{1}{\mu(6B)}\int_{B}|f(x)-m_{B}f|d\mu(x)\leq C_{b}$ (9)
and
$|m_{Q}f-m_{R}f|\leq
C_{b}K_{Q,R}\Big{(}\frac{\mu(6Q)}{\mu(Q)}+\frac{\mu(6R)}{\mu(R)}\Big{)},\ \
\text{for any two balls $Q\subset R$}.$ (10)
3. (c)
There exists some constant $C_{c}$ such that for any doubling ball $B$
$\frac{1}{\mu(B)}\int_{B}|f(x)-m_{B}f|d\mu(x)\leq C_{c}$ (11)
and
$|m_{Q}f-m_{R}f|\leq C_{c}K_{Q,R},\ \ \text{for any two doubling balls
$Q\subset R$}.$ (12)
Moreover, the best constants $C_{b}$ and $C_{c}$ are comparable to the
RBMO$(\mu)$ norm of $f$.
_Proof:_ $(a)\rightarrow(b):$ If $f\in{\rm RBMO}(\mu)$, then (9) and (10) hold
for $C_{b}=C\|f\|_{*}$ for some constant $C$. Indeed, for any ball $B$ we have
$|m_{B}f-m_{\widetilde{B}}f|\leq
m_{Q}(|f-m_{\widetilde{B}}f|)\leq\|f\|_{*}\frac{\mu(6Q)}{Q}.$
Therefore,
$\frac{1}{\mu(6B)}\int_{B}|f(x)-m_{B}f|d\mu(x)\leq\frac{1}{\mu(6B)}\int_{B}(|f-m_{\widetilde{B}}f|+|m_{B}f-m_{\widetilde{B}}f|)\leq
2\|f\|_{*}.$ (13)
On the other hand, for any two balls $Q\subset R$, one has
$|m_{Q}f-m_{R}f|\leq|m_{Q}f-m_{\widetilde{Q}}f|+|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|+|m_{R}f-m_{\widetilde{R}}f|.$
Applying (13) for the first and the third terms, we have
$|m_{Q}f-m_{\widetilde{Q}}f|+|m_{R}f-m_{\widetilde{R}}f|\leq\|f\|_{*}\Big{(}\frac{\mu(6Q)}{Q}+\frac{\mu(6R)}{R}\Big{)}.$
We can follow the argument in [T1] to obtain the estimate for the second term.
Let us remark that for any two balls $Q\subset R$ such that
$\widetilde{Q}\subset\widetilde{R}$, it follows from (8) that
$|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|\leq\|f\|_{*}K_{\widetilde{Q},\widetilde{R}}.$
By Lemma 2.5, we have
$K_{\widetilde{Q},\widetilde{R}}\leq
C(K_{Q,\widetilde{Q}}+K_{Q,R}+K_{R,\widetilde{R}})\leq
C(C_{1}+K_{Q,R}+C_{2})\leq CK_{Q,R}.$
In general, $Q\subset R$ does not imply $\widetilde{Q}\subset\widetilde{R}$.
We consider two cases:
Case 1: If $r_{\widetilde{Q}}\geq r_{\widetilde{R}}$, then
$\widetilde{Q}\subset 3\widetilde{R}$. Setting
$R_{0}=\widetilde{3\widetilde{R}}$, then it follows from Lemma 2.5 and (8)
that
$\displaystyle|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|$
$\displaystyle\leq|m_{\widetilde{Q}}f-m_{R_{0}}f|+m_{R_{0}}f-m_{\widetilde{R}}f$
$\displaystyle\leq(K_{\widetilde{Q},R_{0}}+K_{\widetilde{R},R_{0}})\|f\|_{*}.$
For the term $K_{\widetilde{Q},R_{0}}$ we have
$\displaystyle K_{\widetilde{Q},R_{0}}$ $\displaystyle\leq CK_{Q,R_{0}}$
$\displaystyle\leq C(K_{Q,R}+K_{R,R_{0}})$ $\displaystyle\leq
C(K_{Q,R}+K_{R,\widetilde{R}}+K_{\widetilde{R},3\widetilde{R}}+K_{3\widetilde{R},R_{0}})$
$\displaystyle\leq CK_{Q,R}.$
The remaining term $K_{\widetilde{R},R_{0}}$ is dominated by
$C(K_{\widetilde{R},3\widetilde{R}}+K_{3\widetilde{R},R_{0}})\leq CK_{Q,R}.$
So in this case, we obtain $|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|\leq
CK_{Q,R}\|f\|_{*}$.
Case 2: If $r_{\widetilde{R}}<r_{\widetilde{Q}}$, then $\widetilde{R}\subset
6^{2}\widetilde{Q}$. Obviously, we can find some $m\geq 1$ such that
$r_{\widetilde{R}}\geq\frac{r_{5^{m}Q}}{25}$ and $\widetilde{R}\subset
6^{m}Q\subset 6^{2}\widetilde{Q}$. Therefore, $\widetilde{R}$ and $5^{m}Q$ are
comparable sizes. This implies $K_{\widetilde{R},5^{m}Q}\leq C$. Setting
$Q_{0}=\widetilde{6^{2}\widetilde{Q}}$ we have
$\displaystyle|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|$
$\displaystyle\leq|m_{\widetilde{Q}}f-m_{Q_{0}}f|+|m_{Q_{0}}f-m_{\widetilde{R}}f|$
$\displaystyle\leq(K_{\widetilde{Q},Q_{0}}+K_{\widetilde{R},Q_{0}})\|f\|_{*}.$
Let us estimate $K_{\widetilde{Q},Q_{0}}$. We have
$K_{\widetilde{Q},Q_{0}}\leq
C(K_{\widetilde{Q},6^{2}\widetilde{Q}}+K_{6^{2}\widetilde{Q},Q_{0}})\leq
CK_{Q,R}.$
For the term $K_{\widetilde{R},Q_{0}}$, one has
$\displaystyle K_{\widetilde{R},Q_{0}}$ $\displaystyle\leq
C(K_{\widetilde{R},5^{m}Q}+K_{5^{m}Q,6^{2}\widetilde{Q}}+K_{6^{2}\widetilde{Q},Q_{0}})$
$\displaystyle\leq
C(K_{\widetilde{R},5^{m}Q}+K_{Q,6^{2}\widetilde{Q}}+K_{6^{2}\widetilde{Q},Q_{0}})$
$\displaystyle\leq CK_{Q,R}.$
Therefore, in this case we also obtain
$|m_{\widetilde{Q}}f-m_{\widetilde{R}}f|\leq CK_{Q,R}\|f\|_{*}$.
$(b)\rightarrow(c)$: the proof of this implication it easy and hence we omit
the detail here.
$(c)\rightarrow(a)$: Let $B$ be some ball. We need to show that (7) holds for
$\rho=6$. For any $x\in B$, there exists some $(6^{3},\beta_{0})$-doubling
ball centered $x$ with radius $r_{6^{-2j}B}$ for some $j\in\mathbb{N}$. We
denote by $B_{x}$ the biggest ball satisfying these properties. Let us recall
that by Proposition 2.4, the balls $B_{x},6B_{x}$ and $6^{2}B_{x}$ are three
$(6,\beta_{0})$-doubling balls. Moreover, by Lemma 2.5 we have
$|m_{6B_{x}}f-m_{\widetilde{B}}f|\leq|m_{6B_{x}}f-m_{B_{x}}f|+|m_{B_{x}}f-m_{\widetilde{B}}f|\leq
CC_{c}.$
By Lemma 2.2, we can pick a disjoint subcollection $B_{x_{i}}$, $i\in I$, such
that $B\subset\cup_{i\in I}5B_{x_{i}}\subset\cup_{i\in I}6B_{x_{i}}$. Thus, we
have
$\displaystyle\int_{B}|f-m_{\widetilde{B}}f|d\mu$ $\displaystyle\leq\sum_{i\in
I}\int_{B_{x_{i}}}|f-m_{\widetilde{B}}f|d\mu$ $\displaystyle\leq\sum_{i\in
I}\int_{B_{x_{i}}}(|f-m_{6B_{x_{i}}}f|+|m_{6B_{x_{i}}}f-m_{\widetilde{B}}f|)d\mu$
$\displaystyle\leq\sum_{i\in I}CC_{c}\mu(6B_{x_{i}})$
$\displaystyle\leq\sum_{i\in I}C\beta_{0}C_{c}\mu(B_{x_{i}})$
$\displaystyle\leq C\beta_{0}C_{c}\mu(6B).$
This completes our proof.
## 4 Interpolation results
### 4.1 The sharp maximal operator
Adapting an idea in [T1], we define the sharp maximal operator as follows:
$M^{\sharp}f(x)=\sup_{B\ni
x}\frac{1}{\mu(6B)}\int_{B}|f-m_{\widetilde{B}}f|d\mu+\sup\limits_{(Q,R)\in\Delta_{x}}\frac{|m_{Q}f-m_{R}f|}{K_{Q,R}},$
(14)
here $\Delta_{x}:=\\{(Q,R):x\in Q\subset R\ \text{and}\ Q,R:{\rm
doubling}\\}$.
Note that in our sharp maximal operator, the term $\mu(6B)$ was chosen with
the fixed constant $6$ throughout the paper. It is clear that
$f\in{\rm RBMO}(\mu)\Leftrightarrow M^{\sharp}f\in L^{\infty}(\mu).$
We define, for $\rho\geq 1$, the non-centered maximal operator $M_{(\rho)}$ by
setting
$M_{(\rho)}f(x)=\sup_{x\in Q}\frac{1}{\mu(\rho Q)}\int_{Q}|f|d\mu.$
It was proved that $M_{(\rho)}$ is of weak type $(1,1)$ for $\rho\geq 5$ and
hence $M_{(\rho)}$ is bounded on $L^{p}(\mu)$ for all $p\in(1,\infty]$, see
[Hy, Proposition 3.5]. When $\rho=1$, we write $Mf$ instead of $M_{(1)}f$.
From the boundedness of $M_{(\rho)}$ for $\rho\geq 5$, the non-centered
doubling maximal operator is defined by
$Nf(x)=\sup_{x\in Q:\ {\rm doubling}}\frac{1}{\mu(Q)}\int_{Q}|f|d\mu$
where the supremum is taken over all $(6,\beta_{0})$ doubling balls, is of
weak type $(1,1)$ and hence bounded on $L^{p}(\mu)$ for all $p\in(1,\infty]$.
Note that it is not difficult to show that
$M^{\sharp}f(x)\leq M_{(6)}f(x)+3Nf(x)$
for all $x\in X$. Therefore the operator $M^{\sharp}$ is of type weak $(1,1)$
and bounded on $L^{p}(\mu)$ for all $1<p<\infty$.
###### Lemma 4.1
For $f\in L^{1}_{{\rm loc}}(\mu)$, we have
$M^{\sharp}|f|(x)\leq 5\beta_{0}M^{\sharp}f(x).$
The proof is similar to that of Remark 6.1 in [T1].
We now show that the non-centered doubling maximal operator is dominated by
the sharp maximal operator in the following theorem. Although, some estimates
are inspired from [T1, Theorem 6.2], there are some main differences in our
proof. More specifically, the three consecutive doubling balls argument will
be used to replace the Besicovitch covering lemma.
###### Theorem 4.2
Let $f\in L^{1}_{\rm loc}(\mu)$ with the extra condition $\int fd\mu=0$ if
$\|\mu\|:=\mu(X)<\infty$. Assume that for some $p$, $1<p<\infty$,
$\inf\\{1,Nf\\}\in L^{p}(\mu)$. Then we have
$\|Nf\|_{L^{p}(\mu)}\leq C\|M^{\sharp}f\|_{L^{p}(\mu)}.$
_Proof:_ We assume that $\|\mu\|=\infty$. The proof for $\|\mu\|<\infty$ is
similar. By standard argument, it suffices to prove the following
$\lambda$-good inequality: for some fixed $\nu<1$ and all $\epsilon>0$ there
exists some $\delta>0$ such that for any $\lambda>0$ we have
$\mu\\{x:Nf(x)>(1+\epsilon)\lambda,M^{\sharp}f(x)\leq\delta\lambda\\}\leq\nu\mu\\{x:Nf(x)>\lambda\\}.$
(15)
Setting
$E_{\lambda}=\\{x:Nf(x)>(1+\epsilon)\lambda,M^{\sharp}f(x)\leq\delta\lambda\\}$
and $\Omega_{\lambda}=\\{x:Nf(x)>\lambda\\}$, for $f\in L^{p}(\mu)$. For each
$x\in E_{\lambda}$, we can choose the doubling ball $Q_{x}$ containing $x$
satisfying that $m_{Q_{x}}|f|>(1+\epsilon/2)\lambda$ and if $Q$ is any
doubling ball containing $x$ with $r_{Q}>2r_{Q_{x}}$ then
$m_{Q}|f|\leq(1+\epsilon/2)\lambda$. Such a ball $Q_{x}$ exists due to $f\in
L^{p}(\mu)$.
Let $R_{x}$ be the ball centered $x$ with radius $6r_{Q_{x}}$ and $S_{x}$ be
the smallest $(6^{3},\beta_{0})$-doubling ball in the form $6^{3j}R_{x}$.
Then, by Proposition 2.4, $S_{x},6S_{x}$ and $6^{2}S_{x}$ are three
$(6,\beta_{0})$-doubling balls. Moreover, one has
$K_{Q_{x},6S_{x}}\leq C(K_{Q_{x},R_{x}}+K_{R_{x},S_{x}}+K_{S_{x},6S_{x}})\leq
C.$
Therefore, it follows from Lemma 4.1 that
$|m_{Q_{x}}|f|-m_{6S_{x}}|f||\leq K_{Q_{x},6S_{x}}M^{\sharp}|f|(x)\leq
C\beta_{0}M^{\sharp}f(x)\leq C\beta_{0}\delta\lambda.$
This implies that for sufficiently small $\delta$ we have
$m_{6S_{x}}|f|>\lambda$
and hence $6S_{x}\subset\Omega_{\lambda}$.
Note that by Lemma 2.2, we can pick a disjoint collection
$\\{S_{x_{i}}\\}_{i\in I}$ with $x_{i}\in E_{\lambda}$ and
$E_{\lambda}\subset\cup_{i\in I}5S_{x_{i}}\subset\cup_{i\in I}6S_{x_{i}}$.
Setting $W_{x_{i}}=6S_{x_{i}}$, we will show that
$\mu(6S_{x_{i}}\cap E_{\lambda})\leq C\frac{\nu}{\beta_{0}}\mu(6S_{x_{i}})$
(16)
for all $i\in I$.
Once (16) is proved, (15) follows readily. Indeed, from (16) we have
$\mu(E_{\lambda})\leq\sum_{i\in I}\mu(6S_{x_{i}}\cap
E_{\lambda})\leq\sum_{i\in I}\frac{\nu}{\beta_{0}}\mu(6S_{x_{i}})\leq
C\sum_{i\in I}\nu\mu(S_{x_{i}})\leq C\nu\mu(\Omega_{\lambda}).$
Now we show the proof of (16). Let $y\in W_{x_{i}}\cap E_{\lambda}$. For any
doubling ball $Q\ni y$ satisfying $m_{Q}|f|>(1+\epsilon)\lambda$, it follows
that $r_{Q}\leq r_{W_{x_{i}}}/8$. Indeed, if $r_{Q}>r_{W_{x_{i}}}/8$ then we
have $Q_{x_{i}}\subset W_{x_{i}}\subset\widetilde{16Q}$ and
$|m_{Q}|f|-m_{\widetilde{16Q}}|f||\leq
K_{Q,\widetilde{16Q}}M^{\sharp}|f|(y)\leq
C\delta\lambda\leq\frac{\epsilon}{2}$
for sufficiently small $\delta$. This implies
$m_{\widetilde{16Q}}|f|>(1+\epsilon/2)\lambda$ which is a contradiction to the
choice of $Q_{x_{i}}$. So, $r_{Q}\leq r_{W_{x_{i}}}/8$. This, together with
$m_{Q}|f|>(1+\epsilon)\lambda$, imply
$N(f\chi_{\frac{5}{4}W_{x_{i}}})(y)>(1+\epsilon)\lambda$
and
$m_{\widetilde{\frac{5}{4}W_{x_{i}}}}|f|\leq(1+\epsilon/2)\lambda\
\text{(since $r_{\widetilde{\frac{5}{4}W_{x_{i}}}}>2r_{Q_{x_{i}}}$}).$
This yields,
$N(\chi_{\frac{5}{4}W_{x_{i}}}|f|-m_{\widetilde{\frac{5}{4}W_{x_{i}}}}|f|)(y)>\frac{\epsilon}{2}\lambda.$
Therefore, by using the weak $(1,1)$ boundedness of $N$, we have
$\displaystyle\mu(W_{x_{i}}\cap E_{\lambda})$
$\displaystyle\leq\mu\\{y:N(\chi_{\frac{5}{4}W_{x_{i}}}|f|-m_{\widetilde{\frac{5}{4}W_{x_{i}}}}|f|)(y)>\frac{\epsilon}{2}\lambda\\}$
$\displaystyle\leq\frac{C}{\epsilon\lambda}\int_{\frac{5}{4}W_{x_{i}}}(|f|-m_{\widetilde{\frac{5}{4}W_{x_{i}}}}|f|)d\mu$
$\displaystyle\leq\frac{C}{\epsilon\lambda}\mu(\frac{15}{2}W_{x_{i}})M^{\sharp}|f|(x_{i})$
$\displaystyle\leq\frac{C\delta}{\epsilon}\beta_{0}\mu(6^{3}S_{x_{i}})$
$\displaystyle\leq\frac{C\delta}{\epsilon}\beta_{0}\mu(S_{x_{i}}).$
Thus, (16) holds provided $\delta<\epsilon/C\nu\beta_{0}$.
For the case $f\notin L^{p}(\mu)$, we define the sequence of functions
$\\{f_{k}\\},\ k=1,2,\cdots$ by setting
$f_{k}(x)=\begin{cases}f(x),&|f(x)|\leq k,\\\ k\frac{f(x)}{|f(x)|},\
&|f(x)|>k.\end{cases}$
Then we have $M^{\sharp}f_{k}(x)\leq CM^{\sharp}f(x)$. On the other hand,
$|f_{k}(x)|\leq k\inf\\{1,|f|(x)\\}\leq k\inf(1,Nf)(x)$ and so $f_{k}\in
L^{p}(\mu)$. Hence,
$\|Nf_{k}\|_{L^{p}(\mu)}\leq C\|M^{\sharp}f_{k}\|_{L^{p}(\mu)}\leq
C\|M^{\sharp}f\|_{L^{p}(\mu)}.$
Taking the limit as $k\rightarrow\infty$, we obtain the required result and
the proof is completed.
### 4.2 An Interpolation Theorem for linear operators
###### Theorem 4.3
Let $1<p<\infty$ and let $T$ be a linear operator bounded on $L^{p}(\mu)$ and
from $L^{\infty}(\mu)$ into ${\rm RBMO(\mu)}$. Then $T$ extends to a bounded
operator on $L^{r}(\mu)$ for $p<r<\infty$.
_Proof:_ We consider 2 cases:
Case 1: $\|\mu\|=\infty$: Since $T$ is bounded on $L^{p}(\mu)$, $M^{\sharp}T$
is sublinear bounded on $L^{p}(\mu)$ and on $L^{\infty}(\mu)$. Therefore, by
interpolation, $M^{\sharp}T$ is bounded on $L^{r}(\mu)$ for $p<r<\infty$,
$\|M^{\sharp}Tf\|_{L^{r}(\mu)}\leq C\|f\|_{L^{r}(\mu)}.$
Assume that $f\in L^{r}(\mu)$ is supported in compact set. Then $f\in
L^{p}(\mu)$ and so $Tf\in L^{p}(\mu)$. Hence $Nf\in L^{p}(\mu)$ and
$\inf\\{1,Nf\\}\in L^{r}(\mu)$. By invoking Theorem 4.2,
$\|Tf\|_{L^{r}(\mu)}\leq\|M^{\sharp}Tf\|_{L^{r}(\mu)}\leq
C\|f\|_{L^{r}(\mu)}.$
Case 2: Assume that $\|\mu\|<\infty$. For $f\in L^{r}(\mu)$, set $f=(f-\int
fd\mu)+\int fd\mu=f_{1}+f_{2}$. Since $\int f_{1}d\mu=0$, we can apply the
same argument as for $\|\mu\|=\infty$. It is not difficult to show that
$\|T1\|_{L^{r}(\mu)}\leq C\|1\|_{L^{r}(\mu)}$. This completes the proof.
## 5 Atomic Hardy spaces and their dual spaces
### 5.1 The space $H^{1,\infty}_{at}(\mu)$
For a fixed $\rho>1$, a function $b\in L^{1}_{loc}(\mu)$ is called an atomic
block if
1. (i)
there exists some ball $B$ such that ${\rm supp}b\subset B$;
2. (ii)
$\int bd\mu=0;$
3. (iii)
there are functions $a_{j}$ supported on cubes $B_{j}\subset B$ and numbers
$\lambda_{j}\in\mathbb{R}$ such that
$b=\sum_{j=1}^{\infty}\lambda_{j}a_{j},$ (17)
where the sum converges in $L^{1}(\mu)$, and
$\|a_{j}\|_{L^{\infty}(\mu)}\leq(\mu(\rho B_{j})K_{B_{j},B})^{-1}$ and the
constant $K_{B_{j},B}$ being given in the paragraph before Lemma 2.5.
We denote $|b|_{H^{1,\infty}_{at}(\mu)}=\sum_{j=1}^{\infty}|\lambda_{j}|$. We
say that $f\in H^{1,\infty}_{at}(\mu)$ if there are atomic blocks $b_{i}$ such
that
$f=\sum_{i=1}^{\infty}b_{i}$ (18)
with $\sum_{i=1}^{\infty}|b_{i}|_{H^{1,\infty}_{at}(\mu)}<\infty$. The
$H^{1,\infty}_{at}(\mu)$ norm of $f$ is defined by
$\|f\|_{H^{1,\infty}_{at}(\mu)}:=\inf\sum_{i=1}^{\infty}|b_{i}|_{H^{1,\infty}_{at}(\mu)}$
where the infimum is taken over all the possible decompositions of $f$ in
atomic blocks.
We have the following basic properties of $H_{at}^{1,\infty}(\mu)$.
###### Proposition 5.1
1. (a)
$H_{at}^{1,\infty}(\mu)$ is a Banach space.
2. (b)
$H_{at}^{1,\infty}(\mu)\subset L^{1}(\mu)$ and
$\|f\|_{L^{1}(\mu)}\leq\|f\|_{H_{at}^{1,\infty}(\mu)}$.
3. (c)
The space $H_{at}^{1,\infty}(\mu)$ is independent of the constant $\rho$ when
$\rho>1$.
_Proof:_ The proofs of (a) and (b) are standard and we omit the details here.
The proof of (c): Given $\rho_{1}>\rho_{2}>0$, it is clear that
$H^{1,\infty}_{at,\rho_{1}}\subset H^{1,\infty}_{at,\rho_{2}}$ with
$\|f\|_{H^{1,\infty}_{at,\rho_{2}}}\leq\|f\|_{H^{1,\infty}_{at,\rho_{1}}}$.
Conversely, if $b=\sum_{i=1}^{\infty}\lambda_{i}a_{i}$ is an atomic block with
supp $a_{i}\subset B_{i}\subset B$ in $H^{1,\infty}_{at,\rho_{1}}$, then by
Lemma 2.1 we can cover each $B_{i}$ by
$N\Big{[}\frac{\rho_{1}}{\rho_{2}}\Big{]}^{n}$ balls, says $\\{B_{ik}\\}$,
with the same radius $\frac{\rho_{2}}{\rho_{1}}r_{B}$. Therefore, we can
decompose $a_{i}:=\sum_{k}a_{ik}$ where
$a_{ik}:=a_{i}\frac{\chi_{B_{ik}}}{\sum_{j}\chi_{B_{ij}}}$. It is not
difficult to verify that $b$ is also an atomic block in
$H^{1,\infty}_{at,\rho_{2}}$. This completes our proof.
We now show that the space ${\rm RBMO}(\mu)$ is embedded in the dual space of
$H_{at}^{1,\infty}(\mu).$
###### Lemma 5.2
We have
${\rm RBMO}(\mu)\subset H_{at}^{1,\infty}(\mu)^{*}.$
That is, for $g\in{\rm RBMO}(\mu)$, the linear functional
$L_{g}(f)=\int_{X}fgd\mu$
defines a continuous linear functional $L_{g}$ over $H_{at}^{1,\infty}(\mu)$
with $\|L_{g}\|_{H_{at}^{1,\infty}(\mu)^{*}}\leq C\|g\|_{{\rm RBMO}(\mu)}.$
_Proof:_ Following standard argument, see for example [CW2, p.64], we only
need to check that for an atomic block $b$ and $g\in$RBMO$(\mu)$, we have
$\Big{|}\int bgd\mu\Big{|}\leq C|b|_{H_{at}^{1,\infty}(\mu)}\|g\|_{{\rm
RBMO}(\mu)}.$
Assume that supp$b\subset B$ and $b=\sum_{j}^{\infty}\lambda_{j}a_{j}$, where
$a_{j}$’s are functions satisfying (a) and (b) in the definition of atomic
blocks. If $g\in L^{\infty}$, by using $\int bd\mu=0$, we have
$\Big{|}\int bgd\mu\Big{|}=\Big{|}\int
b(g-g_{B})d\mu\Big{|}\leq\sum_{j}^{\infty}|\lambda_{j}|\|a_{j}\|_{L^{\infty}(\mu)}\int_{B_{i}}|g-g_{B}|d\mu.$
(19)
Since $g\in L^{\infty}(\mu)\subset{\rm RBMO}(\mu)$, we have
$\displaystyle\int_{B_{i}}|g-g_{B}|d\mu$
$\displaystyle\leq\int_{B_{i}}|g-g_{B_{i}}|d\mu+\int_{B_{i}}|g_{B}-g_{B_{i}}|d\mu$
$\displaystyle\leq CK_{B_{i},B}\|g\|_{{\rm RBMO}(\mu)}\mu(\rho B_{j}).$
From (19), we obtain
$\Big{|}\int bgd\mu\Big{|}\leq C|b|_{H_{at}^{1,\infty}(\mu)}\|g\|_{{\rm
RBMO}(\mu)}.$
In general case, if $g\in{\rm RBMO}(\mu)$, define
$g_{N}(x):=\begin{cases}f(x),&\ \ |f(x)|<N,\\\ N\frac{f(x)}{|f(x)|},&\ \
|f(x)|\geq N.\end{cases}$
It can be verified that $\|g_{N}\|_{{\rm RBMO}(\mu)}\leq C\|g\|_{{\rm
RBMO}(\mu)}$. As above, since $g_{N}\in L^{\infty}(\mu)$, we have
$\Big{|}\int fg_{N}d\mu\Big{|}\leq
C\|f\|_{H_{at}^{1,\infty}(\mu)}\|g_{N}\|_{{\rm RBMO}(\mu)}\leq
C\|f\|_{H_{at}^{1,\infty}(\mu)}\|g\|_{{\rm RBMO}(\mu)}.$
Let us denote $L^{\infty}_{0}:=\\{f:f\ \text{in $L^{\infty}(\mu)$ with bounded
support}\\}$ and $D=H_{at}^{1,\infty}(\mu)\cap L^{\infty}_{0}$. So, the
functional $L_{g}:f\mapsto\int gf$ is well-defined on $D$ whenever $g\in{\rm
RBMO}(\mu)$ (since $g\in L^{1}_{loc}(\mu)$). By the dominated convergence
theorem
$\lim_{N\rightarrow\infty}\int fg_{N}d\mu=\int fgd\mu$
for all $f\in D$. We claim that $D$ is dense in $H_{at}^{1,\infty}(\mu)$. To
verify this claim, denote by $H^{1,\infty}_{at,fin}(\mu)$ the set of all
elements in $H_{at}^{1,\infty}(\mu)$ where the sums (17) and (18) are taken
over finite elements. Obviously, $H^{1,\infty}_{at,fin}(\mu)$ is dense in
$H_{at}^{1,\infty}(\mu)$ and each functional $f\in H^{1,\infty}_{at,fin}(\mu)$
is also in $L^{\infty}_{0}$. Therefore , $L_{b}$ is a unique extension on
$H_{at}^{1,\infty}(\mu)$ and hence
$\Big{|}\int fgd\mu\Big{|}\leq C\|f\|_{H_{at}^{1,\infty}(\mu)}\|g\|_{{\rm
RBMO}(\mu)}.$
This completes our proof.
The following lemma can be obtained by the same argument as in [T1, Lemma
4.4].
###### Lemma 5.3
If $g\in{\rm RBMO}(\mu)$, we have
$\|L_{g}\|_{H_{at}^{1,\infty}(\mu)}\approx\|g\|_{{\rm RBMO}(\mu)}.$
### 5.2 The space $H^{1,p}_{at}(\mu)$
For a fixed $\rho>1$, a function $b\in L^{1}_{loc}(\mu)$ is called a
$p$-atomic block, $1<p<\infty$, if
1. (i)
there exists some ball $B$ such that ${\rm supp}b\subset B$;
2. (ii)
$\int bd\mu=0;$
3. (iii)
there are functions $a_{j}$ supported on cubes $B_{j}\subset B$ and numbers
$\lambda_{j}\in\mathbb{R}$ such that
$b=\sum_{j=1}^{\infty}\lambda_{j}a_{j},$ (20)
where the sum converges in $L^{1}(\mu)$, and
$\|a_{j}\|_{L^{p}(\mu)}\leq(\mu(\rho B_{j}))^{1/p-1}K_{B_{j},B}^{-1}.$
We denote $|b|_{H^{1,p}_{at}(\mu)}=\sum_{j=1}^{\infty}|\lambda_{j}|$. We say
that $f\in H^{1,p}_{at}(\mu)$ if there are $p$-atomic blocks $b_{i}$ such that
$f=\sum_{i=1}^{\infty}b_{i}$ (21)
with $\sum_{i=1}^{\infty}|b_{i}|_{H^{1,p}_{at}(\mu)}<\infty$. The
$H^{1,p}_{at}(\mu)$ norm of $f$ is defined by
$\|f\|_{H^{1,p}_{at}(\mu)}:=\inf\sum_{i=1}^{\infty}|b_{i}|_{H^{1,p}_{at}(\mu)}$
where the infimum is taken over all the possible decompositions of $f$ in
$p$-atomic blocks.
Similarly to $H_{at}^{1,\infty}(\mu)$, we have the following basic properties
of $H_{at}^{1,p}(\mu)$
###### Proposition 5.4
1. (a)
$H_{at}^{1,p}(\mu)$ is a Banach space.
2. (b)
$H_{at}^{1,p}(\mu)\subset L^{1}(\mu)$ and
$\|f\|_{L^{1}(\mu)}\leq\|f\|_{H_{at}^{1,p}(\mu)}$.
3. (c)
The space $H_{at}^{1,p}(\mu)$ is independent of the constant $\rho$ when
$\rho>1$.
The proofs of this proposition is in line with Proposition 5.1, so we omit the
details here.
###### Lemma 5.5
We have
${\rm RBMO}(\mu)\subset H_{at}^{1,p}(\mu)^{*}.$
That is, for $g\in{\rm RBMO}(\mu)$, the linear functional
$L_{g}(f)=\int_{X}fgd\mu$
defines a continuous linear functional $L_{g}$ over $H_{at}^{1,p}(\mu)$ with
$\|L_{g}\|_{H_{at}^{1,p}(\mu)^{*}}\leq C\|g\|_{{\rm RBMO}(\mu)}.$
_Proof:_ The proof of this lemma is analogous to that of Lemma 5.2 with minor
modifications. We leave the details to the interested reader.
We remark that a main difference between the Hardy space in Tolsa’s setting
[T1] and our Hardy space in this article is the sense of convergence in the
atomic decomposition. This leads to different approaches in proving the
inclusions RBMO$(\mu)\subset H_{at}^{1,\infty}(\mu)^{*}$ and RBMO$(\mu)\subset
H_{at}^{1,p}(\mu)^{*}$. However, for the inverse inclusion
$H_{at}^{1,p}(\mu)^{*}\subset$ RBMO$(\mu)$, by a careful investigation, Tolsa
[T1] showed that one only needs to consider the sums in (20) and (21) over
finite $p$-atoms and $p$-atomic blocks, hence the sense of convergence in (20)
and (21) does not matter in both settings. This is the reason why we can use
the arguments in [T1] for our setting with minor modifications to obtain the
duality result of $H_{at}^{1,\infty}(\mu)$ and $H_{at}^{1,p}(\mu)$ as in the
next Theorem.
###### Theorem 5.6
For $1<p<\infty,H_{at}^{1,p}(\mu)=H_{at}^{1,\infty}(\mu)$. Also
$H_{at}^{1,p}(\mu)^{*}=H_{at}^{1,\infty}(\mu)^{*}={\rm RBMO}(\mu)$.
As explained above, we omit the details of the proof.
## 6 Calderón-Zygmund decomposition
### 6.1 Calderón-Zygmund decomposition
The following two technical lemmas will be useful for the construction of a
Calderón-Zygmund decomposition on non-homogeneous spaces.
###### Lemma 6.1
Assume that $Q$, $S$ are two concentric balls, $Q\subset R$, such that there
are no $(\alpha,\beta)$-doubling balls with
$\beta>C_{\lambda}^{\log_{2}\alpha}$ in the form $\alpha^{k}Q,k\in\mathbb{N}$
such that $Q\subset\alpha^{k}Q\subset R$. Then we have
$\int_{R\backslash Q}\frac{1}{\lambda(x_{Q},d(x_{Q},x))}d\mu(x)\leq C.$
_Proof:_ Let $N$ be the smallest integer such that $R\subset\alpha^{N}Q$.
Then, $\mu(\alpha^{k}Q)\geq\beta\mu(\alpha^{k-1}Q)$ for all $k=1,\ldots,N$.
Therefore, we have,
$\displaystyle\int_{R\backslash Q}$
$\displaystyle\frac{1}{\lambda(x_{Q},d(x_{Q},x))}d\mu(x)$
$\displaystyle\leq\sum_{k=1}^{N}\int_{\alpha^{k-1}r_{Q}\leq
d(x,y)\leq\alpha^{k}r_{Q}}\frac{1}{\lambda(x_{Q},d(x_{Q},x))}d\mu(x)$
$\displaystyle\leq\sum_{k=1}^{N}\frac{\mu(\alpha^{k}Q)}{\lambda(x_{Q},\alpha^{k-1}r_{Q})}$
$\displaystyle\leq\sum_{k=1}^{N}\frac{\beta^{N-k}\mu(\alpha^{N}Q)}{(C_{\lambda})^{(N-k)\log_{2}\alpha}\lambda(x_{Q},\alpha^{N}r_{Q})}$
$\displaystyle\leq\sum_{k=1}^{N}\Big{[}\frac{\beta}{(C_{\lambda})^{\log_{2}\alpha}}\Big{]}^{N-k}$
$\displaystyle\leq\sum_{j=1}^{\infty}\Big{[}\frac{\beta}{(C_{\lambda})^{\log_{2}\alpha}}\Big{]}^{j}$
$\displaystyle\leq C\ \ \text{(since $\beta>\log_{2}\alpha$)}.$
This completes the proof.
While the Covering Lemma 2.2 for $(X,\mu)$ can be used to replace the
Besicovich covering lemma for $(\mathbb{R}^{n},\mu)$ in certain estimates, the
Calderón-Zygmund decomposition in $(X,\mu)$ will need a covering lemma which
gives the finite overlapping property at all points $x\in X$. This is given in
the next lemma.
###### Lemma 6.2
Every family of balls $\\{B_{i}\\}_{i\in F}$ of uniformly bounded diameter in
a metric space $X$ contains a disjoint sub-family $\\{B_{i}\\}_{i\in E}$ with
$E\subset F$ such that
1. (i)
$\cup_{i\in F}B_{i}\subset\cup_{i\in E}6B_{i},$
2. (ii)
For each $x\in X$, $\sum_{i\in E}\chi_{6B_{i}}<\infty$.
We remark that in (ii), the sum $\sum_{i\in E}\chi_{6B_{i}}<\infty$ at each
$x$ but these sums are not necessarily uniformly bounded on $X$.
_Proof:_ By Lemma 2.2 we can pick a disjoint subfamily
$\\{B_{i}:B_{i}=B(x_{B_{i}},r_{B_{i}})\\}_{i\in E}$ with $E\subset F$
satisfying (i). Moreover, we can assume that for $i,j\in E$, neither
$6B_{i}\subset 6B_{j}$ nor $6B_{j}\subset 6B_{i}$.
To prove (ii), we assume in contradiction that there exists some $x\in X$ such
that there exists an infinite family of balls $\\{B_{i}:i\in I_{x}\subset
E\\}$ such that $x\in B_{i}$ for all $i\in I_{x}$. We will show that
$\lim\inf_{i\in I_{x}}r_{B_{i}}>0$. Otherwise, for any $\epsilon>0$ there
exists $i_{\epsilon}\in I_{x}$ such that $r_{B_{i_{\epsilon}}}<\epsilon$.
Therefore, if $B_{0}$ is any ball in the family $\\{B_{i}:i\in I_{x}\\}$,
there exists $r>0$ such that $B(x,r)\subset 6B_{0}$. For
$\epsilon=\frac{r}{30}$, we have $x\in 6B_{i_{\epsilon}}$ and
$r_{6B_{i_{\epsilon}}}<\frac{r}{4}$. This implies $6B_{i_{\epsilon}}\subset
6B_{0}$ which is a contradiction.
Thus $\lim\inf_{i\in I_{x}}r_{B_{i}}>0$. This together with the uniform
boundedness of diameter of the family of balls shows that there exist $m$ and
$M>0$ such that $m<r_{B_{i}}<M$ for all $i\in I_{x}$. Obviously, $\cup_{i\in
I_{x}}B(x_{B_{i}},m)\subset B(x,2M)$ and the balls $\\{B(x_{B_{i}},m):i\in
I_{x}\\}$ are pairwise disjoint. By Lemma 2.1, there exists a finite family of
balls with radius $\frac{m}{30}$ such that
$B(x,2M)\subset\cup_{i=1}^{K}B(x_{i},\frac{m}{30})$. Therefore, there exist a
ball, says $B_{k}\in\\{B(x_{i},\frac{m}{30}):i\in 1,\ldots K\\}$, and at least
two balls $B_{1}$ and $B_{2}$ in $\\{B_{i}:i\in I_{x}\\}$ such that
$B_{k}\cap\frac{1}{6}B_{1}\notin\emptyset$ and
$B_{k}\cap\frac{1}{6}B_{1}\notin\emptyset$. Since
$\min\\{r_{\frac{1}{6}B_{1}},r_{\frac{1}{6}B_{1}}\\}>\frac{1}{6}m=5r_{B_{k}}$,
we have $B_{k}\subset B_{1}\cap B_{2}$. This is a contradiction, because the
family of balls $\\{B(x_{B_{i}},m):i\in I_{x}\\}$ is pairwise disjoint. Our
proof is completed.
We now give a Calderón-Zygmund decomposition on a non-homogenous space
$(X,\mu)$ which is an extension of a Calderón-Zygmund decomposition on the
non-homogeneous space $(\mathbb{R}^{n},\mu)$ in [T1].
###### Theorem 6.3
(Calderón-Zygmund decomposition) Assume $1\leq p<\infty$. For any $f\in
L^{p}(\mu)$ and any $\lambda>0$ (with $\lambda>\beta_{0}||f||_{p}/||\mu||$ if
$||\mu||<\infty$), the following statements hold.
(a) There exists a family of finite overlapping balls $\\{6Q_{i}\\}_{i}$ such
that $\\{Q_{i}\\}_{i}$ is a pairwise disjoint family and
$\frac{1}{\mu(6^{2}Q_{i})}\int_{Q_{i}}|f|^{p}d\mu>\frac{\lambda^{p}}{\beta_{0}},$
(22) $\frac{1}{\mu(6^{2}\eta Q_{i})}\int_{\eta
Q_{i}}|f|^{p}d\mu\leq\frac{\lambda^{p}}{\beta_{0}},\ \text{for all $\eta>1$},$
(23) $|f|\leq\lambda\ \text{a.e. $(\mu)$ on $X\backslash\bigcup_{i}6Q_{i}$}.$
(24)
(b) For each $i$, let $R_{i}$ be a $(3\times
6^{2},C_{\lambda}^{\log_{2}3\times 6^{2}+1})$-doubling ball concentric with
$Q_{i}$, with $l(R_{i})>6^{2}l(Q_{i})$ and denote
$\omega_{i}=\frac{\chi_{6Q_{i}}}{\sum_{k}\chi_{6Q_{k}}}$. Then there exists a
family of functions $\varphi_{i}$ with constant signs and supp
$(\varphi_{i})\subset R_{i}$ satisfying
$\int\varphi_{i}d\mu=\int_{6Q_{i}}f\omega_{i}d\mu,$ (25)
$\sum_{i}|\varphi_{i}|\leq\kappa\lambda,$ (26)
(where $\kappa$ is some constant which depends only on $(X,\mu)$), and
1. (i)
$||\varphi_{i}||_{\infty}\mu(R_{i})\leq C\int_{X}|w_{i}f|d\mu\ \text{if
$p=1$};$ (27)
2. (ii)
$||\varphi_{i}||_{L^{p}(\mu)}\mu(R_{i})^{1/p^{\prime}}\leq\frac{C}{\lambda^{p-1}}\int_{X}|w_{i}f|^{p}d\mu\
\text{if $1<p<\infty$}.$ (28)
(c) For $1<p<\infty$, if $R_{i}$ is the smallest $(3\times
6^{2},C_{\lambda}^{\log_{2}3\times 6^{2}+1})$-doubling ball of the family
$\\{3\times 6^{2}Q_{i}\\}_{k\geq 1}$, then
$\|b\|_{H_{at}^{1,p}(\mu)}\leq\frac{C}{\lambda^{p-1}}\|f\|_{L^{p}(\mu)}^{p}$
(29)
where $b=\sum_{i}(w_{i}f-\varphi_{i}))$.
_Proof:_ For the sake of simplicity, we only give the proof for the case $p=1$
for (a) and (b). When $p>1$, by setting $g=f^{p}\in L^{1}(\mu)$, we can reduce
to the problem $p=1$. Then, with a simple modification, we will obtain (28)
instead of (27).
(a) Set $E:=\\{x:|f(x)|>\lambda\\}$. For each $x\in E$, there exists some ball
$Q_{x}$ such that
$\frac{1}{\mu(6^{2}Q_{x})}\int_{Q_{x}}|f|d\mu>\frac{\lambda}{\beta_{0}}$ (30)
and such that if $Q^{\prime}_{x}$ is centered at $x$ with
$l(Q^{\prime}_{x})>l(Q_{x})$, then
$\frac{1}{\mu(6^{2}Q^{\prime}_{x})}\int_{Q^{\prime}_{x}}|f|d\mu\leq\frac{\lambda}{\beta_{0}}$
Now we can apply Lemma 6.2 to get a family of balls
$\\{Q_{i}\\}_{i}\subset\\{Q_{x}\\}_{x}$ such that
$\sum_{j}\chi_{6Q_{j}}(x)<\infty$ for all $x\in X$ and (22), (23) and (24) are
satisfied.
(b) Assume first that the family of balls $\\{Q_{i}\\}$ is finite. Without
loss of generality, suppose that $l(R_{i})\leq l(R_{i+1})$. The functions
$\varphi$ will be constructed of the form
$\varphi_{i}=\alpha_{i}\chi_{A_{i}},A_{i}\subset R_{i}$.
First, set $A_{1}=R_{1}$ and $\varphi_{1}=\alpha_{1}\chi_{R_{1}}$ such that
$\int\varphi_{1}=\int_{6Q_{i}}f\omega_{1}$. Assume that
$\varphi_{1},\ldots,\varphi_{k-1}$ have been constructed satisfying (25) and
$\sum_{i=1}^{K-1}\varphi_{i}\leq\kappa\lambda,$
where $\kappa$ is some constant which will be fixed later. There are two
cases:
Case 1: There exists some $i\in\\{1,\ldots,k-1\\}$ such that $R_{i}\cap
R_{k}\neq\emptyset$. Let $R_{s_{1}},\ldots,R_{s_{m}}$ be the family of
$R_{1},\ldots,R_{k-1}$ such that $R_{s_{j}}\cap R_{k}\neq\emptyset$. Since
$l(R_{s_{j}})\leq l(R_{k})$, $R_{s_{j}}\subset 3R_{k}$. By using $R_{k}$ is
$(3\times 6^{2},C_{\lambda}^{\log_{2}3\times 6^{2}+1})$-doubling and (23), we
get
$\displaystyle\sum_{j}|\varphi_{s_{j}}|$
$\displaystyle\leq\sum_{j}\int_{X}|f\omega_{s_{j}}|d\mu$ $\displaystyle\leq
C\sum_{j}\int_{X}\omega_{s_{j}}|f|d\mu\leq C\sum_{j}\int_{3R_{k}}|f|d\mu\leq
C\lambda\mu(3.6^{2}R_{k})\leq C_{1}\lambda\mu(R_{k}).$
Therefore,
$\mu\\{\sum_{j}\|\varphi_{s_{j}}|>2C_{1}\lambda\\}\leq\frac{\mu(R_{k})}{2}.$
Thus,
$\mu(A_{k})\geq\frac{\mu(R_{k})}{2},A_{k}=R_{k}\cap\\{\sum_{j}|\varphi_{s_{j}}|\leq
2C_{1}\lambda\\}.$
The constant $\alpha_{k}$ will be chosen such that
$\int\varphi_{k}=\int_{Q_{k}}f\omega_{k}d\mu$ where
$\varphi_{k}=\alpha_{k}\chi_{A_{k}}$. Then we obtain
$\alpha_{k}\leq\frac{C}{\mu(A_{k})}\int_{X}w_{i}|f|d\mu\leq
C\frac{2}{\mu(R_{k})}\int_{\frac{1}{6^{2}}R_{k}}|f|d\mu\leq C_{2}\lambda\ \
(\text{by using (\ref{cz2})}).$
If we choose $\kappa=2C_{1}+C_{2}$, (26) follows.
Case 2: $R_{i}\cap R_{k}=\emptyset$ for all $i=1,\ldots,k-1$. Set
$A_{k}=R_{k}$ and $\varphi_{k}=\alpha_{k}\chi_{R_{k}}$ such that
$\int\varphi_{k}=\int_{Q_{k}}f\omega_{k}d\mu$. We also get (26).
By the construction of the functions $\varphi_{i}$, it is easy to see that
$\mu(R_{i})\leq 2\mu(A_{k})$. Hence,
$||\varphi_{i}||_{\infty}\mu(R_{i})\leq C\alpha_{i}\mu(A_{k})\leq
C\int_{X}|f\omega_{i}|d\mu.$
When the collection of balls $\\{Q_{i}\\}$ is not finite, we can argue as in
[T1, p.134]. This completes the proofs of (a) and (b).
(c) Since $R_{i}$ is the smallest $(3\times 6^{2},C_{\lambda}^{\log_{2}3\times
6^{2}+1})$-doubling ball of the family $\\{3\times 6^{2}Q_{i}\\}_{k\geq 1}$,
one has $K_{Q_{i},R_{i}}\leq C$. For each $i$, we consider the atomic block
$b_{i}=fw_{i}-\varphi_{i}$ supported in ball $R_{i}$. By (22) and (28) we have
$|b_{i}|_{H_{at}^{1,p}(\mu)}\leq\frac{C}{\lambda^{p-1}}\int_{X}|fw_{i}|^{p}d\mu$
which implies
$|b|_{H_{at}^{1,p}(\mu)}\leq\frac{C}{\lambda^{p-1}}\int_{X}\sum_{i}|fw_{i}|^{p}d\mu\leq\frac{C}{\lambda^{p-1}}\int_{X}(\sum_{i}w_{i})^{p}|f|^{p}d\mu=\frac{C}{\lambda^{p-1}}\int_{X}|f|^{p}d\mu.$
Our proof is completed.
Using the Calderón-Zygmund decomposition and a standard argument, see for
example [J, pp.43-44] (also [T1, p.135]), we obtain the following
interpolation result for a linear operator. For clarity and completeness, we
sketch the proof below.
###### Theorem 6.4
Let $T$ be a linear operator which is bounded from $H_{at}^{1,\infty}(\mu)$
into $L^{1}(\mu)$ and from $L^{\infty}(\mu)$ into ${\rm RBMO}(\mu)$. Then $T$
can be extended to a bounded operator on $L^{p}(\mu)$ for all $1<p<\infty$.
_Proof:_ For simplicity we may assume that $\|\mu\|=\infty$. Let $f$ be a
function in $L^{\infty}(\mu)$ with bounded support satisfying $\int fd\mu=0$.
Let us recall that the set of all such functions is dense in $L^{p}(\mu)$ for
all $1<p<\infty$. For such functions $f$, we need only to show that
$\|M^{\sharp}Tf\|_{L^{p}(\mu)}\leq C\|f\|_{L^{p}(\mu)},\ \ 1<p<\infty.$ (31)
Once (31) is proved, Theorem 6.4 follows from Theorem 4.2.
For such a function $f$ and $\lambda>0$, we can decompose the function $f$ as
in Theorem 6.3
$f:=b+g=\sum_{i}(w_{i}f-\varphi_{i})+g.$
By (24) and (26), we have $\|g\|_{L^{\infty}(\mu)}\leq C\lambda$, and by (29)
$\|b\|_{H^{1,p}_{at}(\mu)}\leq\frac{C}{\lambda^{p-1}}\|f\|^{p}_{L^{p}(\mu)}.$
Since $T$ is bounded from $L^{\infty}(\mu)$ into RBMO$(\mu)$, we have
$\|M^{\sharp}Tg\|_{L^{\infty}(\mu)}\leq C_{0}\lambda.$
Therefore,
$\\{M^{\sharp}Tf>(C_{0}+1)\lambda\\}\subset\\{M^{\sharp}Tb>\lambda\\}.$
The fact that $M^{\sharp}$ is of weak type $(1,1)$ gives
$\mu\\{M^{\sharp}Tb>\lambda\\}\leq C\frac{\|Tb\|_{L^{1}(\mu)}}{\lambda}.$
Moreover, since $T$ is bounded from $H^{1,\infty}_{at}(\mu)$ into
$L^{1}(\mu)$, we have
$\|Tb\|_{L^{1}(\mu)}\leq
C\|b\|_{H^{1,\infty}_{at}(\mu)}\leq\frac{C}{\lambda^{p-1}}\|f\|^{p}_{L^{p}(\mu)}.$
This implies
$\mu\\{M^{\sharp}Tf>(C_{0}+1)\lambda\\}\leq
C\frac{\|f\|^{p}_{L^{p}(\mu)}}{\lambda^{p}}.$
So the sublinear operator $M^{\sharp}T$ is of weak type $(p,p)$ for all
$1<p<\infty$. By Marcinkiewicz interpolation theorem the operator
$M^{\sharp}T$ is bounded for all $1<p<\infty$. This completes our proof.
### 6.2 The weak $(1,1)$ boundedness of Calderón-Zygmund operators
###### Theorem 6.5
If a Calderón-Zygmund operator $T$ is bounded on $L^{2}(\mu)$, then $T$ is of
weak type $(1,1)$.
_Proof:_ Let $f\in L^{1}(\mu)$ and $\lambda>0$. We can assume that
$\lambda>\beta_{0}\|f\|_{L^{1}(\mu)}/\|\mu\|$. Otherwise, there is nothing to
prove. Using the same notations as in Theorem 6.3 with $R_{i}$ which is chosen
as the smallest $(3\times 6^{2},C_{\lambda}^{\log_{2}3\times
6^{2}+1})$-doubling ball of the family $\\{3\times 6^{2}Q_{i}\\}_{k\geq 1}$,
we can write $f=g+b$, with
$g=f\chi_{{}_{X\backslash\cup_{i}6Q_{i}}}+\sum_{i}\varphi_{i}$
and
$b:=\sum_{i}b_{i}=\sum_{i}(w_{i}f-\varphi_{i}).$
Taking into account (22), one has
$\mu(\cup_{i}6^{2}Q_{i})\leq\frac{C}{\lambda}\int_{Q_{i}}|f|d\mu\leq\frac{C}{\lambda}\int_{X}|f|d\mu$
where in the last inequality we use the pairwise disjoint property of family
$\\{Q_{i}\\}_{i}$.
We need only to show that
$\mu\\{x\in
X\backslash\cup_{i}6^{2}Q_{i}:|Tf(x)|>\lambda\\}\leq\frac{C}{\lambda}\int_{X}|f|d\mu.$
We have
$\displaystyle\mu\\{x\in X\backslash\cup_{i}6^{2}Q_{i}:|Tf(x)|>\lambda\\}$
$\displaystyle\leq\mu\\{x\in
X\backslash\cup_{i}6^{2}Q_{i}:|Tg(x)|>\lambda/2\\}$
$\displaystyle~{}~{}~{}+\mu\\{x\in
X\backslash\cup_{i}6^{2}Q_{i}:|Tb(x)|>\lambda/2\\}:=I_{1}+I_{2}.$
Let us estimate the term $I_{1}$ related to the “good part” first. Since
$|g|\leq C\lambda$ then
$\displaystyle\mu\\{x\in
X\backslash\cup_{i}6^{2}Q_{i}:|Tg(x)|>\lambda/2\\}\leq\frac{C}{\lambda^{2}}\int|g|^{2}d\mu\leq\frac{C}{\lambda}\int|g|d\mu.$
Furthermore, we have
$\displaystyle\int|g|d\mu$
$\displaystyle\leq\int_{X\backslash\cup_{i}6Q_{i}}|f|d\mu+\sum_{i}\int_{R_{i}}|\varphi_{i}|$
$\displaystyle\leq\int_{X}|f|d\mu+\sum_{i}\mu(R_{i})\|\varphi_{i}\|_{L^{\infty}(\mu)}$
$\displaystyle\leq\int_{X}|f|d\mu+C\sum_{i}\int_{X}|fw_{i}|d\mu$
$\displaystyle\leq C\int_{X}|f|d\mu\ .$
Therefore,
$\mu\\{x\in
X\backslash\cup_{i}6^{2}Q_{i}:|Tg(x)|>\lambda/2\\}\leq\frac{C}{\lambda}\int|f|d\mu.$
For the term $I_{2}$, we have
$\displaystyle I_{2}$
$\displaystyle\leq\frac{C}{\lambda}\sum_{i}\Big{(}\int_{X\backslash
2R_{i}}|Tb_{i}|d\mu+\int_{2R_{i}}|T\varphi_{i}|d\mu+\int_{2R_{i}\backslash
6^{2}Q_{i}}|Tw_{i}f|d\mu\Big{)}$
$\displaystyle\leq\frac{C}{\lambda}\sum_{i}\Big{(}K_{i1}+K_{i2}+K_{i3}\Big{)}$
Note that $\int b_{i}d\mu=0$ for all $i$. We have, by (3),
$\displaystyle K_{i1}=\int_{X\backslash 2R_{i}}|Tb_{i}|d\mu$
$\displaystyle\leq C\int|b_{i}|d\mu$
$\displaystyle\leq\int_{X}|fw_{i}|d\mu+\int_{R_{i}}|\varphi_{i}|d\mu$
$\displaystyle\leq\int_{X}|fw_{i}|d\mu+\mu(R_{i})\|\varphi_{i}\|_{L^{\infty}(\mu)}$
$\displaystyle\leq C\sum_{i}\int_{X}|fw_{i}|d\mu$ $\displaystyle\leq
C\sum_{i}\int_{X}|f|d\mu.$
On the other hand, by the $L^{2}$ boundedness of $T$ and $R_{i}$ is a
$(3\times 6^{2},C_{\lambda}^{\log_{2}3\times 6^{2}+1})$-doubling ball, we get
$\displaystyle K_{i2}$
$\displaystyle\leq\Big{(}\int_{2R_{i}}|T\varphi_{i}|^{2}\Big{)}^{1/2}(\mu(2R_{i}))^{1/2}$
$\displaystyle\leq\Big{(}\int_{2R_{i}}|\varphi_{i}|^{2}\Big{)}^{1/2}(\mu(2R_{i}))^{1/2}$
$\displaystyle\leq C\|\varphi_{i}\|_{L^{\infty}(\mu)}\mu(2R_{i})$
$\displaystyle\leq C\int|w_{i}f|d\mu.$
Moreover, taking into account the fact that supp$w_{i}f\subset 6Q_{i}$, for
$x\in 2R_{i}\backslash 6^{2}Q_{i}$ we have, by Lemma 6.1,
$\displaystyle K_{i3}$ $\displaystyle\leq C\int_{2R_{i}\backslash
6^{2}Q_{i}}\frac{1}{\lambda(x_{Q_{i}},d(x,x_{Q_{i}}))}d\mu(x)\times\int_{X}|w_{i}f|d\mu.$
Hence we obtain
$I_{2}\leq\frac{C}{\lambda}\sum_{i}\int_{X}|w_{i}f|d\mu\leq\frac{C}{\lambda}\sum_{i}\int_{X}|f|d\mu$
and the proof is completed.
### 6.3 Cotlar inequality
We note that from the weak type $(1,1)$ estimate of $T$, we can obtain a
Cotlar inequality on $T$. More precisely, we have the following result.
###### Theorem 6.6
Assume that $T$ is a Calderón-Zygmund operator and that $T$ is bounded on
$L^{2}(X,\mu)$. Then there exists a constant $C>0$ such that for any bounded
function $f$ with compact support and $x\in X$ we have
$T_{*}f(x)\leq C\Big{(}M_{6,\eta}(Tf)(x)+M_{(5)}f(x)\Big{)}$
where
$M_{p,\rho}f(x)=\sup_{Q\ni x}\Big{(}\frac{1}{\mu(\rho
Q)}\int_{Q}|f|^{p}\Big{)}.$
_Proof:_ For any $\epsilon>0$ and $x\in X$, let $Q_{x}$ be the biggest
$(6,\beta)$-doubling ball centered $x$ of the form $6^{-k}\epsilon,k\geq 1$
and $\beta>6^{n}$. Assume that $Q_{x}=B(x,6^{-k_{0}}\epsilon)$. Then, we can
break $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{\frac{6}{5}Q_{x}}$. Obviously
$T_{*}f_{1}(x)=0$. This follows that
$T_{*}f(x)\leq|Tf_{2}(x)|+\Big{|}\int_{d(x,y)\leq\epsilon}K(x,y)f_{2}(y)d\mu(y)\Big{|}=I_{1}+I_{2}.$
Let us estimate $I_{1}$ first. For any $z\in Q_{x}$, we have
$|Tf_{2}(x)|\leq|Tf_{2}(x)-Tf_{2}(z)|+|Tf(z)|+|Tf_{1}(z)|.$ (32)
On the other hand, it follows from (3) that
$\displaystyle|Tf_{2}(x)-Tf_{2}(z)|$
$\displaystyle\leq\int_{X\backslash\frac{6}{5}Q_{x}}|K(x,y)-K(z,y)||f(y)|d\mu(y)$
$\displaystyle\leq
C\int_{X\backslash\frac{6}{5}Q_{x}}\frac{d(x,z)^{\delta}}{d(x,y)^{\delta}\lambda(x,d(x,y))}|f(y)|d\mu(y)$
$\displaystyle\leq C\int_{X\backslash
Q_{x}}\frac{r_{Q_{x}}^{n+\delta}}{d(x,y)^{\delta}\lambda(x,d(x,y))}|f(y)|d\mu(y)$
$\displaystyle\leq C\sum_{k=0}^{\infty}\int_{6^{k+1}Q_{x}\backslash
6^{k}Q_{x}}\frac{r_{Q_{x}}^{\delta}}{d(x,y)^{\delta}\lambda(x,d(x,y))}|f(y)|d\mu(y)$
$\displaystyle\leq C\sum_{k=0}^{\infty}\int_{6^{k+1}Q_{x}\backslash
6^{k}Q_{x}}\frac{r_{Q_{x}}^{\delta}}{(6^{k}r_{Q_{x}})^{\delta}\lambda(x,6^{k}r_{Q_{x}})}|f(y)|d\mu(y)$
$\displaystyle\leq C\sum_{k=0}^{\infty}6^{-k}\frac{\mu(6\times
6^{k+1}Q_{x})}{\lambda(x,6^{k}r_{Q_{x}})}\frac{1}{\mu(6\times
6^{k+1}Q_{x})}\int_{6^{k+1}Q_{x}}|f(y)|d\mu(y)$ $\displaystyle\leq
C\sum_{k=0}^{\infty}6^{-k}M_{(6)}f(x)=CM_{(6)}f(x).$
This together with (33) implies
$|Tf_{2}(x)|\leq CM_{(6)}f(x)+|Tf_{1}(z)|+|Tf(z)|$ (33)
for all $z\in Q_{x}$.
At this stage, taking the $L^{\eta}(Q_{x},\frac{d\mu(x)}{\mu(Q)})$-norm with
respect to $z$, we have
$|Tf_{2}(x)|\leq
CM_{(6)}f(x)+\Big{(}\frac{1}{\mu(Q_{x})}\int_{Q_{x}}|Tf_{1}(z)|^{\eta}d\mu(z)\Big{)}^{1/\eta}+\Big{(}\frac{1}{\mu(Q_{x})}\int_{Q_{x}}|Tf(z)|^{\eta}d\mu(z)\Big{)}^{1/\eta}.$
By the Kolmogorov inequality and the weak type $(1,1)$ boundedness of $T$, we
have, for $\eta<1$,
$\displaystyle\Big{(}\frac{1}{\mu(Q_{x})}\int_{Q_{x}}|Tf_{1}(z)|^{\eta}d\mu(z)\Big{)}^{1/\eta}$
$\displaystyle\leq\frac{1}{\mu(Q_{x})}\int_{\frac{6}{5}Q_{x}}|f_{1}(z)|d\mu(z)$
$\displaystyle\leq\frac{C}{\mu(15Q_{x})}\int_{\frac{6}{5}Q_{x}}|f_{1}(z)|d\mu(z)\
\ \ \text{(since $Q_{x}$ is $(6,\beta)$- doubling)}$ $\displaystyle\leq
CM_{(5)}f(x).$
Furthermore, since $Q_{x}$ is $(6,\beta)$-doubling,
$\Big{(}\frac{1}{\mu(Q_{x})}\int_{Q_{x}}|Tf(z)|^{\eta}d\mu(z)\Big{)}^{1/\eta}\leq
CM_{\eta,6}M(Tf)(x).$
Therefore, $I_{1}\leq CM_{(6)}f(x)+CM_{\eta,6}M(Tf)(x)$.
For the term $I_{2}$ we have
$\displaystyle I_{2}$
$\displaystyle\leq\int_{d(x,y)\leq\epsilon}|K(x,y)||f_{2}(y)|d\mu(y)$
$\displaystyle\leq C\int_{B(x,\epsilon)\backslash
B(x,6^{-k_{0}}\epsilon)}\frac{1}{\lambda(x,d(x,y))}|f_{2}(y)|d\mu(y)$
$\displaystyle\leq
C\sum_{k=0}^{k_{0}-1}\int_{B(x,6^{k+1-k_{0}}\epsilon)\backslash
B(x,6^{k-k_{0}}\epsilon)}\frac{1}{\lambda(x,d(x,y))}|f_{2}(y)|d\mu(y)$
$\displaystyle\leq
C\sum_{k=0}^{k_{0}-1}\int_{B(x,6^{k+1-k_{0}}\epsilon)}\frac{1}{\lambda(x,6^{k-k_{0}}\epsilon)}|f_{2}(y)|d\mu(y)$
$\displaystyle\leq C\sum_{k=0}^{k_{0}-1}\frac{\mu(x,6\times
6^{k+1-k_{0}}\epsilon)}{\lambda(x,6\times
6^{k+1-k_{0}}\epsilon)}\frac{1}{\mu(x,6\times
6^{k+1-k_{0}}\epsilon)}\int_{B(x,6^{k+1-k_{0}}\epsilon)}|f_{2}(y)|d\mu(y)$
$\displaystyle\leq C\sum_{k=0}^{k_{0}-1}\frac{\mu(x,6\times
6^{k+1-k_{0}}\epsilon)}{\lambda(x,6\times 6^{k+1-k_{0}}\epsilon)}M_{(6)}f(x).$
At this stage, by repeating the argument in the proof of Lemma 6.1, we have
$\sum_{k=0}^{k_{0}-1}\frac{\mu(x,6\times
6^{k+1-k_{0}}\epsilon)}{\lambda(x,6\times 6^{k+1-k_{0}}\epsilon)}\leq C.$
Therefore,
$I_{2}\leq CM_{(6)}f(x).$
This completes our proof.
###### Remark 6.7
From the boundedness of $M_{6,\eta}(\cdot)$ and $M_{(5)}(\cdot)$, the Cotlar
inequality tells us that if $T$ is bounded on $L^{2}(X,\mu)$ then the maximal
operator $T_{*}$ is bounded on $L^{p}(X,\mu)$ for $1<p<\infty$. Note that the
endpoint estimate of $T_{*}$ will be investigated in [AD].
The Calderón-Zygmund decomposition Theorem 6.3 does not require the property
(v) of $\lambda(\cdot,\cdot)$.
## 7 The boundedness of Calderón-Zygmund operators
The main results of this section are Theorems 7.1, 7.3 and 7.6.
### 7.1 The boundedness of Calderón-Zygmund operators from $L^{\infty}$ to
RBMO space
The following result shows that on a non-homogeneous space $(X,\mu)$, a
Calderón-Zygmund operator which is bounded on $L^{2}$ is also bounded from
$L^{\infty}(\mu)$ into the regularized BMO space RBMO$(\mu)$.
###### Theorem 7.1
Assume that $T$ is a Calderón-Zygmund operator and $T$ is bounded on
$L^{2}(\mu)$, then $T$ is bounded from $L^{\infty}(\mu)$ into RBMO$(\mu)$.
Therefore, by interpolation and duality, $T$ is bounded on $L^{p}(\mu)$ for
all $1<p<\infty$.
_Proof:_ We use the RBMO characterizations (9) and (10). The condition (9) can
be obtained by the standard method used in the case of doubling measure. We
omit the details here.
We will check (10). To do this, we have to show that
$|m_{Q}(Tf)-m_{R}(Tf)|\leq
CK_{Q,R}\Big{(}\frac{\mu(6Q)}{\mu(Q)}+\frac{\mu(6R)}{\mu(R)}\Big{)}\|f\|_{L^{\infty}(\mu)}$
for all $Q\subset R$.
Let $N$ be the first integer $k$ such that $R\subset 6^{k}Q$. We denote
$Q_{R}=6^{N+1}Q$. Then for $x\in Q$ and $y\in R$, we set
$\displaystyle Tf(x)-Tf(y)$
$\displaystyle=Tf\chi_{6Q}(x)+Tf\chi_{6^{N}Q\backslash
6Q}(x)+Tf\chi_{X\backslash Q_{R}}(x)$
$\displaystyle~{}~{}~{}-(Tf\chi_{Q_{R}}(y)+Tf\chi_{X\backslash Q_{R}}(y))$
$\displaystyle\leq|Tf\chi_{6Q}(x)|+|Tf\chi_{6^{N}Q\backslash 6Q}(x)|$
$\displaystyle~{}~{}~{}+|Tf\chi_{X\backslash Q_{R}}(x)-Tf\chi_{X\backslash
Q_{R}}(y)|+|Tf\chi_{Q_{R}}(y)|$ $\displaystyle\leq I_{1}+I_{2}+I_{3}+I_{4}.$
Let us estimate $I_{3}$ first. We have
$\displaystyle I_{3}$ $\displaystyle\leq\int\limits_{X\backslash
Q_{R}}|K(x,z)-K(y,z)||f(z)|d\mu(z)$
$\displaystyle\leq\sum_{k=N+1}^{\infty}\int\limits_{6^{k+1}Q\backslash
6^{k}Q}\frac{d(x,y)^{\delta}}{d(x,z)^{\delta}\lambda(x,d(x,z))}|f(z)|d\mu(z)$
$\displaystyle\leq\sum_{k=N+1}^{\infty}6^{-(k-N)\delta}\frac{\mu(6^{k+1}Q)}{\lambda(x,6^{k-1}r_{Q})}\|f\|_{L^{\infty}(\mu)}$
$\displaystyle\leq
C\sum_{k=N+1}^{\infty}6^{-(k-N)\delta}\frac{\mu(6^{k+1}Q)}{\lambda(x,6^{k+1}r_{Q})}\|f\|_{L^{\infty}(\mu)}$
$\displaystyle\leq
C\sum_{k=N+1}^{\infty}6^{-(k-N)\delta}\|f\|_{L^{\infty}(\mu)}=C\|f\|_{L^{\infty}(\mu)},$
where in the last inequality we use the fact that
$\mu(6^{k+1}Q)\leq{\lambda(x,6^{k+1}r_{Q})}$, since $x\in Q\subset 2^{k+1}Q$.
As to the term $I_{2}$, we have
$\displaystyle Tf\chi_{6^{N}Q\backslash 6Q}(x)$
$\displaystyle\leq\int_{6^{N}Q\backslash 6Q}|K(x,y)||f(y)|d\mu(y)$ (34)
$\displaystyle\leq\int_{6^{N}Q\backslash
6Q}\frac{C}{\lambda(x_{Q},d(x_{Q},y))}|f(y)|d\mu(y)$ $\displaystyle\leq
K_{6Q,6^{N}Q}\|f\|_{L^{\infty}(\mu)}.$
Therefore, $I_{2}\leq CK_{Q,R}\|f\|_{\infty}$. So, we get
$\displaystyle Tf(x)-Tf(y)$
$\displaystyle=Tf\chi_{6Q}(x)+CK_{Q,R}\|f\|_{L^{\infty}(\mu)}+|Tf\chi_{Q_{R}}(y)|+C\|f\|_{L^{\infty}(\mu)}.$
Taking the mean over $Q$ and $R$ for $x$ and $y$, respectively, we have
$\displaystyle|m_{Q}(Tf)-m_{R}(Tf)|$ $\displaystyle\leq
m_{Q}(|Tf\chi_{6Q}|)+CK_{Q,R}\|f\|_{L^{\infty}(\mu)}+|Tf\chi_{Q_{R}}(y)|$
$\displaystyle+C\|f\|_{L^{\infty}(\mu)}+m_{R}(Tf\chi_{Q_{R}}).$
For the boundedness on $L^{2}(\mu)$ of $T$, we have
$\displaystyle m_{Q}(|Tf\chi_{6Q}|)$
$\displaystyle\leq\Big{(}\frac{1}{\mu(Q)}\int_{Q}|Tf\chi_{6Q}|^{2}\Big{)}^{1/2}$
$\displaystyle\leq
C\Big{(}\frac{\mu(6Q)}{\mu(Q)}\Big{)}^{1/2}\|f\|_{L^{\infty}(\mu)}$
$\displaystyle\leq
C\Big{(}\frac{\mu(6Q)}{\mu(Q)}\Big{)}\|f\|_{L^{\infty}(\mu)}.$
Next, we write
$m_{R}(Tf\chi_{Q_{R}})\leq m_{R}(Tf\chi_{Q_{R}\cap
6R})+m_{R}(Tf\chi_{Q_{R}\backslash 6R}).$
By similar argument in estimate of $m_{Q}(|Tf\chi_{6Q}|)$, the term
$m_{R}(Tf\chi_{Q_{R}\cap 6R})$ is dominated by
$C\Big{(}\frac{\mu(6R)}{\mu(R)}\Big{)}\|f\|_{L^{\infty}(\mu)}.$
The second term $m_{R}(Tf\chi_{Q_{R}\backslash 6R})$ can be treated as in
(34). Since $r_{Q_{R}}\approx r_{R}$, we have
$m_{R}(Tf\chi_{Q_{R}\backslash 6R})\leq C\|f\|_{L^{\infty}(\mu)}$
To sum up, we have
$\displaystyle|m_{Q}(Tf)-m_{R}(Tf)|$ $\displaystyle\leq
CK_{Q,R}\|f\|_{L^{\infty}(\mu)}+\Big{(}\frac{\mu(6Q)}{\mu(Q)}+\frac{\mu(6R)}{\mu(R)}\Big{)}\|f\|_{L^{\infty}(\mu)}$
$\displaystyle\leq
CK_{Q,R}\Big{(}\frac{\mu(6Q)}{\mu(Q)}+\frac{\mu(6R)}{\mu(R)}\Big{)}\|f\|_{L^{\infty}(\mu)}.$
###### Remark 7.2
By similar argument in [T1, Theorem 2.11], we can replace the assumption of
$L^{2}(\mu)$ boundedness by the weaker assumption: for any ball $B$ and any
function $a$ supported on $B$,
$\int_{B}|Ta|d\mu\leq C\|a\|_{L^{\infty}}\mu(6B)$
uniformly on $\epsilon>0$.
### 7.2 The boundedness of Calderón-Zygmund operators on Hardy spaces
We now show that an $L^{2}$ bounded Calderón-Zygmund operator maps the atomic
Hardy space boundedly into $L^{1}$.
###### Theorem 7.3
Assume that $T$ is a Calderón-Zygmund operator and $T$ is bounded on
$L^{2}(X,\mu)$, then $T$ is bounded from $H_{at}^{1,\infty}(\mu)$ into
$L^{1}(X,\mu)$. Therefore, by interpolation and duality, $T$ is bounded on
$L^{p}(\mu)$ for all $1<p<\infty$.
_Proof:_ By [HoM, Lemm 4.1], it is enough to show that
$\|Tb\|_{L^{1}(\mu)}\leq C|b|_{H_{at}^{1,\infty}(\mu)}$ (35)
for any atomic block $b$ with supp$b\subset B$ and $=\sum_{j}\lambda_{j}a_{j}$
where the $a_{j}$’s are functions satisfying (a) and (b) in definition of
atomic blocks. At this stage we can use the same argument as in [T1, Theorem
4.2] with minor modifications as in Theorem 7.1 to obtain the estimate (35).
We omit the details here.
### 7.3 Commutators of Calderón-Zygmund operators with RBMO functions
In this section we assume that the dominating function $\lambda$ satisfies
$\lambda(x,ar)=a^{m}\lambda(x,r)$ for all $x\in X$ and $a,r>0$. Then, for two
balls $B$, $Q$ such that $B\subset Q$ we can define the coefficient
$K^{\prime}_{B,Q}$ as follows: let $N_{B,Q}$ be the smallest integer
satisfying $6^{N_{B,Q}}r_{B}\geq r_{Q}$, we set
$K^{\prime}_{B,Q}:=1+\sum_{k=1}^{N_{B,Q}}\frac{\mu(6^{k}B)}{\lambda(x_{B},6^{k}r_{B})}.$
(36)
It is not difficult to show that the coefficient $K_{B,Q}\approx
K^{\prime}_{B,Q}$. Note that in the definition of $K^{\prime}_{B,Q}$ we can
replace $6$ by any number $\eta>1$.
To establish the boundedness of commutators of Calderón-Zygmund operators with
RBMO functions, we need the following two lemmas. Note that these lemmas are
similar to those in [T1]. However, due to the difference of choices of
coefficient $K_{Q,R}$, we would like to provide the proof for the first one.
Meanwhile, the proof of Lemma 7.5 is completely analogous to that of Lemma 9.3
in [T1], hence we omit the details.
###### Lemma 7.4
If $B_{i}=B(x_{0},r_{i}),i=1,\ldots,m$ are concentric balls $B_{1}\subset
B_{2}\subset\ldots\subset B_{m}$ with $K_{B_{i},B_{i+1}}>2$ for
$i=1,\ldots,m-1$ then
$\sum_{i=1}^{m-1}K_{B_{i},B_{i+1}}\leq 2K_{B_{1},B_{m}}.$ (37)
_Proof:_ By definition,
$K_{B_{i},B_{i+1}}=1+\int\limits_{r_{i}\leq d(x,x_{0})\leq
r_{i+1}}\frac{1}{\lambda(x_{0},d(x,x_{0}))}d\mu(x).$
Since $K_{B_{i},B_{i+1}}>2$, we have
$K_{B_{i},B_{i+1}}<2\int\limits_{r_{i}\leq d(x,x_{0})\leq
r_{i+1}}\frac{1}{\lambda(x_{0},d(x,x_{0}))}d\mu(x)$
for all $i=1,\ldots,m-1$.
This implies
$\displaystyle\sum_{i=1}^{m-1}K_{B_{i},B_{i+1}}$
$\displaystyle<2\sum_{i=1}^{m-1}\int\limits_{r_{i}\leq d(x,x_{0})\leq
r_{i+1}}\frac{1}{\lambda(x_{0},d(x,x_{0}))}d\mu(x)$ $\displaystyle\leq
2\int\limits_{r_{1}\leq d(x,x_{0})\leq
r_{m}}\frac{1}{\lambda(x_{0},d(x,x_{0}))}d\mu(x)$ $\displaystyle\leq
2K_{B_{1},B_{m}}.$
###### Lemma 7.5
There exists some constant $P_{0}$ such that if $x\in X$ is some fixed point
and $\\{f_{B}\\}_{B\ni x}$ is collection of numbers such that
$|f_{Q}-f_{R}|\leq C_{x}$ for all doubling balls $Q\subset R$ with $x\in Q$
and $K_{Q,R}\leq P_{0}$, then
$|f_{Q}-f_{R}|\leq CK_{Q,R}C_{x}\ \ \text{for all doubling balls $Q\subset R$
with $x\in Q$}.$
###### Theorem 7.6
If $b\in{\rm RBMO}(\mu)$ and $T$ is a Calderón-Zygmund bounded on
$L^{2}(\mu)$, then the commutator $[b,T]$ defined by
$[b,T](f)=bT(f)-T(bf)$
is bounded on $L^{p}(\mu)$ for $1<p<\infty$.
_Proof:_ For $1<p<\infty$ we will show that
$M^{\sharp}([b,T]f)(x)\leq C\|b\|_{{\rm
RBMO}(\mu)}\Big{(}M_{p,5}f(x)+M_{p,6}Tf(x)+T_{*}f(x)\Big{)},$ (38)
where
$M_{p,\rho}f(x)=\sup_{Q\ni x}\Big{(}\frac{1}{\mu(\rho
Q)}\int_{Q}|f|^{p}\Big{)}.$
Once (38) is proved, it follows from the boundedness of $T_{*}$ on
$L^{p}(\mu)$ and $M_{p,\rho}$ on $L^{r}(\mu),r>p$ and $\rho\geq 5$, and from a
standard argument that we can obtain the boundedness of $[b,T]$ on
$L^{p}(\mu)$.
Let $\\{b_{B}\\}$ be a family of numbers satisfying
$\int_{B}|b-b_{B}|d\mu\leq 2\mu(6B)\|b\|_{{\rm RBMO}}$
for balls $B$, and
$|b_{Q}-b_{R}|\leq 2K_{Q,R}\|b\|_{{\rm RBMO}}$
for balls $Q\subset R$. Denote
$h_{Q}:=m_{Q}(T((b-b_{Q})f\chi_{X\backslash\frac{6}{5}Q}).$
We will show that
$\frac{1}{\mu(6B)}\int_{B}|[b,T]f-h_{Q}|d\mu\leq C\|b\|_{{\rm
RBMO}}(M_{p,5}f(x)+M_{p,6}Tf(x))$ (39)
for all $x$ and $B$ with $x\in B$, and
$|h_{Q}-h_{R}|\leq C\|b\|_{{\rm RBMO}}(M_{p,5}f(x)+T_{*}f(x))K^{2}_{Q,R}$ (40)
for all $x\in Q\subset R$.
The proof of (39) is similar to that in Theorem 9.1 in [T1] with minor
modifications and we omit it here.
It remains to check (40). For two balls $Q\subset R$, let $N$ be an integer
such that $(N-1)$ is the smallest number satisfying $r_{R}\leq 6^{N-1}r_{Q}$.
Then, we break the term $|h_{Q}-h_{R}|$ into five terms:
$\displaystyle|m_{Q}(T((b-b_{Q})$ $\displaystyle
f\chi_{X\backslash\frac{6}{5}Q})-m_{R}(T((b-b_{R})f\chi_{X\backslash\frac{6}{5}R})|$
$\displaystyle\leq|m_{Q}(T((b-b_{Q})f\chi_{6Q\backslash\frac{6}{5}Q})|+|m_{Q}(T((b_{Q}-b_{R})f\chi_{X\backslash
6Q})|$ $\displaystyle~{}~{}~{}+|m_{Q}(T((b-b_{R})f\chi_{6^{N}Q\backslash
6Q})|$ $\displaystyle~{}~{}~{}+|m_{Q}(T((b-b_{R})f\chi_{X\backslash
6^{N}Q})-m_{R}(T((b-b_{R})f\chi_{X\backslash 6^{N}Q})|$
$\displaystyle~{}~{}~{}+|m_{R}(T((b-b_{R})f\chi_{6^{N}Q\backslash\frac{6}{5}R})$
$\displaystyle=M_{1}+M_{2}+M_{3}+M_{4}+M_{5}.$
Let us estimate $M_{1}$ first. For $y\in Q$ we have, by Proposition 3.2
$\displaystyle|T((b-b_{Q})$ $\displaystyle
f\chi_{6Q\backslash\frac{6}{5}Q})(x)|$
$\displaystyle\leq\frac{C}{\lambda(x,r_{Q})}\int_{6Q}|b-b_{Q}||f|d\mu$
$\displaystyle\leq\frac{\mu(30Q)}{\lambda(x,30r_{Q})}\Big{(}\frac{1}{\mu(5\times
6Q)}\int_{6Q}|b-b_{Q}|^{p^{\prime}}d\mu\Big{)}^{1/p^{\prime}}\Big{(}\frac{1}{\mu(5\times
6Q)}\int_{6Q}|f|^{p}d\mu\Big{)}^{1/p}$ $\displaystyle C\|b\|_{{\rm
RBMO}}M_{p,5}f(x).$
The term $M_{5}$ can be treated by similar way. So, we have $M_{1}+M_{5}\leq
C\|b\|_{{\rm RBMO}}M_{p,5}f(x).$ For the term $M_{2}$, we have for $x,y\in Q$
$\displaystyle|Tf\chi_{X\backslash 6Q}(y)|$
$\displaystyle=\Big{|}\int_{X\backslash 6Q}K(y,z)f(z)d\mu(z)\Big{|}$
$\displaystyle\leq\Big{|}\int_{X\backslash
6Q}K(x,z)f(z)d\mu(z)\Big{|}+\int_{X\backslash 6Q}|K(y,z)-K(x,z)||f(z)|d\mu(z)$
$\displaystyle\leq T_{*}f(x)+CM_{p,5}f(x).$
This implies
$|m_{Q}(T((b_{Q}-b_{R})f\chi_{X\backslash 6Q})|\leq
CK_{Q,R}(T_{*}f(x)+M_{p,6}f(x)).$
For the term $M_{4}$, we have, for $y,z\in R$
$\displaystyle|T((b-b_{R})$ $\displaystyle f\chi_{X\backslash
6^{N}Q}(y)-T((b-b_{R})f\chi_{X\backslash 6^{N}Q}(z)|$
$\displaystyle\leq\int_{X\backslash
2R}|K(y,w)-K(z,w)||(b(w)-b_{R})||f(x)|d\mu(w)$
$\displaystyle\leq\int_{X\backslash
2R}\frac{d(y,z)^{\delta}}{d(w,y)^{\delta}\lambda(y,d(w,y))}|(b(w)-b_{R})||f(x)|d\mu(w)$
$\displaystyle\leq\sum_{k=1}^{\infty}\int_{2^{k+1}R\backslash
2^{k}R}\frac{d(y,z)^{\delta}}{d(w,y)^{\delta}\lambda(y,d(w,y))}|(b(w)-b_{R})||f(x)|d\mu(w)$
$\displaystyle\leq\sum_{k=1}^{\infty}2^{-k\delta}\frac{1}{\lambda(y,2^{k-1}r_{R})}\int_{2^{k+1}R}|(b(w)-b_{R})||f(x)|d\mu(x)$
By Hölder inequality we have
$\displaystyle|T((b-b_{R})$ $\displaystyle f\chi_{X\backslash
6^{N}Q}(y)-T((b-b_{R})f\chi_{X\backslash 6^{N}Q}(z)|$ $\displaystyle\leq
C\sum_{k=1}^{\infty}2^{-k\delta}\frac{\mu(5\times 2^{k+1}R)}{\lambda(y,5\times
2^{k+1}r_{R})}\Big{(}\frac{1}{\mu(5\times
2^{k+1}R)}\int_{2^{k+1}R}|b-b_{R}|^{p^{\prime}}d\mu\Big{)}^{1/p^{\prime}}$
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{1}{\mu(5\times
2^{k+1}R)}\int_{2^{k+1}R}|f|^{p}d\mu\Big{)}^{1/p}$ $\displaystyle\leq
C\sum_{k=1}^{\infty}2^{-k\delta}\Big{[}\Big{(}\frac{1}{\mu(5\times
2^{k+1}R)}\int_{2^{k+1}R}|b-b_{2^{k+1}R}|^{p^{\prime}}d\mu\Big{)}^{1/p^{\prime}}$
$\displaystyle~{}~{}~{}~{}+\Big{(}\frac{1}{\mu(5\times
2^{k+1}R)}\int_{2^{k+1}R}|b_{R}-b_{2^{k+1}R}|^{p^{\prime}}d\mu\Big{)}^{1/p^{\prime}}\Big{]}\Big{(}\frac{1}{\mu(5\times
2^{k+1}R)}\int_{2^{k+1}R}|f|^{p}d\mu\Big{)}^{1/p}$ $\displaystyle\leq
C\sum_{k=1}^{\infty}C(k+1)2^{-k\delta}\|b\|_{{\rm RBMO}}M_{p,5}f(x)$
$\displaystyle\leq C\|b\|_{{\rm RBMO}}M_{p,5}f(x).$
Taking the mean over $Q$ and $R$ for $y$ and $z$ respectively, we obtain
$M_{4}\leq C\|b\|_{{\rm RBMO}}M_{p,5}f(x).$
Concerning the last estimate for $M_{3}$, we have for $y\in Q$
$\displaystyle|T((b-b_{R})$ $\displaystyle f\chi_{6^{N}Q\backslash 6Q}(y)|$
(41) $\displaystyle\leq
C\sum_{k=1}^{N-1}\frac{1}{\lambda(6^{k}Q)}\int_{6^{k+1Q}\backslash
6^{k}Q}|b-b_{R}||f|d\mu$ $\displaystyle\leq
C\sum_{k=1}^{N-1}\frac{1}{\lambda(y,6^{k}Q)}\Big{[}\int\limits_{6^{k+1}Q\backslash
6^{k}Q}|b-b_{5^{k+1}Q}||f|d\mu+\int\limits_{6^{k+1Q}\backslash
6^{k}Q}|b_{R}-b_{6^{k+1}Q}||f|d\mu\Big{]}$ $\displaystyle\leq
C\sum_{k=1}^{N-1}\frac{\mu(5\times
6^{k+1}Q)}{\lambda(x_{Q},6^{k}Q)}\Big{[}\frac{1}{\mu(6^{k+2}Q)}\int\limits_{6^{k+1}Q\backslash
6^{k}Q}|b-b_{6^{k+1}Q}||f|d\mu$
$\displaystyle~{}~{}~{}~{}+\frac{1}{\mu(5\times
6^{k+1}Q)}\int\limits_{6^{k+1Q}\backslash
6^{k}Q}|b_{R}-b_{6^{k+1}Q}||f|d\mu\Big{]}.$
By Hölder inequality and a similar argument to the estimate of the term
$M_{4}$, we have
$\frac{1}{\mu(5\times 6^{k+2}Q)}\int\limits_{6^{k+1}Q\backslash
6^{k}Q}|b-b_{6^{k+1}Q}||f|d\mu\leq\|b\|_{{\rm RBMO}}M_{p,5}f(x)$
and
$\frac{1}{\mu(5\times 6^{k+1}Q)}\int\limits_{6^{k+1Q}\backslash
6^{k}Q}|b_{R}-b_{6^{k+1}Q}||f|d\mu\leq CK_{Q,R}\|b\|_{{\rm RBMO}}M_{p,5}f(x).$
These two estimates together with (41) give
$|T((b-b_{R})f\chi_{6^{N}Q\backslash 6Q}(y)|\leq CK^{2}_{Q,R}\|b\|_{{\rm
RBMO}}M_{p,5}f(x).$
This implies $M_{3}\leq CK^{2}_{Q,R}\|b\|_{{\rm RBMO}}M_{p,5}f(x).$ From the
estimates $M_{1},M_{2},M_{3},M_{4},M_{5}$, we obtain (40).
To obtain (38) from (39) and (40), we use a trick of [T1]. From (39), if $Q$
is a doubling ball and $x\in Q$, we have
$\displaystyle|m_{Q}([b,T]f)-h_{Q}|$
$\displaystyle\leq\frac{1}{\mu(Q)}\int_{Q}|[b,T]f-h_{Q}|d\mu$ (42)
$\displaystyle\leq C\|b\|_{{\rm RBMO}}(M_{p,5}f(x)+M_{p,6}Tf(x)).$
Also, for any ball $Q\ni x$ (non doubling, in general),
$K_{Q,\widetilde{Q}}\leq C$, and then by (39) and (40) we have
$\displaystyle\frac{1}{\mu(6Q)}$
$\displaystyle\int_{Q}|[b,T]f-m_{\widetilde{Q}}[b,T]f|d\mu$ (43)
$\displaystyle\leq\frac{1}{\mu(6Q)}\int_{Q}|[b,T]f-h_{Q}|d\mu+|h_{Q}-h_{\widetilde{Q}}|+|h_{\widetilde{Q}}-m_{\widetilde{Q}}[b,T]f|$
$\displaystyle\leq C\|b\|_{{\rm
RBMO}(\mu)}\Big{(}M_{p,5}f(x)+M_{p,6}Tf(x)+T_{*}f(x)\Big{)}.$
In addition, for all doubling balls $Q\subset R$ with $x\in Q$ such that
$K_{Q,R}\leq P_{0}$ where $P_{0}$ is a constant in Lemma 7.5, by (40) we have
$|h_{Q}-h_{R}|\leq C\|b\|_{{\rm
RBMO}(\mu)}\Big{(}M_{p,5}f(x)+T_{*}f(x)\Big{)}P_{0}^{2}.$
Due to Lemma 7.5 we get
$|h_{Q}-h_{R}|\leq C\|b\|_{{\rm
RBMO}(\mu)}\Big{(}M_{p,5}f(x)+T_{*}f(x)\Big{)}K_{Q,R},$
for all doubling balls $Q\subset R$ with $x\in Q$. At this stage, applying
(41), we obtain
$\displaystyle m_{Q}([b,T]f)$ $\displaystyle-m_{R}([b,T]f)$ $\displaystyle\leq
C\|b\|_{{\rm
RBMO}(\mu)}\Big{(}M_{p,5}f(x)+M_{p,6}Tf(x)+T_{*}f(x)\Big{)}K_{Q,R}.$
This completes our proof.
###### Remark 7.7
As mentioned earlier in this paper, the results of this article still hold
when $X$ is a quasi-metric space. Indeed, one can see that the main problem in
quasi-metric space setting is that the covering lemma, Lemma 2.1, may not be
true. However, instead of using this covering property, we can adapt the
covering lemma in [FGL, Lemma 3.1] to our situation. This problem is not
difficult and we leave it to the interested reader.
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* [CW2] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. Volume 83 (1977), 569-645.
* [D] G. David, Completely uncrectifiable 1-sets on the plane have vanishing analytic capacity, Prépublications Mathématiques d’Orsay, 1997 (61), 1-94.
* [FGL] G. Di Fazio, C. E. Gutiérrez, and E. Lanconelli, Covering theorems, inequalities on metric spaces and applications to PDE’s, Math. Ann. 341 (2008), no. 2, 255-291.
* [FS] G. Folland and E.M. Stein, Hardy spaces on Homogeneous Groups, Princeton Univ. Press, 1982.
* [J] J.-L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calder on. Lecture Notes in Math. 994, Springer, 1983
* [He] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.
* [HoM] S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no.1, 37-116.
* [Hy] T. Hytönen, A framework for non-homogenous analysis on metric spaces, and RBMO spaces of Tolsa, preprint.
* [HyM] T. Hytönen and H. Martikainen, Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces, preprint.
* [HyYY] T. Hytönen, Dachun Yang and Dongyong Yang, The Hardy Space $H^{1}$ on Non-homogeneous Metric Spaces, preprint.
* [NTV1] F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, Vol 15, 1997, p. 703 - 726.
* [NTV2] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, Vol 9, 1998, p. 463 - 487.
* [NTV3] F. Nazarov, S. Treil and A. Volberg, The $Tb$\- theorem on non-homogeneous spaces, Acta Math., Vol 190, 2003, No 2, p. 151 - 239.
* [MMNO] J. Mateu, P. Mattila, A. Nicolau, J. Orobitg, BMO for non doubling measures, Duke Math. J., 102 (2000), 533-565.
* [St] E.M. Stein, Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, (1993).
* [Tch] E. Tchoundja, Carleson measures for the generalized Bergman spaces via a T (1)-type theorem, Ark. Mat. 46 (2008), no. 2, 377-406.
* [T1] X. Tolsa, BMO, $H^{1}$, and Calderón-Zygmund operators for non doubling measures, Math. Ann. 319 (2001), 89-149.
* [T2] X. Tolsa, A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition, Publ. Mat. 45 (2001), 163-174
* [V] J. Verdera. On the $T(1)$ theorem for the Cauchy integral. Ark. Mat. 38 (2000), 183-199
* [VW] A. Volberg and B. D. Wick, Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball, preprint (2009), arXiv:0910.1142.
Department of Mathematics, Macquarie University, NSW 2109, Australia and
Department of Mathematics, University of Pedagogy, Ho chi Minh city, Vietnam
Email: the.bui@mq.ed.au
Department of Mathematics, Macquarie University, NSW 2109, Australia
Email: xuan.duong@mq.edu.au
|
arxiv-papers
| 2010-09-07T12:48:41 |
2024-09-04T02:49:12.724229
|
{
"license": "Public Domain",
"authors": "The Anh Bui and Xuan Thinh Duong",
"submitter": "The Anh Bui",
"url": "https://arxiv.org/abs/1009.1274"
}
|
1009.1380
|
# Compressive Phase Contrast Tomography
F. R. N. C. Maia National Energy Research Scientific Computing Center,
Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA A.
MacDowell Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA S. Marchesini smarchesini@lbl.gov H. A. Padmore D. Y.
Parkinson Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA J. Pien CUDA consultant, www.jackpien.com A. Schirotzek
Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA,
USA C. Yang Computational Research Division, Lawrence Berkeley National
Laboratory, Berkeley, CA, USA
###### Abstract
When x-rays penetrate soft matter, their phase changes more rapidly than their
amplitude. Interference effects visible with high brightness sources creates
higher contrast, edge enhanced images. When the object is piecewise smooth
(made of big blocks of a few components), such higher contrast datasets have a
sparse solution. We apply basis pursuit solvers to improve SNR, remove ring
artifacts, reduce the number of views and radiation dose from phase contrast
datasets collected at the Hard X-Ray Micro Tomography Beamline at the Advanced
Light Source. We report a GPU code for the most computationally intensive
task, the gridding and inverse gridding algorithm (non uniform sampled Fourier
transform).
Tomography, Compressive Sensing, non uniform FFT, Radon transform, GPU
††preprint: LBNL-3899E
## I INTRODUCTION
Phase contrast tomography allows the imaging of light materials that would
otherwise be transparent to x-rays, and obtaining edge enhancment at higher
SNR for the same dose nugent paganin . Further dose reduction is expected from
the application of compressive sensing reconstruction techniques. This is what
we set out to do in this paper.
The refractive index of X-rays passing through light materials is very close
to a real number; when written in the in the form: $n=1-\delta+i\beta$,
$\delta$ is one to 3 orders of magnitudes larger than $\beta$ (depending on
the energy of the X-rays); exploiting the refractive contrast seems obvious.
While it is not possible to observe phases, experimental methods exist to
observe the Laplacian of the phase changes induced by propagation through an
object. By rotating the object around an axis, one can collect a series of
such projections. Standard tomographic processing of the data yields the
laplacian of the object in the three dimensional spacepaganin .
The phase contrast mechanism is especially useful for defining the boundaries
between composite objects. If the sample is made of large blocks of
components, the boundaries will be thin and far from each other. In other
words, the solution will be sparse.
We tested this concept using experimental phase contrast data from 60 micron
glass spheres at the tomography beamline 8.3.2 at the Advanced Light Source,
Lawrence Berkeley National Lab. The setup is similar to standard tomography
procedures BL832 in that samples are rotated in a monochromatic X-ray beam
and the transmitted X-rays are imaged via a scintillator, magnifying lens and
a digital camera to give an effective voxel size in the reconstructed three-
dimensional image of 1.8 $\mu$m.
Background normalized images are shown in Fig. 1, along with their difference
showing the propagation based phase contrast enhancement, and the
corresponding sinogram. Significant artifacts arise from the residual
fluctuations in the illuminating beam due to vibrations and the high power
beam impinging on a crystal monochromator. Defective detector elements (e.g.
dead pixels in a CCD) with non-linear responses to incoming intensity will
appear in the reconstructions as sharp rings with a width of one pixel.
Similar artifacts also arise from dusty or damaged scintillator screens.
Miscalibrated detector pixels, e.g. due to beam instabilities not completely
taken into account by a normalization correction, give rise to wider and less
marked ringsmunch .
To improve the reconstruction we employed a 3rd order polynomial fit to smooth
the sinogram and reduce the ringing artifacts. Increasing the order of the
polynomial beyond 3 did not improve the image further (Fig. 2).
---
Figure 1: Propagation based phase contrast and tomography of two two glass
balls placed in the microtomography beamline 8.3.2. From left to right: (1)
projection radiograph at short distance from the sample, (2) same as (1) but
with the detector 1 m. downstream. (3) difference between (1) and (2), (4)
sinogram.
---
Figure 2: Significant artifact due to residual fluctuations in the
illuminating beam due to vibrations and the high power beam impinging on a
crystal monochromator. Comparison of image quality using different filters
(polynomial subtraction) (Fig. 2).
---
Figure 3: Comparison of the reconstruction using a filter and using SPGL1
basis pursuit solver with the same filter.
We formulated the final reconstruction problem as an L1-minimization problem,
i.e.
$\min||x||_{1}\,\,\,\,\textnormal{subject to
}||\textnormal{filter}(\textnormal{Radon}\\{x\\}-\textnormal{data})||_{2}<\epsilon$
where $x$ is the phase contrast image to be reconstructed and filter is a
polynomial interpolation operator designed to reduce the ringing artifact in
the reconstruction, and $\epsilon$ is a regularization parameter we choose in
advance. We solve the L1-minimization problem by using the SPGL1 software
developed by E. van den Berg and M. P. Friedlander [1]. The software requires
us to provide a function to perform $y=$ Radon $\\{x\\}$. Due to the large
volume of data produced by at beam line (BL.8.3.2) at the Advanced Light
Source, we would like perform phase contrast tomographic reconstruction in
real time.
---
Figure 4: Fast Radon and inverse Radon transforms implemented on a Tesla GPU
computing engine.
---
Figure 5: Fast Radon and inverse Radon transforms implemented on a Tesla GPU
computing engine.
## II fast radon transform
Fast Radon and inverse Radon transforms form the computational kernels of many
image reconstruction algorithms. The 2D Radon tranform of an image x can be
implemented in a number of ways. One of the most efficient ways to perform
Radon{$x$} is to first perform a 2D FFT of $x$ on a regular grid, and then
interpolate the transform onto a polar grid before 1D inverse Fourier
transforms are applied to interpolated points along the same radial lines.
The interpolation between the Cartesian and polar grid is the key step in this
procedure. It can be carried out using a “gridding” algorithm that maintains
the desired accuracy with low computational complexity. The gridding algorithm
essentially allows us to perform a non-uniform FFT.
The gridding operation requires the convolution between irregular samples and
a kernel calculated at regular sample position and vice versa (Fig. 5 and
jackson ).
A fast GPU implementation requires dividing the problem into each blocks with
similar computational loads. Our approach starts with one bin containing all
the points, and then recursively divides the most computationally intensive
bin into four equally sized bins. The computational intensity is estimated by
multiplying the number of samples in the bin by the number of grid points in
the bin. For the calculation in the GPU we assign one thread block per bin.
The calculation strategy depends on the number of samples in the bin. When the
number of samples does not fit in shared memory we assign all threads to each
grid point and access the samples in a coalesced way. When it does fit in
shared memory we assign one thread per grid point.
Further details of the code will be described in the future. Preliminary tests
indicate 50x speedup on a Tesla GPU C1060 compared to a fast CPU (e,g.: 3 msec
for the FFTs, 6 msec for gridding+inverse gridding for a 1024x1024 grid
sampled with 180 angles, 1024 pixels per angle).
### II.1 Conclusions
In summary we implemented a GPU-accelerated compressive phase contrast
tomography reconstruction using Basis Pursuit solvers for the high throughput
tomography beamline at the Advanced Light Source. The reconstruction procedure
was used to remove ring artifacts but it is expected to also enable lower dose
or smaller datasets for similar image quality as shown for absorption-only (no
phase contrast) tomographic datasets exploting total variation regularization.
### Acknowledgments
This work was supported by the Laboratory Directed Research and Development
Program of Lawrence Berkeley National Laboratory under U.S. Department of
Energy Contract No. DE-AC02-05CH11231. We acknowledge the use of the X-ray
synchrotron micro-tomography beam line (8.3.2) at the Advanced Light Source at
LBNL, supported by the Office of Science of the Department of Energy. Part of
the work was developed using Jacket GPU toolbox for matlab provided by
Accelereyes.
## References
* (1) Nugent, K. A. “Coherent Methods in the X-ray Sciences”, Advances in Physics 59, 1-99, 2010.
* (2) Paganin, D.M. “Coherent X-ray Optics”, Oxford University Press, Oxford, (2006).
* (3) Jackson, J. I. Meyer, C. H. Nishimura, D. G. “Selection of a Convolution Function for Fourier Inversion using Gridding.” IEEE Trans. Med. Imag. 10(3), 473- (1991).
* (4) van den Berg, E. and Friedlander, M. P. “Probing the Pareto frontier for basis pursuit solutions”, SIAM J. on Sci. Comp. 31, 890-912, (2008).
* (5) Kinney, J.H. Nichols, M.C. X-ray tomographic microscopy using synchrotron radiation. Annu Rev Mater Sci 22, 121 (1992).
* (6) Münch, B. Trtik, P. Marone, F. and Stampanoni, M. “Stripe and ring artifact removal with combined wavelet-Fourier filtering”, Optics Express 17(10), pp. 8567-8591 (2009).
|
arxiv-papers
| 2010-09-07T19:55:40 |
2024-09-04T02:49:12.735488
|
{
"license": "Public Domain",
"authors": "F. R. N. C. Maia, A. MacDowell, S. Marchesini, H. A. Padmore, D. Y.\n Parkinson, J. Pien, A. Schirotzek, and C. Yang",
"submitter": "Stefano Marchesini",
"url": "https://arxiv.org/abs/1009.1380"
}
|
1009.1506
|
# Unification of Dark Matter and Dark Energy
in a Modified Entropic Force Model
Zhe Chang1,2111E-mail: changz@ihep.ac.cn Ming-Hua Li1,2222E-mail:
limh@ihep.ac.cn Xin Li2,3333E-mail: lixin@itp.ac.cn 1Institute of High Energy
Physics, Chinese Academy of Sciences, 100049 Beijing, China
2Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
3Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China
###### Abstract
In Verlinde’s entropic force scenario of gravity, Newton’s laws and Einstein
equations can be obtained from the first pinciples and general assumptions.
However, the equipartition law of energy is invalid at very low temperatures.
We show clearly that the threshold of the equipartition law of energy is
related with horizon of the universe. Thus, a one-dimension Debye (ODD) model
in the direction of radius of the modified entropic force (MEF) maybe suitable
in description of the accelerated expanding universe. We present a Friedmann
cosmic dynamical model in the ODD-MEF framework. We examine carefully
constraints on the ODD-MEF model from the Union2 compilation of the Supernova
Cosmology Project (SCP) collaboration, the data from the observation of the
large-scale structure (LSS) and the cosmic microwave background (CMB), i.e.
SNe Ia+LSS+CMB. The combined numerical analysis gives the best-fit value of
the model parameters $\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with
$\chi_{min}^{2}=591.156$. The corresponding age of the universe agrees with
the result of D. Spergel et al.Spergel2003 at $95\%$ confidence level. The
numerical result also yields an accelerated expanding universe without
invoking any kind of dark energy. Taking $\zeta(\equiv 2\pi\omega_{D}/H_{0})$
as a running parameter associated with the structure scale $r$, we obtain a
possible unified scenario of the asymptotic flatness of the radial velocity
dispersion of spiral galaxies, the accelerated expanding universe and the
Pioneer 10/11 anomaly in the entropic force framework of Verlinde.
###### pacs:
95.36.+x,95.35.+d,98.80.-k
## I 1\. Introduction
The law of gravity, which was first discovered by Isaac Newton in the 18th
century and later reformulated by Albert Einstein in the early 20th century,
still remains unclear in the microscopic level nowadays. Enlightened by
Hawking et al.’s workHawking ; Bekenstein ; Hawking2 about the black hole
entropy in 1970’s, JacobsonJacobson got the gravitational field equations as
Einstein’s, starting from the first law of thermodynamics. Further research of
PadmanabhanPadmanabhan1 ; Padmanabhan2 also gives gravity a thermodynamical
interpretation. These results prompt people to take a statistical physics
point of view on gravity. Recently, VerlindeVerlinde reinterpreted gravity as
an entropic force caused by a change of amount of information associated with
the positions of bodies of matter. Newton’s second law could be obtained with
the introduction of Unruh temperatureUnruh . He also got the Newton’s law of
gravity and the Komar’s definition of massKomar in a static curved space in
relativistic case from the holographic principleSusskind and black hole
thermodynamics.
Lots of work have been done to reveal the implications of the entropic force
interpretation of gravity. To name a few: derivation of Friedmann equation in
the entropic force frameworkShu ; Cai1 ; Cai3 ; Cai4 , the corresponding
Newton gravity formulation in loop quantum gravitySmolin , the construction of
holographic actions from black hole entropyMakela ; Caravelli , the entropic
force scenario of holographic dark energyLiM , the generalization of the
Newton’s potential in the relativistic caseTian , and the modified entropic
force(MEF) due to the modification of the equipartition law of energy at very
low temperatureGao , etc.
In Verlinde’s paperVerlinde , the equipartition law of energy of free
particles plays a key role in the derivation of Newton’s laws. GaoGao pointed
out that the equipartition law of energy does not hold at very low
temperatures. He made a modification simply using the three-dimension Debye
model in solid state physics. His work provides an explanation for the
accelerated expanding universe without invoking dark energy. However, we note
that only the vibration in the direction of radius is related with an
observable quantity and the “low temperature” should be fixed by a threshold.
So that, a one-dimension Debye(ODD) function is suitable to be used to revise
the free particle’s equipartition law of energy in the study of cosmologyXin
Li . In fact, such a modification leads us to the famous modification of
Newtonian dynamics(MOND). MOND was constructed to account for the asymptotic
flatness of the rotational velocity curves of spiral galaxiesTrimble ; Rubin ;
TF .
In this paper, we show clearly that the threshold of equipartition law of
energy is related with horizon of the universe. A Friedmann cosmic dynamical
model is set up in the framework of the ODD-MEF. We examine carefully
constraints on the ODD-MEF model from the Union2 compilation of the Supernova
Cosmology Project(SCP) collaboration, the data from the observation of the
large-scale structure(LSS) and the cosmic microwave background(CMB), i.e. SNe
Ia+LSS+CMB. Results yield an accelerated expanding universe without invoking
any kind of dark energy. Furthermore, by taking $\zeta(\equiv
2\pi\omega_{D}/H_{0})$ as a running parameter associated with the structure
scale $r$, we obtain a possible unified scenario of the asymptotic flatness of
the radial velocity dispersion of spiral galaxies, the accelerated expanding
universe and the Pioneer 10/11 anomaly in the entropic force framework of
Verlinde.
The paper is organized as follows. In section 2, a brief review on Verlinde’s
work and the connection between the ODD-MEF model and MOND is presented. We
derive the corresponding Friedmann equation and consider the cosmological
constraints on it in setion 3. In section 4, an alternative approach to the
Pioneer 10/11 anomaly is presented. In section 5, by taking $\zeta$ as a
running parameter associated with the structure scale $r$, we suggest a
unified scenario of the asymptotic flatness of the rotation curves of spiral
galaxies, the accelerated expanding universe and the Pioneer 10/11 anomaly in
the framework of the ODD-MEF. Conclusions and further discussions can be found
in section 6.
## II 2\. From Modified Entropic Force To Mond
Think of a closed holographic screen and a free particle of mass $m$ near it
on the side that spacetime has already emerged. The particle moves towards the
screen, traveling a distance $\triangle x$ before merging into it. According
to BekensteinBekenstein , the change of entropy of the screen associated with
the amount of information stored on it is
$\triangle S=2\pi k_{B}\frac{mc}{\hbar}\triangle x\ ,$ (1)
where $k_{B}$ is the Boltzmann constant and $c$ denotes the speed of light.
The particle will experience an effective macroscopic force due to the
statistical tendency to increase its entropy. This is described by
$F\triangle x=T\triangle S\ ,$ (2)
where $T$ is the temperature of the screen. By introducing Unruh’s relation
between acceleration and temperatureUnruh
$k_{B}T=\frac{1}{2\pi}\frac{\hbar a}{c}\ ,$ (3)
one recovers the second law of Newton
$F=ma\ .$ (4)
Suppose the holographic screen has a radius $R$, the area of which is $A=4\pi
R^{2}$. In theory of emergent space, each fundamental bit occupies one unit
cell, the area of which on the screen is defined as
$L_{p}\equiv\sqrt{G\hbar}$. $L_{p}$ is the Planck length. $G$ is mere a new
constant here, which is later identified with Newton’s constant. Let’s denote
the number of bits by $N$, which is given as
$N\equiv\frac{Ac^{3}}{L_{p}^{2}}=\frac{Ac^{3}}{G\hbar}=\frac{4\pi
R^{2}c^{3}}{G\hbar}\ .$ (5)
Each bit represents a microscopic degree of freedom. The total energy of the
screen is given by the equipartition law
$E=\frac{1}{2}Nk_{B}T\ .$ (6)
Assuming that the energy of the screen is proportional to the mass $M$ that
would emerge in the part of space enclosed by the screen itself as
$E=Mc^{2}\ $ (7)
and using the equation (4), (5), (6) and (7), one obtains the familiar
lawVerlinde :
$F=G\frac{Mm}{R^{2}}\ .$ (8)
Thermodynamics and statistical physics tells us that the equipartition law of
energy is valid only when the temperature is not very low. On the other hand,
the Debye model in solid state physics is found to be successful in describing
experimental results for most of the solid objects at very low
temperaturesDebye . It should be noticed that the threshold of low energy for
each direction of vibration is different, because it is related with the
structure scale in the radial direction. In the case of astronomy and
cosmology, the only temperature threshold is related with horizon of the
universe. Thus, we focus on the ODD-MEF model throughout this paper. The
equipartition law of energy can be rewritten as
$E=\frac{1}{2}Nk_{B}T\mathfrak{D}(x)\ ,$ (9)
where the one-dimension Debye function is defined as
$\mathfrak{D}(x)\equiv\frac{1}{x}\int^{x}_{0}\frac{y}{e^{y}-1}dy\ .$ (10)
$x$ is related to the Debye frequency $\omega_{D}$ and defined as
$x\equiv\frac{\hbar\omega_{D}}{k_{B}T}=\frac{2\pi c\omega_{D}}{a}\ .$ (11)
Combining the equations (4), (5), (7) and (9), we obtain the modified Newton’s
law of gravitation
$\frac{GM}{R^{2}}=a\mathfrak{D}\left(\frac{2\pi c\omega_{D}}{a}\right)\ .$
(12)
There are two limit cases for the equation (12). One is the high temperature
limit, with $x\ll 1$. We have
$\mathfrak{D}(x)\approx\frac{1}{x}\int^{x}_{0}dy=1.$ (13)
Therefore, the modified equipartition law of energy returns to the equation
(6), with which Newton’s law of gravity the equation (8) is recovered.
The other is the low temperature limit, with $x\gg 1$. The Debye function
$\mathfrak{D}(x)$ reduces to
$\mathfrak{D}(x)\approx\frac{1}{x}\int^{\infty}_{0}\frac{y}{e^{y}-1}dy=\frac{\pi^{2}}{6x}.$
(14)
Then the equation (12) reads
$\frac{GM}{R^{2}}=\frac{\pi}{12c\omega_{D}}a^{2}=\frac{a^{2}}{a_{0}}\ ,$ (15)
where the constant $a_{0}$ is defined as
$a_{0}\equiv\frac{12c\omega_{D}}{\pi}.$ (16)
Thus, $x$ can be rewritten as
$x=\frac{2\pi c\omega_{D}}{a}\ =\frac{\pi^{2}a_{0}}{6a}\ $ (17)
and the equation (10) turns out to be
$\displaystyle\mathfrak{D}(x)=\frac{6}{\pi^{2}}\frac{a}{a_{0}}\int_{0}^{\frac{\pi^{2}a_{0}}{6a}}\frac{y}{e^{y}-1}dy\
.$ (18)
Defining the function
$\mu(t)=\frac{6}{\pi^{2}}t\int_{0}^{\frac{\pi^{2}}{6t}}\frac{y}{e^{y}-1}dy\ ,$
(19)
where $t=a/a_{0}$, one can immediately obtain the gravity law of MONDXin Li :
$\frac{GM}{R^{2}}=a\mathfrak{D}\left(\frac{2\pi
c\omega_{D}}{a}\right)=a\mu(x)\ .$ (20)
Note that the definition of $\mu(x)$ in the equation (19) has the
propertyMilgrom
$\lim_{x\gg 1}\mu(x)=1~{}~{}~{}{\rm and}~{}~{}~{}\lim_{x\ll 1}\mu(x)=x\ $ (21)
as demanded by MOND.
The equation (20) implies the modified Poisson equation for the gravitational
potential $\phi$
$\nabla\cdot(\mu(|\nabla\phi|/a_{0})\nabla\phi)=4\pi G\rho\ ,$ (22)
where $\rho$ is the energy density of matter sources.
In MOND, in order to explain the observed rotational velocity curves of spiral
galaxies, $a_{0}$ is suggested to be of the order of
$a_{0}\sim~{}10^{-8}cm/s^{2}\ .$ (23)
MilgromMilgrom ; Milgrom1 found that
$2\pi a_{0}\approx cH_{0}\ .$ (24)
Combining the equation (16) and (24), we get that
$H_{0}\approx 24\omega_{D}\ .$ (25)
Thus, the corresponding wave length of $\omega_{D}$ is
$\lambda<c/\omega_{D}=c/24H_{0}\ .$ (26)
It really should be less than the cosmological horizon $L_{h}\sim c/H_{0}$.
Thus, Milgrom’s relation between the threshold of the acceleration $a_{0}$ and
the Hubble constant $H_{0}$ has a deep origin.
## III 3\. The Cosmological Constraints
### III.1 3.1 The Modified Friedmann Model
For convenience, we adopt the natural unit $c=k_{B}=1$ in this section.
The $(3+1)$-demension Friedmann-Robertson-Walker(FRW) metric can be written as
$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega_{2}^{2}\ ,$ (27)
where $\tilde{r}=a(t)r$ and the two-dimension metric $h_{ab}={\rm
diag}(-1,a^{2}/(1-kr^{2}))$, with $k=0,-1,1$ refers to a flat, open and closed
universe respectively. Given
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$, one gets the radius of
the apparent horizon(AH)Hawking3 :
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}\ .$ (28)
$H\equiv\dot{a}/a$ is the Hubble parameter and the dot “ $\cdot$ ” denotes the
derivative with respect to time $t$. The area of the AH is
$A=4\pi{\tilde{r}_{A}}^{2}$. The number of bits of information stored on the
AH is
$N=4\pi\tilde{r}_{A}^{2}/L_{p}^{2}\ .$ (29)
Suppose from time $t$ to $t+dt$, the radius of the AH changes from
$\tilde{r}_{A}$ to $\tilde{r}_{A}+d\tilde{r}_{A}$. The full derivative of the
identity (9) gives the change of the total energy on the screen as
$dE=\frac{1}{2}N\mathfrak{D}(x)dT+\frac{1}{2}T\mathfrak{D}(x)dN+\frac{1}{2}NTd\mathfrak{D}(x)\
.$ (30)
Let’s first do formal reductions of the above identity. The temperature of the
AH, the so-called Hawking temperatureHawking2 ; Cai2 , is
$T_{A}=\hbar/(2\pi\tilde{r}_{A})\ .$ (31)
The change of $T_{A}$ is of the form
$dT_{A}=-\frac{\hbar}{2\pi\tilde{r}_{A}^{2}}d\tilde{r}_{A}\ .$ (32)
From the equation (29), we get
$dN=\frac{8\pi\tilde{r}_{A}}{L_{p}^{2}}d\tilde{r}_{A}\ .$ (33)
The full derivative of the one-dimension Debye function takes the form
$d\mathfrak{D}(x)=\left[-\frac{1}{x}\mathfrak{D}(x)+\frac{1}{e^{x}-1}\right]dx\
.$ (34)
The identity (11) gives
$dx=-\frac{x}{T}dT\ .$ (35)
Thus, the equation (34) can be rewritten as
$d\mathfrak{D}(x)=\left[\mathfrak{D}(x)-\frac{x}{e^{x}-1}\right]\frac{dT_{A}}{T_{A}}\
.$ (36)
Making use of the equations (29), (31), (32), (33) and (36 ), one can rewrite
the equation (30) as
$\displaystyle dE=$
$\displaystyle\left[\frac{1}{2}\cdot\frac{4\pi\tilde{r}_{A}^{2}}{L_{p}^{2}}\cdot\left(-\frac{\hbar}{2\pi\tilde{r}_{A}^{2}}d\tilde{r}_{A}\right)+\frac{1}{2}\cdot\left(\frac{\hbar}{2\pi\tilde{r}_{A}}\right)\cdot\left(\frac{8\pi\tilde{r}_{A}}{L_{p}^{2}}d\tilde{r}_{A})\right)\right]\cdot\mathfrak{D}(x)$
$\displaystyle+\frac{1}{2}NT_{A}\cdot\left[\mathfrak{D}(x)-\frac{x}{e^{x}-1}\right]\frac{dT_{A}}{T_{A}}$
$\displaystyle=$
$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{d\tilde{r}_{A}}{G}\ .$ (37)
From the definition of the AH (28) and remembering $H\equiv\dot{a}/a$, we have
$d\tilde{r}_{A}=-H\tilde{r}_{A}^{3}\left(\dot{H}-\frac{k}{a^{2}}\right)dt\ .$
(38)
Thus, we get
$dE=-H\tilde{r}_{A}^{3}\left(\dot{H}-\frac{k}{a^{2}}\right)\left(\frac{x}{e^{x}-1}\right)\frac{dt}{G}\
.$ (39)
Next, we consider the energy-momentum tensor of a perfect isotropic fluid
$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}$, where $\rho$ and $p$ are
respectively the energy density and the pressure. This is just the case of the
matter in the universe. The energy flow of the matter isShu
$dE=4\pi\tilde{r}_{A}^{2}T_{\mu\nu}k^{\mu}k^{\nu}dt=4\pi\tilde{r}_{A}^{3}(\rho+p)Hdt,$
(40)
where the Killing vector of the horizon $k^{\mu}=(1,-Hr,0,0)$.
Identifying the equation (39) and (40), one obtains
$\left(\frac{x}{e^{x}-1}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-4\pi
G(\rho+p).$ (41)
Combining the equation (41) with the energy conservation equation
$\dot{\rho}+3H(\rho+p)=0\ ,$ (42)
we get
$\left(\frac{x}{e^{x}-1}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-\frac{4\pi
G}{3H}\dot{\rho}\ .$ (43)
Notice that we consider a spatially flat, matter dominated universe throughout
this paper, namely $k=0$ and
$\rho=\rho_{m}=\rho_{m0}\,(1+z)^{3}=\rho_{m0}\,a^{-3}$. The equation (III.1)
multiplying by $2Hdt/H_{0}^{2}$ makes
$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{2H\dot{H}}{H_{0}^{2}}dt$
$\displaystyle=$ $\displaystyle-\frac{8\pi G}{3H_{0}^{2}}\dot{\rho}dt\ ,$
$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{d(H^{2})}{H_{0}^{2}}$
$\displaystyle=$ $\displaystyle-\frac{8\pi G}{3H_{0}^{2}}d\rho\ .$ (44)
With the notations
$\Omega_{m0}\equiv\frac{8\pi
G\rho_{m0}}{3H_{0}^{2}}\,,~{}~{}~{}~{}~{}~{}~{}E\equiv\frac{H}{H_{0}}=\frac{\zeta}{x}\,,~{}~{}~{}~{}~{}~{}~{}\zeta\equiv\frac{H_{D}}{H_{0}}\equiv\frac{2\pi\omega_{D}}{H_{0}}\,,$
(45)
one obtains
$\left(\frac{x}{e^{x}-1}\right)dE^{2}=\Omega_{m0}~{}da^{-3}\ .$ (46)
$E$ is also called the reduced Hubble parameter.
Instead of numerically solving the differential equation (46) to get the
modified Friedmann equation of the MEF model, we need to find an approximated
expression of it. Note that $E(z)$ grows rapidly as the redshift $z$
increases. Suppose that $x=\zeta/E(z)$ is a small quantity today, i.e. $x\ll
1$, so that we can expand $x/(e^{x}-1)$ in powers of $x$ near $x=0$:
$\frac{x}{e^{x}-1}=1-\frac{x}{2}+{\cal
O}(x^{2})~{},~{}~{}~{}~{}~{}~{}~{}~{}x\ll 1\ .$ (47)
Then we get the approximated expression of the equation (46)
$\displaystyle\Omega_{m0}~{}da^{-3}$ $\displaystyle\thickapprox$
$\displaystyle\left(2E-\zeta\right)dE\ .$ (48)
Integrating the equation (48), we have
$E^{2}-\frac{3}{2}\,\zeta E=\Omega_{m0}\,a^{-3}+{\rm const.}\ ,$ (49)
where “ const. ” is an integral constant and should be determined by
$E(a=1)=1$. Finally, one obtains a quadratic equation for $E$,
$E^{2}-\zeta E=\Omega_{m0}a^{-3}+(1-\zeta-\Omega_{m0})\ .$ (50)
Note that $E$ only allows of a positive value by its definition (45). Solving
the equation (50), one gets the Friedmann equation in the ODD-MEF model,
$E(a)=\frac{1}{2}\zeta+\frac{1}{2}\left[(\zeta-2)^{2}+4\Omega_{m0}(a^{-3}-1)\right]^{1/2}\
,$ (51)
or equivalently,
$E(z)=\frac{1}{2}\zeta+\frac{1}{2}\left[(\zeta-2)^{2}+4\Omega_{m0}((1+z)^{3}-1)\right]^{1/2}\
.$ (52)
As long as $\zeta\rightarrow 0$, the ODD-MEF model reduces to the
$\Lambda$-CDM model, in which
$E(a)=\left[\Omega_{m0}a^{-3}+\left(1-\Omega_{m0}\right)\right]^{1/2}\ .$ (53)
In astronomy, distance modulus is defined asCarroll
$\mu_{th}(z)\equiv 5\log_{10}[d_{L}(z)({\rm Mpc})]+25\ ,$ (54)
where $d_{L}$ is the luminosity distance:
$\displaystyle d_{L}(z)$ $\displaystyle=$
$\displaystyle\frac{(1+z)}{H_{0}}\int_{(1+z)^{-1}}^{1}\frac{da}{a^{2}E(a)}$
(55) $\displaystyle=$
$\displaystyle\frac{(1+z)}{H_{0}}\int_{(1+z)^{-1}}^{1}\frac{2da}{a^{2}\cdot\left\\{\zeta+\left[(\zeta-2)^{2}+4\Omega_{m0}(a^{-3}-1)\right]^{1/2}\right\\}}\
.$
Finally, we give the specific form of the deceleration parameter $q$ as a
function of $E(z)$. Remembering $H\equiv\dot{a}/a$, $a=1/(1+z)$ and
$\displaystyle\frac{d}{dt}=\frac{da}{dt}\cdot\frac{dz}{da}\cdot\frac{d}{dz}=-\frac{H}{a}\frac{d}{dz}\
,$ (56)
one can easily get the following identities
$q\equiv-\frac{\ddot{a}}{aH^{2}}=-1+(1+z)\,E^{-1}\frac{dE}{dz}\ .$ (57)
### III.2 3.2 The Numerical Study
In this subsection, we consider the cosmological constraints on the ODD-MEF
model from the observational data. First, we limit ourselves to the 557 SNe Ia
from the Union2 compilation of the Supernova Cosmology Project (SCP)
collaboration . Next, we include the data from the observation of the large-
scale structure (LSS) and the cosmic microwave background (CMB) as useful
complements for the SNe Ia data to put a joint constraint on the parameters of
the model.
The Union2 compilation consists of 557 SNe Ia, which is the largest published
and spectroscopically confirmed sample of SNe Ia so farSCP . The $\chi^{2}$
statistic of the sample is given by
$\chi_{SN}^{2}(\zeta,\Omega_{m0})=\sum\limits_{i=1}^{557}\frac{\left[\mu_{obs}(z_{i})-\mu_{th}(z_{i};\zeta,\Omega_{m0})\right]^{2}}{\sigma^{2}(z_{i})}\
,$ (58)
where $\mu_{obs}(z_{i})$ and $\sigma(z_{i})$ are respectively the observed
value and the $1\sigma$ uncertainty of the distance modulus of 557 Union2 SNe
Ia. Given the redshift $z_{i}$ and the parameter values of $\zeta$ and
$\Omega_{m0}$, one can get the theoretical, model-related value of the
distance modulus $\mu_{th}$ according to the equation (54). By minimizing the
$\chi_{SN}^{2}$, one obtains the best-fit values of the parameters:
$\zeta\simeq 10^{-7}$ and $\Omega_{m0}=0.192$, with a minimum value of
$\chi^{2}$ as $\chi_{min}^{2}=574.583$.
In addition, we consider the cosmic age in the ODD-MEF model. The age of the
universe is given by
$t_{0}=H_{0}^{-1}\int_{0}^{\infty}\frac{dz}{(1+z)E(z)}\ ,$ (59)
where $H_{0}^{-1}$ represents the Hubble time, with the value
$H_{0}^{-1}=9.778h^{-1}$Gyr. We take $h$ to be $0.72$. According to D. Spergel
et al.Spergel2003 , the age of the universe is $t_{0}=13.7\pm 0.2$Gyr. This
can also be written in the dimensionless age parameter $H_{0}t_{0}\simeq
0.99$, with $1\sigma$ confidence level range $0.96\lesssim H_{0}t_{0}\lesssim
1.05$Zhang . We show these in FIG.1, together with the $70\%$, $95\%$ and
$99\%$ confidence level contours in the $\Omega_{m0}$-$\zeta$ parameter plane.
As we see, the SNe Ia data alone are not enough to put a very strong
constraint on the parameters. The parameter $\zeta$ can still take a value
close to $1$. This fact invalidates the approximation (47). To recognize this,
one should notice that $E(z)\equiv H/H_{0}=\zeta/x=1$ today. A large $x$
implies a large $\zeta$. Moreover, one can see that the best-fit result of the
SNe Ia data is almost ruled out by the age of the universe at $1\sigma$
confidence level ($0.96\lesssim H_{0}t_{0}\lesssim 1.05$). So in what follows,
we use the data from the observation of the large-scale structure (LSS) and
the cosmic microwave background (CMB) to enforce a more rigorous constraint on
the model parameters.
Here, we only take into account the most conservative and robust informations
from the LSS and the CMB observationsZhang . For the LSS, we use the distance
parameter ${\cal A}$. It can be obtained from a spectroscopic sample of 46,748
luminous red galaxies of the Sloan Digital Sky Survey (SDSS). It is defined
asSDSS ; Eisenstein
${\cal
A}\equiv\Omega_{m0}^{1/2}\,E(z_{b})^{-1/3}\left[\frac{1}{z_{b}}\int_{0}^{z_{b}}\frac{dz}{E(z)}\right]^{2/3}\
,$ (60)
where $z_{b}=0.35$. $E(z)$ is given by equation (52). The value of ${\cal A}$
has been determined to be ${\cal A}_{0}=0.469\pm\sigma_{\cal A}$, where the
$1\sigma$ uncertainty $\sigma_{\cal A}=0.017$Eisenstein .
For the CMB, we use the measurement of shift parameter $\cal R$ alone. It is
defined asWang ; Tegmark97
${\cal R}\equiv\Omega_{m0}^{1/2}\int_{0}^{z_{\ast}}{dz\over E(z)}~{},$ (61)
where the redshift of recombination is found to be ${z_{\ast}}=1091.3$
according to the WMAP 7-year (WMAP7) dataWMAP7 . The value of $\cal R$ has
been updated to ${\cal R}_{0}=1.725\pm\sigma_{\cal R}$, where the $1\sigma$
uncertainty $\sigma_{\cal R}=0.018$WMAP7 .
To include both the LSS and the CMB observation as well as the 557 Union2 SNe
Ia data to perform a combined numerical analysis of the parameters, we use a
$\chi^{2}$ statistic as
$\chi^{2}=\chi_{SN}^{2}+\chi_{LSS}^{2}+\chi_{CMB}^{2}\ ,$ (62)
where $\chi_{SN}^{2}$ is given by the identity (58). $\chi_{CMB}^{2}$ and
$\chi_{LSS}^{2}$ are the contributions from the CMB and the LSS data, which
are defined as
$\chi^{2}_{LSS}=\frac{({\cal A}-{\cal A}_{0})^{2}}{\sigma_{\cal
A}^{2}}~{}~{}~{}and~{}~{}~{}\chi^{2}_{CMB}=\frac{({\cal R}-{\cal
R}_{0})^{2}}{\sigma_{\cal R}^{2}}\ ,$ (63)
where ${\cal A}$ and $\cal R$ are given by the identity (60) and (61)
respectively. By minimizing such a $\chi^{2}$, one gets the best-fit parameter
values $\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with
$\chi_{min}^{2}=591.156$. These results are shown in FIG.2. For comparison, we
also present the result for the SNe Ia+LSS fit in FIG.3, of which we use a
$\chi^{2}$ statistic
$\chi^{2}=\chi_{SN}^{2}+\chi_{LSS}^{2}\ .$ (64)
The best-fit parameter values are $\zeta\simeq 10^{-8}$ and
$\Omega_{m0}=0.213$, with $\chi_{min}^{2}=584.831$.
One can see that after using the LSS and the CMB data, we successfully put a
much more rigorous constraint on $\zeta$. It is within the range
$0\leq\zeta\leq 0.09$ at $95\%$ confidence level. This result confirms that
our approximation (47) is reasonable. And it is more than one order smaller
than Wei’s predictionWei . In addition, the best-fit result of the combined
analysis is in accordance with the age of the universe given by D. Spergel et
al.Spergel2003 at $1\sigma$ confidence level.
The corresponding distance modulus $\mu_{th}(z)$ is shown in FIG.4. And the
corresponding reduced Hubble parameter $E(z)$ and the deceleration parameter
$q(z)$ are plotted in FIG.5 and FIG.6. Notice that $q=0$ at the redshift
$z_{t}=0.906$ for the SNe Ia+LSS+CMB fit. $z_{t}$ is called the transition
redshift. This fact implies that the universe goes from a decelerated
expanding period to an accelerated expanding period at a time point in the
early epoch, in the absence of any dark matter or dark energy.
## IV 4\. The Pioneer 10/11 Anomaly
Last, but not the least, let us consider the Pioneer 10/11 anomaly in the ODD-
MEF model. The Pioneer 10 and 11 spacecraft, which were launched in 1972 and
1973, are the two most accurately navigated vehicles in the solar system. The
radio-metric data received by them at the heliocentric distances ranging from
$20$-$70$ AU has consistently indicated the existence of a small, anomalous
frequency drift changing with a rate of $6\times 10^{-9}$ Hz/sAnderson , which
was later interpreted as a constant, sunward accelerationTuryshev1
$a_{P}=(8.74\pm 1.33)\times 10^{-10}~{}m/s^{2}\ .$ (65)
This apparent violation of the Newton’s law of gravity has been known as the
Pioneer anomaly, the nature of which still remains unclear to date.
Several mechanisms have been developed to explain for the anomaly. To name a
fewTuryshev2 , it could be attributed to the possible systematic errors due to
the gas leaks from the propulsion system, the gravitational effect of the
Kuiper Belt Objects or dust, or the influence of the expansion of the
universe, etc. In our previous workXin Li2 , we have also provided a possible
explanation of the Pioneer 10/11 anomaly in the framework of Finsler geometry.
Here, we present another possible interpretation of the Pioneer anomaly in the
ODD-MEF model. Starting from the identity (12) and using the approximation
(47), we have
$\displaystyle\frac{GM}{R^{2}}=a\mathfrak{D}(x)$ $\displaystyle=$
$\displaystyle a\cdot\frac{1}{x}\int^{x}_{0}\frac{y}{e^{y}-1}dy$ (66)
$\displaystyle=$ $\displaystyle a-\frac{\pi cw_{D}}{2}\ .$
We may rewrite it as
$a=\frac{GM}{R^{2}}+\frac{\pi cw_{D}}{2}\ .$ (67)
In the above identity, $GM/R^{2}$ is the conventional Newtonian prediction of
the gravitational acceleration at a distance $R$ from the source with mass
$M$. We take the left hand side of the equation (67), i.e. $a$, to be the
acceleration that the spacecraft actually experiences. Then the second term at
the right hand side should be interpreted as the abnormal acceleration
$a_{P}$, namely
$a=\frac{GM}{R^{2}}+a_{P}\ ,$ (68)
where
$a_{P}\equiv\frac{\pi cw_{D}}{2}\ .$ (69)
Suppose that the parameter takes a value as $\zeta\approx 5$ over the
heliocentric distance ranging from $20$ to $70$ AU. This is the structure
scale where the abnormal acceleration $a_{P}$ has been detected. Using the
definition (45) of $\zeta$, we get the magnitude of $\omega_{D}$
$w_{D}=\frac{H_{0}}{2\pi}\zeta\approx 2.0\times 10^{-18}~{}{\rm Hz}\ .$ (70)
With the identity (69), one gets the abnormal acceleration predicted by the
ODD-MEF model
$a_{P}\equiv\frac{\pi cw_{D}}{2}~{}\approx~{}8.74\times 10^{-10}~{}m/s^{2}\ .$
(71)
It agrees with the detected result (65). The plus sign in the equation (68)
illustrates that the direction of $a_{P}$ is in the same direction as the
conventional Newtonian prediction $GM/R^{2}$, which is pointing towards the
sun. This is in accordance with the detection.
## V 5\. A Unified Scenario of Anomalies in Gravity
In section 4, we found that $\zeta\approx 5$ provides a possible explanation
for the Pioneer 10/11 anomaly in the solar system. On the other hand, to
account for the observations of the SNe Ia, the LSS and the CMB, $\zeta$ is
found to be $\zeta\sim 10^{-9}$. This was demonstrated in section 3.
In the following, we will find out the magnitude of $\zeta$ corresponding to
MOND. Making use of
$w_{D}=\frac{H_{0}}{2\pi}\zeta$ (72)
and the identity (16), we get
$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{6cH_{0}}{\pi^{2}}\zeta\ .$ (73)
The above identity multiplying $2\pi$ makes
$2\pi a_{0}=\frac{12\zeta}{\pi}cH_{0}\ .$ (74)
Note that in the equation (24), $2\pi a_{0}\approx cH_{0}$. So we have
$\frac{12\zeta}{\pi}\approx 1\ ,$ (75)
which implies
$\zeta\approx 0.25\ .$ (76)
In summary, we obtain three different magnitudes of $\zeta$ with respect to
three different structure scales:
$\zeta\sim\begin{cases}10^{-9}\ ,&10^{8}\sim
10^{10}~{}$pc~{}~{}~{}~{}~{}$(\text{SNe Ia+LSS+CMB})\\\ 0.25\ ,&10^{3}\sim
10^{4}~{}$pc~{}~{}~{}~{}~{}~{}$(\text{MOND})\\\ 5\ ,&10^{-4}\sim
10^{-3}~{}$pc~{}~{}~{}$(\text{Pioneer 10/11 anomaly})\end{cases}\ ~{}~{}~{},$
(77)
where $1~{}$pc$=2.06\times 10^{5}$ AU$=3.09\times 10^{16}$ m$=3.26$ l.y.. One
can see that $\zeta$ is found to be of a declining value while the relevant
structure scale increases.
## VI 6\. Conclusions and Remarks
VerlindeVerlinde interpreted gravity as an entropic force due to the tendency
of the system to increase its entropy, which is associated with the change of
positions of material bodies. With the holographic principle, the total
entropy of the system can be calculated via the area $A$ and the temperature
$T$ of its boundary. With Unruh’s relation of the acceleration and the
temperature experienced by the accelerated observer, one can easily recover
the laws of Newton. However, at very low temperatures, the equipartition law
of energy is no longer valid. We noted that the threshold of low temperature
is related to the specific dimension of the direction. In the case of
astronomy and cosmology, the only threshold comes from the limited radius of
the solar system , the galaxies and the universe. We made a modification by
revising the equipartition law of energy with the one-dimension Debye function
(10). It was referred as the ODD-MEF model.
With the definition (16), we deduced the famous formula of MOND. To consider
the cosmological constraints on the model, we derived the Friedmann equation
(52). We made use of the Union2 compilation of the Supernova Cosmology
Project(SCP) collaboration, the data from the observation of the large-scale
structure(LSS) and the cosmic microwave background(CMB), i.e. SNe Ia+LSS+CMB,
to put a joint constraint on the parameters $\zeta$ and $\Omega_{m0}$. The
combined numerical analysis gives the best-fit values of the model parameters
$\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with $\chi_{min}^{2}=591.156$.
In FIG.2, one can see that the best-fit result agrees with the age prediction
of the universe by D. Spergel et al.Spergel2003 and the WMAP collaboration at
$1\sigma$ confidence level. And the parameter $\zeta$ is successfully limited
to a narrow range $0\leq\zeta\leq 0.09$ at $95\%$ confidence level. It is more
than one order smaller than Wei’s prediction $0\leq\zeta\leq 0.2$Wei .
In the ODD-MEF model, we introduced a dimensionless, structure scale-related
parameter $\zeta$. Phenomenological study indicated that different values of
$\zeta$ corresponds to dynamics at different distance scales, as in the
identity (77). Therefore, we are allowed to suggest a unified scenario of the
asymptotic flatness of the rotation curves of spiral galaxies, an accelerated
expanding universe, and the Pioneer 10/11 anomaly in the solar system in a
single framework. Thus, we got a possible unification of dark matter and dark
energy in the one-dimensional Debye model of the modified entropic force.
Since $\zeta$ is a running parameter associated with the structure scale $r$,
a specific form of the function $\zeta(r)$ should be given. As one foresees,
this function should have the monotonically decreasing property as shown in
(77). Such a running $\zeta$ also implies that, other than the ODD-MEF model,
there exists a more fundamental theory, which taking the model as its
effective scenario. To find the specific $\zeta(r)$ and such a fundamental
theory is one of the main subjects of our future investigation.
###### Acknowledgements.
## VII Acknowledgments
This work was supported by the National Natural Science Fund of China under
Grant No. 10575106 and No. 10875129.
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Figure 1: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration.
The dark gray dashed line denotes constraints from the age of the universe at
the $1\sigma$ confidence level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$, with a
central value $H_{0}t_{0}\simeq 0.99$. It can be seen that the best-fit result
of the SNe Ia data is almost excluded by the age of the universe at the
$1\sigma$ confidence level. The isochrones are labeled for the case of
$H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick solid lines
indicate the age of the universe $t_{0}$ in the unit of Hubble time
$H_{0}^{-1}$. The dot with the coordinate $(0.192,10^{-7})$ represents the
best-fit value, with $\chi_{min}^{2}=574.583$. The best-fit result is almost
excluded by E. Komatsu et al.WMAP7 in $1\sigma$ confidence level.
Figure 2: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration
plus the data from the observation of the large-scale structure(LSS) and the
cosmic microwave background(CMB), i.e. SNe Ia+LSS+CMB. The dark gray dashed
lines denote constraints from the age of the universe at $1\sigma$ confidence
level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$, with a central value
$H_{0}t_{0}\simeq 0.99$. The isochrones are labeled for the case of
$H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick solid lines
indicate the age of the universe $t_{0}$ in the unit of Hubble time
$H_{0}^{-1}$. The dot with the coordinate $(0.224,10^{-9})$ represents the
best-fit value, with $\chi_{min}^{2}=591.156$. The best-fit result is in
accordance with D. Spergelet al.Spergel2003 at $1\sigma$ confidence level. A
small $\zeta$, $0\leq\zeta\leq 0.09$ within $95\%$ confidence level, is
obtained.
Figure 3: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration
and the data from the observation of the large-scale structure(LSS), i.e. SNe
Ia+LSS. The dark gray dashed lines denote constraints from the age of the
universe at $1\sigma$ confidence level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$,
with a central value $H_{0}t_{0}\simeq 0.99$. The isochrones are labeled for
the case of $H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick
solid lines indicate the age of the universe $t_{0}$ in the unit of Hubble
time $H_{0}^{-1}$. The dot with the coordinate $(0.213,10^{-8})$ represents
the best-fit value, with $\chi_{min}^{2}=584.831$. The $95\%$ confidence level
of the parameter $\zeta$, i.e. $0\leq\zeta\leq 0.17$, is much smaller than the
SNe Ia-only case.
Figure 4: The distance modulus $\mu$ versus redshift $z$ for the Union2
compilation of the Supernova Cosmology Project(SCP) collaboration plus the
data from the observation of the large-scale structure(LSS) and the cosmic
microwave background(CMB), i.e. SNe Ia+LSS+CMB. The observation consists of
557 SNIa data points with $1\sigma$ error bars. The theoretically computed
result $\mu_{th}$ with the best-fit parameter values $\zeta=10^{-9}$ and
$\Omega_{m0}=0.224$ are plotted in black solid line for the one-dimension
Debye model of modified entropic force.
Figure 5: The reduced Hubble parameter $E$ versus redshift $z$ for three fits,
namely SNe Ia-only, SNe Ia+LSS, and SNe Ia+LSS+CMB, denoted by the dark gray
dotted line, dashed line, and the solid line, respectively. It can be seen
that $E(z)$ grows rapidly as the redshift $z$ increases.
Figure 6: The deceleration parameter $q$ versus redshift $z$ for three fits,
namely SNe Ia-only, SNe Ia+LSS, and SNe Ia+LSS+CMB. Note that $q=0$ at the
redshift $z_{t}=1.034$ for SNe Ia only, $z_{t}=0.948$ for SNe Ia+LSS, and
$z_{t}=0.906$ for SNe Ia+LSS+CMB. $z_{t}$ is called the transition redshift,
which implies that the universe goes from a decelerated expanding period to an
accelerated expanding period at a time point in the early epoch.
|
arxiv-papers
| 2010-09-08T11:31:21 |
2024-09-04T02:49:12.744487
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang, Ming-Hua Li, and Xin Li",
"submitter": "Ming-Hua Li",
"url": "https://arxiv.org/abs/1009.1506"
}
|
1009.1509
|
# CONSTRAINTS FROM TYPE IA SUPERNOVAE ON
$\Lambda$-CDM MODEL IN RANDERS-FINSLER SPACE
Zhe Chang1,2 zchang@ihep.ac.cn Ming-Hua Li1,2 limh@ihep.ac.cn 1Institute of
High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China Xin
Li2,3 lixin@itp.ac.cn 2Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences
3Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China
###### Abstract
Gravitational field equations in Randers-Finsler space of approximate Berwald
type are investigated. A modified Friedmann equation and a new luminosity
distance-redshift relation is proposed. A best-fit to the Type Ia supernovae
(SNe) observations yields that the $\Omega_{\Lambda}$ in the $\Lambda$-CDM
model is suppressed to almost zero. This fact indicates that the astronomical
observations on the Type Ia SNe can be described well without invoking any
form of dark energy. The best-fit age of the universe is given. It is in
agreement with the age of our galaxy.
###### pacs:
## I I. Introdution
The phenomenon that our universe is expanding was first observed by E. Hubble
Hubble in 1929, soon after the birth of General Relativity. Since 1990s,
rapid progress on the observations of the Type Ia SNe have been made riess
Perlmutter Bennett Kowalski . In order to explain the observations, dark
energy with the property of negative pressure must be involved in the
framework of standard cosmological model. Multiple kinds of models have been
proposed in the past decade Copeland . Some of them suggested an evolving
canonical scalar field with a potential peebles earlyqu Carroll:1998zi .
Others tried to modify the General Relativity to make up for the gap between
the theory and the observations Capo Carroll:2003wy odintsovl . One of the
most famous candidates of dark energy is the cosmological constant $\Lambda$.
However, it requires “fine tuning” in an early epoch of the universe
Steinhardt .
_Finsler geometry_ , which includes the Riemann geometry as its special case,
supplies a new systematic approach to the problems mentioned above. Gravity in
a Finsler space has been studied for a long time Takano Tavakol Bogoslovsky1 .
A theory of the gauge transformations in the context of Finsler space has been
discussed by G. Asanov Asanov1 Asanov2 Asanov3 and S. Ikeda Ikeda1 ; Ikeda2 ,
and its applications to General Relativity has been suggested by R. Beil Beil1
; Beil2 . The corresponding gravitational field equation was derived from the
Riemannian osculating metric in Asanov1 . Considering consistency with the
Bianchi identity in Finsler geometry and the general covariance principle of
Einstein, we gave a new gravitational field equation in a Berwald-Finsler
space Lixin . In the framework of Finsler geometry, a modified Newton s
gravity has been proposed, which agrees quite well with the asymptotically
flat rotation curves of spiral galaxies without invoking dark matter Finsler
DM .
In this paper, we test possible constraints from the Type Ia SNe on cosmology
in Finsler geometry. Based on preceding work Lixin ; Modified Friedmann model
, we present a new relation between the luminosity distance and redshift. A
best-fit to the Type Ia SNe data indicates that our universe is matter-
dominated in Finsler space. A reasonable age of our universe is proposed,
which is consistent with that obtained from radioactive dating of isotopes in
stars and from white dwarfs in our galaxy.
The paper is organized as follows. In section II, a brief introduction of
Finsler geometry is presented. In section III, we discuss the gravitational
field equation in a Randers-Finsler space of approximate Berwald type. A new
luminosity-redshift relation is set up in section IV. It is one of the keys to
understand the Type Ia SNe observations without invoking dark energy
hypothesis. The numerical study is carried out in section V. In section VI, we
give concluding remarks and the comparison bewtween our age prediction and
that of other models.
## II II. Finsler Geometry
Let us first introduce basic concepts and notions in Finsler geometry. We use
$T_{x}M$ to denote the _tangent space_ at a point $x$ on a manifold $M$, i.e.
$x\in M$. $TM$ is the _tangent bundle_ of $M$. Each element of $TM$ is
described by $(x,y)$, where $x\in M$ and $y\in T_{x}M$. The _natural
projection_ $\pi:TM\rightarrow M$ is defined as $\pi(x,y)\equiv x$.
_Finsler geometry_ has its genesis in integrals of the form
$\displaystyle\int^{b}_{a}F\left(x^{1},\cdots,x^{n};y^{1},\cdots,y^{n}\right)dt\
,$ (1)
where $x^{i}$ stands for _position_ and $y^{i}\equiv dx^{i}/{dt}$ for
_velocity_. The integrand $F$ is called a _Finsler structure_.
A _Finsler structure_ of $M$ Book by Bao
$\displaystyle F:TM\rightarrow[0,\infty)$
is a function with the following properties:
(i) _Regularity_ : F is $C^{\infty}$ on the entire _slit tangent bundle_
$TM\backslash 0$.
(ii) _Positive homogeneity_ : $F(x,\lambda y)=\lambda F(x,y)$ for all
$\lambda>0$.
(iii) _Strong convexity_ : The $n\times n$ Hessian matrix
$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right)$ (2)
is positive-definite at every point of $TM\backslash 0$. And the lowering and
raising of indices in this paper are carried out by the _fundamental tensor_
$g_{\mu\nu}$ defined above and its inverse $g^{\mu\nu}$. The _Carten tensor_
$A_{\lambda\mu\nu}$, which is defined as
$\displaystyle A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})\ ,$ (3)
is regarded as a measurement of deviation from the Riemannian manifold.
According to Chern’s work Chern2 , each Finsler manifold admits a unique
linear connection, called the _Chern connection_. It is torsion-free and
almost metric-compatible. The _connection coefficients_ take a form as
$\displaystyle\Gamma^{\alpha}_{~{}\mu\nu}=\gamma^{\alpha}_{~{}\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{~{}\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{~{}\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{~{}\mu}}{F}\right),$
(4)
where $\gamma^{\alpha}_{~{}\mu\nu}$ is the _formal Christoffel symbols_ of the
second kind
$\displaystyle\gamma^{\alpha}_{~{}\mu\nu}\equiv\frac{1}{2}g^{\alpha
s}\left(\frac{\partial g_{s\mu}}{\partial x^{\nu}}-\frac{\partial
g_{\mu\nu}}{\partial x^{s}}+\frac{\partial g_{\nu s}}{\partial
x^{\mu}}\right)\ .$ (5)
$N^{\mu}_{~{}\nu}$ is defined as
$\displaystyle
N^{\mu}_{~{}\nu}\equiv\gamma^{\mu}_{~{}\nu\alpha}y^{\alpha}-\frac{A^{\mu}_{~{}\nu\lambda}}{F}\gamma^{\lambda}_{~{}\alpha\beta}y^{\alpha}y^{\beta}\
.$ (6)
The _curvature tensor_ of a Finsler space is given as Book by Bao
$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\delta\Gamma^{\lambda}_{~{}\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{~{}\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{~{}\alpha\mu}\Gamma^{\alpha}_{~{}\kappa\nu}-\Gamma^{\lambda}_{~{}\alpha\nu}\Gamma^{\alpha}_{~{}\kappa\mu}\
,$ (7)
where
$\displaystyle\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{~{}\mu}\frac{\partial}{\partial y^{\mu}}\ .$ (8)
The _Ricci scalar_ is defined as
$\displaystyle Ric\equiv g^{\mu\nu}R_{\mu\nu}\ ,$ (9)
where
$\displaystyle
R^{\mu}_{~{}\nu}\equiv\frac{y^{\lambda}}{F}R^{~{}\mu}_{\lambda~{}\nu\kappa}\frac{y^{\kappa}}{F}=\frac{y^{\lambda}}{F}\left(\frac{\delta}{\delta
x^{\nu}}\frac{N^{\mu}_{~{}\lambda}}{F}-\frac{\delta}{\delta
x^{\lambda}}\frac{N^{\mu}_{~{}\nu}}{F}\right)\ .$ (10)
The _Ricci tensor_ $Ric_{\mu\nu}$, first introduced by Akbar-Zadeh Akbar , is
$\displaystyle
Ric_{\mu\nu}\equiv\frac{\partial^{2}(\frac{1}{2}F^{2}Ric)}{\partial
y^{\mu}\partial y^{\nu}}=\left[\frac{1}{2}F^{2}Ric\right]_{y^{\mu}y^{\nu}}\ .$
(11)
A _Randers space_ is a specific type of Finsler space, whose Finsler structure
takes the form Randers
$\displaystyle F(x,y)\equiv\alpha(x,y)+\beta(x,y)\ ,$ (12)
where
$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}\ ,$ (13)
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}\ .$ (14)
The $\tilde{a}_{\mu\nu}$ are the components of a Riemannian metric and the
$\tilde{b}_{\mu}$ are those of a 1-form. Lower case Greek indices run from
‘$0$’ to ‘$3$’. Specifically, the lowering and raising of indices for the
terms decorated with a tilde are carried out by $\tilde{a}_{\mu\nu}$ and its
inverse $\tilde{a}^{\mu\nu}$ instead of the fundamental tensor.
A Finsler structure $F$ is said to be of _Berwald type_ if the Chern
connection coefficients $\Gamma^{\alpha}_{~{}\mu\nu}$ in natural coordinates
have no $y$ dependence. A Randers space is said to be of Berwald type if and
only if Kikuchi
$\displaystyle\tilde{b}_{\mu|\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\tilde{b}_{\kappa}\tilde{\gamma}^{\kappa}_{~{}\mu\nu}=0\ ,$ (15)
where $\tilde{\gamma}^{\kappa}_{~{}\mu\nu}$ is the Christoffel symbols of a
Riemannian metric $\tilde{\alpha}$. After some tedious calculations, one
obtains that
$\displaystyle\Gamma^{\kappa}_{~{}\mu\nu}=\tilde{\gamma}^{\kappa}_{~{}\mu\nu}\
.$ (16)
So the curvature tensor of a Randers space of Berwald type is given as
$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\partial\tilde{\gamma}^{\lambda}_{~{}\kappa\nu}}{\partial
x^{\mu}}-\frac{\partial\tilde{\gamma}^{\lambda}_{~{}\kappa\mu}}{\partial
x^{\nu}}+\tilde{\gamma}^{\lambda}_{~{}\alpha\mu}\tilde{\gamma}^{\alpha}_{~{}\kappa\nu}-\tilde{\gamma}^{\lambda}_{~{}\alpha\nu}\tilde{\gamma}^{\alpha}_{~{}\kappa\mu}\
,$ (17)
and the corresponding Ricci tensor $Ric_{\mu\nu}$ is
$\displaystyle
Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu})\
.$ (18)
## III III. The Friedmann Model in the framework of Finsler Geometry
In order to investigate the Friedmann-Robertson-Walker (FRW) cosmology, we
take $\tilde{\alpha}_{\mu\nu}$ to be the form
$\displaystyle\tilde{a}_{\mu\nu}={\rm
diag}\left(1,-\frac{R^{2}(t)}{1-kr^{2}},-R^{2}(t)r^{2},-R^{2}(t)r^{2}\sin^{2}\theta\right)\
,$ (19)
where $k=0,+1,-1$ stands for a flat, closed or open universe respectively.
With the condition (15) in mind and assuming that the space of our universe is
almost homogeneous and isotropic, we take
$\displaystyle\tilde{b}_{\mu}=(\tilde{b}_{0},0,0,0)\ ,$ (20)
where $\tilde{b}_{0}$ is a small constant.
Using the identities (2), (17) and (18), one may directly calculates the Ricci
tensor in the Randers space of approximate Berwald type. Nonzero components
are listed below:
$\displaystyle Ric_{00}$ $\displaystyle=$
$\displaystyle-3\frac{\ddot{R}}{R}\tilde{a}_{00},$ $\displaystyle Ric_{ij}$
$\displaystyle=$
$\displaystyle-\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\tilde{a}_{ij}\
.$ (21)
The trace of the Ricci tensor $Ric_{\mu\nu}$ gives the _scalar curvature_
$S\equiv g^{\mu\nu}Ric_{\mu\nu}$ ,
$\displaystyle S$ $\displaystyle=$
$\displaystyle-6\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)-3\frac{\ddot{R}}{R}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)$
(22)
$\displaystyle-\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right).$
A new gravitational field equation in the Berwald-Finsler space is given as
Lixin
$\displaystyle\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}\ ,$ (23)
where
$\displaystyle
B_{\mu\nu\alpha\beta}=-A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F\
.$ (24)
$T^{\mu}_{\nu}$ is the energy-momenta tensor as $T^{\mu}_{\nu}={\rm
diag}(\rho,-p,-p,-p)$, where $p\equiv p(x)$ and $\rho\equiv\rho(x)$ is
respectively the pressure and the energy density of the cosmic fluid.
In a Randers space of approximate Berwald type, the gravitational field
equation (23) reduces to
$\displaystyle\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]=8\pi
GT_{\mu\nu}\ ,$ (25)
because the terms $B^{~{}\alpha}_{\alpha~{}\mu\nu}$ and
$B^{~{}\alpha}_{\mu~{}\nu\alpha}$ are zero. $Ric_{\mu\nu}$ and $S$ are given
by the identities (21) and (22).
For the sake of simplicity, we introduce two parameters $A$ and $B$ as
$\displaystyle
A\equiv\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)$
(26)
and
$\displaystyle
B\equiv\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right)\
.$ (27)
The $0$-$0$ component of the field equation (25) gives the modified Friedmann
equation
$\displaystyle\frac{\alpha}{F}\left(\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)-\frac{1}{2}\frac{\alpha}{F}\frac{\ddot{R}}{R}A+\frac{1}{6}\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+2\frac{k}{R^{2}}\right)B=\frac{8\pi
G}{3}\rho$ (28)
and the $i$-$j$ component of (25) gives
$\displaystyle
3\frac{\alpha}{F}\left(2\frac{\ddot{R}}{R}+\frac{\dot{R}}{R^{2}}+\frac{k}{R^{2}}\right)+\frac{9}{2}\frac{\ddot{R}}{R}\frac{\alpha}{F}A+\frac{1}{2}\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}}{R^{2}}+2\frac{k}{R^{2}}\right)B=8\pi
G(-3p)\ .$ (29)
Subtracting the above two equations, we have
$\displaystyle\frac{\alpha}{F}\frac{\ddot{R}}{R}(1+A)=-\frac{4\pi
G}{3}(\rho+3p)\ .$ (30)
The $0$-$0$ component of the field equation (28) can be rewritten into the
form
$\displaystyle\frac{\alpha}{F}\left(\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)\left(1+\frac{B}{3}\right)\left(1+A\right)=\frac{8\pi
G}{3}\rho\left(1+A\right)+\frac{4\pi
G}{3}\left(\rho+3p\right)\left(-\frac{A}{2}+\frac{B}{6}\right)\ .$ (31)
Assuming that $A$ and $B$ are both time-independent, implementation of time
derivative $\frac{d}{dt}$ on both sides of the equation (31) and using (30)
again leads us to
$\displaystyle-\frac{\dot{R}}{R}\left[\left(1+\frac{B}{3}\right)(\rho+3p)+\rho\left(2+\frac{3A}{2}+\frac{B}{6}\right)+p\left(-\frac{3A}{2}+\frac{B}{2}\right)\right]=\
$ $\displaystyle\dot{\rho}\left(1+\frac{3A}{4}+\frac{B}{12}\right)$ (32)
$\displaystyle+\dot{p}\left(-\frac{3A}{4}+\frac{B}{4}\right)\ .$
With the equations of state $p_{i}=w_{i}\rho_{i}$ of each individual component
$i$ (where the constant $w_{i}=0,-1,-1/3$ corresponds to _matter_ , _vacuum_
and ‘ _curvature_ ’ respectively), the equation (32) can be solved,
$\displaystyle\rho_{i}\propto
R^{-\displaystyle{3(1+w_{i})+\frac{3}{2}(1-w_{i})A+\frac{1}{2}(1+3w_{i})B\over
1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}}\ .$ (33)
## IV IV. A New Luminosity-Redshift Relation
We adopt the conventional definitions
$\displaystyle H(a)\equiv\frac{\dot{R}}{R}\ ,\qquad\rho_{\rm
crit0}\equiv{{3H_{0}^{2}}\over{8\pi G}}\
,\qquad\Omega_{i0}\equiv{{\rho_{i0}}\over{\rho_{\rm crit0}}}=\left({{8\pi
G}\over 3H_{0}^{2}}\right)\rho_{i0}\ ,$ (34)
and
$\displaystyle\rho_{k}\equiv-{{3k}\over{8\pi GR_{0}^{2}a^{-2}}}\
,\qquad\rho_{\rm vac}=\rho_{\Lambda}\equiv{{\Lambda}\over{8\pi G}}\ .$ (35)
Combining the equations (31) and (33), one obtains
$\displaystyle
H(a)=H_{0}\left[\sum_{i(k)}\Omega_{i0}f_{i}(w_{i},A,B)a^{-n_{i}(w_{i},A,B)}\right]^{1/2}\
,$ (36)
where
$\displaystyle
f_{i}(w_{i},A,B)=\frac{1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}{(1+\frac{B}{3})(1+A)}\
,$ (37) $\displaystyle
n_{i}(w_{i},A,B)=-\frac{3(1+w_{i})+\frac{3}{2}(1-w_{i})A+\frac{1}{2}(1+3w_{i})B}{1+\frac{3}{4}(1-w_{i})A+\frac{1}{12}(1+3w_{i})B}\
.$ (38)
Here $\Omega_{\rm k}=1-\Omega_{\rm M}-\Omega_{\Lambda}$ . The notation
$\sum_{i(k)}$ denotes that the sum includes $\Omega_{k}$. As an approximation,
we do not take the radiation term into account due to its little influence on
predictions of the $\Lambda$-CDM model when using the Type Ia SNe data alone.
The _luminosity distance_ $d_{L}$ as a function of the redshift $z$ of a
supernova is Sean M. Carroll
$\displaystyle\begin{array}[]{rcl}d_{L}(z)=&\displaystyle{{(1+z)\over{H_{0}\sqrt{|\Omega_{k0}|}}}\,{\rm
sinn}\left[H_{0}\sqrt{|\Omega_{k0}|}\int^{1}_{1/(1+z)}{{da}\over{a^{2}H(a)}}\right]}\
,\end{array}$ (40)
where ‘$sinn$’ stands for ‘$sin$’ (if $k>0$), ‘$1$’ (if $k=0$) or ‘$sinh$’ (
if $k<0$ ) under certain circumstances. Substituting the equation (36) into
(40), we get a luminosity distance-redshift relation that looks like
$\displaystyle d_{L}(z)=d_{L}(z;\Omega_{\rm M},\Omega_{\Lambda},A,B)\ .$ (41)
## V V. Numerical Study
The _distance modulus_ $\mu$ is related to the luminosity distance via
$\displaystyle\mu\equiv m-M=5\log_{10}[d_{L}({\rm Mpc})]+25\ ,$ (42)
where $m$ is the _apparent magnitude_ of the source and $M$ its _absolute
magnitude_. And it is $\mu$ and $z$ that the Supernova Project measured. A
total uncertainty of $\mu$ (denoted by ‘$\sigma_{\mu}$’) and the corresponding
redshift $z$ were presented in the reference Kowalski .
The $\chi^{2}$ statistic in our fit is
$\displaystyle\chi_{\rm SN}^{2}(\Omega_{\rm
M},\Omega_{\Lambda})\equiv\sum_{i=1}^{307}{[\mu_{\rm obs}(z_{i})-\mu_{\rm
th}(z_{i};\Omega_{\rm
M},\Omega_{\Lambda})]^{2}\over{{\sigma_{\mu}(z_{i})}^{2}}}\ ,$ (43)
where $\mu_{\rm th}(z_{i})$ is obtained by the equation(42), while $\mu_{\rm
obs}(z_{i})$ and $\sigma_{\mu}(z_{i})$ come from the observations. We employ
the Markov Chain Monte-Carlo (MCMC) techniques Antony Lewis to explore the
parameter space. The likelihood function looks like ${\cal L}\propto
e^{-\chi_{\rm SN}^{2}(\Omega_{\rm M},\Omega_{\Lambda})/2}$. For simplicity, we
take $A=-3$ and $B=-1$, leaving $\Omega_{\Lambda}$(or $\Omega_{\rm M}$) to be
the _only free parameter_ in our model with the constraint $\Omega_{\rm
M}+\Omega_{\Lambda}=1$ of a $k=0$ flat universe.
The marginalized posterior and the mean likelihood distributions of the
density parameter $\Omega_{\rm M}$ are shown in Fig.[1]. The two contours in
Fig.[2] and Fig.[3] respectively line out the $68\%$ and $95\%$ confidence
regions of the marginalized distribution in the
$\Omega_{\Lambda}$-$\Omega_{\rm M}$ and Age-$\Omega_{\rm M}$ planes. Best-fit
values of $\Omega_{\rm M}=0.9997_{-0.0009}^{+0.0003}$ and
$\Omega_{\Lambda}=0.0003_{-0.0003}^{+0.0001}$ are obtained with $-ln{\cal
L}=182.1819$ for a total number of $307$ data points. The almost vanished
$\Omega_{\Lambda}$ indicates that, in our model, there is no need to invoke
the $\Omega_{\Lambda}$ term in the Einstein’s field equation to account for
the supernova observations. The best-fit age of the universe is
$18.298_{-0.013}^{+0.102}$ Gyr.
## VI VI. Conclusions
In this paper, we have initiated an exploration on the possibility of a
modified Friedmann model in a Randers-Finsler space of approximate Berwald
type as an alternative to the dark energy hypothesis. Wondering whether the
space-time of our universe is a Randers-Finslerian manifold instead of a
Riemannian one, we have rewritten the Einstein’s field equation in such a
space and the new Friedmann equation was also given. A best-fit to the Type Ia
SNe data suppresses the effective density parameter $\Omega_{\Lambda}$ in the
$\Lambda$-CDM model to almost zero. This fact demonstrates that a Randers-
Finsler geometrical explanation of the ‘accelerated’ expanding universe
without invoking dark energy is possible.
Moreover, the best-fit age of the universe in our model is consistent with the
10 to 20 Gyr estimate obtained from radioactive dating of isotopes in stars
schramm truran and the 6.5 to 10 Gyr minimum age given by the white dwarfs in
our Galactic disk oswalt96 bergeron . However, the change from the old Riemann
space-time to a new Randers-Finsler one may call for a redefinition of not
only the luminosity distance, but also probably other metric-related
quantities. This will be the subject of our future investigation.
###### Acknowledgements.
## VII Acknowledgments
Our work was supported by the NSF of China under Grant No. 10575106 and No.
10875129.
## References
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* (11) S. Carroll, V. Duvvuri, M. Trodden, and M. Turner, Phys. Rev. D 70, 043528 (2004).
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* (28) S. S. Chern, Finsler geometry is just Riemann geometry without the quadratic restrictions, Notice of AMS, 959 (1996).
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Figure 1: The posterior constraints of $\Omega_{\rm M}$ with $A=-3$ and $B=-1$
using all data. The solid line indicates the fully marginalized posterior of
$\Omega_{\rm M}$. The dotted line shows the mean likelihood of the samples.
The fact that the two matching well implies that the mean likelihood is well
constrained and our result is justifiable. We take the Hubble constant to be
$H_{0}=70.5$ $\rm km\cdot s^{-1}\cdot Mpc^{-1}$ instead of a base variable
parameter, because the Type Ia supernova data alone cannot put a well
constraint on it. The center value lies at $\Omega_{\rm M}=0.9997$. The best-
fit likelihood ${\cal L}$ for a total number of $307$ data points is $-ln{\cal
L}({\cal L}\propto e^{-\chi_{\rm SN}^{2}(\Omega_{\rm
M},\Omega_{\Lambda})/2})=182.1819$. Corresponding to a high best-fit
$\Omega_{\rm M}$, a low best-fit $\Omega_{\Lambda}=0.0003$ should be
anticipated, which indicates a matter-dominated universe.
Figure 2: Best-fit $68\%$ and $95\%$ confidence regions from the marginalized
posterior distributions in the $\Omega_{\Lambda}$-$\Omega_{\rm M}$ plane for
the Finsler cosmological model with $A=-3$ and $B=-1$. The inner contour
denotes the $68\%$ confidence limit and the outer one denotes the $95\%$ one.
The cross ‘$+$’ at the upper left corner denotes the best-fit values of
$(\Omega_{\Lambda},\Omega_{\rm M})=(0.0003,0.9997)$ in the modified Friedmann
model. Compared to the $(\Omega_{\Lambda},\Omega_{\rm M})=(0.28,0.72)$ in the
reference spergel , our result indicates that in a Randers-Finsler universe,
no dark energy but only matter components exist.
Figure 3: Best-fit $68\%$ and $95\%$ confidence regions from the marginalized
posterior distributions in the Age-$\Omega_{\rm M}$ plane for the Finsler
cosmological model with $A=-3$ and $B=-1$. The inner contour denotes the
$68\%$ confidence limit and the outer one denotes the $95\%$ one. The cross
‘$+$’ at the upper right corner in the above figure denotes the best-fit value
of the age of the universe in our modified Friedmann model is $18.298$ Gyr.
This prediction is consistent with the $10$ to $20$ Gyr estimate obtained from
radioactive dating of isotopes in stars schramm truran and the $6.5$ to $10$
Gyr minimum age given by the white dwarfs in our Galactic disk oswalt96
bergeron . Thus, the Type Ia SNe data could be well explained by a Randers-
Finsler space-time without invoking any form of dark energy.
|
arxiv-papers
| 2010-09-08T11:41:56 |
2024-09-04T02:49:12.753786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang, Ming-Hua Li, and Xin Li",
"submitter": "Ming-Hua Li",
"url": "https://arxiv.org/abs/1009.1509"
}
|
1009.1609
|
# Bright source of spectrally uncorrelated polarization-entangled photons with
nearly single-mode emission
P. G. Evans evanspg@ornl.gov Center for Quantum Information Science,
Computing and Computational Sciences Directorate, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA J. Schaake Department of Physics
and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA R. S.
Bennink Center for Quantum Information Science, Computing and Computational
Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA W. P. Grice Center for Quantum Information Science, Computing and
Computational Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA T. S. Humble Center for Quantum Information Science,
Computing and Computational Sciences Directorate, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA
(December 13, 2010)
###### Abstract
We present results of a bright polarization-entangled photon source operating
at 1552 nm via type-II collinear degenerate spontaneous parametric down-
conversion in a periodically poled potassium titanyl phosphate crystal. We
report a conservative inferred pair generation rate of 123,000 pairs/s/mW into
collection modes. Minimization of spectral and spatial entanglement was
achieved by group velocity matching the pump, signal and idler modes and
through properly focusing the pump beam. By utilizing a pair of calcite beam
displacers, we are able to overlap photons from adjacent down-conversion
processes to obtain polarization-entanglement visibility of 94.7 +/- 1.1% with
accidentals subtracted.
###### pacs:
42.50.Dv 03.67.Bg 42.50.Ex 42.65.Lm
Spontaneous parametric down-conversion (SPDC) is the leading mechanism for
realizing photonic quantum states. For many applications, particularly those
that involve the interference of photons from multiple SPDC sources, the
individual photon pairs are required to be spectrally and spatially pure. This
is because multimode down-converted photons carry distinguishing information
that undermines the interference central to such applications pittman2003 . It
has been shown previously that spectral entanglement can be eliminated through
source engineering. This approach has led to an experimental demonstration of
heralded generation of spectrally pure single photons from type-II SPDC
mosley2008 ; migdall2010 , and has also inspired similar approaches of type-I
SPDC vicent2010 ; uren2005 and spontaneous four-wave mixing in fibers
soller2010 ; cohen2009 ; halder2009 .
In this Letter, we demonstrate the first bright SPDC source for polarization-
entangled biphoton states that are uncorrelated in both the spectral and
spatial degrees of freedom. As in the previous works, our source is engineered
for spectral purity through the selection of the pump properties and the
phase-matching characteristics of the SPDC medium. In addition, special
attention is paid to the spatial entanglement. Specifically, our source is
based on collinear type-II SPDC in noncritically phase-matched periodically
poled potassium titanyl phosphate (PPKTP), a geometry that eliminates walk-off
effects and maximizes the overlap of the pump, signal, and idler fields
fedrizzi2007 . In this configuration, a properly focused pump causes nearly
all of the photon to be emitted into a single spatial mode grice2010 ;
bennink2010 . The combination of these techniques for controlling the SPDC
output gives rise to a single-mode emission rate of 123,000 pairs/s/mW. The
source is further configured in a novel arrangement of calcite beam displacers
that separate and then recombine photons generated by either of the two
parallel pump beams.
The spectral and spatial properties of SPDC photons are determined by the pump
field and by the dispersive properties of the crystal. The two-photon
probability amplitude is the product of a pump function and a phase-matching
function. Entanglement can be eliminated only if this product yields no
correlations between the signal and idler photon properties. By careful
selection of the wavelength, pulse duration, and focus of the pump, as well as
the crystal material and length, it is possible to minimize entanglement in
these degrees of freedom. In general, the spectral and spatial properties are
not independent, but for the purposes of illustrating the design principles,
it is sufficient here to treat them separately.
To eliminate spectral entanglement, the shapes of the pump function and phase-
matching function must be chosen correctly grice2001 . The pump function,
which describes the range of energies available for down-conversion, generally
yields negatively correlated photon energies — the signal and idler energies
must sum to an energy somewhere in the pump spectrum, so a longer wavelength
for one photon is necessarily accompanied by a shorter wavelength for the
other. This correlation is strongest for a monochromatic pump, for which there
is but a single pump energy. Therefore, one requirement for the elimination of
spectral entanglement is a broad pump spectrum. However, this alone will not
eliminate spectral entanglement if the phase-matching function also leads to
negatively correlated energies.
Whereas the pump function describes the range of energies available for down-
conversion, the phase-matching function describes the ways that the pump
energies may be distributed to the signal and idler photons. The influence of
the phase-matching function on the spectral properties of the photons is
revealed by noting that the function has appreciable value only for
$\Delta{k}L\simeq 0$, where $L$ is the crystal length and where
$\Delta{k}=k_{p}-k_{s}-k_{i}$ is the wavevector mismatch. Using the
approximation $k\simeq k_{0}+\nu k^{\prime}$ and the fact that
$k_{p0}-k_{s0}-k_{i0}=0$ for a phase-matched interaction, we have
$\Delta{k}\simeq(\nu_{s}+\nu_{i})k_{p}^{\prime}-\nu_{s}k_{s}^{\prime}-\nu_{i}k_{i}^{\prime}$.
Here $\nu=\omega-\omega_{0}$ and $k^{\prime}=\partial{k}/\partial{\omega}$.
Imposing the requirement that $\Delta{k}\simeq 0$ yields
$\nu_{s}=-\nu_{i}\frac{k_{p}^{\prime}-k_{i}^{\prime}}{k_{p}^{\prime}-k_{s}^{\prime}}$
(1)
such that the phase-matching function leads to positively correlated photon
energies only if $k_{p}^{\prime}$ lies between $k_{s}^{\prime}$ and
$k_{i}^{\prime}$ or, since the group velocity $v_{g}=1/k^{\prime}$, if the
group velocity of the pump lies between the group velocities of the signal and
idler photons.
It is difficult to satisfy the group velocity matching condition as defined in
Eq. (1) for visible wavelengths with most materials since normal material
dispersion results in lower group velocities for the bluer pump wavelengths.
However, dispersion is more accommodating at longer wavelengths, and solutions
can be found for several type-II materials grice2001 . In particular, group
velocity matching can be achieved with type-II SPDC in KTP with a pump
wavelength range of 650-900 nm for degenerate down-conversion to 1.3-1.8
$\mu$m. Once the material and pump wavelength have been specified, the widths
of the pump and phase-matching functions must be chosen so that the resulting
probability amplitude exhibits neither positive nor negative correlations in
the photon energies. This requirement leads to a specific relationship between
the pump bandwidth and the crystal length. For the 20-mm PPKTP crystal used in
our source, our calculations predict that the spectral entanglement will be
minimized, with a spectral Schmidt number of 1.06, using a 776 nm pump with a
bandwidth corresponding to a transform-limited pulse duration of 1.3 ps. We
note that, with a longer crystal and/or a broader pump bandwidth, it is
possible to generate photon pairs having positive energy correlations, as has
been shown previously shimizu2009 .
The factors that must be considered to minimize the spatial (transverse
momentum) entanglement are similar to those pertaining to the spectral
entanglement. As in the spectral domain, the pump function leads to a tendency
toward negatively correlated transverse momenta — the signal and idler momenta
must sum to a momentum somewhere in the pump spectrum, so the emission
directions of the photons are negatively correlated. This correlation is
strongest when the pump transverse momentum spectrum is narrow, i.e., when the
pump is collimated. A necessary condition for the elimination of spatial
entanglement, therefore, is a pump with a broad transverse momentum spectrum,
a requirement that is easily met by focusing.
The requirement that the phase-matching function yield toward positively
correlated transverse momenta is satisfied in most down-conversion materials,
particularly when there is not spatial walk-off. This is the case for the
noncritically phase-matched PPKTP. All that remains then is to choose the
widths of the pump and phase-matching functions so as to eliminate the spatial
entanglement. For the 20-mm PPKTP crystal used in our source, calculations
predict the best performance with the pump having a divergence of 13.1 mrad
and focused at the center of the crystal. It is shown elsewhere grice2010
that the conditions that minimize spatial entanglement are the same conditions
that maximize coupling to single-mode collection optics. In our case, the
signal and idler photons are predicted to be emitted collinearly into single
modes with divergences of 18.4 and 18.1 mrad, respectively.
Our approach provides several advantages in comparison to traditional
multiphoton entanglement experiments: namely
* •
the pump wavelength of 776 nm is accessible with a tunable pulsed Ti:Sapphire
laser, without the requirement for a second-harmonic generation crystal to
double the pump wavelength as in the usual UV $\rightarrow$ Vis down-
conversion schemes,
* •
the minimization of spectral and spatial entanglement by the source removes
any need for interference and spatial filters to be used, with SPDC photons
emitted into a single spatial mode for optimal coupling to collection optics
and
* •
776 nm $\rightarrow$ 1552 nm SPDC occurs in the technologically important
telecom band where optical fibers exhibit minimal attenuation and standard
telecoms equipment is readily available.
The experimental setup, which is a modification of the scheme first presented
in fiorentino2008 , is illustrated in Fig. 1. A 776 nm pump beam from a
Coherent Mira Ti:Sapphire laser is incident upon a lens, half-wave plate
(HWP1), birefringent wedge pair (BWP), and beam displacer (BD1). BD1 displaces
the orthogonal polarizations of the pump components by 4.2 mm; the vertically
polarized pump component is passed through the half-wave plate HWP2 oriented
with the fast axis at 45∘. The two pump beams incident on the PPKTP crystal
are horizontally polarized with a focus of 13.1 mrad to satisfy type-II phase
matching and to minimize spatial entanglement as described above. Both pump
waists are located midway along the length of the PPKTP crystal.
(a) Experimental setup
(b) Detail of beam displacer pair inset
Figure 1: (Color online) (a) Experimental setup. (b) Detail of the BD pair
showing the beam configuration on entry to BD2 (left diagram), the action of
the half-wave plates between BD2 and BD3 (middle diagram), and upon exit from
BD3 (right diagram). LPF: long-pass filter; PBS: polarization beam splitter.
Upon emergence from the PPKTP crystal, the signal and idler photons from the
two down-conversion regions are incident upon beam displacer BD2, which acts
to displace the signal and idler photons vertically, creating four beam paths,
shown in detail in Fig. 1b. Thin half-wave plates rotate the photon
polarizations in two of the paths before all four beams are incident on beam
displacer BD3, which acts to recombine photons such that both signal photons
are emergent in the upper path and both idler photons are emergent in the
lower path. Lenses following the beam displacers are used to match the spatial
mode of the signal and idler photons to the collection optics. A pair of long-
pass filters remove the residual pump from each arm and the photons are
coupled into single-mode fibers. Wave plates and polarizers are placed in the
two beams for analysis of the polarization-entangled state.
Photon detection is accomplished using two fiber-coupled idQuantique id200
InGaAs/InP avalanche photodiodes with a reported detection efficiency of
approximately 10%. Both detectors are set to use a 2.5 ns gate width and are
triggered at 4.75 MHz synchronized with the 76 MHz pulse train from the
Ti:Sapphire. Output pulses from the id200s, corresponding to photon detection
events, are input into ORTEC counting, delay and coincidence logic circuits
for computer readout.
The singles and coincidence counts are measured as a function of incident pump
power and are displayed in Fig. 2. The half-wave plates and polarization beam
splitters were removed from both collection arms for this measurement. The
single counts are linear with respect to increasing pump power; the
coincidence counts exhibit a slight nonlinear trend that we attribute to
multiple-pair generation at higher pump power wong2008 . By taking into
account Fresnel losses from the uncoated PPKTP crystal and calcite beam
displacers, plus attenuation by the pair of long-pass filters in each arm, we
estimate 38% transmission at 1552 nm through the system. In conjunction with a
conservative 50% free-space to SMF coupling efficiency and 10% detection
efficiency for each id200 detector, our detection efficiency for 1552 nm
photons is 1.9%. Given the actual pair generation rate is equal to the
measured coincidence rate divided by the singles detection efficiency squared,
we conservatively infer the pair generation rate of our source to be 123,000
pairs/s/mW pump. To the best of our knowledge, this is the brightest narrow-
band source operating in the telecom band reported to date.
Figure 2: (Color online) Singles $\sqrt{A.B}$ (left axis) and coincidence
counts (right axis) vs incident pump power. The dashed lines serve as guides
to the eye.
Polarization correlation measurements were carried out with the half-wave
plates and polarization beam splitters reinserted in both arms. Figure 3 shows
polarization correlation plots in the $\pm$45∘ basis. The incident pump power
was set to 16 mW and accidentals were subtracted. Curve fitting to the data
yields a visibility of 94.7 $\pm$ 1.1%.
Figure 3: Coincidence counts vs analyzer orientation angle in the $\pm$45∘
polarization basis showing a visibility of 94.7 $\pm$ 1.1%. The incident pump
power is 16 mW and accidental counts have been subtracted.
In order to examine the effect of multipair generation on the $\pm$45∘ basis
visibility, we conducted several polarization correlation measurements with
various incident pump powers. The results are presented in Fig. 4. The open
circles show the uncorrected data and exhibit a linear decrease with respect
to increasing pump power, providing evidence of multipair generation at higher
pump power. The filled circles represent the corrected data, i.e., raw data
with accidentals subtracted, which average 94% and are constant with respect
to incident pump power as one would expect. We considered several explanations
that would lead to our visibility being capped to 94%: differences in the
optical path lengths of the two down-conversion processes, small differences
in signal and idler photon wavelengths due to poling inhomogeneities, and
errors in wave plate orientation. Optical path differences were minimized by
positioning the birefringent wedge pair BWP to maximize visibility. We
measured the spectra of all four photons using a monochromator and found that
any wavelength differences were on the order of the monochromator resolution.
Thus we conclude that the likely source of the reduced visibility is
orientation error for the two wave plates placed between BD2 and BD3.
Figure 4: Raw ($\circ$) and subtracted accidentals ($\bullet$) visibility in
the $\pm$45∘ polarization basis as a function of incident pump power. The
dashed lines indicate linear fits to the data.
We performed joint spectral intensity measurements on our source using a
customized dual-slit scanning monochromator with 0.3 nm resolution. By
controlling the pump laser mode locking, we were able to adjust the pump pulse
duration and measure the joint spectral intensity at several different pump
pulse durations. Figure 5 shows the measured joint spectral intensity taken
with an incident pump power of 25 mW and 200 s counts per measurement and with
accidental coincidences subtracted, for an optimal pump laser pulse duration
of 1.3 ps [Fig. 5(a)] and 1.9 ps [Fig. 5(b)] respectively. Analysis of the raw
data yields spectral Schmidt numbers of 1.07 and 1.16, respectively, in
agreement to 1.06 as predicted by our calculations for the optimal case.
Furthermore, changes in the joint spectral intensity can clearly be seen
between the different pump pulse durations.
(a) 1.3 ps pump
(b) 1.9 ps pump
Figure 5: (Color online) Joint spectral intensities using two pump pulse
durations, 1.3 ps (a) and 1.9 ps (b). Analysis yields spectral Schmidt numbers
of 1.07 and 1.16, respectively.
In summary, we have demonstrated a unique source insofar as entanglement in
the spatial and spectral degrees of freedom has been minimized by appropriate
pump focusing and careful selection of pump bandwidth, wavelength, and phase
matching with type-II PPKTP. As a result, photons are emitted into single
spectral and spatial modes which require no spectral or spatial filtering to
observe high multiphoton visibility of 94.7 $\pm$ 1.1% with an inferred pair
generation rate of 123,000/s/mW pump. By using a novel arrangement of beam
displacers to recombine the signal and idler photons into distinct paths, we
have demonstrated a source design that can be extended to the construction of
multiphoton ($\geq$ 4) polarization-entangled states.
We thank M. Fiorentino and A. Migdall for enlightening discussions. Research
is sponsored by the Laboratory Directed Research and Development Program of
Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U. S.
Department of Energy under Contract No. De-AC05-00OR22725.
## References
* (1) T. B. Pittman et al., IEEE J. Sel. Topics in Quantum Electron. 9, 1478 (2003).
* (2) P. J. Mosley et al., Phys. Rev. Lett. 100, 133601 (2008).
* (3) Z. H. Levine et al., Opt. Express 18, 4, 3708-3718 (2010).
* (4) A. B. U’Ren et al., Laser Phys. 15, 146 (2005).
* (5) L. E. Vicent et al., New J. Phys. 12, 093027 (2010).
* (6) C. Söller et al., Phys. Rev. A 81, 031801(R) (2010).
* (7) O. Cohen et al., Phys. Rev. Lett. 102, 123603 (2009).
* (8) M. Halder et al., Opt. Express 17, 4670-4676 (2009).
* (9) A. Fedrizzi et al., Opt. Express 15, 23, 15377-15386 (2007).
* (10) W. P. Grice et al., Submitted to Phys. Rev. A.
* (11) R. S. Bennink, Phys. Rev. A 81, 053805 (2010).
* (12) W. P. Grice, A. B. U’Ren, and I. A. Walmsley, Phys. Rev. A 64, 063815 (2001).
* (13) R. Shimizu, and K. Edamatsu, Opt. Express 17, 16385-16393 (2009).
* (14) M. Fiorentino and R. Beausoleil, Opt. Express 16, 20149 (2008).
* (15) O. Kuzucu and F. N. C. Wong, Phys. Rev. A 77, 032314 (2008).
|
arxiv-papers
| 2010-09-08T19:11:25 |
2024-09-04T02:49:12.760461
|
{
"license": "Public Domain",
"authors": "P. G. Evans, J. Schaake, R. S. Bennink, W. P. Grice, T. S. Humble",
"submitter": "Philip Evans",
"url": "https://arxiv.org/abs/1009.1609"
}
|
1009.1631
|
# Point mass insertion on the real line and non-exponential perturbation of
the recursion coefficients
Manwah Lilian Wong
(Date: June 3, 2009)
###### Abstract.
We present the construction of a probability measure $d\gamma$ with compact
support on $\mathbb{R}$ such that adding a discrete pure point results in
changes in the recursion coefficients without exponential decay.
###### Key words and phrases:
point perturbation, bounded variation, asymptotics of orthogonal polynomials.
###### 2000 Mathematics Subject Classification:
28A35, 42C05, 05E35
∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.
E-mail: wongmw@caltech.edu
## 1\. Introduction
Suppose $d\mu$ is a probability measure on the unit circle
$\partial\mathbb{D}$. We define an inner product and a norm on
$L^{2}(\partial\mathbb{D},d\mu)$ respectively as follows:
$\displaystyle\left\langle f,g\right\rangle$ $\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\mathbb{D}}\overline{f(e^{i\theta})}g(e^{i\theta})d\mu(\theta)$
(1.1) $\displaystyle\|f\|_{d\mu}$ $\displaystyle=$
$\displaystyle\left(\displaystyle\int_{\partial\mathbb{D}}|f(e^{i\theta})|^{2}d\mu(\theta)\right)^{1/2}$
(1.2)
Using the inner product defined above, we can orthogonalize $1,z,z^{2},\dots$
to obtain the family of monic orthogonal polynomials associated with the
measure $d\mu$, namely, $(\Phi_{n}(z,d\mu))_{n}$. We denote the normalized
family as $(\varphi_{n}(z,d\mu))_{n}$.
Closely related to $\Phi_{n}(z)$ is the family of reversed polynomials,
defined as $\Phi_{n}^{*}(z)=z^{n}\overline{\Phi_{n}(1/\overline{z})}$. They
obey the well-known Szegő recursion relation
$\Phi_{n+1}(z)=z\Phi_{n}(z)-\overline{\alpha_{n}}\Phi_{n}^{*}(z)$ (1.3)
and $\alpha_{n}$ is known as the $n$-th Verblunsky coefficient. The Szegő
recursion relations for the normalized family of orthogonal polynomials is
$\displaystyle\varphi_{n+1}(z)$
$\displaystyle=(1-|\alpha_{n}|^{2})^{-1/2}(z\varphi_{n}(z)-\overline{\alpha_{n}}\varphi_{n}^{*}(z))$
(1.4)
These recursion relations will be useful later in this paper.
Now we turn to a probability measure $d\gamma$ on $\mathbb{R}$. We can define
an inner product and norm on $L^{2}(\mathbb{R},d\gamma)$ as in (1.1) and
(1.2), except that in this case it does not involve any conjugation. By the
Gram–Schmidt process, we can orthogonalize $1,x,x^{2},\dots$ and form the
family of monic orthogonal polynomials, $(P_{n}(x))_{n=0}^{\infty}$. Upon
normalization, we obtain the family of orthonormal polynomials,
$(p_{n}(x))_{n=0}^{\infty}$. These polynomials satisfy the following three-
term recursion relation
$xp_{n}(x)=a_{n+1}p_{n+1}(x)+b_{n+1}p_{n}(x)+a_{n}p_{n-1}(x)$ (1.5)
where $a_{n}$ and $b_{n}$ are real numbers with $a_{n}>0$. They are called the
recursion coefficients of $d\gamma$.
The main result of this paper is as follows:
###### Theorem 1.1.
There exists a purely absolutely continuous measure $d\gamma_{0}$ supported on
$[-2,2]$ with no eigenvalues outside of $[-2,2]$, such that if we add a pure
point $x_{0}\in\mathbb{R}\backslash[-2,2]$ in the following manner
$d\tilde{\gamma}(x)=(1-\beta)d\gamma_{0}(x)+\beta\delta_{x_{0}}\quad\quad\beta>0$
(1.6)
it will result in non-exponential perturbation of the recursion coefficients
$a_{n}(d\gamma_{0})$ and $b_{n}(d\gamma_{0})$.
This example is of particular interest because of the following history: back
in 1946, Borg [1] proved a well-known result concerning the Sturm–Liouville
problem that in general, a single spectrum is insufficient to determine the
potential. Later, Gel’fand–Levitan [8] showed that in order to recover the
potential one also needs the norming constants.
Norming constants correspond to the weights of pure points and it is known
that in the short range case (in orthogonal polynomials language,
$a_{n}-1,b_{n}\to 0$ fast), varying the norming constants will result in
exponential change in the potential.
Moreover, when considering the effect of varying the weight of discrete point
masses on orthogonal polynomials (both on $\mathbb{R}$ and
$\partial\mathbb{D}$), Simon proved that it will result in exponential
perturbation of the recursion coefficients (see Corollary 24.4 and Corollary
24.3 of [15]).
All the results mentioned above lend to a few the intuition that if the
recursion coefficients $a_{n}\to 1$ and $b_{n}\to 0$ fast, then adding a pure
point will result in exponential change in the recursion coefficients.
However, it turned out not to be the case!
## 2\. Tools Involved in the Proof
### 2.1. The Szegő Mapping
It turns out that one can relate measures supported on $[-2,2]$ with a certain
class of measures on $\partial\mathbb{D}$.
Note that the map $\theta\mapsto 2\cos\theta$ is a two-one map from
$\partial\mathbb{D}$ to $[-2,2]$. Therefore, given a non-trivial probability
measure $d\xi$ on $\partial\mathbb{D}$ that is invariant under
$\theta\rightarrow-\theta$, we can define a measure
$d\gamma={\rm Sz}(d\xi)$ (2.1)
using what is known as the Szegő map, such that for $g$ measurable on
$[-2,2]$,
$\displaystyle\int g(2\cos\theta)d\xi(\theta)=\displaystyle\int
g(x)d\gamma(x)$ (2.2)
Conversely, if we have a probability measure $\beta$ supported on $[-2,2]$, we
can obtain a probability measure
$\nu={\rm Sz^{-1}}(d\gamma)$ (2.3)
on $\partial\mathbb{D}$ by what is known as the Inverse Szegő Mapping, such
that for $h(z)$ measurable on $\partial\mathbb{D}$,
$\displaystyle\int h(\theta)d\nu(\theta)=\int
h\left(\cos^{-1}\frac{x}{2}\right)d\gamma(x)$ (2.4)
There are many interesting results about the Szegő mapping (see Chapter 13 of
[14]), but the only relevant one for this paper is the following by Geronimus
[9] (see also Theorem 13.1.7 of [14]):
###### Theorem 2.1 (Geronimus [10]).
Let $d\xi$ be a probability measure on $\partial\mathbb{D}$ which is invariant
under $\theta\rightarrow-\theta$ and let $d\gamma={\rm Sz}(\xi)$. Let
$\alpha_{n}\equiv\alpha_{n}(d\xi)$, $a_{n}\equiv a_{n}(d\gamma)$ and
$b_{n}\equiv b_{n}(d\gamma)$. Then for $n=0,1,2,\dots$,
$\displaystyle a_{n+1}^{2}$ $\displaystyle=$
$\displaystyle(1-\alpha_{2n-1})(1-\alpha_{2n})^{2}(1+\alpha_{2n+1})$ (2.5)
$\displaystyle b_{n+1}$ $\displaystyle=$
$\displaystyle(1-\alpha_{2n-1})\alpha_{2n}-(1+\alpha_{2n-1})\alpha_{2n-2}$
(2.6)
with the convention that $\alpha_{-1}=-1$.
### 2.2. The Point Mass Formuula
We add a point mass $\zeta=e^{i\omega}\in\partial\mathbb{D}$ with weight
$0<\beta<1$ to $d\mu$ in the following manner:
$d\nu=(1-\beta)d\mu+\beta\delta_{\omega}$ (2.7)
Our goal is to investigate $\alpha_{n}(d\nu)$.
Point mass perturbation has a long history (see the Introduction of [17]). One
of the classic results is the following theorem:
###### Theorem 2.2 (Geronimus [10, 11]).
Suppose the probability measure is defined as in (2.7). Then
$\Phi_{n}(z,d\nu)=\Phi_{n}(z)-\displaystyle\frac{\varphi_{n}(\zeta)K_{n-1}(z,\zeta)}{(1-\beta)\beta^{-1}+K_{n-1}(\zeta,\zeta)}$
(2.8)
where
$K_{n}(z,\zeta)=\displaystyle\sum_{j=0}^{n}\overline{\varphi_{j}(\zeta)}\varphi_{j}(z)$
(2.9)
and all objects without the label $(d\nu)$ are associated with the measure
$d\mu$.
Since $\Phi_{n}(0)=-\overline{\alpha_{n-1}}$, by putting $z=0$ into (2.8) one
gets a formula relating the Verblunsky coefficients of $d\mu$ and $d\nu$. For
more on the formula (2.8), the reader may refer to Nevai [13, 12], and
Cachafeiro–Marcellán [3, 2, 6, 4, 5].
Using a totally different approach, Simon [14] found the following formula for
OPUC:
$\alpha_{n}(d\nu)=\alpha_{n}-q_{n}^{-1}\beta\overline{\varphi_{n+1}(\zeta)}\left(\displaystyle\sum_{j=0}^{n}\alpha_{j-1}\frac{\|\Phi_{n+1}\|}{\|\Phi_{j}\|}\varphi_{j}(\zeta)\right)$
(2.10)
where $q_{n}=(1-\beta)+\beta K_{n}(\zeta);\alpha_{-1}=-1$.
Simon’s result lays the foundation for the point mass formula. In [16, 17],
Wong applied the Christoffel–Darboux formula to (2.10) and within a few steps
from (2.10) proved the following formula for $\alpha_{n}(d\nu)$:
$\alpha_{n}(d\nu)=\alpha_{n}(d\mu)+\Delta_{n}(\zeta)$ (2.11)
where
$\Delta_{n}(\zeta)=\displaystyle\frac{(1-|\alpha_{n}|^{2})^{1/2}\overline{\varphi_{n+1}(\zeta)}\varphi_{n}^{*}(\zeta)}{(1-\beta)\beta^{-1}+K_{n}(\zeta)};\quad
K_{n}(\zeta)=\displaystyle\sum_{j=0}^{n}|\varphi_{j}(\zeta)|^{2}$ (2.12)
Formula (2.12) turns out to be very useful (see for example, [16, 17, 18]).
## 3\. Outline of the Proof
### 3.1. Case 1: $x_{0}>2$
We construct a measure $d\gamma_{0}$ with recursion coefficients $(a_{n})$ and
$(b_{n})$ satisfying
$\displaystyle a_{n}\nearrow 1\quad\quad b_{n}\equiv 0$ (3.1)
$\displaystyle\displaystyle\sum_{n}|a_{n}-1|^{2}=\infty$ (3.2)
The measure $d\gamma_{0}$ is purely absolutely continuous and symmetrically
supported on $[-2,2]$, with no pure points outside $[-2,2]$. We scale it by a
factor $0<y<2$ to form the measure $d\gamma_{y}$ supported on
$[-y,y]\subset[-2,2]$ (we will show the connection between $y$ and $x_{0}$ a
bit later; see (3.3)).
Then we use the Inverse Szegő map on $d\gamma_{y}$ to obtain the measure
$d\mu_{y}$. By looking at the Direct Geronimus Relations (2.5) and (2.6), we
find necessary conditions for $\alpha_{n}(d\mu_{y})$ so that both (3.1) and
(3.2) hold.
Since $d\gamma_{y}$ is supported on $[-y,y]\subset[-2,2]$, we know that
$d\mu_{y}$ is supported on two identical bands. Besides, $d\mu_{y}$ is
symmetric along both the $x$-and $y$-axes because of the symmetry of
$d\gamma_{y}$ and the Szegő map.
We add a pure point at $z=1$ to $d\mu_{y}$ to form the measure
$d\tilde{\mu}_{y}$ and compute the perturbed Verblunsky coefficients
$\alpha_{n}(d\tilde{\mu}_{y})$.
Then we use the Szegő map on $d\tilde{\mu}_{y}$ to obtain the probability
measure $d\tilde{\gamma}_{y}$ on $\mathbb{R}$. Finally, we scale
$d\tilde{\gamma}_{y}$ to form the measure $d\tilde{\gamma}$.
Note that if we have chosen $y$ such that
$\displaystyle\frac{y}{2}=\displaystyle\frac{2}{|x_{0}|}$ (3.3)
then we have $d\tilde{\gamma}=(1-\beta)d\gamma_{0}+\beta\delta_{x_{0}}$.
As the final step, we show that for some constants $C_{x_{0}},D_{x_{0}}$ (both
dependent on $x_{0}$) such that
$\displaystyle a_{n}(d\tilde{\gamma})$ $\displaystyle=$ $\displaystyle
a_{n}(d\gamma_{0})+\displaystyle\frac{C_{x_{0}}}{n^{3/2}}+o\left(\displaystyle\frac{1}{n^{3/2}}\right)$
(3.4) $\displaystyle b_{n}(d\tilde{\gamma})$ $\displaystyle=$ $\displaystyle
b_{n}(d\gamma_{0})+\displaystyle\frac{D_{x_{0}}}{n^{3/2}}+o\left(\displaystyle\frac{1}{n^{3/2}}\right)$
(3.5)
### 3.2. Case 2: $x_{0}<-2$
Everything in Case 1 will follow except that we add a point $z=-1$ to
$d\mu_{y}$ instead. As we shall see later in the proof, $d\mu_{y}$ is
symmetric both along the $x-$ and $y-$ axes. Therefore, adding a pure point at
$z=-1$ is the same as adding a pure point at $z=1$ and then rotating the
measure by an angle of $\pi$.
For the convenience of the reader, here is a diagram showing all the measures
involved. We will start from the measure $d\mu_{y}$, and move along two
directions:
$d\gamma_{0}\overset{\text{scaling}}{\longleftarrow}d\gamma_{y}\overset{\text{Sz}^{-1}}{\longleftarrow}d\mu_{y}\overset{\text{add}\,\,z=1}{\longrightarrow}d\tilde{\mu}_{y}\overset{\text{Sz}^{-1}}{\longrightarrow}d\tilde{\gamma}_{y}\overset{\text{scaling}}{\longrightarrow}d\tilde{\gamma}$
(3.6)
## 4\. The Proof
Let $d\gamma_{0}$ be a probability measure on $\mathbb{R}$ with recursion
coefficients satisfying (3.1) and (3.2).
This measure, supported on $[-2,2]$, is purely absolutely continuous, and has
no eigenvalues outside $[-2,2]$. Moreover, if we write
$d\gamma_{0}(x)=f(x)dx$, $f(x)$ is symmetric.
Now we scale $d\gamma_{0}$ to form the measure $d\gamma_{y}$ defined by
$d\gamma_{y}(x)=d\gamma\left(2xy^{-1}\right)\quad 0<y<2$ (4.1)
The measure $d\gamma_{y}$, supported on $[-y,y]\subset[-2,2]$, is purely
absolutely continuous and the a.c. part of $d\gamma_{y}(x)$ is
$f_{y}(x)=f(2xy^{-1})\chi_{[-y,y]}$ (4.2)
which is also symmetric.
It is well known that scaling has the following effects on the recursion
coefficients
$a_{n}(d\gamma_{y})=\displaystyle\left(\displaystyle\frac{y}{2}\right)a_{n}(d\gamma_{0})\quad
b_{n}(d\gamma_{y})=\displaystyle\left(\displaystyle\frac{y}{2}\right)b_{n}(d\gamma_{0})$
(4.3)
Now we apply the inverse Szegő map to $d\gamma_{y}$ to form the probability
measure $d\mu_{y}$ on $\partial\mathbb{D}$, see figure below:
Figure 1. Graph of $\text{\rm{supp}}(d\mu)$
The measure $d\mu_{y}$ is supported on two arcs, $[\theta_{y},\pi-\theta_{y}]$
and $[\pi+\theta_{y},2\pi-\theta_{y}]$, with a.c. part
$w_{y}(\theta)=w_{y}(\theta)|_{[\theta_{y},\pi-\theta_{y}]}+w_{y}(\theta)|_{[\pi+\theta_{y},2\pi-\theta_{y}]}$
(4.4)
where
$\displaystyle w_{y}(\theta)$ $\displaystyle=$ $\displaystyle
2\pi|\sin(\theta)|f_{y}(2\cos\theta)\chi_{[\theta_{y},\pi-\theta_{y}]}(\theta)$
(4.5) $\displaystyle\theta_{y}$ $\displaystyle=$
$\displaystyle\cos^{-1}\left(\frac{y}{2}\right)\in\left(0,\frac{\pi}{2}\right)$
(4.6)
By Corollary 13.1.8 of [14], $b_{n}(d\gamma_{y})\equiv 0$ if and only if
$\alpha_{2n}(d\mu_{y})\equiv 0$. Therefore, we can express the Verblunsky
coefficients of $d\mu_{y}$ as
$0,\tau_{0},0,\tau_{1},0,\tau_{2},\dots$ (4.7)
with $\tau_{j}=\alpha_{2j+1}$. Moreover, by Theorem 13.1.7 of [14], we know
that
$\begin{array}[]{ll}a_{n+1}^{2}(d\gamma_{y})&=(1-\alpha_{2n-1}(d\mu_{y}))(1-\alpha_{2n}(d\mu_{y})^{2})(1+\alpha_{2n+1}(d\mu_{y}))\\\
&=(1-\tau_{n-1})(1+\tau_{n})\end{array}$ (4.8)
Now we will choose a suitable family of $\tau_{n}\in\mathbb{R}$ such that the
corresponding $a_{n}(d\gamma_{y})$ satisfy both (3.1) and (4.8).
Observe that by (4.8) above,
$a_{n+1}(d\gamma_{y})^{2}-a_{n}(d\gamma_{y})^{2}=(1-\tau_{n-1})(\tau_{n}-\tau_{n-1})+(1+\tau_{n-1})(\tau_{n-1}-\tau_{n-2})$
(4.9)
Therefore, if we have an increasing family of $\tau_{n}<0$ such that
$\tau_{n}\nearrow\tau_{\infty}=-\sqrt{1-\left(\displaystyle\frac{y}{2}\right)^{2}}<0$
(4.10)
then $a_{n}(d\gamma_{y})\nearrow y/2$ and the corresponding measure
$d\mu_{y}$.
In particular, if we let
$\tau_{k}=\displaystyle\tau_{\infty}-\frac{1}{\sqrt{k}}$ (4.11)
then the goal is achieved.
Next, we prove the following lemma:
###### Lemma 4.1.
Let $d\mu_{y}$ be the measure on $\partial\mathbb{D}$ with Verblunsky
coefficients as in (4.7) where for all large $n$,
$\tau_{n}=\tau_{\infty}-\displaystyle\frac{1}{\sqrt{n}}\quad\quad-1<\tau_{\infty}<0$
(4.12)
We add a pure point at $z=1$ to $d\mu_{y}$ as in (2.7) to form
$d\tilde{\mu}_{y}$. Then for $n=2m$ or $2m+1$, $\Delta_{n}(1)$ has the
following expansion
$\Delta_{n}(1)=-\tau_{\infty}+\frac{1}{\sqrt{m}}+0+\left(1+\displaystyle\frac{1}{2\tau_{\infty}}\right)\displaystyle\frac{1}{m^{3/2}}+o\left(\displaystyle\frac{1}{m^{3/2}}\right)$
(4.13)
Therefore,
$\alpha_{n}(d\mu_{y})=\begin{cases}-\tau_{m}+\displaystyle\left(1+\frac{1}{2\tau_{\infty}}\right)\displaystyle\frac{1}{m^{3/2}}+e_{m}&n=2m\\\
\displaystyle\left(1+\frac{1}{2\tau_{\infty}}\right)\displaystyle\frac{1}{m^{3/2}}+e_{m}&n=2m+1\end{cases}$
(4.14)
where $e_{m}=o\left(m^{-3/2}\right)$.
###### Proof.
Since all the Verblunsky coefficients of $d\mu_{y}$ are real, by induction on
the recursion relation (1.4),
$\begin{array}[]{ll}\varphi_{n}(1)&=\displaystyle\prod_{j=0}^{n-1}\sqrt{\displaystyle\frac{1-\alpha_{j}}{1+\alpha_{j}}}\end{array}$
(4.15)
By (4.7), when $n=2m$ or $2m+1$,
$\varphi_{n}^{*}(1)=\varphi_{n}(1)=\displaystyle\prod_{j=0}^{m-1}\sqrt{\displaystyle\frac{1-\tau_{j}}{1+\tau_{j}}}$
(4.16)
This formula will play a crucial role in the computation below.
### 4.1. n is even
First, we compute $\Delta_{n}(1)$ when $n=2m$ using the point mass formula
(2.12). Let
$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\overline{\varphi_{n+1}(1)}\varphi_{n}^{*}(1)$ (4.17)
$\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle(1-\gamma)\gamma^{-1}+K_{n}(1,1)$ (4.18)
Then
$\lim_{m\to\infty}\Delta_{2m}(1)=\lim_{m\to\infty}(1-|\alpha_{2m}|^{2})^{1/2}\,\displaystyle\frac{A_{2m}}{B_{2m}}=\lim_{m\to\infty}\displaystyle\frac{A_{2m}}{B_{2m}}$
(4.19)
because $\alpha_{2m}=0$ for all $m$. However, instead of computing this
directly, we use the Stolz–Cesàro theorem (see [7]), which reads as follows
###### Theorem 4.1 (Stolz–Cesàro [7]).
Let $(\Gamma_{k})_{k},(\Theta_{k})_{k}$ be two sequences of numbers such that
$\Theta_{n}$ is positive, strictly increasing and tends to infinity. If the
following limit exists,
$\displaystyle\lim_{k\to\infty}\frac{\Gamma_{k}-\Gamma_{k-1}}{\Theta_{k}-\Theta_{k-1}}$
(4.20)
then it is equal to $\lim_{k\to\infty}\Gamma_{k}/\Theta_{k}$.
First, note that $\tau_{k}\to\tau_{\infty}<0$. Thus,
$B_{m}\approx K_{n}(1,1)>|\varphi_{n}(1)|^{2}\to\infty$ (4.21)
by (4.16). Hence, it is legitimate for us to use Theorem 4.1 above.
Let $K_{n}\equiv K_{n}(1,1)$ and $\varphi_{n}\equiv\varphi_{n}(1)$. Observe
that $\varphi_{2m+1}=\varphi_{2m}$. Therefore, by (4.16),
$B_{2(m+1)}-B_{2m}=\varphi_{2(m+1)}^{2}+\varphi_{2m}^{2}=\displaystyle\frac{2\varphi_{2m}^{2}}{1+\tau_{m}}$
(4.22)
and
$A_{2(m+1)}-A_{2m}=\varphi_{2(m+1)}^{2}-\varphi_{2m}^{2}=\left(\displaystyle\frac{-2\tau_{m}}{1+\tau_{m}}\right)\varphi_{2m}^{2}$
(4.23)
As a result,
$\lim_{m\to\infty}\Delta_{2m}(1)=\displaystyle\lim_{m\to\infty}\frac{A_{2(m+1)}-A_{2m}}{B_{2(m+1)}-B_{2m}}=-\tau_{\infty}$
(4.24)
Next, we will prove that the rate of convergence is
$\Delta_{2m}(1)=-\tau_{\infty}+\displaystyle\frac{1}{\sqrt{m}}+o\left(\displaystyle\frac{1}{\sqrt{m}}\right)$
(4.25)
by computing the following limit
$\displaystyle\lim_{m\to\infty}m\left({\Delta_{2m}(1)+\tau_{\infty}}\right)=1$
(4.26)
Recall the definition of $\Delta_{n}(1)$ and the facts $\alpha_{2m}\equiv 0$
and $\varphi_{2m+1}\varphi_{2m}=\varphi_{2m}^{2}$. Thus, the left hand side of
(4.26) can be expressed as $X_{n}/Y_{n}$, where
$\displaystyle X_{m}$ $\displaystyle=$
$\displaystyle\sqrt{m}\left[\varphi_{2m}^{2}+\tau_{\infty}K_{2m}\right]$
(4.27) $\displaystyle Y_{m}$ $\displaystyle=$ $\displaystyle K_{2m}\to\infty$
(4.28)
We use the Stolz–Cesàro Theorem again. First, observe that
$Y_{m+1}-Y_{m}=\displaystyle\frac{1-\tau_{m}}{1+\tau_{m}}+1=\displaystyle\frac{2}{1+\tau_{m}}\varphi_{2m}^{2}$
(4.29)
Then we compute
$X_{2(m+1)}-X_{2m}\\\
=\underbrace{\left[\sqrt{m+1}\displaystyle\frac{1-\tau_{m}}{1+\tau_{m}}-\sqrt{m}\right]\varphi_{2m}^{2}}_{\text{(I)}}+\underbrace{\tau_{\infty}\left[\sqrt{m+1}K_{2(m+1)}-\sqrt{m}K_{2m}\right]}_{\text{(II)}}$
(4.30)
Consider each term in (4.30) above.
$\displaystyle\frac{\text{(I)}}{Y_{2(m+1)}-Y_{2m}}=\displaystyle\frac{\sqrt{m+1}\frac{1-\tau_{m}}{1+\tau_{m}}-\sqrt{m}}{\frac{2}{1+\tau_{m}}}=\displaystyle\frac{\sqrt{m+1}(1-\tau_{m})-\sqrt{m}(1+\tau_{m})}{2}$
(4.31)
Moreover,
$\text{(II)}=\tau_{\infty}\left[\sqrt{m+1}(K_{2(m+1)}-K_{2m})+(\sqrt{m+1}-\sqrt{m})K_{2m}\right]$
(4.32)
which implies that
$\displaystyle\frac{\text{(II)}}{Y_{m+1}-Y_{m}}=\tau_{\infty}\left[\sqrt{m+1}+(\sqrt{m+1}-\sqrt{m})\displaystyle\frac{1+\tau_{m}}{2}\frac{K_{2m}}{\varphi_{2m}^{2}}\right]$
(4.33)
Next, we show that $\lim_{m\to\infty}K_{2m}/\varphi_{2m}^{2}=-1/\tau_{\infty}$
by the Stolz–Cesàro Theorem.
$\begin{array}[]{ll}\displaystyle\lim_{m\to\infty}\frac{K_{2m}}{\varphi_{2m}^{2}}&=\left(\displaystyle\lim_{m\to\infty}\displaystyle\frac{\varphi_{2(m+1)}^{2}-\varphi_{2m}^{2}}{K_{2(m+1)}-K_{2m}}\right)^{-1}\\\
&=\displaystyle\lim_{m\to\infty}\displaystyle\left({\frac{1-\tau_{m}}{1+\tau_{m}}-1}\right)^{-1}\left({\frac{1-\tau_{m}}{1+\tau_{m}}+1}\right)\\\
&=-\displaystyle\frac{1}{\tau_{\infty}}\end{array}$ (4.34)
Combining (4.31), (4.33) and (4.34), we obtain
$\displaystyle\lim_{m\to\infty}m\left({\Delta_{2m}(1)-(\tau_{\infty})}\right)=1$
(4.35)
Next, we are going to show that
$\Delta_{2m}(1)=-\tau_{\infty}+\displaystyle\frac{1}{\sqrt{m}}+o\left(\displaystyle\frac{1}{m}\right)$
(4.36)
by computing the second-order term. We do so by proving that
$L_{2}\equiv\displaystyle\lim_{m\to\infty}m\left(\Delta_{2m}-(-\displaystyle\tau_{\infty})-\displaystyle\frac{1}{\sqrt{m}}\right)=0$
(4.37)
Let
$P_{m}=m\varphi_{2m}^{2}+m\tau_{\infty}K_{2m}-\sqrt{m}K_{2m}$ (4.38)
Then
$P_{m+1}-P_{m}=\left[(m+1)\displaystyle\frac{1-\tau_{m}}{1+\tau_{m}}-m\right]\varphi_{2m}^{2}\\\
+(m+1)\tau_{\infty}\left[K_{2(m+1)}-K_{2m}\right]+\left[(m+1)-m\right]\tau_{\infty}K_{2m}\\\
-\sqrt{m+1}\left[K_{2(m+1)}-K_{2m}\right]-(\sqrt{m+1}-\sqrt{m})K_{2m}$ (4.39)
Combining with previous results about $Y_{m+1}-Y_{m}$ and
$K_{2m}/\varphi_{2m}^{2}$, we have
$L_{2}=\displaystyle\lim_{m\to\infty}\frac{P_{m+1}-P_{m}}{Y_{m+1}-Y_{m}}=0$
(4.40)
which proves (4.36).
Next, we will obtain the third-order term by computing
$L_{3}=\displaystyle\lim_{m\to\infty}m^{3/2}\left(\Delta_{2m}-(-\tau_{\infty})-\displaystyle\frac{1}{\sqrt{m}}\right)$
(4.41)
Let
$J_{m}=m^{3/2}\varphi_{2m}^{2}+m^{3/2}\tau_{\infty}K_{2m}-mK_{2m}$ (4.42)
By a similar argument as in (4.34),
$J_{m+1}-J_{m}=\left[(m+1)^{3/2}\displaystyle\frac{1-\tau_{m}}{1+\tau_{m}}-m^{3/2}\right]\varphi_{2m}^{2}\\\
+(m+1)^{3/2}\tau_{\infty}\left[K_{2(m+1)}-K_{2m}\right]+\left[(m+1)^{3/2}-m^{3/2}\right]\tau_{\infty}K_{2m}\\\
-(m+1)\left[K_{2(m+1)}-K_{2m}\right]-(m+1-m)K_{2m}$ (4.43)
which implies that
$L_{3}=\displaystyle\lim_{m\to\infty}\displaystyle\frac{J_{m}}{Y_{m}}=1+\frac{1}{2\tau_{\infty}}$
(4.44)
### 4.2. when n is odd
We compute $\Delta_{n}(1)$ when $n=2m+1$ using the point mass formula (2.12).
Let $A_{n}$ and $B_{n}$ be defined as in (4.17) and (4.18). Then
$\lim_{m\to\infty}\Delta_{2m+1}(1)=(1-|\tau_{\infty}|^{2})^{1/2}\lim_{m\to\infty}\,\displaystyle\frac{A_{2m+1}}{B_{2m+1}}$
(4.45)
We will use the Stolz–Cesàro Theorem again. Note that
$A_{2(m+1)+1}-A_{2m+1}=\left(\displaystyle\sqrt{\frac{1-\tau_{m+1}}{1+\tau_{m+1}}}\frac{1-\tau_{m}}{1+\tau_{m}}-\displaystyle\sqrt{\frac{1-\tau_{m}}{1+\tau_{m}}}\right)\varphi_{2m}^{2}$
(4.46)
and because $\varphi_{2m+3}=\varphi_{2m+2}$,
$B_{2(m+1)+1}-B_{2m+1}=2\varphi_{2m+2}^{2}=2\left(\frac{1-\tau_{m}}{1+\tau_{m}}\right)\varphi_{2m}^{2}$
(4.47)
Therefore,
$\displaystyle\lim_{m\to\infty}\Delta_{2m+1}(1)=\displaystyle\frac{-\tau_{\infty}(1-|\tau_{\infty}|^{2})^{1/2}}{\sqrt{(1+\tau_{\infty})(1-\tau_{\infty})}}=-\tau_{\infty}$
(4.48)
Next, we prove the rate of convergence by computing
$\displaystyle\lim_{m\to\infty}\sqrt{m}\left({\Delta_{2m+1}(1)+\tau_{\infty}}\right)=1$
(4.49)
Since $\alpha_{n}\in\mathbb{R}$, the recursion relation becomes
$(1-|\alpha_{n}|^{2})^{1/2}\varphi_{n+1}=\varphi_{n}-\overline{\alpha_{n}}\varphi_{n}^{*}=(1-\alpha_{n})\varphi_{n}$
(4.50)
Therefore,
$\Delta_{2m+1}(1)=\displaystyle\frac{(1-\alpha_{2m+1})\varphi_{2m+1}^{2}}{K_{2m+1}}=(1-\tau_{m})\displaystyle\frac{\varphi_{2m}^{2}}{K_{2m+1}}$
(4.51)
Let
$\displaystyle P_{m}$ $\displaystyle=$
$\displaystyle\sqrt{m}\left[(1-\tau_{m})\varphi_{2m}^{2}+\tau_{\infty}K_{2m+1}\right]$
(4.52) $\displaystyle Q_{m}$ $\displaystyle=$ $\displaystyle
K_{2m+1}\to\infty$ (4.53)
Note that
$Q_{m+1}-Q_{m}=K_{2m+3}-K_{2m+1}=2\varphi_{2(m+1)}^{2}$ (4.54)
and
$P_{m+1}-P_{m}=\underbrace{\left[\sqrt{m+1}(1-\tau_{m+1})\varphi_{2(m+1)}^{2}-\sqrt{m}(1-\tau_{m})\varphi_{2m}^{2}\right]}_{\text{(I)}}\\\
+\tau_{\infty}\sqrt{m+1}\left[K_{2m+3}-K_{2m+1}\right]+\underbrace{(\sqrt{m+1}-\sqrt{m})\tau_{\infty}K_{2m+1}}_{\text{(II)}}$
(4.55)
Since $(1-\tau_{m})\varphi_{2m}^{2}=(1+\tau_{m})\varphi_{2(m+1)}^{2}$,
$\begin{array}[]{ll}\displaystyle\frac{\text{(I)}}{Q_{m+1}-Q_{m}}&=\displaystyle\frac{\sqrt{m+1}(1-\tau_{m+1})-\sqrt{m}(1+\tau_{m})}{2}\end{array}$
(4.56)
Next, consider (II). We compute
$\displaystyle\lim_{m\to\infty}\displaystyle\frac{\varphi_{2(m+1)}^{2}-\varphi_{2m}^{2}}{K_{2m+1}-K_{2m-1}}=\displaystyle\frac{\left(\frac{1-\tau_{m}}{1+\tau_{m}}-1\right)\varphi_{2m}^{2}}{2\varphi_{2m}^{2}}=\displaystyle\frac{-\tau_{\infty}}{1+\tau_{\infty}}$
(4.57)
which implies
$\displaystyle\frac{\text{(II)}}{Q_{m+1}-Q_{m}}=-(1+\tau_{\infty})(\sqrt{m+1}-\sqrt{m})$
(4.58)
Therefore,
$\displaystyle\lim_{m\to\infty}m\left(\Delta_{2m+1}(1)-(-\tau_{\infty})\right)=\displaystyle\lim_{m\to\infty}\frac{P_{m}}{Q_{m}}=1$
(4.59)
Next, we will prove that
$\Delta_{2m+1}=-\tau_{\infty}+\displaystyle\frac{1}{\sqrt{m}}+o\left(\displaystyle\frac{1}{m}\right)$
(4.60)
by showing
$L_{2}^{\prime}\equiv\displaystyle\lim_{m\to\infty}m\left(\Delta_{2m+1}+\tau_{\infty}-\frac{1}{\sqrt{m}}\right)=0$
(4.61)
As explained in (4.51), it suffices to consider
$H_{m}=m(1-\tau_{m})\varphi_{2m}^{2}+m\tau_{\infty}K_{2m+1}-\sqrt{m}K_{2m+1}$
(4.62)
$H_{m+1}-H_{m}=\underbrace{(m+1)(1-\tau_{m+1})\varphi_{2(m+1)}^{2}-m(1-\tau_{m})\varphi_{2m}^{2}}_{\text{(I)}}\\\
\underbrace{+(m+1)\tau_{\infty}K_{2m+3}-m\tau_{\infty}K_{2m+1}}_{\text{(II)}}\underbrace{-\sqrt{m+1}K_{2m+3}+\sqrt{m}K_{2m+1}}_{\text{(III)}}$
(4.63)
Since $(1-\tau_{m})\varphi_{2m}^{2}=(1+\tau_{m})\varphi_{2(m+1)}^{2}$, we have
$\displaystyle\frac{\text{(I)}}{Q_{m+1}-Q_{m}}=\displaystyle\frac{(m+1)(1-\tau_{m+1})-m(1+\tau_{m})}{2}$
(4.64)
$\displaystyle\frac{\text{(II)}}{Q_{m+1}-Q_{m}}=\tau_{\infty}(m+1)+(m+1-m)\tau_{\infty}\displaystyle\frac{K_{2m+1}}{2\varphi_{2(m+1)}^{2}}$
(4.65)
$\displaystyle\frac{\text{(III)}}{Q_{m+1}-Q_{m}}=\displaystyle\frac{(-\sqrt{m+1}+\sqrt{m})K_{2m+3}}{2\varphi_{2(m+1)}^{2}}+(-\sqrt{m})$
(4.66)
This proves that $L^{\prime}=0$ and thus (4.60).
Next, we compute
$L_{3}^{\prime}=\displaystyle\lim_{m\to\infty}m^{3/2}\left(\Delta_{2m+1}+\tau_{\infty}-\displaystyle\frac{1}{\sqrt{m}}\right)$
(4.67)
By similar arguments as in (4.64), (4.65) and (4.66), we conclude that
$L_{3}^{\prime}=1+\displaystyle\frac{1}{2\tau_{\infty}}$ (4.68)
This concludes the proof of Lemma 4.1 ∎
Finally, we apply the Szegő map to this perturbed measure $d\tilde{\mu}_{y}$
to form the perturbed measure $d\tilde{\gamma}_{y}$ on $[-2,2]$, which is
defined by
$\tilde{\gamma}_{y}(x)=(1-\gamma)d\gamma_{y}(x)+\gamma\delta_{x=2}$ (4.69)
For the sake of convenience, we temporarily denote
$\alpha_{n}\equiv\alpha_{n}(d\tilde{\mu}_{y})$. Since
$b_{n}(d\gamma_{y})\equiv 0$, if suffices to consider
$b_{n+1}(d\gamma_{y})=\displaystyle\frac{1}{\sqrt{n}}-\displaystyle\frac{1}{\sqrt{n-1}}+o\left(\displaystyle\frac{1}{n^{3/2}}\right)=\frac{-1}{2n^{3/2}}+o\left(\frac{1}{n^{3/2}}\right)$
(4.70)
It is more complicated with $a_{n}(d\tilde{\gamma_{y}})$. Recall that
$a_{n+1}(d\tilde{\gamma}_{y})^{2}=(1-\alpha_{2n-1})(1-\alpha_{2n}^{2})(1+\alpha_{2n+1})\\\
$ (4.71)
and we know that
$a_{n+1}(d\gamma_{y})^{2}=(1-\tau_{n-1})(1+\tau_{n})$ (4.72)
Therefore, upon solving the algebra, we obtain
$a_{n+1}(d\tilde{\gamma}_{y})^{2}-a_{n+1}(d\gamma_{y})^{2}=\displaystyle\frac{1}{2(1+\tau_{\infty})m^{3/2}}+o\left(\displaystyle\frac{1}{m^{3/2}}\right)$
(4.73)
Upon scaling, we have
$a_{n+1}^{2}(d{\gamma})-a_{n+1}^{2}(d\gamma_{0})=\displaystyle\frac{2}{y^{2}(1+\tau_{\infty})m^{3/2}}+o\left(\displaystyle\frac{1}{m^{3/2}}\right)$
(4.74)
## 5\. acknowledgement
I would like to thank Professor Barry Simon for suggesting this problem and
for all the very helpful discussions.
## References
* [1] G. Borg, _Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte_ , Acta Math. 78 (1946), 1–96.
* [2] A. Cachafeiro and F. Marcellán, _Asymptotics for the ratio of the leading coefficients of orthogonal polynomials associated with a jump modification_ , in ”Approximation and Optimization” (Havana, 1987), 111–117, Lecture Notes in Math., 1354, Springer, Berlin, 1988.
* [3] A. Cachafeiro and F. Marcellán, _Orthogonal polynomials and jump modifications_ , in ”Orthogonal Polynomials and Their Applications”, (Segovia, 1986), pp. 236–240, Lecture Notes in Math., 1329, Springer, Berlin, 1988.
* [4] A. Cachafeiro and F. Marcellán, _Perturbations in Toeplitz matrices_ , in Orthogonal Polynomials and Their Applications , (Laredo, 1987), pp. 139–146, Lecture Notes in Pure and Applied Math., 117, Marcel Dekker, New York, 1989.
* [5] A. Cachafeiro and F. Marcellán, _Perturbations in Toeplitz matrices: Asymptotic properties_ , J. Math. Anal. Appl. 156 (1991) 44–51.
* [6] A. Cachafeiro and F. Marcellán, _Modifications of Toeplitz matrices: jump functions_ , Rocky Mountain J. Math. 23 (1993), 521–531.
* [7] E. Cesàro and O. Stolz, _http://en.wikipedia.org/wiki/Stolz-Cesàro_theorem_.
* [8] I. M. Gel’fand and B. M. Levitan, _On the determination of a differential equation from its spectral function_ , Amer. Math. Soc. Transl. (2) 1 (1955), 253–304; Russian original in Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309–360.
* [9] Ya. L. Geronimus, _On the trigonometric moment problem_ , Ann. of Math. (2) 47 (1946), 742–761.
* [10] Ya. L. Geronimus, _Polynomials Orthogonal on a Circle and Their Applications_ , Amer. Math. Soc. Translation 1954 (1954), no. 104, 79pp.
* [11] Ya. L. Geronimus, _Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval_ , Consultants Bureau, New York, 1961.
* [12] P. Nevai, _Orthogonal polynomials, measures and recursions on the unit circle_ , Trans. Amer. Math. Soc. 300 (1987), 175–189.
* [13] P. Nevai, _Orthogonal Polynomials_ , Mem. Amer. Math. Soc. 18 (1979), no. 213.
* [14] B. Simon, _Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory_ , AMS Colloquium Series, Amer. Math. Soc, Providence, RI, 2005.
* [15] B. Simon, _The Christoffel–Darboux kernel_ , to appear in Proc. Sympos. Pure Math. 79 (2008), 295–335.
* [16] M.-W. L. Wong, _A formula for inserting point masses_ , to appear in J. Comput. and Appl. Math.
* [17] M.-W. L. Wong, _Generalized bounded variation and inserting point masses_ , to appear in Const. Approx.
* [18] M.-W. L. Wong, _Asymptotics of orthogonal polynomials and point perturbation in a gap_ , preprint.
|
arxiv-papers
| 2010-09-08T20:02:26 |
2024-09-04T02:49:12.765610
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Manwah Lilian Wong",
"submitter": "Manwah Wong",
"url": "https://arxiv.org/abs/1009.1631"
}
|
1009.1698
|
arxiv-papers
| 2010-09-09T07:41:42 |
2024-09-04T02:49:12.771963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Philip Ilten",
"submitter": "Philip Ilten",
"url": "https://arxiv.org/abs/1009.1698"
}
|
|
1009.1756
|
# Conductance and Eigenvalue
Girish Varma
girish@tcs.tifr.res.in
School of Technology and Computer Science
Tata Institute of Fundamental Research, Mumbai
###### Abstract
We show the following.
###### Theorem.
Let $M$ be an finite-state ergodic time-reversible Markov chain with
transition matrix $P$ and conductance $\phi$. Let $\lambda\in(0,1)$ be an
eigenvalue of $P$. Then, $\phi^{2}+\lambda^{2}\leq 1$ This strengthens the
well-known [4, 3, 2, 1, 5] inequality $\lambda\leq 1-\phi^{2}/2$. We obtain
our result by a slight variation in the proof method in [5, 4]; the same
method was used earlier in [6] to obtain the same inequality for random walks
on regular undirected graphs.
A _Markov chain_ is a sequence of random variables $\\{X_{i}\\}_{i\geq 1}$
taking values in a finite set such that
$\displaystyle\Pr[X_{t}=i\mid
X_{t-1}=j,X_{t-2}=x_{t-2},\cdots,X_{0}=x_{0}]=\Pr[X_{t}=i\mid X_{t-1}=j].$
Let the state space of the Markov chain be $[n]$ and let $P=(P_{ij})$ be its
$n\times n$ transition matrix: $P_{ij}=\Pr[X_{t}=i\mid X_{t-1}=j]$. We will
assume that the Markov chain is ergodic, that is, irreducible( for every pair
of states $i,j\in[n]$, $P^{s}_{ij}>0$ for some $s$) and aperiodic(for any
state $i\in[n]$, $\text{gcd}\\{s\ :\ P^{s}_{ii}>0\\}=1$). Then, the Markov
chain has a unique stationary distribution $\pi$: $P\pi=\pi$. We say that the
Markov chain is time-reversible if it satisfies the following detailed balance
condition:
$\forall i,j\in[n],~{}P_{ij}\pi_{j}=P_{ji}\pi_{i}$ (1)
All Markov chains considered in this note will be assumed to be finite-state
ergodic and time-reversible. The _conductance_ of a Markov chain with state
space $[n]$ is defined to be
$\phi=\min_{S\subset[n]:\sum_{i\in S}\pi_{i}\leq 1/2}\frac{\sum_{i\in
S,j\notin S}P_{ji}\pi_{i}}{\sum_{i\in S}\pi_{i}}$
The following theorem plays a central role in the theory of rapidly mixing
Markov chains.
###### Theorem ([5]).
Let $\lambda<1$ be an eigenvalue of the transition matrix of an ergodic time-
reversible Markov chain with conductance $\phi$. Then, $\lambda\leq
1-\frac{\phi^{2}}{2}$.
In this note we strengthen this inequality slightly.
###### Theorem.
Let $\lambda\in(0,1)$ be an eigenvalue of the transition matrix of an ergodic
time-reversible Markov chain with conductance $\phi$. Then,
$\phi^{2}+\lambda^{2}\leq 1$
Such an inequality was derived by Radhakrishnan and Sudan [6] for the special
case of random walks on regular undirected graphs. The purpose of this note is
to show that their arguments (which were a slight variation on the arguments
in [5, 4]) apply to finite-state ergodic time-reversible Markov chains as
well.
###### Proof.
Let $\pi$ be the stationary distribution of the chain with transition matrix
$P$. Let $f,g\in\mathbb{R}^{n}$. We will be thinking of $f,g,\pi$ as vectors
in $\mathbb{R}^{n}$. Let
$\langle f,g\rangle=\sum_{i\in[n]}f_{i}\pi_{i}g_{i}$
and $||f||=\sqrt{\langle f,f\rangle}$. $f$ is said to be _proper_ if
$f\neq 0~{}\text{ and }~{}\forall i\in[n],~{}f_{i}\geq 0~{}\text{ and
}\sum_{i\in[n]:f_{i}>0}\pi_{i}\leq\frac{1}{2}$
We have the following two claims.
###### Claim 1.
For any proper $f$,
$\phi^{2}||f||^{4}\leq||f||^{4}-\langle f,P^{T}f\rangle^{2}$ (2)
###### Claim 2.
For $\lambda\in(0,1)$, there exists a proper $f$ such that
$\langle f,P^{T}f\rangle\geq\lambda||f||^{2}$ (3)
Using (2) and (3), we obtain
$\phi^{2}||f||^{4}\leq||f||^{4}-\lambda^{2}||f||^{4}$
from which the theorem follows.
Proof of Claim 1. Permute the co-ordinates of $f$ such that $f_{1}\geq
f_{2}\geq\cdots\geq f_{r}>0\text{ and }f_{r+1}=\cdots=f_{n}=0$. (Note that
$\sum_{i\in[r]}\pi_{i}\leq 1/2$.) We show that
$\phi^{2}||f||^{4}\leq\left[\sum_{i<j}P_{ij}\pi_{j}(f_{i}^{2}-f_{j}^{2})\right]^{2}\leq||f||^{4}-\langle
f,P^{T}f\rangle^{2}$
To see the first inequality, we observe that
$\displaystyle\sum_{i<j}P_{ij}\pi_{j}(f_{i}^{2}-f_{j}^{2})$
$\displaystyle=\sum_{i<j}P_{ij}\pi_{j}\sum_{i\leq k<j}(f_{k}^{2}-f_{k+1}^{2})$
$\displaystyle=\sum_{k\in[r]}(f_{k}^{2}-f_{k+1}^{2})\sum_{i\in[k],j\notin[k]}P_{ij}\pi_{j}$
$\displaystyle\geq\phi\sum_{k\in[r]}(f^{2}_{k}-f^{2}_{k+1})\left(\sum_{i\in[k]}\pi_{i}\right)$
$\displaystyle=\phi\sum_{k\in[r],i\in[k]}(f^{2}_{k}-f^{2}_{k+1})\pi_{i}$
$\displaystyle=\phi\sum_{i\in[r]}\pi_{i}f^{2}_{i}$
$\displaystyle=\phi||f||^{2}$
Secondly
$\displaystyle\sum_{i<j}P_{ij}\pi_{j}(f_{i}^{2}-f_{j}^{2})$
$\displaystyle=\sum_{i<j}\sqrt{P_{ij}\pi_{j}}(f_{i}-f_{j})\sqrt{P_{ij}\pi_{j}}(f_{i}+f_{j})$
$\displaystyle\leq\left[\sum_{i<j}P_{ij}\pi_{j}(f_{i}-f_{j})^{2}\sum_{i<j}P_{ij}\pi_{j}(f_{i}+f_{j})^{2}\right]^{\frac{1}{2}}$
(by the Cauchy-Schwarz inequality)
The calculations up to this point are identical to those in [5, 4]; the
calculations below are similar to those in [6].
$\displaystyle=$
$\displaystyle\left[\sum_{i<j}P_{ji}\pi_{i}(f_{i}^{2}+f_{j}^{2}-2f_{i}f_{j})\sum_{i<j}P_{ji}\pi_{i}(f_{i}^{2}+f_{j}^{2}+2f_{i}f_{j})\right]^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left[\left(\sum_{ij}P_{ji}\pi_{i}f_{i}^{2}-\sum_{ij}P_{ji}\pi_{i}f_{i}f_{j}\right)\left(\sum_{ij}P_{ji}\pi_{i}f_{i}^{2}+\sum_{ij}P_{ji}\pi_{i}f_{i}f_{j}-2\sum_{i}P_{ii}\pi_{i}f_{i}^{2}\right)\right]^{\frac{1}{2}}(\text{using
}\ref{det-bal})$ $\displaystyle\leq$
$\displaystyle\left[\left(\sum_{i}\pi_{i}f_{i}^{2}-\sum_{ij}P_{ji}\pi_{i}f_{i}f_{j}\right)\left(\sum_{i}\pi_{i}f_{i}^{2}+\sum_{ij}P_{ji}\pi_{i}f_{i}f_{j}\right)\right]^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left[\left(\sum_{i}\pi_{i}f_{i}^{2}\right)^{2}-\left(\sum_{ij}P_{ji}\pi_{i}f_{i}f_{j}\right)^{2}\right]^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\left[||f||^{4}-\langle
f,P^{T}f\rangle^{2}\right]^{\frac{1}{2}}$
Proof of Claim 2. Let $g\in\mathbb{R}^{n}$ be a right eigenvector of $P^{T}$
with eigenvalue $\lambda\in(0,1)$ . We may assume
$\sum_{i:g(i)>0}\pi_{i}\leq\frac{1}{2}$ (otherwise consider $-g$). By renaming
the co-ordinates we may assume that $g_{1}\geq g_{2}\geq\cdots\geq g_{r}>0\geq
g_{r+1}\geq g_{r+2}\geq\cdots\geq g_{n}$. Let $f$ be such that $f_{i}=g_{i}$
for $i\in[r]$ and $0$ otherwise. Then
$\forall i\in[r],~{}(P^{T}f)_{i}\geq(P^{T}g)_{i}=\lambda g_{i}=\lambda f_{i}$
Then, $\langle
f,P^{T}f\rangle=\sum_{i\in[r]}\pi_{i}f_{i}(P^{T}f)_{i}\geq\lambda\sum_{i\in[r]}\pi_{i}f_{i}^{2}=\lambda||f||^{2}$.
∎
## References
* [1] N Alon. Eigen values and expanders. Combinatorica, 6(2):83–96, 1986.
* [2] N. Alon and V. D. Milman. $\lambda_{1}$, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38(1):73 – 88, 1985\.
* [3] Jozef Dodziuk. Difference equations, isoperimetric inequality and transience of certain random walks. Transactions of the American Mathematical Society, 284(2):pp. 787–794, 1984.
* [4] Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S, 43:439–561, 2006.
* [5] Alistair Sinclair and Mark Jerrum. Approximate counting, uniform generation and rapidly mixing markov chains. Inf. Comput., 82(1):93–133, 1989.
* [6] J Radhakrishnan M Sudan. On Dinur’s proof of the PCP theorem. Bull. Amer. Math. Soc., 44:19–61, 2007.
|
arxiv-papers
| 2010-09-09T12:39:41 |
2024-09-04T02:49:12.777629
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Girish Varma",
"submitter": "Girish Varma",
"url": "https://arxiv.org/abs/1009.1756"
}
|
1009.1896
|
# Detailed X-Ray Line Properties of in Quiescence
Arik W. Mitschang11affiliation: Smithsonian Astrophysical Observatory (SAO),
Cambridge, MA , Norbert S. Schulz22affiliation: MIT Kavli Institute for
Astrophysics and Space Research, Cambridge, MA , David P.
Huenemoerder22affiliation: MIT Kavli Institute for Astrophysics and Space
Research, Cambridge, MA , Joy S. Nichols11affiliation: Smithsonian
Astrophysical Observatory (SAO), Cambridge, MA , Paola Testa11affiliation:
Smithsonian Astrophysical Observatory (SAO), Cambridge, MA
###### Abstract
We investigate X-ray emission properties of the peculiar X-ray source in the
Orion trapezium region using more than 500 ksec of HETGS spectral data in the
quiescent state. The amount of exposure provides tight constraints on several
important diagnostics involving O, Ne, Mg, and Si line flux ratios from He-
like ion triplets, resonance line ratios of the H- and He-like lines and line
widths. Accounting for the influence of the strong UV radiation field of the
O9.7V star we can now place the He-like line origin well within two stellar
radii of the O-star’s surface. The lines are resolved with average line widths
of $341\pm$38 km s-1 confirming a line origin relatively close to the stellar
surface. In the framework of standard wind models this implies a rather weak,
low opacity wind restricting wind shocks to temperatures not much larger than
2$\times 10^{6}$ K. The emission measure distribution of the X-ray spectrum,
as reported previously, includes very high temperature components which are
not easily explained in this framework. The X-ray properties are also not
consistent with coronal emissions from an unseen low-mass companion nor with
typical signatures from colliding wind interactions. The properties are more
consistent with X-ray signatures observed in the massive Trapezium star
$\theta^{1}$ Ori C which has recently been successfully modeled with a
magnetically confined wind model.
stars: magnetic fields – stars: winds, outflows – X-rays: stars – stars:
individual ()
## 1 Introduction
$\theta^{2}$ Ori A is a triple star system at the heart of the Orion Nebula
Cluster (ONC), Its massive primary has been identified as a 5th magnitude O9.5
V star (Abt et al., 1991) with a mass of 25 M⊙ (Preibisch et al., 1999),
making it the second most massive star in the ONC next to the 45 M⊙ O5.5 V
star of $\theta^{1}$ Ori C. A more recent photometric study provides an
optical identification of O9 V and a total system mass of 39$\pm 14$ M⊙
(Simón-Díaz et al., 2006). The studies of Abt et al. (1991) and Preibisch et
al. (1999) show that this system includes two close intermediate mass
companions at 174 AU and 0.47 AU separation with mass estimates between 7 and
9 M⊙ for each.
$\theta^{2}$ Ori A has been extensively monitored in X-rays with The Chandra
X-ray Observatory and has shown its fair share of odd behavior. Observations
in 2000 found that the X-ray source exhibited unusual and dramatic variability
with a 50$\%$ flux drop in less than 12 hours accompanied by multiple small
flares with only a few hours durations (Feigelson et al., 2002). Such behavior
in an early type stellar system is surprising since this can not be explained
by the standard wind shock models for X-rays in early type stars (Lucy, 1982;
Owocki et al., 1988), nor by the magnetically confined wind models (MCWMs;
Babel & Montmerle (1997)). While the MCWM can produce hard X-ray emission like
observed in $\theta^{1}$ Ori C (Schulz et al., 2000, 2003; Gagné et al., 2005)
and $\tau$ Sco (Cohen et al., 2003)., it does not explain the observed
variability in . At the time, the suggestion was made that such emission could
be the result of magnetic reconnection events. To add to this excitement, a
specifically powerful X-ray flare from , seen with the Chandra High Energy
Transmission Grating Spectrometer (HETGS), surprised observers in 2004 (Schulz
et al., 2006) and produced a total power output exceeding 1037 ergs s-1.
Considering the orbital phase of the close spectroscopic companion, the low
He-like forbidden/intercombination line ratios, and the fact that all lines
remained unresolved led to the argument that these events are triggered by
magnetic interactions with the close companion. A sub-pixel re-analysis of a
similar flare event which appeared during observations in the Chandra Orion
Ultradeep Project (COUP) (Stelzer et al., 2005; Schulz et al., 2006) in 2003,
however, seem to indicate that these events may originate from the companion
instead (M. Gagne, priv. comm.). An unseen T Tauri companion appears unlikely
due to the observed peculiar line properties.
In contrast to that observed in the elevated states, the quiescent spectrum of
exhibits temperatures above 25 MK and has line ratios which suggest that the
X-ray emitting plasma is close enough to the stellar surface of the massive
star to argue for some form of magnetic confinement (Schulz et al., 2006). The
argument is strengthened by the fact that the line widths, quite in contrast
to the narrow line widths observed during the outbursts, seem broadened to the
order of 300 km s-1. These properties are very reminiscent of the MCWM results
obtained in $\theta^{1}$ Ori C (Gagné et al., 2005), where, through detailed
simulations, it was demonstrated that the bulk of the emitting plasma is close
to the photosphere, or within $\sim$2 R⋆, and line widths are $\leq$400 km
s-1. However, in spite of these apparent differences, the properties of the
quiescent state remained fairly unconstrained with respect to precise line
ratios and widths. $\theta^{2}$ Ori A’s X-ray luminosity is about an order of
magnitude lower than than that observed during outburst and the study remained
statistically limited.
The Chandra Data Archive (CDA)111http://cxc.harvard.edu/cda/ now contains an
additional $\sim$ 300 ks on $\theta^{2}$ Ori A between 2004 and 2008 and in
this paper we present a full analysis of the quiescent spectrum allowing us to
derive much better constrained line properties. The results are also used to
test the hypothesis that the X-ray emission from $\theta^{2}$ Ori A is
consistent with predictions from the MCWM. The paper is structured as follows;
in Section 2 we discuss the observations and analysis methods, in Section 3 we
discuss the results of our emission line measurements, and finally we
summarize our findings in Section 4
## 2 Observations and Analysis
We have retrieved Chandra HETGS data in the vicinity of the ONC, which were
originally observed as a part of the HETG Orion Legacy Project (Schulz et al.,
2008), from the CDA. There are now seventeen separate Chandra observations
which include within an off-axis angle suitable for extraction. See Table 1
for a list of the included observations and selected properties. Noting that
this study is focused on the quiescent state spectrum and that ObsID 4474 was
not included in any analysis in the current study due to the substantially
elevated count rate during its entire exposure, we have accumulated 520 ks of
exposure time on in the quiescent state. Figure 1 shows the total combined
counts spectrum using the 520 ks on .
Figure 1: Counts spectrum from the total combined data on (MEG+HEG). Table 1: Observation Log Sequence | ObsID | Start Date | Start Time | Exposure | Offsetaa zeroth order position offset from nominal pointing | Phase RangebbAssuming a 20.974 day period and periastron passage at HJD∘=2440581.27 (Abt et al., 1991)
---|---|---|---|---|---|---
Number | | (UT) | (UT) | ($ks$) | arcmin |
200001 | 3 | 1999-10-31 | 05:58:56 | 49.6 | 2.42 | 0.76-0.79
200002 | 4 | 1999-11-24 | 05:39:24 | 30.9 | 2.28 | 0.92-0.99
200175 | 2567 | 2001-12-28 | 12:25:56 | 46.4 | 1.98 | 0.99-1.01
200176 | 2568 | 2002-02-19 | 20:29:42 | 46.3 | 2.10 | 0.53-0.55
200242 | 4473 | 2004-11-03 | 01:48:04 | 49.1 | 1.26 | 1.00-1.03
200243 | 4474cc4474 is included here only for reference, no analysis herein utilized it due to the extremely elevated count rate during its entirety | 2004-11-23 | 07:48:38 | 50.8 | 1.39 | 0.96-0.99
200420 | 7407 | 2006-12-03 | 19:07:48 | 24.6 | 1.64 | 0.27-0.29
200423 | 7410 | 2006-12-06 | 12:11:37 | 13.1 | 3.02 | 0.40-0.41
200421 | 7408 | 2006-12-19 | 14:17:30 | 25.0 | 2.08 | 0.02-0.04
200422 | 7409 | 2006-12-23 | 00:47:40 | 27.1 | 2.30 | 0.19-0.21
200424 | 7411 | 2007-07-27 | 20:41:22 | 24.6 | 3.94 | 0.53-0.54
200425 | 7412 | 2007-07-28 | 06:16:09 | 25.2 | 4.39 | 0.55-0.56
200462 | 8568 | 2007-08-06 | 06:54:08 | 36.1 | 2.53 | 0.98-1.00
200462 | 8589 | 2007-08-08 | 21:30:35 | 50.7 | 2.53 | 0.10-0.13
200478 | 8897 | 2007-11-15 | 10:03:16 | 23.7 | 3.37 | 0.80-0.81
200477 | 8896 | 2007-11-30 | 21:58:33 | 22.7 | 2.34 | 0.54-0.55
200476 | 8895 | 2007-12-07 | 03:14:07 | 25.0 | 1.74 | 0.84-0.85
As noted, none of these observations were targeted at ; indeed no Chandra
gratings observations have ever targeted . However using the suite of advanced
extraction tools provided by the Chandra Transmissions Grating Catalog and
Archive (TGCat; Huenemoerder et al. (2010); Mitschang et al.
(2010))222http://tgcat.mit.edu, extraction of the dispersed counts of off-axis
X-ray source positions proved to be trivial.
Grating spectra were extracted and responses computed using TGCat software to
locate the optimal centroid position of and apply proper calibration for each
observation. In a crowded field such as the Orion Trapezium, careful attention
must be made during analysis to contamination from other zeroth order counts
lying close to or on top of dispersion counts and dispersion arms crossing one
another at critical points. To this end, we reviewed order sorting images
(ACIS CCD event energy vs. gratings order $\times$ wavelength, or specifically
FITS-file columns $\mathrm{TG\\_MLAM}$ vs. $\mathrm{ENERGY}$) for each
observation and identified potential contamination. In this view, the source
traces two hyperbolas centered on $m\lambda=0$ (e.g. see Chandra
POG333http://cxc.harvard.edu/proposer/POG/ Fig 8.13); traces from confusing
sources show as offset hyperbolas (dispersed) or vertical lines (zeroth
order). We found no significant source of contamination in the regions used
for line fitting; See Table 2 for details on the locations of these regions.
Similarly when fitting the continuum we used a set of wavelength ranges
containing few lines, the “line free regions”, in which we found little
contamination. See Section 2.1 for a more detailed discussion on the continuum
modeling and Section 2.2 on line fitting. Line width analysis is treated
separately in Section 2.3.
All fitting of data was done using the Interactive Spectral Interpretation
System (ISIS; Houck & Denicola (2000))444http://space.mit.edu/CXC/isis, along
with the Astrophysical Plasma Emission Database (APED; Smith et al.
(2001))555http://cxc.harvard.edu/atomdb/sources_aped.html for line
emissivities and continuum modeling.
### 2.1 Continuum
The continuum emission of was modeled by fitting a single temperature APED
model to the combined MEG+HEG counts for all observations to improve
statistics. In order to fit only the continuum emission, we selected a set of
narrow bands, considered free of significant line emission, specifically
2.00-2.95Å, 4.4-4.6Å, 5.3-6.0Å, 7.5-7.8Å, 12.5-12.7Å and 19.1-20 Å (e.g. see
Testa et al. (2007)). We assumed a hydrogen column density (NH) of $2\times
10^{21}$ cm-2. Potential contamination resulting from cross-dispersion or
zeroth order confusion was mitigated in these regions by simply ignoring the
affected region of an individual order during the computation of the fit. The
resulting continuum model was then used when fitting lines.
### 2.2 Line Fluxes & Ratios
Figure 2: Fit to the fir triplet $\mathcal{G}$\- and $\mathcal{R}$-ratios for,
clockwise from top left, Si XIII, Mg XI, O VII, Ne IX showing the predicted
line profile in red, data and errors in black and gray respectively. The line
centroid positions for each $fir$ component are given here for clarity
[$r$,$i$,$f$]: Si XIII [6.65,6.68,6.74], Mg XI [9.17,9.24,933], O VII
[21.60,21.80,22.11] and Ne IX [13.45,13.55,13.69].
Figure 3: Confidence contours for measured $\mathcal{G}$\- and
$\mathcal{R}$-ratios for, clockwise from top left, Si XIII, Mg XI, O VII, Ne
IX. The red inner contours show 1$\sigma$, green middle contours show
2$\sigma$ and outer blue contours show 3$\sigma$ confidences.
The fir (forbidden, intercombination, and resonance) line ratios given by
$\mathcal{R}$$=f/i$ and $\mathcal{G}$$=(f+i)/r$ have been shown to be probes
of both density ($\mathcal{R}$) and temperature ($\mathcal{G}$) (Gabriel &
Jordan, 1969) in X-ray emitting plasmas, and in the presence of a strong UV
radiation field, such as is typical in O stars like , Waldron & Cassinelli
(2001) demonstrated that the $\mathcal{R}$ value rather acts as a proxy for
the radial distance of X-ray emission from the stellar surface. Specifically
for the $\mathcal{R}$-ratio, it is also important to make the comparison
between the observed ratio and that of the low density limit. Blumenthal et
al. (1972) showed that
$\mathcal{R}=\frac{\mathcal{R}_{\circ}}{1+\frac{\phi}{\phi_{c}}+\frac{n}{n_{c}}}$
(1)
where $\phi/\phi_{c}$ is a measure of the photo-excitation, $n/n_{c}$ is a
measure of the density and $\mathcal{R}$o, $\phi_{c}$, and $n_{c}$ depend only
on atomic parameters and temperature. It is easily seen from Eq. 1 that,
ignoring photo-excitation, $\mathcal{R}$o=$\mathcal{R}$ when $n/n_{c}\ll 1$
and thus represents the low density limit. We have computed $\mathcal{R}$o
using emissivities in APED and temperatures derived from the $\mathcal{G}$
ratio given in Table 3 for each $fir$ triplet and list them in Table 2. In
order to test the MCWM predictions we derive the radial distance
($R/R_{\star}$) using these $fir$ ratios, a surface temperature of 30,000 K
for the 09.7V star and photo-excitation and decay rates from Blumenthal et al.
(1972). Figure 4 shows the dependence curves with measured values and 90%
confidence intervals over-plotted.
In cases where there were significant contributions from other lines or line
groups, as in the case of Ne IX $fir$ triplet where Fe XIX and Fe XXI converge
and blend, those lines were included in the model. A special case is Ne X
which is unresolvably blended with FeXVII. In this case we assumed the Fe
component contributed flux equaling 13% of the flux of a prominent FeXVII line
at 15.01Å (e.g. see Walborn et al. (2009)). Additionally, the Mg Ly-series
converges at the centroid position of the Si XIII f-line where we assumed,
based on the theoretical relative line strengths, the observed flux was
overestimated by 10% of the measured flux of the isolated H-Like Mg XII
Ly$\alpha$ line.
When fitting the He-like $fir$ triplet lines, the relative separation of the
lines was fixed and the positions of the resonance lines were constrained by
their rest positions. Where available, we fit using both MEG and HEG counts,
where MEG counts were rebinned onto the HEG grid whose intrinsic channel size
is half that of the MEG. Fits were performed by applying Gaussian functions
for each contributing line. Several contaminating lines known to be in the
vicinity were included as well. The $\mathcal{G}$\- and $\mathcal{R}$-ratios
were computed directly during the fitting procedure and the $fir$ fluxes were
treated co-dependently. The instrumental profile was included as calibration
data while the excess width was included as a gaussian turbulent broadening
term ($v_{\mathrm{turb}}$). In Figures 2 and 3 the triplet regions are shown
with residuals, over-plotted models, and computed confidence contours.
### 2.3 Line Widths
Due to degradation of Chandra image quality at off-axis angles, the HETG
resolving power likewise decreases. Though the PSF is well defined across the
ACIS detector, this degradation becomes a problem for gratings because, owing
to the complexity of modeling, responses are only calibrated for zeroth order
positions at the instrument nominal pointing.
This effect can be critical in line width measurements which may include a
significant instrumental broadening signature. Our flux measurements are
unaffected by the broadening, and we have utilized as much available data as
possible to improve statistics. Four of our observations are at off-axis
angles greater than the others, in particular is greater than 3′ off-axis in
obsids 7410, 7411, 7412, and 8897. We have chosen to ignore counts in these
obsids during computation of line width parameters. There are two exceptions.
Si XIV and Mg XII where statistics are too poor in the absence of extra counts
to obtain reasonable measurements. In these cases we provide upper limits on
the line widths.
The average offset of our data is 2′.1 which is around the location where
degradation becomes noticeable. Based on analysis of ACIS zeroth order Line
Response Functions (LRFs) at large axial offsets (e.g. see Chandra POG), we
estimate that our reported line widths are on the order of up to $\sim$5%
broader than that of identical on-axis profiles.
Table 2: Line Measurements ION | $\lambda$aaMeasured position of resonance line for He-like triplet line groups | fluxbbflux is that of the resonance line only for He-like triplet line groups | Line Ratiosccfor H-Like Ly$\alpha$ lines this is the ratio of the H-Like Ly$\alpha$ flux to He-Like resonance line flux of the corresponding ion | $v_{\mathrm{turb}}$
---|---|---|---|---
| (Å) | ($10^{-6}\mathrm{phots^{-1}cm^{-2}}$) | $\mathcal{G}$ | $\mathcal{R}$ | $\mathcal{R}$oddComputed from APED emissivities according to Eq. 1 at $\mathcal{G}$-ratio derived temperatures (see Table 3) | ($\mathrm{kms^{-1}}$)
He-Like Lines
Si XIII | 6.650 | 1.3$\pm$0.3 | 1.1$\pm$0.3 | 1.7$\pm$0.4 | 2.4 | 491$\pm$120
Mg XI | 9.171 | 4.8$\pm$0.6 | 1.0$\pm$0.1 | 0.3$\pm$0.1 | 3.1 | 432$\pm$53
Ne IX | 13.448 | 28.9$\pm$3.3 | 1.2$\pm$0.1 | 0.1$\pm$0.04 | 2.8 | 228$\pm$34
O VII | 21.602 | 147.3$\pm$40.9 | 0.9$\pm$0.3 | $<$0.09 | 4.1 | 274$\pm$83
H-Like Ly$\alpha$ Lines $\frac{\mathrm{H\ Ly}\alpha}{\mathrm{He\ Ly}\alpha}$
Si XIV | 6.187 | 0.4$\pm$0.2 | 0.3$\pm$0.2 | $<$686
Mg XII | 8.423 | 0.8$\pm$0.3 | 0.2$\pm$0.1 | $<$518
Ne X | 12.133 | 20.0$\pm$2.9 | 0.7$\pm$0.1 | 315$\pm$43
O VIII | 18.971 | 164.7$\pm$22.3 | 1.1$\pm$0.4 | 327$\pm$53
## 3 Discussion
The exposure obtained from the Chandra archive of represents the deepest
combined high resolution spectroscopic dataset on this young massive O-star to
date. The long exposure provides high statistics in critical emission lines,
allowing to diagnose its X-ray stellar wind properties beyond the $3\sigma$
level. In a previous study Schulz et al. (2006) provided some preliminary
results for the quiescent state for less than half of the current exposure.
This limited measurements of critical line fluxes and widths to uncertainties
larger than 50$\%$. Our new analysis greatly reduces these uncertainties to
the order of 20$\%$. For example, while the previous analysis could only
speculate about possible line broadening of the order of 300 km s-1, we now
clearly resolve the lines to values between 228$\pm$34 km s-1 for Ne IX and
491$\pm$120 km s-1 for Si XIII, with an average of all lines of 341$\pm$38 km
s-1. Likewise critical line ratios such as the $\mathcal{R}$-ratios are
significantly improved, specifically for the cases of Mg XI with 0.3$\pm$0.09
and Si XIII with 1.7$\pm$0.4; for the case of O VII since its $f$-line was not
detected, we now also have an upper limit.
The measured $\mathcal{R}$-ratios are significantly less than $\mathcal{R}$o
(Table 2). In early type stars this is due to the substantial UV radiation
field provided by blackbody radiation of the hot surface temperature (Kahn et
al., 2001; Gabriel & Jordan, 1969), which for the O9.7V star in is about
30,000 K. In this case the $\mathcal{R}$ ratio maps the distance of emission
from the stellar surface and we utilize this relation to show that the X-ray
line emission from is indeed located close to the O-star’s surface. Schulz et
al. (2006) estimated that the emissions could be within several stellar radii,
Table 2 and Figure 4 show emission origins within two stellar radii (dotted
line in Figure 4) for Si XIII, Mg XI, and Ne IX with their 90$\%$
uncertainties.
Table 3: Derived Temperatures ION | Log T(H/He) | Log T(G)
---|---|---
O VII | 6.41 (6.36, 6.44) | 6.3 (6.0, 6.5)
Ne IX | 6.62 (6.60, 6.64) | 6.1 (6.0, 6.3)
Mg XI | 6.70 (6.65, 6.72) | 6.5 (6.3, 6.6)
Si XIII | 6.96 (6.87, 7.02) | 6.4 (6.0, 6.8)
Figure 4: Dependence of $\mathcal{R}$ ratios on the distance to the stellar
surface of the emission from Si XIII, Mg XI, Ne IX, O VII. The diamonds show
the best fit and highlighted lines show 90% confidence limits projected on to
dependence curves computed for a stellar surface temperature of 30,000 K. The
vertical dashed line at 2R⋆ represents the approximate theoretical limit for
generation of X-rays under the MCWM.
Another important result of our analysis is that the measured line centroid
positions shown in Table 2 are, with quite high accuracy, at the expected ion
rest wavelengths indicating that there are no line shifts within the Chandra
sensitivity. This is an important result because any shift would indicate fast
outward moving sources in a high density wind. The line profiles appear
symmetric, supporting a low density wind assumption even though at such low
broadening, profile deviations are almost impossible to trace even at our data
quality.
In the case of $\zeta$ Ori, Waldron & Cassinelli (2001) find that lines are
resolved with comparatively low Doppler velocities of around 900 km s-1,
$\mathcal{R}$-ratios that are characteristic of several stellar radii, an
extremely small Si XIII $\mathcal{R}$-ratio, and symmetric and unshifted
lines. Except for the extremely small Si XIII $\mathcal{R}$-ratio, our results
seem very similar, if not more extreme with respect to line widths and
$\mathcal{R}$-ratios. Our line widths indicate an even lower shock jump
velocity than in the case of $\zeta$ Ori making the formation of the observed
ionization states even more difficult. At 350 km s-1 shock temperatures are
expected to not exceed 2$\times 10^{6}$ K. This discrepancy is supported by
the emissivity distribution of the spectrum which includes X-ray temperatures
greater than 25 MK (Schulz et al., 2006). These results are difficult to
reconcile within the standard wind model. In this respect we conclude that a
picture of a low density wind with shocks produced near its onset is not
particularly convincing.
There are not many scenarios left which could explain our findings. We can
rule out significant contributions of unseen low-mass pre-main sequence
companions by the level of the line broadening. Standard coronal emission
would show unresolved lines or moderate broadening due to orbital motion
(Brickhouse et al., 2001; Huenemoerder et al., 2006); neither is the case
here. Colliding winds are ruled out simply by the fact this would require an
unseen massive companion with a much earlier type than the O9.7, which would
be impossible to hide.
We find, however, quite strong similarities to the most massive star in the
Orion Trapezium $\theta^{1}$ Ori C (Schulz et al., 2003; Gagné et al., 2005).
In the magnetically confined wind scenario, field lines of the magnetic dipole
act to channel emitted material from either pole toward the magnetic equator.
Simulations by Gagné et al. (2005) demonstrate that these two components meet
at the magneto-equator and wind plasma with high tangential velocities
reaching up to 1000 km s-1 collides generating strong shocks and elevate gas
temperatures to tens of millions of degrees, thus producing the observed hard
X-ray emission. Gagné et al. (2005) further demonstrate that the conditions
for X-ray production are quite specific; the post shock in-falling material is
rather cool, and the outflowing material’s density is too low to produce
sufficient X-rays. This places a relatively tight constraint on the location
of the hard X-ray emission around R$\leq$2R⋆.
Another result of the simulations by Gagné et al. (2005) states that the post
shock-heated material is moving slowly, thus generating observed line profiles
much narrower than expected for non-magnetic shock-heated X-ray production in
O stars (Lucy, 1982; Waldron & Cassinelli, 2001). In order to quantify the
expected broadening, Gagné et al. (2005) recreated emission measure and line
profiles from the simulations and found that the turbulent broadening is
expected to be on the order of 250 km s-1, with little to no blueshift in the
line centroid position, which is very close to what we observe in .
## 4 Conclusion
We have analyzed high resolution X-ray spectra from Chandra on the young
massive O star , totaling over 500 ks in the quiescent state, and computed
line widths and $fir$ line ratios for a series of prominent emission lines
appearing in its spectrum. The resulting measurements show relatively narrow
lines at an average width of 341$\pm$38 km s-1 and $\mathcal{R}$-ratio derived
X-ray emitting origin within 2 stellar radii. Comparing these results to the
simulation results of Gagné et al. (2005) for $\theta^{1}$ Ori C, we argue
that the X-ray production mechanism in is most likely via magnetic confinement
of its stellar wind outflows.
We have explored other possibilities, including standard O-star wind models
and close companions, for the the generation of X-rays in but find that none
of these are ideal for explaining the observed spectral properties. Observed
line widths are too low, while shock temperatures too high to satisfy model
predictions in most of these cases.
Finally, we note that this is a comparative analysis, and sets up a case for a
more rigorous analysis specifically aimed at magneto-hydrodynamical modeling
(MHD) using the MCWM similar to that under-taken for $\theta^{1}$ Ori C.
This research has made use of data obtained from the Chandra Data Archive and
software provided by the Chandra X-ray Center (CXC) in the application package
CIAO. This research also made use of the Chandra Transmission Grating Catalog
and archive http://tgcat.mit.edu. Chandra is operated by the Smithsonian
Astrophysical Observatory under NASA contract NAS 8-03060. This work was
supported by NASA through the Smithsonian Astrophysical Observatory (SAO)
contracts NAS 8-03060 and SV3-73016 for the Chandra X-Ray Center and Science
Instruments. Facilities: Chandra
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|
arxiv-papers
| 2010-09-09T22:00:50 |
2024-09-04T02:49:12.784092
|
{
"license": "Public Domain",
"authors": "Arik W. Mitschang (1), Norbert S. Schulz (2), David P. Huenemoerder\n (2), Joy S. Nichols (1), Paola Testa (1) ((1) Smithsonian Astrophysical\n Observatory (2) MIT Kavli Institute for Astrophysics and Space Research)",
"submitter": "Arik Mitschang",
"url": "https://arxiv.org/abs/1009.1896"
}
|
1009.2043
|
# Sampling and Recovery of Multidimensional Bandlimited Functions via Frames
Benjamin Bailey Department of Mathematics, Texas A&M University
College Station, TX 77843, USA abailey@math.tamu.edu
###### Abstract.
In this paper, we investigate frames for $L_{2}[-\pi,\pi]^{d}$ consisting of
exponential functions in connection to oversampling and nonuniform sampling of
bandlimited functions. We derive a multidimensional nonuniform oversampling
formula for bandlimited functions with a fairly general frequency domain. The
stability of said formula under various perturbations in the sampled data is
investigated, and a computationally managable simplification of the main
oversampling theorem is given. Also, a generalization of Kadec’s $1/4$ Theorem
to higher dimensions is considered. Finally, the developed techniques are used
to approximate biorthogonal functions of particular exponential Riesz bases
for $L_{2}[-\pi,\pi]$, and a well known theorem of Levinson is recovered as a
corollary.
## 1\. Introduction
The subject of recovery of bandlimited signals from discrete data has its
origins in the Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem (stated
below), historically the first and simplest such recovery formula. Without
loss of generality, the formula recovers a function with a frequency band of
$[-\pi,\pi]$ given the function’s values at the integers. The WKS theorem has
drawbacks. Foremost, the recovery formula does not converge given certain
types of error in the sampled data, as Daubechies and DeVore mention in [7].
They use oversampling to derive an alternative recovery formula which does not
have this defect. Additionally for the WKS theorem, the data nodes have to be
equally spaced, and nonuniform sampling nodes are not allowed. As discussed in
[15, pages 41-42], nonuniform sampling of bandlimited functions has its roots
in the work of Paley, Wiener, and Levinson. Their sampling formulae recover a
function from nodes $(t_{n})_{n}$, where $(e^{it_{n}x})_{n}$ forms a Riesz
basis for $L_{2}[-\pi,\pi]$. More generally, frames have been applied to
nonuniform sampling, particularly in the work of Benedetto and Heller in [2]
and [3]; see also [15, chapter 10].
In Section 3, we derive a multidimensional oversampling formula, (see equation
(4)), for nonuniform nodes and bandlimited functions with a fairly general
frequency domain; Section 4 investigates the stability of equation (4) under
perturbation of the sampled data. Section 5 presents a computationally
feasible version of equation (4) in the case where the nodes are
asymptotically uniformly distributed. Kadec’s theorem gives a criterion for
the nodes $(t_{n})_{n}$ so that $(e^{it_{n}x})_{n}$ forms a Riesz basis for
$L_{2}[-\pi,\pi]$. Generalizations of Kadec’s 1/4 theorem to higher dimensions
are considered in Section 6, and an asymptotic equivalence of two
generalizations is given. Section 7 investigates approximation of the
biorthogonal functionals of Riesz bases. Additionally, we give a simple proof
of a theorem of Levinson.
This paper forms a portion of the author’s doctoral thesis, which is being
prepared at Texas A & M University under the direction of Thomas Schlumprecht
and N. Sivakumar.
## 2\. Preliminaries
We use the $d$-dimensional $L_{2}$ Fourier transform
$\mathcal{F}(f)(\cdotp)=\int_{\mathbb{R}^{d}}f(\xi)e^{-i\langle\cdotp,\xi\rangle}d\xi,\quad
f\in L_{2}(\mathbb{R})^{d},$
where the inverse transform is given by
$\mathcal{F}^{-1}(f)(\cdotp)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}f(\xi)e^{i\langle\cdotp,\xi\rangle}d\xi,\quad
f\in L_{2}(\mathbb{R})^{d}.$
This is an abuse of notation. The integral is actually a principal value where
the limit is in the $L_{2}$ sense. This map is an onto isomorphism from
$L_{2}(\mathbb{R}^{d})$ to itself.
###### Definition 2.1.
Given a bounded measurable set $E$ with positive measure, we define
$PW_{E}:=\\{f\in
L_{2}(\mathbb{R}^{d})\arrowvert\mathrm{supp}(\mathcal{F}^{-1}(f))\subset
E\\}$. Functions in $PW_{E}$ are said to be bandlimited.
###### Definition 2.2.
The function $\mathrm{sinc}:\mathbb{R}\rightarrow\mathbb{R}$ is defined by
$\mathrm{sinc}(x)=\frac{\sin(x)}{x}.$ We also define the multidimensional sinc
function $\mathrm{SINC}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ by
$\mathrm{SINC}(x)=\mathrm{sinc}(x_{1})\cdot\ldots\cdot\mathrm{sinc}(x_{d})$,
$x=(x_{1},\ldots,x_{d})$.
We recall some basic facts about $PW_{E}$:
1) $PW_{E}$ is a Hilbert space consisting of entire functions, though in this
paper we only regard the functions as having real arguments.
2) In $PW_{E}$, $L_{2}$ convergence implies uniform convergence. This is an
easy consequence of the Cauchy-Schwarz inequality.
3) The function $\mathrm{sinc}(\pi(x-y)))$ is a reproducing kernel for
$PW_{[-\pi,\pi]}$, that is, if $f\in PW_{[-\pi,\pi]}$, then we have
(1) $f(t)=\int_{-\infty}^{\infty}f(\tau)\mathrm{sinc}(\pi(t-\tau))d\tau,\quad
t\in\mathbb{R}.$
4) The WKS sampling theorem (see for example [14, page 91]): If $f\in
PW_{[-\pi,\pi]}$, then
$f(t)=\sum_{n\in\mathbb{Z}}f(n)\mathrm{sinc}(\pi(t-n)),\quad t\in\mathbb{R},$
where the sum converges in $PW_{[-\pi,\pi]}$, and hence uniformly.
If $(f_{n})_{n\in\mathbb{N}}$ is a Schauder basis for a Hilbert space $H$,
then there exists a unique set of functions $(f_{n}^{*})_{n\in\mathbb{N}}$,
(the biorthogonals of $(f_{n})_{n\in\mathbb{N}}$) such that $\langle
f_{n},f_{m}^{*}\rangle=\delta_{nm}.$ The biorthogonals also form a Schauder
basis for $H$. Note that biorthogonality is preserved under a unitary
transformation.
###### Definition 2.3.
A sequence $(f_{n})_{n}\subset H$ such that the map $Le_{n}=f_{n}$ is an onto
isomorphism is called a Riesz basis for $H$.
The following definitions and facts concerning frames are found in [6, section
4].
###### Definition 2.4.
A frame for a separable Hilbert space $H$ is a sequence $(f_{n})_{n}\subset H$
such that for some $0<A<B,$
(2) $A\Arrowvert f\Arrowvert^{2}\leq\sum_{n}|\langle f,f_{n}\rangle|^{2}\leq
B\Arrowvert f\Arrowvert^{2},\quad\forall f\in H.$
The numbers $A$ and $B$ in the equation (2) are called the lower and upper
frame bounds.
Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n}$. The following
conditions are equivalent to $(f_{n})_{n}\subset H$ being a frame for $H$.
1) The map $L:H\rightarrow H$ defined by $Le_{n}=f_{n}$ is bounded linear and
onto. This map is called the preframe operator.
2) The map $L^{*}:H\rightarrow H$ (the adjoint of the preframe operator) given
by $f\mapsto\sum_{n}\langle f,f_{n}\rangle e_{n}$ is an isomorphic embedding.
Given a frame $(f_{n})_{n}$ with preframe operator $L$, the map $S=LL^{*}$
given by $Sf=\sum_{n}\langle f,f_{n}\rangle f_{n}$ is an onto isomorphism. $S$
is called the frame operator associated to the frame. It follows that $S$ is
positive and self-adjoint.
The basic connection between frames and sampling theory of bandlimited
functions (more generally in a reproducing kernel Hilbert space) is
straightforward. If $(e^{it_{n}(\cdot)})_{n}$ is a frame for $f\in
PW_{[-\pi,\pi]}$ with frame operator $S$, and $f\in PW_{[-\pi,\pi]}$, then
$\displaystyle
S(\mathcal{F}^{-1}(f))=\sum_{n}\langle\mathcal{F}^{-1}(f),f_{n}\rangle
f_{n}=\sum_{n}\mathcal{F}(\mathcal{F}^{-1}(f))(t_{n})f_{n}=\sum_{n}f(t_{n})f_{n},$
implying that $\mathcal{F}^{-1}(f)=\sum_{n}f(t_{n})S^{-1}f_{n}$, so that
$f=\sum_{n}f(t_{n})\mathcal{F}(S^{-1}f_{n}).$ Note that in the case when
$t_{n}=n$, we recover the WKS theorem.
###### Definition 2.5.
A sequence $(f_{n})_{n}$ satisfying the second inequality in equation (2) is
called a Bessel sequence.
###### Definition 2.6.
An exact frame is a frame which ceases to be one if any of its elements is
removed.
It can be shown that the notions of Riesz bases, exact frames, and
unconditional Schauder bases coincide.
###### Definition 2.7.
A subset $S$ of $\mathbb{R}^{d}$ is said to be uniformly separated if
$\inf_{x,y\in S,x\neq y}\Arrowvert x-y\Arrowvert_{2}>0.$
###### Definition 2.8.
If $S=(x_{k})_{k}$ is a sequence of real numbers and $f$ is a function with
$S$ in its domain, then $f_{S}$ denotes the sequence $(f(x_{k}))_{k}$.
## 3\. The multidimensional oversampling theorem
In [7], Daubechies and DeVore derive the following formula:
(3)
$f(t)=\frac{1}{\lambda}\sum_{n\in\mathbb{Z}}f\big{(}\frac{n}{\lambda}\big{)}g\big{(}t-\frac{n}{\lambda}\big{)},\quad
t\in\mathbb{R},$
where $g$ is infinitely smooth and decays rapidly. Thus oversampling allows
the representation of bandlimited functions as combinations of integer
translates of $g$ rather than the sinc function. In this sense equation (3) is
a generalization of the WKS theorem. The rapid decay of $g$ yields a certain
stability in the recovery formula, given bounded perturbations in the sampled
data [7].
In this section we derive a multidimensional version of equation (3), (Theorem
3.1) for unequally spaced sample points, and the corresponding non-
oversampling version of the WKS theorem is given in Theorem 3.2.
Daubechies and DeVore regard $\mathcal{F}^{-1}(f)$ as an element of
$L_{2}[-\lambda\pi,\lambda\pi]$ for some $\lambda>1$. In their proof the
obvious fact that $[-\pi,\pi]\subset[-\lambda\pi,\lambda\pi]$ allows for the
construction of the bump function $\mathcal{F}^{-1}(g)\in
C^{\infty}(\mathbb{R})$ which is $1$ on $[-\pi,\pi]$ and $0$ off
$[-\lambda\pi,\lambda\pi]$. If their result is to be generalized to a sampling
theorem for $PW_{E}$ in higher dimensions, a suitable condition for $E$
allowing the existence of a bump function is necessary. If
$E\subset\mathbb{R}^{d}$ is chosen to be compact such that for all
$\lambda>1$, $E\subset\mathrm{int}(\lambda E)$, then Lemma 8.18 in [9, page
245], a $C^{\infty}$-version of the Urysohn lemma, implies the existence of a
smooth bump function which is $1$ on $E$ and $0$ off $\lambda E$. It is to
such regions that we generalize equation (3):
###### Theorem 3.1.
Let $0\in E\subset\mathbb{R}^{d}$ be compact such that for all $\lambda>1$,
$E\subset\mathrm{int}(\lambda E).$ Choose
$S=(t_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$(f_{n})_{n\in\mathbb{N}}$, defined by
$f_{n}(\cdotp)=e^{i\langle\cdotp,t_{n}\rangle}$, is a frame for $L_{2}(E)$
with frame operator $S$. Let $\lambda_{0}>1$ with
$\mathcal{F}^{-1}(g):\mathbb{R}^{d}\rightarrow\mathbb{R}$,
$\mathcal{F}^{-1}(g)\in C^{\infty}$ where
$\mathcal{F}^{-1}(g)\arrowvert_{E}=1$ and
$\mathcal{F}^{-1}(g)\arrowvert_{(\lambda_{0}E)^{c}}$=0. If
$\lambda\geq\lambda_{0}$ and $f\in PW_{E}$, then
(4)
$f(t)=\frac{1}{\lambda^{d}}\sum_{k\in\mathbb{N}}\Big{(}\sum_{n\in\mathbb{N}}B_{kn}f\big{(}\frac{t_{n}}{\lambda}\big{)}\Big{)}g\big{(}t-\frac{t_{k}}{\lambda}\big{)},\quad
t\in\mathbb{R}^{d},$
where $B_{kn}=\langle S^{-1}f_{n},S^{-1}f_{k}\rangle_{E}$. Convergence of the
sum is in $L_{2}(\mathbb{R}^{d})$, hence also uniform. Further, the map
$B:\ell_{2}(\mathbb{N})\rightarrow\ell_{2}(\mathbb{N})$ defined by
$(y_{k})_{k\in\mathbb{N}}\mapsto\big{(}\sum_{n\in\mathbb{N}}B_{kn}y_{n}\big{)}_{k\in\mathbb{N}}$
is bounded linear, and is an onto isomorphism iff $(f_{n})_{n\in\mathbb{N}}$
is a Riesz basis for $L_{2}(E)$.
###### Proof.
Define $f_{\lambda,n}(\cdotp)=f_{n}\big{(}\frac{\cdotp}{\lambda}\big{)}$. Note
that $(f_{\lambda,n})_{n}$ is a frame for $L_{2}(\lambda E)$ with frame
operator $S_{\lambda}.$
Step 1: We show that
(5)
$f=\sum_{n}f\Big{(}\frac{t_{n}}{\lambda}\Big{)}\mathcal{F}[(S^{-1}_{\lambda}f_{\lambda,n})\mathcal{F}^{-1}(g)],\quad
f\in PW_{E}.$
We know $\mathrm{supp}(\mathcal{F}^{-1}(f))\subset E\subset\lambda E$, so we
may work with $\mathcal{F}^{-1}(f)$ via its frame decomposition. We have
$\mathcal{F}^{-1}(f)=S_{\lambda}^{-1}S_{\lambda}(\mathcal{F}^{-1}(f))=\sum_{n}\langle\mathcal{F}^{-1}(f),f_{\lambda,n}\rangle_{\lambda
E}S^{-1}_{\lambda}f_{\lambda,n},\quad\mathrm{on}\quad\lambda E.$
This yields
$\mathcal{F}^{-1}(f)=\sum_{n}\langle\mathcal{F}^{-1}(f),f_{\lambda,n}\rangle_{\lambda
E}(S^{-1}_{\lambda}f_{\lambda,n})\mathcal{F}^{-1}(g),\quad\mathrm{on}\quad\mathbb{R}^{d},$
since $\mathrm{supp}\mathcal{F}(g)\subset\lambda E$. Taking Fourier transforms
we obtain
(6) $f=\sum_{n}\langle\mathcal{F}^{-1}(f),f_{\lambda,n}\rangle_{\lambda
E}\mathcal{F}[(S^{-1}_{\lambda}f_{\lambda,n})\mathcal{F}^{-1}(g)],\quad\mathrm{on}\quad\mathbb{R}^{d}.$
Now
$\displaystyle\langle\mathcal{F}^{-1}(f),f_{\lambda,n}\rangle_{\lambda
E}=\int_{\lambda
E}\mathcal{F}^{-1}(f)(\xi)e^{-i\langle\xi,\frac{t_{n}}{\lambda}\rangle}d\xi=f\Big{(}\frac{t_{n}}{\lambda}\Big{)}$
which, when substituted into equation (6), yields (5).
Step 2: We show that
(7)
$f(\cdotp)=\sum_{n}f\Big{(}\frac{t_{n}}{\lambda}\Big{)}\Big{[}\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}g\big{(}\cdotp-\frac{t_{k}}{\lambda}\big{)}\Big{]},$
where convergence is in $L_{2}$.
We compute $\mathcal{F}[(S^{-1}_{\lambda}f_{\lambda,n})\mathcal{F}^{-1}(g)]$.
For $h\in L_{2}(\lambda E)$ we have
$\displaystyle h=S_{\lambda}(S_{\lambda}^{-1}h)=\sum_{k}\langle
S_{\lambda}^{-1}h,f_{\lambda,k}\rangle_{\lambda
E}f_{\lambda,k}=\sum_{k}\langle
h,S_{\lambda}^{-1}f_{\lambda,k}\rangle_{\lambda E}f_{\lambda,k}.$
Letting $h=S^{-1}_{\lambda}f_{\lambda,n}$,
$S^{-1}_{\lambda}f_{\lambda,n}=\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}f_{\lambda,k}.$
This gives
$\displaystyle\mathcal{F}[(S^{-1}_{\lambda}f_{\lambda,n})F^{-1}(g)](\cdotp)$
$\displaystyle=$ $\displaystyle\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}\mathcal{F}[f_{\lambda,k}\mathcal{F}^{-1}(g)](\cdotp)$ $\displaystyle=$
$\displaystyle\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}\int_{\lambda
E}e^{i\langle\xi,\frac{t_{k}}{\lambda}\rangle}\mathcal{F}^{-1}(g)(\xi)e^{-i\langle\xi,\cdotp\rangle}d\xi$
$\displaystyle=$ $\displaystyle\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}\int_{\lambda
E}\mathcal{F}^{-1}(g)(\xi)e^{-i\langle\cdotp-\frac{t_{k}}{\lambda},\xi\rangle}d\xi$
$\displaystyle=$ $\displaystyle\sum_{k}\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}g\big{(}\cdotp-\frac{t_{k}}{\lambda}\big{)},$
so (7) follows from (5).
Step 3: We show that
(8) $\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}=\frac{1}{\lambda^{d}}\langle
S^{-1}f_{n},S^{-1}f_{k}\rangle_{E},\quad\mathrm{for}\quad n,k\in\mathbb{N}.$
First we show
$(S^{-1}_{\lambda}f_{\lambda,n})(\cdotp)=\frac{1}{\lambda^{d}}(S^{-1}f_{n})(\frac{\cdotp}{\lambda})$,
or equivalently that
$f_{\lambda,n}=\frac{1}{\lambda^{d}}S_{\lambda}\big{(}(S^{-1}f_{n})(\frac{\cdotp}{\lambda})\big{)}$.
We have for any $g\in L_{2}(\lambda E),$
$\displaystyle\langle g,f_{\lambda,k}\rangle_{\lambda E}$ $\displaystyle=$
$\displaystyle\int_{\lambda
E}g(\xi)e^{-i\langle\frac{\xi}{\lambda},t_{k}\rangle}d\xi=\lambda^{d}\int_{E}g(\lambda
x)e^{-i\langle x,t_{k}\rangle}dx=\lambda^{d}\langle
g(\lambda(\cdot)),f_{k}\rangle_{E}.$
By definition of the frame operator $S_{\lambda}$,
$S_{\lambda}g=\sum_{k\in\mathbb{N}}\langle g,f_{\lambda,k}\rangle_{\lambda
E}f_{\lambda,k},$
which then becomes
$S_{\lambda}g=\lambda^{d}\sum_{k}\langle
g(\lambda(\cdot)),f_{k}\rangle_{E}f_{\lambda,k}.$
Substituting $g=\frac{1}{\lambda^{d}}(S^{-1}f_{n})(\frac{\cdotp}{\lambda})$
into the equation above we obtain
$\displaystyle\frac{1}{\lambda^{d}}S_{\lambda}\big{(}(S^{-1}f_{n})\big{(}\frac{\cdot}{\lambda}\big{)}\big{)}$
$\displaystyle=$ $\displaystyle\sum_{k}\langle
S^{-1}f_{n},f_{k}\rangle_{E}f_{\lambda,k}=\big{(}S(S^{-1}f_{n})\big{)}\big{(}\frac{\cdot}{\lambda}\big{)}=f_{\lambda,n}.$
We now compute the desired inner product:
$\displaystyle\langle
S^{-1}_{\lambda}f_{\lambda,n},S^{-1}_{\lambda}f_{\lambda,k}\rangle_{\lambda
E}$ $\displaystyle=$ $\displaystyle\frac{1}{\lambda^{2d}}\int_{\lambda
E}(S^{-1}f_{n})\big{(}\frac{x}{\lambda}\big{)}\overline{(S^{-1}f_{k})\big{(}\frac{x}{\lambda}\big{)}}dx$
$\displaystyle=$
$\displaystyle\frac{\lambda^{d}}{\lambda^{2d}}\int_{E}(S^{-1}f_{n})(x)\overline{(S^{-1}f_{k})(x)}dx=\frac{1}{\lambda^{d}}\langle
S^{-1}f_{n},S^{-1}f_{k}\rangle_{E}.$
Note that equation (7) becomes
(9)
$f(\cdotp)=\frac{1}{\lambda^{d}}\sum_{n}f\big{(}\frac{t_{n}}{\lambda}\big{)}\Big{[}\sum_{k}\langle
S^{-1}f_{n},S^{-1}f_{k}\rangle
g\big{(}\cdotp-\frac{t_{k}}{\lambda}\big{)}\big{]}.$
Step 4: The map $V:\ell_{2}(\mathbb{N})\rightarrow\ell_{2}(\mathbb{N})$ given
by
$x=(x_{k})_{k\in\mathbb{N}}\mapsto\big{(}\sum_{n}B_{kn}x_{n}\big{)}_{k\in\mathbb{N}}=Bx$
is bounded linear and self-adjoint.
Let $(d_{k})_{k\in\mathbb{N}}$ be the standard basis for
$\ell_{2}(\mathbb{N})$, and let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal
basis for $L_{2}(E)$. Then
$\displaystyle Vd_{j}$ $\displaystyle=$
$\displaystyle(B_{kj})_{k\in\mathbb{N}}=\sum_{k}B_{kj}d_{k}=\sum_{k}\langle
S^{-1}f_{j},S^{-1}f_{k}\rangle d_{k}=\sum_{k}\langle
L^{*}(S^{-1})^{2}Le_{j},e_{k}\rangle d_{k},$
where $L$ is the preframe operator, i.e., $S=LL^{*}$. Define
$\phi:\ell^{2}(\mathbb{N})\rightarrow L_{2}(E)$ by $\phi(d_{k})=e_{k}$,
$k\in\mathbb{N}$. Clearly $\phi$ is unitary. It follows that
$V=\phi^{-1}L^{*}(S^{-1})^{2}L\phi$, which concludes Step 4. From here on we
identify $V$ with $B$. Clearly $B$ is an onto isomorphism iff $L$ and $L^{*}$
are both onto, i.e., iff the map $Le_{n}=f_{n}$ is an onto isomorphism.
Step 5: Verification of equation (4). Recalling Definition 2.8,
$f_{S/\lambda}=\big{(}f\big{(}\frac{t_{n}}{\lambda}\big{)}\big{)}_{n\in\mathbb{N}}$;
for each $t\in\mathbb{R}^{d}$, let
$g_{\lambda}(t)=\big{(}g\big{(}t-\frac{t_{n}}{\lambda}\big{)}\big{)}_{n\in\mathbb{N}}$.
Noting that
$f\big{(}\frac{\cdot}{\lambda}\big{)},g\big{(}t-\frac{\cdot}{\lambda}\big{)}\in
L_{2}(\lambda E)$, and recalling that $(f_{\lambda,n})_{n}$ is a frame for
$L_{2}(\lambda E)$, we have
(10)
$\sum_{n}\big{|}f\big{(}\frac{t_{n}}{\lambda}\big{)}\big{|}^{2}=\sum|\langle\mathcal{F}^{-1}(f),f_{\lambda,n}\rangle_{\lambda
E}|^{2}\leq A_{\lambda}\Arrowvert\mathcal{F}^{-1}(f)\Arrowvert^{2},$
and
$\sum_{n}\big{|}g\big{(}t-\frac{t_{n}}{\lambda}\big{)}\big{|}^{2}=\sum|\langle\mathcal{F}^{-1}\big{(}g\big{(}t-\frac{\cdot}{\lambda}\big{)}\big{)},f_{\lambda,n}\rangle_{\lambda
E}|^{2}\leq
A_{\lambda}\Arrowvert\mathcal{F}^{-1}\big{(}g\big{(}t-\frac{\cdot}{\lambda}\big{)}\big{)}\Arrowvert^{2}.$
Note that equation (9) becomes
$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{n}f\big{(}\frac{t_{n}}{\lambda}\big{)}\Big{[}\sum_{k}B_{kn}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}\Big{]}=\frac{1}{\lambda^{d}}\sum_{n}f\big{(}\frac{t_{n}}{\lambda}\big{)}\overline{\Big{[}\sum_{k}B_{nk}\overline{g\big{(}t-\frac{t_{k}}{\lambda}\big{)}}\Big{]}}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{n}(f_{S/\lambda})_{n}(B\overline{g_{\lambda}(t)})_{n}=\frac{1}{\lambda^{d}}\langle
f_{S/\lambda},B\overline{g_{\lambda}(t)}\rangle=\frac{1}{\lambda^{d}}\langle
Bf_{S/\lambda},\overline{g_{\lambda}(t)}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{k}(Bf_{S/\lambda})_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}=\frac{1}{\lambda^{d}}\sum_{k\in\mathbb{N}}\Big{(}\sum_{n\in\mathbb{N}}B_{kn}f\big{(}\frac{t_{n}}{\lambda}\big{)}\Big{)}g\big{(}t-\frac{t_{k}}{\lambda}\big{)},$
which proves (4).
Step 6: We verify that convergence in equation (4) is in $L_{2}(\mathbb{R})$
(hence uniform). Define
$f_{n}(t)=\frac{1}{\lambda^{d}}\sum_{1\leq k\leq
n}(Bf_{S/\lambda})_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}$
and
$f_{m,n}(t)=\frac{1}{\lambda^{d}}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}.$
Then
$\displaystyle[\mathcal{F}^{-1}(f_{m,n})](\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}\mathcal{F}^{-1}\big{[}g\big{(}\cdotp-\frac{t_{n}}{\lambda}\big{)}\big{]}$
$\displaystyle=$ $\displaystyle\frac{1}{\lambda^{d}}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}\mathcal{F}^{-1}(g)(\xi)e^{i\langle\xi,\frac{t_{k}}{\lambda}\rangle},$
so
$\displaystyle\Arrowvert[\mathcal{F}^{-1}(f_{m,n})]\Arrowvert^{2}_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{\lambda^{d}}\int_{\lambda
E}|\mathcal{F}^{-1}(g)(\xi)|^{2}\Big{|}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}e^{i\langle\xi,\frac{t_{k}}{\lambda}\rangle}\Big{|}^{2}d\xi$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{d}}\Big{\Arrowvert}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}f_{\lambda,k}\Big{\Arrowvert}^{2}_{2}.$
If $(h_{n})_{n}$ is a orthonormal basis for $L_{2}(\lambda E)$, then the map
$Th_{k}=f_{\lambda,k}$ (the preframe operator) is bounded linear, so
$\displaystyle\Arrowvert[\mathcal{F}^{-1}(f_{m,n})]\Arrowvert^{2}_{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{d}}\Big{\Arrowvert}T\Big{(}\sum_{m\leq k\leq
n}(Bf_{S/\lambda})_{k}h_{k}\Big{)}\Big{\Arrowvert}^{2}_{2}\leq\frac{1}{\lambda^{d}}\Arrowvert
T\Arrowvert^{2}\sum_{m\leq k\leq n}|(Bf_{S/\lambda})_{k}|^{2}.$
But $Bf_{S/\lambda}\in\ell^{2}(\mathbb{N})$, so
$\Arrowvert[\mathcal{F}^{-1}(f_{m,n})]\Arrowvert_{2}\rightarrow 0$ as
$m,n\rightarrow\infty$. As $\mathcal{F}^{-1}$ is an onto isomorphism, we have
$\Arrowvert f_{m,n}\Arrowvert\rightarrow 0$, implying that $\Arrowvert
f-f_{n}\Arrowvert\rightarrow 0$ as $n\rightarrow\infty$. ∎
Note that equation (3.1) is conveniently written as
(11)
$f(t)=\frac{1}{\lambda^{d}}\sum_{k}(Bf_{S/\lambda})_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)},\quad
t\in\mathbb{R}^{d}.$
Remark: There is a geometric characterization of sets $E\subset\mathbb{R}^{d}$
such that $E\subset\mathrm{int}(\lambda E)$ for all $\lambda>0$. Intuitively,
$E$ must be a “continuous radial stretching of the closed unit ball”. This is
precisely formulated in the following proposition (whose proof is omitted).
###### Proposition 3.2.
If $0\in E\subset\mathbb{R}^{d}$ is compact, then the following are
equivalent:
1) $E\subset\mathrm{int}(\lambda E)$ for all $\lambda>1.$
2) There exists a continuous map $\phi:S^{d-1}\rightarrow(0,\infty)$ such that
$E=\\{ty\phi(y)|y\in S^{d-1},t\in[0,1]\\}$.
The following is a simplified version of Theorem 3.1, which is proven in a
similiar fashion:
###### Theorem 3.3.
Choose $(t_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$(f_{n})_{n\in\mathbb{N}}$, defined by
$f_{n}(\cdotp)=\frac{1}{(2\pi)^{d/2}}e^{i\langle\cdotp,t_{n}\rangle}$, is a
frame for $L_{2}([-\pi,\pi]^{d})$. If $f\in PW_{E}$, then
(12)
$f(t)=\sum_{k\in\mathbb{N}}\Big{(}\sum_{n\in\mathbb{N}}B_{kn}f(t_{n})\Big{)}\mathrm{SINC}(\pi(t-t_{k})),\quad
t\in\mathbb{R}^{d}.$
The matrix $B$ and the convergence of the sum are as in Theorem 3.1.
Equation (4) generalizes equation (12) in the same way that equation (3)
generalizes the WKS equation.
We can write equation (12) as
(13) $f(t)=\sum_{k\in\mathbb{N}}(Bf_{S})_{k}\mathrm{SINC}(\pi(t-t_{k})).$
The preceding result is similar in spirit to Theorem 1.9 in [4, page 19].
Frames for $L_{2}(E)$ satisfying the conditions in Theorems 3.1 and 3.3 occur
in abundance. The following result is due to Beurling in [5, see Theorem 1,
Theorem 2, and (38)].
###### Theorem 3.4.
Let $\Lambda\subset\mathbb{R}^{d}$ be countable such that
$\displaystyle r(\Lambda)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\inf_{\lambda,\mu\in\Lambda,\lambda\neq\mu}\Arrowvert\lambda-\mu\Arrowvert_{2}>0$
$\displaystyle\mathrm{and}\quad R(\Lambda)$ $\displaystyle:=$
$\displaystyle\sup_{\xi\in\mathbb{R}^{d}}\inf_{\lambda\in\Lambda}\Arrowvert\lambda-\mu\Arrowvert_{2}<\frac{\pi}{2}.$
If $E$ is a subset of the closed unit ball in $\mathbb{R}^{d}$ and $E$ has
positive measure, then
$\\{e^{i\langle\cdot,\lambda\rangle}|\lambda\in\Lambda\\}$ is a frame for
$L_{2}(E)$.
## 4\. Remarks regarding the stability of Theorem 3.1
A desirable trait in a recovery formula is stability given error in the
sampled data. Suppose we have sample values
$\tilde{f}_{n}=f\big{(}\frac{n}{\lambda}\big{)}+\epsilon_{n}$ where
$\sup_{n}|\epsilon_{n}|=\epsilon$. If in equation (3) we replace
$f\big{(}\frac{n}{\lambda}\big{)}$ by $\tilde{f}_{n}$, and call the resulting
expression $\tilde{f}$, then we have
$|f(t)-\tilde{f}(t)|\leq\epsilon\frac{1}{\lambda}\sum_{n\in\mathbb{Z}}\Big{|}g\big{(}t-\frac{n}{\lambda}\big{)}\Big{|}\leq\epsilon(\lambda^{-1}\Arrowvert
g^{\prime}\Arrowvert_{L_{1}}+\Arrowvert g\Arrowvert_{L_{1}}).$
It follows that equation (3) is certainly stable under $\ell_{\infty}$
perturbations in the data, while the WKS sampling Theorem is not. For a more
detailed discussion see [7].
Such a stability result is not immediately forthcoming for equation (4), as
the following example illustrates.
Restricting to $d=1$, let $(t_{n})_{n\in\mathbb{Z}}$ satisfy
$t_{0}=D\notin\mathbb{Z}$, and $t_{n}=n$ for $n\neq 0$. The forthcoming
discussion in Section 5 shows that $(f_{n})_{n\in\mathbb{Z}}$ is a Riesz basis
for $L_{2}[-\pi,\pi]$.
Note that when $(f_{n})_{n}$ is a Riesz basis, the sequence
$(S^{-1}f_{n})_{n}$ is its biorthogonal sequence. We matrix $B$ associated to
this basis is computed as follows.
The biorthogonal functions $(G_{n})_{n\in\mathbb{Z}}$ for
$(\mathrm{sinc}(\pi(\cdot-n)))_{n\in\mathbb{Z}}$ are
$\displaystyle G_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}n(t-D)\mathrm{sinc}(\pi t)}{(n-D)(t-n)},\quad
n\neq 0,\quad\mathrm{and}$ $\displaystyle G_{0}(t)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{sinc}(\pi t)}{\mathrm{sinc}(\pi D)}.$
That these functions are in $PW_{[-\pi,\pi]}$ is verified by applying the
Paley-Wiener Theorem [14, page 85], and the biorthogonality condition is
verified by applying equation (1). Again using equation (1), we obtain
$\displaystyle i)$ $\displaystyle\quad B_{m0}=\langle
G_{0},G_{m}\rangle=\frac{D(-1)^{m}}{\mathrm{sinc}(\pi D)(m-D)},\quad m\neq 0,$
$\displaystyle ii)$ $\displaystyle\quad B_{00}=\langle
G_{0},G_{0}\rangle=\frac{1}{\mathrm{sinc}^{2}(\pi D)},$ $\displaystyle iii)$
$\displaystyle\quad B_{mn}=\langle
G_{n},G_{m}\rangle=\delta_{nm}+\frac{D^{2}(-1)^{n+m}}{(n-D)(m-D)},\quad\mathrm{else}.$
Note that the rows of $B$ are not in $\ell_{1}$, so that as an operator acting
on $\ell_{\infty}$, $B$ does not act boundedly. Consequently, the equation
(14)
$\tilde{f}(t)=\frac{1}{\lambda}\sum_{k}(B\tilde{f}_{S/\lambda})_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}$
is not defined for all perturbed sequences $\tilde{f}_{S/\lambda}$ where
$(\tilde{f}_{S/\lambda})_{n}=(f_{S/\lambda})_{n}+\epsilon_{n}$ where
$\sup_{n}|\epsilon_{n}|=\epsilon$.
Despite the above failure, the following shows that there is some advantage of
equation (4) over equation (12).
If $\tilde{f}_{S/\lambda}$ is some perturbation of $f_{S/\lambda}$ such that
$\Arrowvert
B\tilde{f}_{S/\lambda}-Bf_{S/\lambda}\Arrowvert_{\infty}\leq\epsilon$, then
(15)
$\sup_{t\in\mathbb{R}^{d}}|f(t)-\tilde{f}(t)|\leq\epsilon\sum_{k}\Big{|}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}\Big{|}.$
## 5\. Restriction of the sampling Theorem to the case where the exponential
frame is a Riesz basis
From here on, we focus on the case where $(t_{n})_{n\in\mathbb{N}}$ is an
$\ell_{\infty}$ perturbation of the lattice $\mathbb{Z}^{d}$, and
$(f_{n})_{n\in\mathbb{N}}$ is a Riesz basis for $L_{2}[-\pi,\pi]^{d}$. In this
case, under the additional constraint that the sample nodes are asymptotically
the integer lattice, the following theorem gives a computationally feasible
version of equation (4) . The summands in equation (4) involves an infinite
invertible matrix $B$, though under the constraints mentioned above, we show
that $B$ can be replaced by a related finite-rank operator which can be
computed concretely. Precisely, one has the following.
###### Theorem 5.1.
Let $(n_{k})_{k\in\mathbb{N}}$ be an enumeration of $\mathbb{Z}^{d}$, and
$S=(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$\lim_{k\rightarrow\infty}\Arrowvert n_{k}-t_{k}\Arrowvert_{\infty}=0.$
Define $e_{k},f_{k}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ by
$e_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle n_{k},x\rangle}$ and
$\frac{1}{(2\pi)^{d/2}}e^{i\langle t_{k},x\rangle}$, and let $(h_{k})_{k}$ be
the standard basis for $\ell_{2}(\mathbb{N})$. Let
$P_{l}:\ell_{2}(\mathbb{N})\rightarrow\ell_{2}(\mathbb{N})$ be the orthogonal
projection onto $\mathrm{span}\\{h_{1},\cdots,h_{l}\\}$. If
$(f_{k})_{k\in\mathbb{N}}$ is a Riesz basis for $L_{2}[-\pi,\pi]^{d}$, then
for all $f\in PW_{[-\pi,\pi]^{d}}$, we have
(16)
$f(t)=\lim_{l\rightarrow\infty}\frac{1}{\lambda^{d}}\sum_{k=1}^{l}[(P_{l}B^{-1}P_{l})^{-1}f_{S/\lambda}]_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)},\quad
t\in\mathbb{R}^{d},$
where convergence is in $L_{2}$ and uniform. Furthermore,
$(P_{l}B^{-1}P_{l})_{nm}=\left\\{\begin{array}[]{lr}\mathrm{sinc}\pi(t_{n,1}-t_{m,1})\cdot\ldots\cdot\mathrm{sinc}\pi(t_{n,d}-t_{m,d}),&1\leq
n,m\leq l\\\ 0,&\mathrm{otherwise.}\end{array}\right.$
Convergence of the sum is in $L_{2}$ and also uniform.
There is a slight abuse of notation in the formula above. The matrix
$P_{l}B^{-1}P_{l}$ is clearly not invertible as an operator on $\ell_{2}$, and
it should be interpreted as the inverse of an $l\times l$ matrix acting on the
first $l$ coordinates of $f_{S/\lambda}$.
The following version of Theorem 5.1 avoids oversampling. Its proof is similar
to that of Theorem 5.1.
###### Theorem 5.2.
Under the hypotheses of Theorem 5.1,
(17)
$f(t)=\lim_{l\rightarrow\infty}\sum_{k=1}^{l}[(P_{l}B^{-1}P_{l})^{-1}f_{S}]_{k}\mathrm{SINC}(t-t_{k}),\quad
t\in\mathbb{R}^{d},$
where convergence of the sum is both $L_{2}$ and uniform.
The following lemma forms the basis of the proof of the preceding theorems, as
well as the other results in the paper.
###### Lemma 5.3.
Let $(n_{k})_{k\in\mathbb{N}}$ be an enumeration of $\mathbb{Z}^{d}$, and let
$(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$. Define
$e_{k},f_{k}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ by
$e_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle n_{k},x\rangle}$ and
$f_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle t_{k},x\rangle}$. Then for any
$r,s\geq 1$, and any finite sequence $(a_{k})_{k=r}^{s}$, we have
(18)
$\Bigg{\Arrowvert}\sum_{k=r}^{s}\Big{(}\frac{a_{k}}{(2\pi)^{d/2}}e^{i\langle(\cdot),n_{k}\rangle}-\frac{a_{k}}{(2\pi)^{d/2}}e^{i\langle(\cdot),t_{k}\rangle}\Big{)}\Bigg{\Arrowvert}_{2}\leq\Big{(}e^{\pi
d\big{(}{\sup\atop{r\leq k\leq s}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}\big{)}}-1\Big{)}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{1/2}.$
###### Proof.
Let $\delta_{k}=t_{k}-n_{k}$ where
$\delta_{k}=(\delta_{k1},\cdots,\delta_{kd})$. Then
(19)
$\displaystyle\phi_{r,s}(x):=\sum_{k=r}^{s}\frac{a_{k}}{(2\pi)^{d/2}}\big{[}e^{i\langle
n_{k},x\rangle}-e^{i\langle
t_{k},x\rangle}\big{]}=\sum_{k=r}^{s}\frac{a_{k}}{(2\pi)^{d/2}}e^{i\langle
n_{k},x\rangle}\big{[}1-e^{i\langle\delta_{k},x\rangle}\big{]},$
Now for any $\delta_{k}$,
$\displaystyle 1-e^{i\langle\delta_{k},x\rangle}$ $\displaystyle=$
$\displaystyle 1-e^{i\delta_{k1}x_{1}}\cdot\ldots\cdot
e^{i\delta_{kd}x_{d}}=1-\Big{(}\sum_{j_{1}=0}^{\infty}\frac{(i\delta_{k1}x_{1})^{j_{1}}}{j_{1}!}\Big{)}\cdot\ldots\cdot\Big{(}\sum_{j_{d}=0}^{\infty}\frac{(i\delta_{kd}x_{d})^{j_{d}}}{j_{d}!}\Big{)}$
$\displaystyle=$ $\displaystyle 1-\sum_{(j_{1},\cdots,j_{d})\atop j_{i}\geq
0}\frac{(i\delta_{k1}x_{1})^{j_{1}}\cdot\ldots\cdot(i\delta_{kd}x_{d})^{j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}$ $\displaystyle=$ $\displaystyle-\sum_{(j_{1},\cdots,j_{d})\in
J}i^{j_{1}+\ldots+j_{d}}\frac{(\delta_{k1}x_{1})^{j_{1}}\cdot\ldots\cdot(\delta_{kd}x_{d})^{j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!},$
where $J=\\{(j_{1},\cdots,j_{d})\in\mathbb{Z}^{d}|j_{i}\geq
0,(j_{1},\cdots,j_{d})\neq 0\\}$. Then equation (19) becomes
$\displaystyle\phi_{r,s}(x)$ $\displaystyle=$
$\displaystyle-\sum_{k=r}^{s}\frac{a_{k}}{(2\pi)^{d/2}}e^{i\langle
n_{k},x\rangle}\Big{[}\sum_{(j_{1},\cdots,j_{d})\in
J}i^{j_{1}+\ldots+j_{d}}\frac{(\delta_{k1}x_{1})^{j_{1}}\cdot\ldots\cdot(\delta_{kd}x_{d})^{j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Big{]}$ $\displaystyle=$ $\displaystyle-\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{x_{1}^{j_{1}}\cdot\ldots\cdot x_{d}^{j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}i^{j_{1}+\ldots+j_{d}}\sum_{k=r}^{s}\frac{a_{k}}{(2\pi)^{d/2}}\delta_{k1}^{j_{1}}\cdot\ldots\cdot\delta_{kd}^{j_{d}}e^{i\langle
n_{k},x\rangle},$
so
$\displaystyle|\phi_{r,s}(x)|\leq\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{\pi^{j_{1}+\ldots+j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Big{|}\sum_{k=r}^{s}a_{k}\delta_{k1}^{j_{1}}\cdot\ldots\cdot\delta_{kd}^{j_{d}}\frac{e^{i\langle
n_{k},x\rangle}}{(2\pi)^{d/2}}\Big{|}.$
For brevity denote the outer summand above by $h_{j_{1},\ldots,j_{d}}(t)$.
Then
$\displaystyle\bigg{(}\int_{[-\pi,\pi]^{d}}|\phi_{r,s}(x)|^{2}dt\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\bigg{(}\int_{[-\pi,\pi]^{d}}\Big{|}\sum_{(j_{1},\cdots,j_{d})\in
J}h_{j_{1},\ldots,j_{d}}(x)\Big{|}^{2}dx\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{(j_{1},\cdots,j_{d})\in
J}\bigg{(}\int_{[-\pi,\pi]^{d}}\Big{|}h_{j_{1},\ldots,j_{d}}(x)\Big{|}^{2}dx\bigg{)}^{\frac{1}{2}},$
so that
$\displaystyle\Arrowvert\phi_{r,s}\Arrowvert_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{\pi^{j_{1}+\cdot\ldots\cdot+j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Big{(}\int_{[-\pi,\pi]^{d}}\bigg{|}\sum_{k=r}^{s}a_{k}\delta_{k1}^{j_{1}}\cdot\ldots\cdot\delta_{kd}^{j_{d}}\frac{e^{i\langle
n_{k},x\rangle}}{(2\pi)^{d/2}}\bigg{|}^{2}dx\Big{)}^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{\pi^{j_{1}+\cdot\ldots\cdot+j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}|\delta_{k1}^{j_{1}}|^{2}\cdot\ldots\cdot|\delta_{kd}^{j_{d}}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{\pi^{j_{1}+\cdot\ldots\cdot+j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Bigg{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{(}{\sup\atop{r\leq k\leq
s}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}\Big{)}^{2(j_{1}+\ldots+j_{d})}\Bigg{)}^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\sum_{(j_{1},\cdots,j_{d})\in
J}\frac{\Big{(}\pi{\sup\atop{r\leq k\leq s}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}\Big{)}^{j_{1}+\cdot\ldots\cdot+j_{d}}}{j_{1}!\cdot\ldots\cdot
j_{d}!}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\bigg{[}\prod_{l=1}^{d}\bigg{(}\sum_{j_{\ell}=0}^{\infty}\frac{\big{(}\pi{\sup\atop{r\leq
k\leq s}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}\big{)}^{j_{\ell}}}{j_{\ell}!}\bigg{)}-1\bigg{]}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\Big{(}e^{\pi d\big{(}{\sup\atop{r\leq k\leq
s}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}\big{)}}-1\Big{)}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}.$
∎
###### Corollary 5.4.
Let $(n_{k})_{k\in\mathbb{N}}$ be an enumeration of $\mathbb{Z}^{d}$, and let
$(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$\sup_{k\in\mathbb{N}}\Arrowvert n_{k}-t_{k}\Arrowvert_{\infty}=L<\infty.$
Define $e_{k},f_{k}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ by
$e_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle n_{k},x\rangle}$ and
$\frac{1}{(2\pi)^{d/2}}e^{i\langle t_{k},x\rangle}$. Then the map
$T:L_{2}[-\pi,\pi]^{d}\rightarrow L_{2}[-\pi,\pi]^{d}$, defined by
$Te_{n}=e_{n}-f_{n}$, satisfies the following estimate:
(20) $\Arrowvert T\Arrowvert\leq e^{\pi Ld}-1.$
###### Proof.
Lemma (5.3) shows that $T$ is uniformly continuous on a dense subset of the
ball in $L_{2}(E)$, so $T$ is bounded on $L_{2}[-\pi,\pi]^{d}$. The inequality
(20) follows immediately. ∎
###### Corollary 5.5.
Let $(n_{k})_{k\in\mathbb{N}}$,
$(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$, and let $e_{k}$, $f_{k}$ and
$T$ be defined as in Corollary 5.4. For each $l\in\mathbb{N}$, define $T_{l}$
by $T_{l}e_{k}=e_{k}-f_{k}$ for $1\leq k\leq l$, and $T_{l}e_{k}=0$ for $l<k$.
If $\lim_{k\rightarrow\infty}\Arrowvert n_{k}-t_{k}\Arrowvert_{\infty}=0$,
then $\lim_{l\rightarrow\infty}T_{l}=T$ in the operator norm. In particular,
$T$ is a compact operator.
###### Proof.
As
$\displaystyle(T-T_{l})\big{(}\sum_{k=1}^{\infty}a_{k}e_{k}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}a_{k}(e_{k}-f_{k})-\sum_{k=1}^{l}a_{k}(e_{k}-f_{k})$
$\displaystyle=$
$\displaystyle\sum_{k=l+1}^{\infty}a_{k}(e_{k}-f_{k})=T\big{(}\sum_{k=l+1}^{\infty}a_{k}e_{k}\big{)},$
the estimate derived in lemma (5.3) yields
$\displaystyle\big{\Arrowvert}(T-T_{l})\big{(}\sum_{k=1}^{\infty}a_{k}e_{k}\big{)}\big{\Arrowvert}_{2}$
$\displaystyle=$
$\displaystyle\big{\Arrowvert}T\big{(}\sum_{k=l+1}^{\infty}a_{k}e_{k}\big{)}\big{\Arrowvert}_{2}\leq\big{(}e^{\pi
d{\sup\atop k\geq
l+1}\Arrowvert\delta_{k}\Arrowvert_{\infty}}-1\big{)}\big{\Arrowvert}\sum_{k=1}^{\infty}a_{k}e_{k}\big{\Arrowvert}_{2},$
so $\big{\Arrowvert}(T-T_{l})\big{\Arrowvert}_{2}\rightarrow 0$ as
$l\rightarrow\infty$. As $T_{l}$ has finite rank, we deduce that $T$ is
compact. ∎
We are ready for the proof of Theorem 5.1.
###### Proof.
Step 1: $B$ is a compact perturbation of the identity map, namely
(21) $B=I+\lim_{l\rightarrow\infty}(-P_{l}+(P_{l}B^{-1}P_{l})^{-1}).$
Since $(f_{k})_{k\in\mathbb{N}}$ is a Riesz basis for $L_{2}[-\pi,\pi]^{d}$,
$L^{*}=(I-T)$ is an onto isomorphism where $Te_{k}=e_{k}-f_{k}$; so $B$
simplifies to $(I-T)^{-1}(I-T^{*})^{-1}$. We examine
$\displaystyle B^{-1}=(I-T^{*})(I-T)=I+(T^{*}T-T-T^{*}):=I+\Delta,$
where $\Delta$ is a compact operator. If an operator $\Delta:H\rightarrow H$
is compact then so is $\Delta^{*}$, hence $P_{l}\Delta P_{l}\rightarrow\Delta$
in the operator norm because
$\displaystyle\Arrowvert P_{l}\Delta P_{l}-\Delta\Arrowvert$
$\displaystyle\leq$ $\displaystyle\Arrowvert P_{l}\Delta
P_{l}-P_{l}\Delta\Arrowvert+\Arrowvert
P_{l}\Delta-\Delta\Arrowvert\leq\Arrowvert\Delta
P_{l}-\Delta\Arrowvert+\Arrowvert P_{l}\Delta-\Delta\Arrowvert$
$\displaystyle=$ $\displaystyle\Arrowvert
P_{l}\Delta^{*}-\Delta^{*}\Arrowvert+\Arrowvert
P_{l}\Delta-\Delta\Arrowvert\rightarrow 0.$
We have
$\displaystyle B^{-1}$ $\displaystyle=$
$\displaystyle\lim_{l\rightarrow\infty}(I+P_{l}\Delta
P_{l})=\lim_{l\rightarrow\infty}(I+P_{l}(B^{-1}-I)P_{l})=\lim_{l\rightarrow\infty}(I-P_{l}+P_{l}B^{-1}P_{l}).$
Now $(P_{l}B^{-1}P_{l})$ restricted to the first $l$ rows and columns is the
Grammian matrix for the set $(f_{1},\cdots,f_{l})$ which can be shown (in a
straightforward manner) to be linearly independent. We conclude that
$P_{l}B^{-1}P_{l}$ is invertible as an $l\times l$ matrix. By
$(P_{l}B^{-1}P_{l})^{-1}$ we mean the inverse as an $l\times l$ matrix and
zeroes elsewhere. Observing that the ranges of $P_{l}B^{-1}P_{l}$ and
$(P_{l}B^{-1}P_{l})^{-1}$ are in the kernel of $I-P_{l}$, and that the range
of $I-P_{l}$ is in the kernels of $P_{l}B^{-1}P_{l}$ and
$(P_{l}B^{-1}P_{l})^{-1}$, we easily compute
$(I-P_{l}+(P_{l}B^{-1}P_{l})^{-1})^{-1}=I-P_{l}+P_{l}B^{-1}P_{l},$
so that
$B^{-1}=\lim_{l\rightarrow\infty}(I-P_{l}+(P_{l}B^{-1}P_{l})^{-1})^{-1},$
implying
$\displaystyle B$ $\displaystyle=$
$\displaystyle\lim_{l\rightarrow\infty}(I-P_{l}+(P_{l}B^{-1}P_{l})^{-1}):=\lim_{l\rightarrow\infty}B_{l}=I+\lim_{l\rightarrow\infty}(-P_{l}+(P_{l}B^{-1}P_{l})^{-1}).$
Step 2: We verifiy equation (16) and its convergence properties. Recalling
equation (11), we have
$\displaystyle f(t)$ $\displaystyle-$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{k=1}^{\infty}[(I-P_{l}+(P_{l}B^{-1}P_{l})^{-1})f_{S/\lambda}]_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}=\frac{1}{\lambda^{d}}\sum_{k=1}^{\infty}[(B-B_{l})f_{S/\lambda}]_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}$
implying
$\displaystyle f(t)$ $\displaystyle-$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{k=1}^{l}[(P_{l}B^{-1}P_{l})^{-1}f_{S/\lambda}]_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\sum_{k=1}^{\infty}[(B-B_{l})f_{S/\lambda}]_{k}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}+\frac{1}{\lambda^{d}}\sum_{k=l+1}^{\infty}f\big{(}\frac{t_{k}}{\lambda}\big{)}g\big{(}t-\frac{t_{k}}{\lambda}\big{)}.$
Therefore,
$\displaystyle\Big{\Arrowvert}$ $\displaystyle
f(\cdot)-\frac{1}{\lambda^{d}}\sum_{k=1}^{l}[(P_{l}B^{-1}P_{l})^{-1}f_{S/\lambda}]_{k}g\big{(}\cdot-\frac{t_{k}}{\lambda}\big{)}\Big{\Arrowvert}_{2}=$
$\displaystyle=$
$\displaystyle\Big{\Arrowvert}\frac{1}{\lambda^{d}}\sum_{k=1}^{\infty}[(B-B_{l})f_{S/\lambda}]_{k}g\big{(}\cdot-\frac{t_{k}}{\lambda}\big{)}+\frac{1}{\lambda^{d}}\sum_{k=l+1}^{\infty}f\big{(}\frac{t_{k}}{\lambda}\big{)}g\big{(}\cdot-\frac{t_{k}}{\lambda}\big{)}\Big{\Arrowvert}_{[-\lambda\pi,\lambda\pi]^{d}}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{d}}\Big{\Arrowvert}\mathcal{F}^{-1}(g)(\cdot)\Big{(}\sum_{k=1}^{\infty}[(B-B_{l})f_{S/\lambda}]_{k}e^{i\langle\cdot,\frac{t_{k}}{\lambda}\rangle}+\sum_{k=l+1}^{\infty}f\big{(}\frac{t_{k}}{\lambda}\big{)}e^{i\langle\cdot,\frac{t_{k}}{\lambda}\rangle}\Big{)}\Big{\Arrowvert}_{[-\lambda\pi,\lambda\pi]^{d}}$
after taking the inverse Fourier transform. Now
$\displaystyle\Big{\Arrowvert}$ $\displaystyle
f(\cdot)-\frac{1}{\lambda^{d}}\sum_{k=1}^{l}[(P_{l}B^{-1}P_{l})^{-1}f_{S/\lambda}]_{k}g\big{(}\cdot-\frac{t_{k}}{\lambda}\big{)}\Big{\Arrowvert}_{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{d}}\Big{\Arrowvert}\sum_{k=1}^{\infty}[(B-B_{l})f_{S/\lambda}]_{k}e^{i\langle\cdot,\frac{t_{k}}{\lambda}\rangle}\Big{\Arrowvert}_{[-\lambda\pi,\lambda\pi]^{d}}+\frac{1}{\lambda^{d}}\Big{\Arrowvert}\sum_{k=l+1}^{\infty}f\big{(}\frac{t_{k}}{\lambda}\big{)}e^{i\langle\cdot,\frac{t_{k}}{\lambda}\rangle}\Big{\Arrowvert}_{[-\lambda\pi,\lambda\pi]^{d}}$
$\displaystyle\leq$
$\displaystyle\frac{M}{\lambda^{d}}\Big{\Arrowvert}(B-B_{l})f_{S/\lambda}\Big{\Arrowvert}_{\ell_{2}(\mathbb{N})}+\frac{M}{\lambda^{d}}\Big{(}\sum_{k=l+1}^{\infty}\big{|}f\big{(}\frac{t_{k}}{\lambda}\big{)}\big{|}^{2}\Big{)}^{\frac{1}{2}},$
since $\big{(}f_{k}\big{(}\frac{\cdot}{\lambda}\big{)}\big{)}_{k}$ is a Riesz
basis for $L_{2}[-\lambda\pi,\lambda\pi]^{d}$. Since $B_{l}\rightarrow B$ as
$l\rightarrow\infty$ and
$\big{(}f\big{(}\frac{t_{k}}{\lambda}\big{)}\big{)}_{k}\in\ell_{2}(\mathbb{N})$,
the last two terms in the inequality above tend to zero, which proves the
required result.
Finally, to compute $(P_{l}B^{-1}P_{l})_{nm}$, recall that
$B^{-1}=(I-T^{*})(I-T)$. Proceeding in a manner similar to the proof of
equation (3), we obtain
$\displaystyle B^{-1}_{mn}$ $\displaystyle=$ $\displaystyle\langle
LL^{*}e_{n},e_{m}\rangle=\langle L^{*}e_{n},L^{*}e_{m}\rangle=\langle
f_{n},f_{m}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{sinc}\pi(t_{n,1}-t_{m,1})\cdot\ldots\cdot\mathrm{sinc}\pi(t_{n,d}-t_{m,d}).$
The entries of $P_{l}B^{-1}P_{l}$ agree with those of $B^{-1}$ when $1\leq
n,m\leq l$. ∎
One generalization of Kadec’s $1/4$ theorem given by Pak and Shin in [12]
(which is actually a special case of Avdonin’s theorem) is:
###### Theorem 5.6.
Let $(t_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ be a sequence of distinct
points such that
$\limsup_{|n|\rightarrow\infty}|n-t_{n}|=L<\frac{1}{4}.$
Then the sequence of functions $(f_{k})_{k\in\mathbb{Z}}$, defined by
$f_{k}(x)=\frac{1}{\sqrt{2\pi}}e^{it_{k}x}$, is a Riesz basis for
$L_{2}[-\pi,\pi]$.
Theorem 5.6 shows that in the univariate case of Theorem 5.1, the restriction
that $(f_{k})_{k\in\mathbb{N}}$ is a Riesz basis for $L_{2}[-\pi,\pi]$ can be
dropped. The following example shows that the multivariate case is very
different
Let $(e_{n})_{n}$ be an orthonormal basis for a Hilbert space $H$. Let
$f_{1}\in H$ with $\Arrowvert f_{1}\Arrowvert=1$, then
$(f_{1},e_{2},e_{3},\cdots)$ is a Riesz basis for $H$ iff $\langle
f_{1},e_{1}\rangle\neq 0$. Verifying that the map $T$, given by $e_{k}\mapsto
e_{k}$ for $k>1$ and $e_{1}\mapsto f_{1}$, is a continuous bijection is
routine, so $T$ is an isomorphism via the Open Mapping Theorem. In the
language of Theorem 5.1, $(f_{1},e_{2},e_{3},\cdots)$ is a Riesz basis for
$L_{2}[-\pi,\pi]$ iff
$0\neq\mathrm{sinc}(\pi t_{1,1})\cdot\ldots\cdot\mathrm{sinc}(\pi t_{1,d}),$
that is, iff $t_{1}\in(\mathbb{R}\setminus\\{\pm 1,\pm 2,\cdots\\})^{d}$.
## 6\. Generalizations of Kadec’s 1/4 Theorem
Corollary 5.4 yields the following generalization of Kadec’s Theorem in $d$
dimensions.
###### Corollary 6.1.
Let $(n_{k})_{k\in\mathbb{N}}$ be an enumeration of $\mathbb{Z}^{d}$, and let
$(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
(22) $\sup_{k\in\mathbb{N}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}=L<\frac{\ln(2)}{\pi d}.$
Then the sequence $(f_{k})_{k\in\mathbb{N}}$ defined by
$f_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,t_{k}\rangle}$ is a Riesz basis
for $L_{2}[-\pi,\pi]^{d}$.
The proof is immediate. Note that equation (20) implies that the map $T$ given
in Corollary 5.4 has norm less than $1$. We conclude that the map
$(I-T)e_{k}=f_{k}$ is invertible by considering its Neumann series.
The proof of Corollary (5.4) and Corollary (6.1) are straightforward
generalizations of the univariate result proved by Duffin and Eachus [8].
Kadec improved the value of the constant in the inequality (22) (for $d=1$)
from $\frac{\ln(2)}{\pi}$ to the optimal value of 1/4; this is his celebrated
“1/4 theorem” [10].
Kadec’s method of proof is to expand $e^{i\delta x}$ with respect to the
orthogonal basis
$\\{1,\cos(nx),\sin\big{(}n-\frac{1}{2}\big{)}x\\}_{n\in\mathbb{N}}$
for $L_{2}[-\pi,\pi]$, and use this expansion to estimate the norm of $T$. In
the proof of Corollary (5.4) and Corollary (6.1) we simply used a Taylor
series. Unlike the estimates in Kadec’s Theorem, the estimate in equation (20)
can be used for any sequence $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$
such that $\sup_{k\in\mathbb{N}}\Arrowvert
n_{k}-t_{k}\Arrowvert_{\infty}=L<\infty$, not only those for which the
exponentials $(e^{it_{n}x})_{n}$ form a Riesz basis. An impressive
generalization of Kadec’s 1/4 theorem when $d=1$ is Avdonin’s “1/4 in the
mean” theorem, [1].
Sun and Zhou (see [13] second half of Theorem 1.3) refined Kadec’s argument to
obtain a partial generalization of his result in higher dimensions:
###### Theorem 6.2.
Let $(a_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ such that
$0<L<\frac{1}{4},$ $D_{d}(L):=\Big{(}1-\cos\pi L+\sin\pi L+\frac{\sin\pi
L}{\pi L}\Big{)}^{d}-\Big{(}\frac{\sin\pi L}{\pi
L}\Big{)}^{d},\quad\mathrm{and}$ $\Arrowvert a_{n}-n\Arrowvert_{\infty}\leq
L,\quad n\in\mathbb{Z}^{d}.$
If $D_{d}(L)<1$, then $\big{(}\frac{1}{(2\pi)^{d}}e^{i\langle
a_{n},(\cdot)\rangle}\big{)}$ is a Riesz basis for $L_{2}[-\pi,\pi]^{d}$ with
frame bounds $(1-D_{d}(L))^{2}$ and $(1+D_{d}(L))^{2}$.
In the one-dimensional case, Kadec’s theorem is recovered exactly from Theorem
6.2, When $d>1$, the value $x_{d}$ satisfying $0<x_{d}<1/4$ and
$D_{d}(x_{d})=1$ is an upper bound for any value of $L$ satisfying $0<L<1/4$
and $D_{d}(L)<1$. The value of $x_{d}$ is not readily apparent, whereas the
constant in Corollary 6.1 is $\frac{\ln 2}{\pi d}$. A relationship between
this number and $x_{d}$ is given in the following theorem (whose proof is
omitted).
###### Theorem 6.3.
Let $x_{d}$ be the unique number satisfying $0<x_{d}<1/4$ and
$D_{d}(x_{d})=1$. Then
$\lim_{d\rightarrow\infty}\frac{x_{d}-\frac{\ln 2}{\pi d}}{\frac{(\ln
2)^{2}}{12\pi d^{2}}}=1.$
Thus, for sufficiently large $d$, Theorem 6.2 and Corollary 6.1 are
essentially the same.
## 7\. A method of approximation of biorthogonal functions and a recovery of
a theorem of Levinson
In this section we apply the techniques developed in the previously to
approximate the biorthogonal functions to Riesz bases
$\big{(}\frac{1}{\sqrt{2\pi}}e^{it_{n}(\cdot)}\big{)}$ for which the preframe
operator is small perturbation of the identity. This is the content of Theorem
7.1. A well known theorem of Levinson (see [11, pages 47-67]), follows as a
corollary to Theorem 7.1.
###### Definition 7.1.
A Kadec sequence is a sequence $(t_{n})_{n\in\mathbb{Z}}$ of real numbers
satisfying
$\sup_{n\in\mathbb{Z}}|t_{n}-n|=D<1/4.$
###### Theorem 7.2.
Let $(t_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ be a sequence (with $t_{n}\neq
0$ for $n\neq 0$) such that
$(f_{n})_{n}=\big{(}\frac{1}{\sqrt{2\pi}}e^{it_{n}(\cdot)}\big{)}_{n}$ is a
Riesz basis for $L_{2}[-\pi,\pi]$, and let $(e_{n})_{n}$ be the standard
exponential orthonormal basis for $L_{2}[-\pi,\pi]$. If the map $L$ given by
$Le_{n}=f_{n}$ satisfies the estimate $\Arrowvert I-L\Arrowvert<1,$ then the
biorthogonals $G_{n}$ of
$\frac{1}{\sqrt{2\pi}}\mathcal{F}(f_{n})(\cdot)=\mathrm{sinc}(\pi(\cdot-
t_{n}))$ in $PW_{[-\pi,\pi]}$ are
(23) $G_{n}(t)=\frac{H(t)}{(t-t_{n})H^{{}^{\prime}}(t_{n})},\quad
n\in\mathbb{Z},$
where
(24)
$H(t)=(t-t_{0})\prod_{n=1}^{\infty}\Big{(}1-\frac{t}{t_{n}}\Big{)}\Big{(}1-\frac{t}{t_{-n}}\Big{)}.$
###### Definition 7.3.
Let $(t_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ be a sequence such that
$(f_{n})_{n}=\big{(}\frac{1}{\sqrt{2\pi}}e^{it_{n}(\cdot)}\big{)}_{n}$ is a
Riesz basis for $L_{2}[-\pi,\pi]$. If $l\geq 0$, the $l$-truncated sequence
$(t_{l,n})_{n\in\mathbb{Z}}$ is defined by $t_{l,n}=t_{n}$ if $|n|\leq l$ and
$t_{l,n}=n$ otherwise. Define
$f_{l,n}=\frac{1}{\sqrt{2\pi}}e^{it_{l,n}(\cdot)}$ for $n\in\mathbb{Z},$
$l\geq 0$.
Let $P_{l}:L_{2}[-\pi,\pi]\rightarrow L_{2}[-\pi,\pi]$ be the orthogonal
projection onto $\mathrm{span}\\{e_{-l},\ldots,e_{l}\\}$.
###### Proposition 7.4.
Let $(t_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ be a sequence such that
$(f_{n})_{n}$ (defined above) is a Riesz basis for $L_{2}[-\pi,\pi]$. If
$(e_{n})_{n}$ is the standard exponential orthonormal basis for
$L_{2}[-\pi,\pi]$ and the map $L$ (defined above) satisfies the estimate
$\Arrowvert I-L\Arrowvert=\delta<1$, then the following are true:
1) For $l\geq 0$ , the sequence $(f_{l,n})_{n}$ is a Riesz basis for
$L_{2}[-\pi,\pi]$.
2) For $l\geq 0$, the map $L_{l}$ defined by $L_{l}e_{n}=f_{l,n}$ satisfies
$\Arrowvert L_{l}^{-1}\Arrowvert\leq\frac{1}{1-\delta}.$
###### Proof.
If $(c_{n})_{n}\in\ell_{2}(\mathbb{Z})$, then
$\displaystyle(I-L_{l})\big{(}\sum_{n}c_{n}e_{n}\big{)}$ $\displaystyle=$
$\displaystyle\sum_{n}c_{n}(e_{n}-L_{l}e_{n})=\sum_{|n|\leq
l}(e_{n}-f_{n})=(I-L)P_{l}\big{(}\sum_{n}c_{n}e_{n}\big{)},$
so that
(25) $(I-L_{l})=(I-L)P_{l}.$
From this, $\Arrowvert I-L_{l}\Arrowvert\leq\delta$, which implies 1) and 2).
∎
Define the biorthogonal functions of $(f_{l,n})_{n}$ to be
$(f_{l,n}^{*})_{n}$. Passing to the Fourier transform, we have
$\frac{1}{\sqrt{2\pi}}\mathcal{F}(f_{l,n})(t)=\mathrm{sinc}(\pi(t-t_{l,n}))$
and $G_{l,n}(t):=\frac{1}{\sqrt{2\pi}}\mathcal{F}(f_{l,n}^{*})(t)$. Define the
biorthogonal functions of $(f_{n})_{n}$ similarly.
###### Lemma 7.5.
If $(t_{n})_{n}\subset\mathbb{R}$ satisfies the hypotheses of proposition 7.4,
then
$\lim_{l\rightarrow\infty}G_{l,n}=G_{n}$
in $PW_{[-\pi,\pi]}$.
###### Proof.
Note that
$\displaystyle\delta_{nm}=\langle f_{l,n},f_{l,m}^{*}\rangle=\langle
L_{l}e_{n},f_{l,m}^{*}\rangle=\langle e_{n},L_{l}^{*}f_{l,m}^{*}\rangle$
so that for all $m$, $f_{l,m}^{*}=(L_{l}^{*})^{-1}e_{m}$. Similarly,
$f_{m}^{*}=(L^{*})^{-1}e_{m}$. We have
$\displaystyle f_{l,m}^{*}-f_{m}^{*}$ $\displaystyle=$
$\displaystyle((L_{l}^{*})^{-1}-(L^{*})^{-1})e_{m}=(L_{l}^{*})^{-1}(L^{*}-L_{l}^{*})(L^{*})^{-1}e_{m}.$
Now equation (25) implies $L-L_{l}=(L-I)(I-P_{l})$, so that
$\displaystyle
f_{l,m}^{*}-f_{m}^{*}=(L_{l}^{*})^{-1}(I-P_{l})(L^{*}-I)(L^{*})^{-1}e_{m}.$
Applying proposition 7.4 yields
$\displaystyle\Arrowvert
f_{l,m}^{*}-f_{m}^{*}\Arrowvert\leq\frac{1}{1-\delta}\Arrowvert(I-P_{l})(L^{*}-I)(L^{*})^{-1}e_{m}\Arrowvert,$
which for fixed $m$ goes to $0$ as $l\rightarrow\infty$. We conclude
$\lim_{l\rightarrow\infty}f_{l,m}^{*}=f_{m}^{*}$, which, upon passing to the
Fourier transform, yields $\lim_{l\rightarrow\infty}G_{l,m}=G_{m}$. ∎
Proof of Theorem 7.2.
We see that $\delta_{nm}=\langle G_{l,m},S_{l,n}\rangle$, where
$S_{l,n}(t)=\mathrm{sinc}(\pi(t-t_{n}))$ when $|n|\leq l$ and
$S_{l,n}(t)=\mathrm{sinc}(\pi(t-n))$ when $|m|>l$. Without loss of generality,
let $|m|<l$. Equation (1) implies that $G_{l,m}(k)=0$ when $|k|>l$. By the WKS
theorem we have
$\displaystyle G_{l,m}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=-l}^{k=l}G_{l,m}(k)\mathrm{sinc}(\pi(t-k))=\Big{(}\sum_{k=-l}^{k=l}\frac{tG_{l,m}(k)}{k-t}\Big{)}\mathrm{sinc}(\pi
t)$ $\displaystyle=$
$\displaystyle\frac{w_{l}(t)}{\prod_{k=1}^{l}(k-t)(-k-t)}\mathrm{sinc}(\pi
t),$
where $w_{l}$ is a polynomial of degree at most $2l$. Noting that
$\mathrm{sinc}(\pi
t)=\prod_{k=1}^{\infty}\big{(}1-\frac{t^{2}}{k^{2}}\big{)}\quad\mathrm{and}\quad\prod_{k=1}^{l}(k-t)(-k-t)=(-1)^{l}(l!)^{2}\prod_{k=1}^{l}\big{(}1-\frac{t^{2}}{k^{2}}\big{)},$
we have
$\displaystyle
G_{l,m}(t)=\frac{(-1)^{l}w_{l}(t)}{(l!)^{2}}\prod_{k=l+1}^{\infty}\big{(}1-\frac{t^{2}}{k^{2}}\big{)}.$
Again by equation (1), $\delta_{nm}=G_{l,m}(t_{n})$ when $|n|\leq l$ so that
$\delta_{nm}=\frac{(-1)^{l}}{(l!)^{2}}w_{l}(t_{n})\prod_{k=l+1}^{\infty}\big{(}1-\frac{t_{n}^{2}}{k^{2}}\big{)}.$
This determines the zeroes of $w_{l}$. We deduce that
$w_{l}(t)=\frac{c_{l}\prod_{k=1}^{k=l}(t-t_{k})(t-t_{-k})}{t-t_{m}}$
for some constant $c_{l}$. Absorbing constants, we have
$G_{l,m}(t)=\frac{c_{l}H_{l}(t)}{t-t_{m}},$
where
$H_{l}(t):=(t-t_{0})\prod_{k=1}^{l}\big{(}1-\frac{t}{t_{k}}\big{)}\big{(}1-\frac{t}{t_{-k}}\big{)}\prod_{l+1}^{\infty}\big{(}1-\frac{t^{2}}{k^{2}}\big{)}.$
Now $0=H_{l}(t_{m})$, so
$G_{l,m}(t)=c_{l}\frac{H_{l}(t)-H_{l}(t_{m})}{t-t_{m}}$. Taking limits,
$c_{l}=\frac{1}{(H_{l})^{\prime}(t_{m})}$. This yields
$\\\ G_{l,m}(t)=\frac{H_{l}(t)}{(t-t_{m})H_{l}^{\prime}(t_{m})}.$
Define
$H(t)=(t-t_{0})\prod_{k=1}^{\infty}\big{(}1-\frac{t}{t_{k}}\big{)}\big{(}1-\frac{t}{t_{-k}}\big{)}.$
Basic complex analysis shows that $H$ is entire, and $H_{l}\rightarrow H$ and
$H_{l}^{\prime}\rightarrow H^{\prime}$ uniformly on compact subsets of
$\mathbb{C}$. Furthermore, $H^{\prime}(t_{k})\neq 0$ for all $k$, since each
$t_{k}$ is a zero of $H$ of multiplicity one. Together we have
$\lim_{l\rightarrow\infty}G_{l,m}(t)=\frac{H(t)}{(t-t_{m})H^{\prime}(t_{m})},\quad
t\in\mathbb{R}.$
By the foregoing lemma, $G_{l,m}\rightarrow G_{m}$. Observing that convergence
in $PW_{[-\pi,\pi]}$ implies pointwise convergence yields the desired result.
Levinson proved a version of Theorem 7.2 in the case where
$(t_{n})_{n\in\mathbb{Z}}$ is a Kadec sequence. His original proof is found in
[11, pages 47-67]). We recall that if $(f_{n})_{n}$ is a Riesz basis arising
from a Kadec sequence, then the preframe operator $L$ satisfies $\Arrowvert
I-L\Arrowvert<1$. Levinson’s theorem is then recovered from Theorem 7.2.
## References
* [1] S. A. Avdonin. On the question of Riesz bases of exponential functions in $L^{2}$. Vestnik Leningrad Univ. Ser. Mat., 13, 5-12 (Russian); English translation in Vestnik Leningrad Univ. Math, 7, (1979) 203-211.
* [2] J. Benedetto, Irregular sampling and frames, Wavelets - A Tutorial in Theory and Applications, C. Chui, Ed., Academic Press, New York, (1991) 1-63.
* [3] J. Benedetto and W. Heller, Irregular sampling and the theory of frames, Part I, Note Mathematica, Suppl. 1, (1990) 103-125.
* [4] J. J. Benedetto, Paulo J. S. G. Ferriera, Modern Sampling Theory, Birkhauser, (2001).
* [5] A. Beurling, Local harmonic analysis with some applications to differential operators, in “Some Recent Advances in the Basic Sciences, Vol. 1” (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1962-1964), (1966) 109-125.
* [6] Peter G. Casazza, The Art of Frames, Taiwanese Journal of Mathematics, Vol. 4, No. 2, (2001) 129-201.
* [7] I. Daubechies, R. DeVore, Approximating a bandlimited function using very coursely quantized data: a family of stable sigma-delta modulators of arbitrary order, Annals of Mathematics 158, (2003) 679-710.
* [8] Duffin. R. J , and Eachus, J. J. Some notes on an expansion theorem of Paley and Wiener. Bull. Am. Math. Soc. 44, (1942) 850-855.
* [9] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, (1999) 245.
* [10] Kadec, M. I. The exact value of the Paley-Wiener constant. Sov. Math. Dokl. 5, (1964) 559-561.
* [11] N. Levinson, Gap and Density Theorems, American Mathematical Society, (1940).
* [12] H. Pak and C. Shin, Perturbation of Nonharmonic Fourier Series and Nonuniform Sampling Theorem, Bulletin of the Korean Mathematical Society 44, (2007) No.2, 351-358.
* [13] W. Sun, and X. Zhou, On the Stability of Multivariate Trigonometric Systems, Journal of Mathematical Analysis and Applications, 235, (1999) 159-167.
* [14] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, (1980).
* [15] A. Zayed, Advances in Shannon’s Sampling Theory, CRC Press, (2000).
|
arxiv-papers
| 2010-09-10T16:33:38 |
2024-09-04T02:49:12.791959
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Benjamin Aaron Bailey",
"submitter": "Benjamin Bailey Mr.",
"url": "https://arxiv.org/abs/1009.2043"
}
|
1009.2045
|
11institutetext: B. A. Bailey 22institutetext: Department of Mathematics,
Texas A&M University, College Station, TX 77843-3368, 22email:
abailey@math.tamu.edu
# An asymptotic equivalence between two frame perturbation theorems
B. A. Bailey
###### Abstract
In this paper, two stability results regarding exponential frames are
compared. The theorems, (one proven herein, and the other in SZ ), each give a
constant such that if $\sup_{n\in\mathbb{Z^{d}}}\|\epsilon_{n}\|_{\infty}<C$,
and $(e^{i\langle\cdot,t_{n}\rangle})_{n\in\mathbb{Z}^{d}}$ is a frame for
$L_{2}[-\pi,\pi]^{d}$, then
$(e^{i\langle\cdot,t_{n}+\epsilon_{n}\rangle})_{n\in\mathbb{Z}^{d}}$ is a
frame for $L_{2}[-\pi,\pi]^{d}$. These two constants are shown to be
asymptotically equivalent for large values of $d$.
## 1 The perturbation theorems
We define a frame for a separable Hilbert space $H$ to be a sequence
$(f_{n})_{n}\subset H$ such that for some $0<A\leq B,$
$A^{2}\|f\|^{2}\leq\sum_{n}|\langle f,f_{n}\rangle|^{2}\leq
B^{2}\|f\|^{2},\quad f\in H.$
The best $A^{2}$ and $B^{2}$ satisfying the inequality above are said to be
the frame bounds for the frame. If $(e_{n})_{n}$ is an orthonormal basis for
$H$, the synthesis operator $Le_{n}=f_{n}$ is bounded, linear, and onto, iff
$(f_{n})_{n}$ is a frame. Equivalently, $(f_{n})_{n}$ is a frame iff the
operator $L^{*}$ is an isomorphic embedding, (see CA ). In this case, $A$ and
$B$ are the best constants such that
$A\|f\|\leq\|L^{*}f\|\leq B\|f\|,\quad f\in H.$
The simplest stability result regarding exponential frames for
$L_{2}[-\pi,\pi]$ is the theorem below, which follows immediately from (Y, ,
Theorem 13, p 160).
###### Theorem 1.1
Let $(t_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ be a sequence such that
$(h_{n})_{n\in\mathbb{Z}}:=\big{(}\frac{1}{\sqrt{2\pi}}e^{it_{n}x}\big{)}_{n\in\mathbb{Z}}$
is a frame for $L_{2}[-\pi,\pi]$ with frame bounds $A^{2}$ and $B^{2}$. If
$(\tau_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ and
$(f_{n})_{n\in\mathbb{Z}}:=\big{(}\frac{1}{\sqrt{2\pi}}e^{i\tau_{n}x}\big{)}_{n\in\mathbb{Z}}$
is a sequence such that
$\sup_{n\in\mathbb{Z}}|\tau_{n}-t_{n}|<\frac{1}{\pi}\ln\left(1+\frac{A}{B}\right),$
(1)
then the sequence $(f_{n})_{n\in\mathbb{Z}}$ is also a frame for
$L_{2}[-\pi,\pi]$.
The following theorem is a very natural generalization of Theorem 1.1 to
higher dimensions.
###### Theorem 1.2
Let $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a sequence such that
$(h_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{\langle(\cdot),t_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
is a frame for $L_{2}[-\pi,\pi]^{d}$ with frame bounds $A^{2}$ and $B^{2}$. If
$(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ and
$(f_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),\tau_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
is a sequence such that
$\sup_{k\in\mathbb{N}}\Arrowvert\tau_{k}-t_{k}\Arrowvert_{\infty}<\frac{1}{\pi
d}\ln\left(1+\frac{A}{B}\right),$ (2)
then the sequence $(f_{k})_{k\in\mathbb{N}}$ is also a frame for
$L_{2}[-\pi,\pi]^{d}$.
The proof of Theorem 1.2 relies on the following lemma:
###### Lemma 1
Choose $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$(h_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{\langle(\cdot),t_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
satisfies
$\Big{\|}\sum_{k=1}^{n}a_{k}h_{k}\Big{\|}_{L_{2}[-\pi,\pi]^{d}}\leq
B\Big{(}\sum_{k=1}^{n}|a_{k}|^{2}\Big{)}^{1/2},\quad\mathrm{for\
all}\quad(a_{k})_{k=1}^{n}\subset\mathbb{C}.$
If $(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$, and
$(f_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),\tau_{k}\rangle}\big{)}_{k\in\mathbb{N}}$,
then for all $r,s\geq 1$ and any finite sequence $(a_{k})_{k}$, we have
$\Bigg{\Arrowvert}\sum_{k=r}^{s}a_{k}(h_{k}-f_{k})\Bigg{\Arrowvert}_{L_{2}[-\pi,\pi]^{d}}\\\
{}\leq B\Big{(}e^{\pi d\big{(}\sup\limits_{r\leq k\leq
s}\Arrowvert\tau_{k}-t_{k}\Arrowvert_{\infty}\big{)}}-1\Big{)}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}.$
This lemma is a slight generalization of Lemma 5.3, proven in BB using simple
estimates. Lemma 1 is proven similarly. Now for the proof of Theorem 1.2.
###### Proof
Define
$\delta=\sup_{k\in\mathbb{N}}\Arrowvert\tau_{k}-t_{k}\Arrowvert_{\infty}$.
Lemma 1 shows that the map $\tilde{L}e_{n}=f_{n}$ is bounded and linear, and
that
$\|L-\tilde{L}\|\leq B\big{(}e^{\pi d\delta}-1\big{)}:=\beta A$
for some $0\leq\beta<1$. This implies
$\|L^{*}f-\tilde{L}^{*}f\|\leq\beta A,\quad\mathrm{when}\quad\|f\|=1.$ (3)
Rearranging, we have
$A(1-\beta)\leq\|\tilde{L}^{*}f\|,\quad\mathrm{when}\quad\|f\|=1.$
By the previous remarks regarding frames, $(f_{k})_{k\in\mathbb{N}}$ is a
frame for $L_{2}[-\pi,\pi]^{d}$.
Theorem 1.3, proven in SZ , is a more delicate frame perturbation result with
a more complex proof:
###### Theorem 1.3
Let $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a sequence such that
$(h_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{\langle(\cdot),t_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
is a frame for $L_{2}[-\pi,\pi]^{d}$ with frame bounds $A^{2}$ and $B^{2}$.
For $d\geq 1$, define
$D_{d}(x):=\Big{(}1-\cos\pi x+\sin\pi x+\frac{\sin\pi x}{\pi
x}\Big{)}^{d}-\Big{(}\frac{\sin\pi x}{\pi x}\Big{)}^{d},$
and let $x_{d}$ be the unique number such that $0<x_{d}\leq 1/4$ and
$D_{d}(x_{d})=\frac{A}{B}$. If
$(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ and
$(f_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),\tau_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
is a sequence such that
$\sup_{k\in\mathbb{N}}\|\tau_{k}-t_{k}\|_{\infty}<x_{d},$ (4)
then the sequence $(f_{k})_{k\in\mathbb{N}}$ is also a frame for
$L_{2}[-\pi,\pi]^{d}$.
## 2 An asymptotic equivalence
It is natural to ask how the constants $x_{d}$ and $\frac{1}{\pi
d}\ln\big{(}1+\frac{A}{B}\big{)}$ are related. Such a relationship is given in
the following theorem.
###### Theorem 2.1
If $x_{d}$ is the unique number satisfying $0<x_{d}<1/4$ and
$D_{d}(x_{d})=\frac{A}{B}$, then
$\lim_{d\rightarrow\infty}\frac{x_{d}-\frac{1}{\pi
d}\ln\big{(}1+\frac{A}{B}\big{)}}{\frac{\big{[}\ln\big{(}1+\frac{A}{B}\big{)}\big{]}^{2}}{6\pi\big{(}1+\frac{B}{A}\big{)}d^{2}}}=1.$
We prove the theorem with a sequence of propositions.
###### Proposition 1
Let $d$ be a positive integer. If
$\displaystyle f(x)$ $\displaystyle:=$ $\displaystyle
1-\cos(x)+\sin(x)+\mathrm{sinc}(x),$ $\displaystyle g(x)$ $\displaystyle:=$
$\displaystyle\mathrm{sinc}(x),$
then
$\displaystyle 1)$ $\displaystyle\quad f^{\prime}(x)+g^{\prime}(x)>0,\quad
x\in(0,\pi/4),$ $\displaystyle 2)$ $\displaystyle\quad g^{\prime}(x)<0,\quad
x\in(0,\pi/4),$ $\displaystyle 3)$ $\displaystyle\quad
f^{\prime\prime}(x)>0,\quad x\in(0,\Delta)\quad\mathrm{for\ some}\quad
0<\Delta<1/4.$
The proof of Proposition 1 involves only elementary calculus and is omitted.
###### Proposition 2
The following statements hold:
1) For $d>0$, $D_{d}(x)$ and $D^{\prime}_{d}(x)$ are positive on $(0,1/4)$.
2) For all $d>0$, $D^{\prime\prime}_{d}(x)$ is positive on $(0,\Delta)$.
###### Proof
Note $D_{d}(x)=f(\pi x)^{d}-g(\pi x)^{d}$ is positive. This expression yields
$D^{\prime}_{d}(x)/(d\pi)=f(\pi x)^{d-1}f^{\prime}(\pi x)-g(\pi
x)^{d-1}g^{\prime}(\pi x)>0\quad\mathrm{on}\quad(0,1/4)$
by Proposition 1. Differentiating again, we obtain
$\displaystyle D^{\prime\prime}_{d}(x)/(d\pi^{2})$ $\displaystyle=$
$\displaystyle(d-1)\big{[}f(\pi x)^{d-2}(f^{\prime}(\pi x))^{2}-g(\pi
x)^{d-2}(g^{\prime}(\pi x))^{2}\big{]}+$ $\displaystyle+$ $\displaystyle[f(\pi
x)^{d-1}f^{\prime\prime}(\pi x)-g(\pi x)^{d-1}g^{\prime\prime}(\pi
x)]\quad\text{on}\quad(0,1/4).$
If $g^{\prime\prime}(\pi x)\leq 0$ for some $x\in(0,1/4)$, then the second
bracketted term is positive. If $g^{\prime\prime}(\pi x)>0$ for some
$x\in(0,1/4)$, then the second bracketted term is positive if
$f^{\prime\prime}(\pi x)-g^{\prime\prime}(\pi x)>0$, but
$f^{\prime\prime}(\pi x)-g^{\prime\prime}(\pi x)=\pi^{2}(\cos(\pi x)-\sin(\pi
x))$
is positive on $(0,1/4)$.
To show the first bracketted term is positive, it suffices to show that
$f^{\prime}(\pi x)^{2}>g^{\prime}(\pi x)^{2}=(f^{\prime}(\pi x)+g^{\prime}(\pi
x))(f^{\prime}(\pi x)-g^{\prime}(\pi x))>0$
on $(0,\Delta)$. Noting $f^{\prime}(\pi x)-g^{\prime}(\pi x)=\pi(\cos(\pi
x)+\sin(\pi x))>0$, it suffices to show that $f^{\prime}(\pi x)+g^{\prime}(\pi
x)>0$, but this is true by Proposition 1.
Note that Proposition 2 implies $x_{d}$ is unique.
###### Corollary 1
We have $\lim_{d\rightarrow\infty}x_{d}=0$.
###### Proof
Fix $n>0$ with $1/n<\Delta$, then $\lim_{d\rightarrow\infty}D_{d}(1/n)=\infty$
(since $f$ increasing implies
$0<-\cos(\pi/n)+\sin(\pi/n)+\mathrm{sinc}(\pi/n)).$ For sufficiently large
$d$, $D_{d}(1/n)>\frac{A}{B}$. But $\frac{A}{B}=D_{d}(x_{d})<D_{d}(1/n)$, so
$x_{d}<1/n$ by Proposition 2.
###### Proposition 3
Define $\omega_{d}=\frac{1}{\pi d}\ln\big{(}1+\frac{A}{B}\big{)}$. We have
$\displaystyle\lim_{d\rightarrow\infty}d\Big{(}\frac{A}{B}-D_{d}(\omega_{d})\Big{)}=\frac{A}{6B}\Big{[}\ln\Big{(}1+\frac{A}{B}\Big{)}\Big{]}^{2},$
$\displaystyle\lim_{d\rightarrow\infty}\frac{1}{d}D^{\prime}_{d}(\omega_{d})=\pi\Big{(}1+\frac{A}{B}\Big{)},$
$\displaystyle\lim_{d\rightarrow\infty}\frac{1}{d}D^{\prime}_{d}(x_{d})=\pi\Big{(}1+\frac{A}{B}\Big{)}.$
###### Proof
1) For the first equality, note that
$D_{d}(\omega_{d})=\Big{[}(1+h(x))^{\ln(c)/x}-g(x)^{\ln(c)/x}\Big{]}\Big{|}_{x=\frac{\ln(c)}{d}}$
(5)
where $h(x)=-\cos(x)+\sin(x)+\mathrm{sinc}(x)$, $g(x)=\mathrm{sinc}(x)$, and
$c=1+\frac{A}{B}$. L’Hospital’s rule implies that
$\lim_{x\rightarrow
0}(1+h(x))^{\ln(c)/x}=c\quad\mathrm{and}\quad\lim_{x\rightarrow
0}g(x)^{\ln(c)/x}=1.$
Looking at the first equality in the line above, another application of
L’Hospital’s rule yields
$\lim_{x\rightarrow
0}\frac{(1+h(x))^{\ln(c)/x}-c}{x}=c\ln(c)\Bigg{[}\frac{\frac{h^{\prime}(x)}{1+h(x)}-1}{x}-\frac{\ln(1+h(x))-x}{x^{2}}\Bigg{]}.$
(6)
Observing that $h(x)=x+x^{2}/3+O(x^{3}))$, we see that
$\lim_{x\rightarrow 0}\frac{\frac{h^{\prime}(x)}{1+h(x)}-1}{x}=-\frac{1}{3}.$
L’Hospital’s rule applied to the second term on the right hand side of
equation (6) gives
$\lim_{x\rightarrow 0}\frac{(1+h(x))^{\ln(c)/x}-c}{x}=\frac{-c\ln(c)}{6}.$ (7)
In a similar fashion,
$\lim_{x\rightarrow 0}\frac{g(x)^{\ln(c)/x}-1}{x}=\ln(c)\lim_{x\rightarrow
0}\Bigg{[}\frac{\frac{g^{\prime}(x)}{g(x)}}{x}-\frac{\ln(g(x))}{x^{2}}\Bigg{]}.$
(8)
Observing that $g(x)=1-x^{2}/6+O(x^{4})$, we see that
$\lim_{x\rightarrow 0}\frac{\frac{g^{\prime}(x)}{g(x)}}{x}=-\frac{1}{3}.$
L’Hospital’s rule applied to the second term on the right hand side of
equation (8) gives
$\lim_{x\rightarrow 0}\frac{g(x)^{\ln(c)/x}-1}{x}=-\frac{\ln(c)}{6}.$ (9)
Combining equations (5) (7), and (9), we obtain
$\lim_{d\rightarrow\infty}d\Big{(}\frac{A}{B}-D_{d}(\omega_{d})\Big{)}=\frac{A}{6B}\Big{[}\ln\Big{(}1+\frac{A}{B}\Big{)}\Big{]}^{2}.$
2) For the second equality we have, (after simplification),
$\frac{1}{d}D^{\prime}_{d}(\omega_{d})=\pi\Bigg{[}\frac{\Big{(}1+h\big{(}\frac{\ln(c)}{d}\big{)}\Big{)}^{\big{(}\ln(c)\big{)}/\big{(}\frac{\ln(c)}{d}\big{)}}}{1+h\Big{(}\frac{\ln(c)}{d}\Big{)}}-\frac{g\Big{(}\frac{\ln(c)}{d}\Big{)}^{\big{(}\ln(c)\big{)}/\big{(}\frac{\ln(c)}{d}\big{)}}}{g\Big{(}\frac{\ln(c)}{d}\Big{)}}g^{\prime}\Big{(}\frac{\ln(c)}{d}\Big{)}\Bigg{]}.$
In light of the previous work, this yields
$\lim_{d\rightarrow\infty}\frac{1}{d}D^{\prime}_{d}(\omega_{d})=\pi\Big{(}1+\frac{A}{B}\Big{)}.$
3) To derive the third equality, note that $(1+h(\pi
x_{d}))^{d}=\frac{A}{B}+g(\pi x_{d})^{d}$ yields
$\frac{1}{d}D^{\prime}_{d}(x_{d})=\pi\Bigg{[}\frac{\frac{A}{B}+g(\pi
x_{d})^{d}}{1+h(\pi x_{d})}h^{\prime}(\pi x_{d})-\frac{g(\pi x_{d})^{d}}{g(\pi
x)}g^{\prime}(\pi x_{d})\Bigg{]}.$ (10)
Also, the first inequality in propostion 3 yields that, for sufficiently large
$d$ (also large enough so that $x_{d}<\Delta$ and $\omega_{d}<\Delta$), that
$D_{d}(\omega_{d})<\frac{A}{B}=D_{d}(x_{d})$. This implies $\omega_{d}<x_{d}$
since $D_{d}$ is increasing on $(0,1/4)$. But $D_{d}$ is also convex on
$(0,\Delta)$, so we can conclude that
$D^{\prime}_{d}(\omega_{d})<D^{\prime}_{d}(x_{d}).$ (11)
Combining this with equation (10), we obtain
$\displaystyle\Bigg{[}\frac{1}{d}D^{\prime}_{d}(\omega_{d})+\frac{\pi g(\pi
x_{d})^{d}}{g(\pi x_{d})}g^{\prime}(\pi x_{d})\Bigg{]}\Big{(}\frac{1+h(\pi
x_{d})}{h^{\prime}(\pi x_{d})}\Big{)}<\pi\Big{(}\frac{A}{B}+g(\pi
x_{d})^{d}\Big{)}<\pi\Big{(}1+\frac{A}{B}\Big{)}.$
The limit as $d\rightarrow\infty$ of the left hand side of the above
inequality is $\pi\Big{(}1+\frac{A}{B}\Big{)}$, so
$\lim_{d\rightarrow\infty}\pi\Big{(}\frac{A}{B}+g(\pi
x_{d})^{d}\Big{)}=\pi\Big{(}1+\frac{A}{B}\Big{)}.$
Combining this with equation (10), we obtain
$\lim_{d\rightarrow\infty}\frac{1}{d}D^{\prime}_{d}(x_{d})=\pi\Big{(}1+\frac{A}{B}\Big{)}.$
Now we complete the proof of Theorem 2.1.
For large $d$, the mean value theorem implies
$\displaystyle\frac{D_{d}(x_{d})-D_{d}(\omega_{d})}{x_{d}-\omega_{d}}=D^{\prime}_{d}(\xi),\quad\xi\in(\omega_{d},x_{d}),$
so that
$\displaystyle
x_{d}-\omega_{d}=\frac{\frac{A}{B}-D_{d}(\omega_{d})}{D^{\prime}_{d}(\xi)}.$
For large $d$, convexity of $D_{d}$ on $(0,\Delta)$ implies
$\frac{d\Big{(}\frac{A}{B}-D_{d}(\omega_{d})\Big{)}}{\frac{1}{d}D^{\prime}_{d}(x_{d})}<d^{2}(x_{d}-\omega_{d})<\frac{d\Big{(}\frac{A}{B}-D_{d}(\omega_{d})\Big{)}}{\frac{1}{d}D^{\prime}_{d}(\omega_{d})}.$
Applying Proposition 3 proves the theorem.
## References
* (1) Bailey, B.A.: Sampling and recovery of multidimensional bandlimited functions via frames. J. Math. Anal. Appl. 367, Issue 2 374–388 (2010)
* (2) Casazza, P.G.: The art of frames. Taiwanese J. Math. 4. No. 2 129–201 (2001)
* (3) Sun, W., Zhou, X.: On the stability of multivariate trigonometric systems. J. Math. Anal. Appl. 235, 159–167 (1999)
* (4) Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press (2001)
###### Acknowledgements.
This research was supported in part by the NSF Grant DMS0856148.
|
arxiv-papers
| 2010-09-10T16:35:48 |
2024-09-04T02:49:12.801692
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. A. Bailey",
"submitter": "Benjamin Bailey Mr.",
"url": "https://arxiv.org/abs/1009.2045"
}
|
1009.2047
|
# Multivariate polynomial interpolation and sampling in Paley-Wiener spaces
B. A. Bailey Department of Mathematics, Texas A&M University
College Station, TX 77843, USA abailey@math.tamu.edu
###### Abstract.
In this paper, an equivalence between existence of particular exponential
Riesz bases for multivariate bandlimited functions and existence of certain
polynomial interpolants for these bandlimited functions is given. For certain
classes of unequally spaced data nodes and corresponding $\ell_{2}$ data, the
existence of these polynomial interpolants allows for a simple recovery
formula for multivariate bandlimited functions which demonstrates $L_{2}$ and
uniform convergence on $\mathbb{R}^{d}$. A simpler computational version of
this recovery formula is also given, at the cost of replacing $L_{2}$ and
uniform convergence on $\mathbb{R}^{d}$ with $L_{2}$ and uniform convergence
on increasingly large subsets of $\mathbb{R}^{d}$. As a special case, the
polynomial interpolants of given $\ell_{2}$ data converge in the same fashion
to the multivariate bandlimited interpolant of that same data. Concrete
examples of pertinant Riesz bases and unequally spaced data nodes are also
given.
This research was supported in part by the NSF Grant DMS0856148
## 1\. Introduction
Approximation of bandlimited functions as limits of polynomials has a long
history, as the following question illustrates: if $(\mathrm{sinc}\pi(\cdot-
t_{n}))_{n\in\mathbb{Z}}$ forms a Riesz basis for $PW_{[-\pi,\pi]}$, what are
the canonical product expansions of the biorthogonal functions for this Riesz
basis? The first results along these lines were given by Paley and Wiener in
[8], and improved upon by Levinson in [5, pages 47-67]), while Levin extends
these results to different classes of Riesz bases in [4]. A complete solution
is given by Lyubarskii and Seip in [6] and Pavlov in [9]. In particular, they
prove the following theorem:
###### Theorem 1.1.
Let $(t_{n})_{n}\subset\mathbb{R}$, where $t_{n}\neq 0$ when $n\neq 0$, be a
sequence such that the family of functions $(\mathrm{sinc}\pi(\cdot-
t_{n}))_{n}$ is a Riesz basis for $PW_{[-\pi,\pi]}$, then the function
$S(z)=\lim_{r\rightarrow\infty}(z-t_{0})\prod_{\\{t_{n}\ :\ |t_{n}|<r,n\neq
0\\}}\Big{(}1-\frac{z}{t_{n}}\Big{)}$
is entire, where convergence is uniform on compacta, and the biorthogonal
functions $(G_{n})_{n}$ of
$(\mathrm{sinc}\pi((\cdot)-t_{n}))_{n}$
are given by
$G_{n}(z)=\frac{S(z)}{(z-t_{n})S^{\prime}(t_{n})}.$
The following is a readily proven corollary of Theorem 1.1:
###### Corollary 1.2.
Let $(t_{n})_{n}\subset\mathbb{R}$ and $(G_{k})_{k}$ be defined as in Theorem
1.1, then for each $k$, there exists a sequence of polynomials
$(\Phi_{N,k})_{N}$ such that
1) $\Phi_{N,k}(t_{n})=G_{k}(t_{n})$ when $|t_{n}|<N$.
2) $\lim_{N\rightarrow\infty}\Phi_{N,k}=G_{k}$ uniformly on compacta.
Corollary 1.2 raises two questions:
1) Does every multivariate bandlimited function, (not just biorthogonal
functions associated with a particular exponential Riesz basis), have a
corresponding sequence of polynomial interpolants?
2) If such polynomial interpolants for a multivariate bandlimited function
exist, can these interpolants be used to be approximate the function in some
simple and straightforward way?
Let $(t_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ be a sequence such
that the family of exponentials
$\big{(}e^{i\langle\cdot,t_{n}\rangle}\big{)}_{n\in\mathbb{Z}^{d}}$ is a
uniformly invertible Riesz basis for $L_{2}([-\pi,\pi]^{d})$ (defined in
section 4). Under this condition, Theorem 4.2 answers the first question
affirmatively by showing that multivariate bandlimited functions can be
approximated globally, both uniformly and in $L_{2}$, by a rational function
times a multivariate sinc function. Stated informally,
(1) $f(t)\simeq\Psi_{\ell}(t)\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)},\quad\ell>0,$
where $(\Psi_{\ell}))_{\ell\in\mathbb{N}}$ is a particular sequence of
interpolating polynomials and $(Q_{d,\ell})_{d,\ell}$ is a sequence of
polynomials which removes the zeros of the SINC function. This gives a partial
answer to the second question, but the fraction in the approximants above
becomes more complex as $\ell$ increases. Theorem 5.1 gives a more
satisfactory answer to question 2) by using
$e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}},\quad\ell>0$
in lieu of the SINC function in expression (1). The exponent in the above
expression is simply a rational function of $\ell$. This simplicity
necessitates replacing global $L_{2}$ and uniform convergence with a more
local (though not totally local) convergence. Corollary 5.2 is of particular
interest as a multivariate analogue of Theorem 1.1, stately informally as
$f(t)\simeq\Psi_{\ell}(t),\quad\ell>0.$
The author is unaware of any other multivariate polynomial approximation
theorem which applies to exponential Riesz bases which are not necessarily
tensor products of single-variable Riesz bases, or that demonstrate
convergence stronger than uniform convergence on compacta. As a note, Theorems
4.2, 5.1, and Corollary 5.2 do not, at this point, recover Corollary 1.2 in
its generality of allowable sequences $(t_{n})_{n}\subset\mathbb{R}$; however,
the comments above show that their value is primarily due to their
multidimensional nature and convergence properties.
This paper is outlined as follows. Section 2 covers the necessary preliminary
and background material regarding bandlimited functions, and section 3
outlines some basic properties of uniformly invertible operators. Theorems 4.2
and Theorem 5.1 are proven in sections 4 and 5 respectively, along with
pertinant corollaries. Section 6 gives explicit examples of sequences
$(t_{n})_{n\in\mathbb{Z}^{d}}$ to which Theorems 4.2 and 5.1 apply. Section 7
(as an appendix) addresses the optimality of growth rates appearing in Theorem
5.1.
## 2\. Preliminaries
###### Definition 2.1.
A reproducing kernel Hilbert space is a Hilbert space $H$ of functions on $X$
such that there exists $K:X\times X\rightarrow\mathbb{C}$ satisfying the
following:
1) For all $y\in X$, $K(\cdot,y)\in H$.
2) $f(x)=\langle f(\cdot),K(\cdot,x)\rangle$ for all $x\in X$ and $f\in H.$
###### Definition 2.2.
A Riesz basis for a Hilbert space $H$ is a sequence $(f_{n})_{n\in\mathbb{N}}$
which is isomorphically equivalent to an orthonormal basis of $H$.
Equivalently, a Riesz basis is an unconditional Schauder basis.
If $(f_{n})_{n\in\mathbb{N}}$ is a Schauder (Riesz) basis for a Hilbert space
$H$, then there exists a unique set of functions
$(f_{n}^{*})_{n\in\mathbb{N}}$, (the biorthogonals of
$(f_{n})_{n\in\mathbb{N}}$) such that $\langle
f_{n},f_{m}^{*}\rangle=\delta_{nm}.$ The biorthogonals also form a Schauder
(Riesz) basis for $H$. Note that biorthogonality is preserved under a unitary
transformation.
We use the $d$-dimensional $L_{2}$ isometric Fourier transform
$\mathcal{F}(f)(\cdotp)=\mathrm{P.V.\
}\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}f(\xi)e^{-i\langle\cdotp,\xi\rangle}d\xi,\quad
f\in L_{2}(\mathbb{R}^{d}),$
where the inverse transform is given by
$\mathcal{F}^{-1}(f)(\cdotp)=\mathrm{P.V.\
}\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}f(\xi)e^{i\langle\cdotp,\xi\rangle}d\xi,\quad
f\in L_{2}(\mathbb{R}^{d}).$
###### Definition 2.3.
We define $PW_{[-\pi,\pi]^{d}}:=\\{f\in
L_{2}(\mathbb{R}^{d})\arrowvert\mathrm{supp}(\mathcal{F}^{-1}(f))\subset[-\pi,\pi]^{d}\\}$,
with the inherited $L_{2}(\mathbb{R}^{d})$ norm. Functions in
$PW_{[-\pi,\pi]^{d}}$ are also called bandlimited functions.
Here are facts concerning $PW_{[-\pi,\pi]^{d}}$ which will be used
ubiquitously.
1) $PW_{[-\pi,\pi]^{d}}$ is isometric to $L_{2}([-\pi,\pi]^{d})$ by way of the
Fourier transform.
2) $PW_{[-\pi,\pi]^{d}}$ consists of entire functions, though in this paper we
restrict the domain to $\mathbb{R}^{d}$.
3) $PW_{[-\pi,\pi]^{d}}$ is a reproducing kernel Hilbert space with
reproducing kernel
$K(x,y)=\mathrm{SINC}\pi(x-y)$
where
$\mathrm{SINC}(x):=\mathrm{sinc}(x(1))\cdot\ldots\cdot\mathrm{sinc}(x(d)),\quad\mathrm{sinc}(x):=\frac{\sin(x)}{x}.$
4) $\big{(}\mathrm{SINC}\pi((\cdot)-n)\big{)}_{n\in\mathbb{Z}^{d}}$ is an
orthonormal basis for $PW_{[-\pi,\pi]^{d}}$. This follows from
$\mathcal{F}\big{(}\frac{1}{\sqrt{2\pi}}e^{i\tau(\cdot)}\big{)}(t)=\mathrm{sinc}\pi(t-\tau).$
5) In $PW_{[-\pi,\pi]^{d}}$, $L_{2}$ convergence implies uniform convergence.
6) If $f\in PW_{[-\pi,\pi]^{d}}$, then
$\lim_{\|x\|_{\infty}\rightarrow\infty}f(x)=0.$
This follows from the $d$-dimensional Riemann-Lebesgue Lemma.
7) The following result [10, Theorem 19.3] due to Paley and Wiener
characterizes single-variable bandlimited functions.
###### Theorem 2.4.
A function $f$ is in $PW_{[-\pi,\pi]}$ if and only if the following statements
hold.
1) $f$ is entire.
2) There exists $M\geq 0$ such that $|f(z)|\leq Me^{\pi|z|}$ for
$z\in\mathbb{C}$.
3) $f\big{|}_{\mathbb{R}}\in L_{2}(\mathbb{R})$.
## 3\. Uniform invertibility of operators and Riesz bases
###### Definition 3.1.
Let $A:\ell_{2}(\mathbb{N})\rightarrow\ell_{2}(\mathbb{N})$ be an onto
isomorphism. Regard $A$ as a matrix map with respect to the unit vector basis
of $\ell_{2}(\mathbb{N})$. Let $\pi_{k}$ be the orthogonal projection onto the
span of the first $k$ terms of the unit vector basis. If
(2) $\sup_{j\in\mathbb{N}}\|(\pi_{k_{j}}A\pi_{k_{j}})^{-1}\|<\infty,$
for an increasing sequence $(k_{j})_{j\in\mathbb{N}},$ then $A$ is said to be
uniformly invertible as a matrix map with respect to the projections
$(\pi_{k_{j}})_{j\in\mathbb{N}}$. The terms in inequality (2) should be
interpreted as standard matrix norms and inverses of finite dimensional
matrices.
Let $S$ be an orthonormal basis for a Hilbert space $H$. Let
$(S_{n})_{n\in\mathbb{N}}$ be a sequence of sets such that
1) $\emptyset\neq S_{1}\subsetneq S_{2}\subsetneq\cdots\subset S$, and
2) $\bigcup_{n=1}^{\infty}S_{n}=S$.
Define $P_{\ell}$ to be the orthogonal projection onto
$\mathrm{span}\\{e_{k}\\}_{e_{k}\in S_{\ell}}$. Note that
(3) $\lim_{\ell\rightarrow\infty}P_{\ell}x=x,\quad x\in H.$
Linearly order $S=(e_{n})_{n\in\mathbb{N}}$ such that, if $e_{n}\in
S_{k}\setminus S_{k-1}$, and $e_{m}\in S_{k-1}$, then $m<n$.
###### Definition 3.2.
Let $(v_{k})_{k\in\mathbb{N}}$ be the unit vector basis for
$\ell_{2}(\mathbb{N})$ and define $\phi$ by $\phi e_{k}=v_{k}$. Let
$L:H\rightarrow H$ be an onto isomorphism. $L$ is said to be uniformly
invertible with respect to the projections $(P_{\ell})_{\ell\in\mathbb{N}}$ if
$\phi L\phi^{-1}$ is uniformly invertible as a matrix map with respect to the
projections $(\pi_{|S_{\ell}|})$.
We define the following notation:
(4) $(P_{\ell}LP_{\ell})^{-1}:=(\pi_{|s_{\ell}|}(\phi
L\phi^{-1})\pi_{|s_{\ell}|})^{-1}.$
By saying $P_{\ell}LP_{\ell}$ is invertible, we mean that the right hand side
of equation (4) is well defined. If $L$ is defined on
$\mathrm{span}(e_{n})_{n\in\mathbb{N}}$, but perhaps not on $H$, we define
“$P_{\ell}LP_{\ell}$ is invertible” in the same way.
If $L$ is an operator on $H$ (perhaps densely defined), and
$(P_{\ell})_{\ell\in\mathbb{N}}$ is a sequence of projections defined above,
define the operator $L_{\ell}=LP_{\ell}+I-P_{\ell}.$
###### Definition 3.3.
Let $(v_{k})_{k\in\mathbb{N}}$ be a Riesz basis for $H$. We define
$(v_{k})_{k\in\mathbb{N}}$ to be uniformly invertible with respect to the
projections $(P_{\ell})_{\ell\in\mathbb{N}}$ if the corresponding isomorphism
$Le_{k}=v_{k}$ is uniformly invertible with respect to the projections
$(P_{\ell})_{\ell\in\mathbb{N}}$.
We can now state and prove the following lemmas:
###### Lemma 3.4.
Let $(e_{n})_{n\in\mathbb{N}}$ be an orthonormal basis for $H$, let
$(f_{n})_{n\in\mathbb{N}}\subset H$, and let $P_{\ell}$ be the orthogonal
projection onto $\mathrm{span}(e_{n})_{n\leq\ell}$. Define
$L:\mathrm{span}\\{e_{n}\\}_{n\in\mathbb{N}}\rightarrow H$ by $Le_{n}=f_{n}$.
For each $\ell>0$, the following statements are equivalent:
1) $(f_{n})_{n\leq\ell}\cup(e_{n})_{n>\ell}$ is a Riesz basis for $H$.
2) $P_{\ell}LP_{\ell}$ is invertible.
###### Proof of Lemma 3.4.
1) $\Longrightarrow$ 2): From the definition of $L_{\ell}$ we know that it is
an onto isomorphism. This yields $P_{\ell}=P_{\ell}LP_{\ell}L_{\ell}^{-1}$,
implying $P_{\ell}=(P_{\ell}LP_{\ell})(P_{\ell}L_{\ell}^{-1}P_{\ell})$.
2) $\Longrightarrow$ 1): Let $A_{\ell}$ be the unique square matrix such that
$P_{\ell}LP_{\ell}A_{\ell}=A_{\ell}P_{\ell}LP_{\ell}=P_{\ell}$. We need to
show that $L_{\ell}$ is an onto isomorphism.
First we show that $L_{\ell}$ is one to one. Say
$0=L_{\ell}x=LP_{\ell}x+(I-P_{\ell})x$, then $0=P_{\ell}LP_{\ell}x$, so that
$0=A_{\ell}P_{\ell}LP_{\ell}x=P_{\ell}x$. We conclude that $x=(I-P_{\ell})x$.
This implies $0=L_{\ell}(I-P_{\ell})x=(I-P_{\ell})x=x$.
Next we show that $L_{\ell}$ is onto. Note
$L_{\ell}(I-P_{\ell})x=(I-P_{\ell})x$, so we only need to show that for all
$x$, $P_{\ell}x$ is in the range of $L_{\ell}$. Define
$y=P_{\ell}A_{\ell}P_{\ell}x+P_{\ell}x-LP_{\ell}A_{\ell}P_{\ell}x.$
We have $L_{\ell}(P_{\ell}A_{\ell}P_{\ell}x)=LP_{\ell}A_{\ell}P_{\ell}x$ and
$\displaystyle L_{\ell}(P_{\ell}x-LP_{\ell}A_{\ell}P_{\ell}x)$
$\displaystyle=$ $\displaystyle
LP_{\ell}x-L(P_{\ell}LP_{\ell})(A_{\ell}P_{\ell})x+P_{\ell}x-LP_{\ell}A_{\ell}P_{\ell}x$
$\displaystyle=$ $\displaystyle P_{\ell}x-LP_{\ell}A_{\ell}P_{\ell}x,$
from which $L_{\ell}y=P_{\ell}x$.
$L_{\ell}$ is a continuous bijection between Hilbert spaces, and hence is an
onto isomorphism by the open mapping theorem. ∎
###### Lemma 3.5.
Define $L$ as in Lemma 3.4. For all $\ell>0$, $L_{\ell}$ is an onto
isomorphism iff it is one to one.
###### Proof of Lemma 3.5.
We only need to show that $P_{\ell}LP_{\ell}$ is one to one on $P_{\ell}H$
when $L_{\ell}$ is one to one on $H$, and apply Lemma 3.4. Let
$(P_{\ell}LP_{\ell})P_{\ell}x=0.$ We have
$\displaystyle L_{\ell}[P_{\ell}x-(I-P_{\ell})LP_{\ell}x)]$ $\displaystyle=$
$\displaystyle L_{\ell}P_{\ell}x-L_{\ell}(I-P_{\ell})LP_{\ell}x$
$\displaystyle=$ $\displaystyle L_{\ell}P_{\ell}x-(I-P_{\ell})LP_{\ell}x$
$\displaystyle=$ $\displaystyle L_{\ell}P_{\ell}x-LP_{\ell}x=0,$
where the last equality follows from $L_{\ell}P_{\ell}=LP_{\ell}$. Since
$L_{\ell}$ is one to one, we have that $P_{\ell}x=(I-P_{\ell})LP_{\ell}x$, so
that $P_{\ell}x=0.$ ∎
###### Lemma 3.6.
Let $(e_{n})_{n\in\mathbb{N}}$ be an orthonormal basis for $H$, and
$(f_{n})_{n\in\mathbb{N}}$ be a Riesz basis for $H$, and
$(k_{\ell})_{\ell\in\mathbb{N}}\subset\mathbb{N}$ be an increasing sequence.
Let $P_{\ell}$ be the orthogonal projection onto
$\mathrm{span}\\{e_{n}\\}_{n\leq k_{\ell}}$, then the following are
equivalent:
1) The operator $L$ is uniformly invertible with respect to
$(P_{\ell})_{\ell\in\mathbb{N}}$.
2) For all $\ell>0$, $L_{\ell}$ is an onto isomorphism, and
$\sup_{\ell\in\mathbb{N}}\|L_{\ell}^{-1}\|<\infty.$
###### Proof of Lemma 3.6.
1) $\Longrightarrow$ 2): By Lemma 3.4, we only need to show that that
$\sup_{\ell\in\mathbb{N}}\|L_{\ell}^{-1}\|<\infty.$
This follows from the identity
(6) $L_{\ell}^{-1}=[I-(I-P_{\ell})L](P_{\ell}LP_{\ell})^{-1}+I-P_{\ell},$
which is hereby demonstrated:
$\displaystyle[I-(I-P_{\ell})L](P_{\ell}LP_{\ell})^{-1}+I-P_{\ell}$
$\displaystyle=$
$\displaystyle[I-(I-P_{\ell})L]P_{\ell}L_{\ell}^{-1}P_{\ell}+I-P_{\ell}$
$\displaystyle=$ $\displaystyle
P_{\ell}L_{\ell}^{-1}P_{\ell}-LP_{\ell}L_{\ell}^{-1}P_{\ell}+I$
$\displaystyle=$ $\displaystyle(I-L)P_{\ell}L_{\ell}^{-1}P_{\ell}+I.$
We have $(I-L)P_{\ell}=I-L_{\ell}$, so
$\displaystyle[I-(I-P_{\ell})L](P_{\ell}LP_{\ell})^{-1}+I-P_{\ell}$
$\displaystyle=$ $\displaystyle(I-L)L_{\ell}^{-1}P_{\ell}+I$ $\displaystyle=$
$\displaystyle L_{\ell}^{-1}P_{\ell}-P_{\ell}+I.$
Noting that $L_{\ell}(I-P_{\ell})=I-P_{\ell}$, we obtain
$L_{\ell}^{-1}P_{\ell}-P_{\ell}+I=L_{\ell}^{-1}$, which proves the identity.
2) $\Longrightarrow$ 1): Noting that
$(P_{\ell}LP_{\ell})^{-1}=P_{\ell}L_{\ell}^{-1}P_{\ell}$ yields the result.
∎
## 4\. The first main result
We begin with some necessary definitions:
###### Definition 4.1.
Define $C_{\ell,d}=\\{-\ell,\ldots,\ell\\}^{d}$, and
$e_{n}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,n\rangle}$ for
$n\in\mathbb{Z}^{d}$. Let $P_{\ell}:L_{2}([-\pi,\pi]^{d})\rightarrow
L_{2}([-\pi,\pi]^{d})$ be the orthogonal projection from
$L_{2}([-\pi,\pi]^{d})$ onto $\mathrm{span}(e_{n})_{n\in C_{\ell,d}}$.
Let $(f_{n})_{n\in\mathbb{Z}^{d}}$ be an exponential Riesz basis. In the
following sections, we abbreviate the statement
“$(f_{n})_{n\in\mathbb{Z}^{d}}$ is a uniformly invertible Riesz basis with
respect to the projections $(P_{\ell})_{\ell\in\mathbb{N}}$ defined in
definition 4.1” by “$(f_{n})_{n\in\mathbb{Z}^{d}}$ is a uniformly invertible
Riesz basis”.
To avoid confusion of indices, we write $t\in\mathbb{R}^{d}$ as
$t=(t(1),\cdots,t(d))$.
For $\ell,d\in\mathbb{N}$ define the multivariate polynomial
$Q_{d,\ell}(t)=\prod_{k_{1}=1}^{\ell}\Big{(}1-\frac{t(1)^{2}}{k_{1}^{2}}\Big{)}\cdot\ldots\cdot\prod_{k_{d}=1}^{\ell}\Big{(}1-\frac{t(d)^{2}}{k_{d}^{2}}\Big{)},\quad
t=(t(1),\cdots,t(d)).$
Here is the first main result of this paper.
###### Theorem 4.2.
Let $(t_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$, and define
$f_{n}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,t_{n}\rangle}$ for
$n\in\mathbb{Z}^{d}$. Let $(f_{n})_{n\in\mathbb{Z}^{d}}$ be a uniformly
invertible Riesz basis, then for all $f\in PW_{[-\pi,\pi]^{d}}$, there exists
a unique sequence of polynomials $(\Psi_{\ell})_{\ell\in\mathbb{N}}$,
$\Psi_{\ell}:\mathbb{R}^{d}\rightarrow\mathbb{R},$ such that
(a) $\Psi_{\ell}$ has coordinate degree at most $2\ell$.
(b) $\Psi_{\ell}(t_{n})=f(t_{n})$ for all $n\in C_{\ell,d}$.
(c) $f(t)=\lim_{\ell\rightarrow\infty}\Psi_{\ell}(t)\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)}$, where the limit is both $L_{2}$ and uniform.
Note: The expression in statement (c) of Theorem 4.2 has removable
singularities, but these can be evaluated by
$\lim_{t\rightarrow n}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}=\frac{(\ell!)^{2}}{(\ell+n)!(\ell-n)!},\quad
n\in\\{-\ell,\ldots\ell\\}.$
The proof of Theorem 4.2 requires several lemmas, beginning with the following
equivalence between the existence of particular Riesz bases and a polynomial
interpolation condition:
###### Lemma 4.3.
Let $(t_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ where
$f_{n}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,t_{n}\rangle}$ and
$e_{n}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,n\rangle}$. The sequence
$(f_{\ell,n})_{n\in\mathbb{Z}^{d}}:=(f_{n})_{n\in
C_{\ell,d}}\cup(e_{n})_{n\notin C_{\ell,d}}$ is a Riesz basis for
$L_{2}([-\pi,\pi]^{d})$ iff the following conditions hold.
1) For all $n\in C_{\ell,d}$,
$t_{n}\notin(\mathbb{Z}\setminus\\{-\ell,\cdots,\ell\\})^{d}$.
2) For any sequence $(c_{k})_{k\in C_{\ell,d}}$, there exists a unique
polynomial $\Psi_{\ell}$ with coordinate degree at most $2\ell$ such that
$\Psi_{\ell}(t_{k})=c_{k}$ for $k\in C_{\ell,d}$.
###### Proof of Lemma 4.3.
Suppose that the sequence $(f_{n})_{n\in\mathbb{Z}^{d}}$ is a Riesz basis for
$L_{2}([-\pi,\pi]^{d})$. We compute the biorthogonal functions of
$(f_{\ell,n})_{n\in\mathbb{Z}^{d}}$ when $n\in C_{\ell,d}$:
$\displaystyle f_{\ell,n}^{*}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}^{d}}\langle f_{\ell,n}^{*},e_{k}\rangle
e_{k}=\sum_{k\in C_{\ell,d}}\langle f_{\ell,n}^{*},e_{k}\rangle
e_{k}+\sum_{k\notin C_{\ell,d}}\langle f_{\ell,n}^{*},e_{k}\rangle e_{k}$
$\displaystyle=$ $\displaystyle\sum_{k\in C_{\ell,d}}\langle
f_{\ell,n}^{*},e_{k}\rangle e_{k}.$
Passing to the Fourier transform and defining
$G_{\ell,n}=\mathcal{F}(f_{\ell,n}^{*})$, we have
$\displaystyle G_{\ell,n}(t)$ $\displaystyle=$ $\displaystyle\sum_{k\in
C_{\ell,d}}G_{\ell,n}(k)\mathrm{SINC}\pi(t-k)$ $\displaystyle=$
$\displaystyle\bigg{(}\sum_{k\in
C_{\ell,d}}\frac{G_{\ell,n}(k)(-1)^{k(1)+\ldots+k(d)}t(1)\cdot\ldots\cdot
t(d)}{(t(1)-k(1))\cdot\ldots\cdot(t(d)-k(d))}\bigg{)}\mathrm{SINC}(\pi
t),\quad t\in\mathbb{R}^{d}.$
Denote the $k^{th}$ summand in equation (4) by $A_{k}$, then
$\displaystyle A_{\ell,n,k}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!A_{\ell,n,k}\frac{\prod_{j_{1}=-\ell\atop j_{1}\neq
k(1)}^{\ell}(t(1)-j_{1})\cdot\ldots\cdot\prod_{j_{d}=-\ell\atop j_{d}\neq
k(d)}^{\ell}(t(d)-j_{d})}{\prod_{j_{1}=-\ell\atop j_{1}\neq
k(1)}^{\ell}(t(1)-j_{1})\cdot\ldots\cdot\prod_{j_{d}=-\ell\atop j_{d}\neq
k(d)}^{\ell}(t(d)-j_{d})}$ $\displaystyle=$
$\displaystyle\\!\\!\frac{G_{\ell,n}(k)(-1)^{k(1)+\ldots+k(d)}t(1)\cdot\ldots\cdot
t(d)\prod_{j_{1}=-\ell\atop j_{1}\neq
k(1)}^{\ell}(t(1)-j_{1})\cdot\ldots\cdot\prod_{j_{d}=-\ell\atop j_{d}\neq
k(d)}^{\ell}(t(d)-j_{d})}{\prod_{j_{1}=-\ell}^{\ell}(t(1)-j_{1})\cdot\ldots\cdot\prod_{j_{d}=-\ell}^{\ell}(t(d)-j_{d})}$
$\displaystyle=$
$\displaystyle\\!\\!\frac{G_{\ell,n}(k)\frac{1}{(\ell!)^{2}}(-1)^{k(1)+\ldots+k(d)+\ell
d}\prod_{j_{1}=-\ell\atop j_{1}\neq
k(1)}^{\ell}(t(1)-j_{1})\cdot\ldots\cdot\prod_{j_{d}=-\ell\atop j_{d}\neq
k(d)}^{\ell}(t(d)-j_{d})}{\prod_{j_{1}=1}^{\ell}\Big{(}1-\frac{t(1)^{2}}{j_{1}^{2}}\Big{)}\cdot\ldots\cdot\prod_{j_{d}=1}^{\ell}\Big{(}1-\frac{t(d)^{2}}{k_{d}^{2}}\Big{)}}$
$\displaystyle=$ $\displaystyle\\!\\!\frac{p_{\ell,n,k}(t)}{Q_{d,\ell}(t)},$
where $p_{\ell,n,k}$ is some polynomial with coordinate degree at most
$2\ell$. Substituting into equation (4), we obtain
$G_{\ell,n}(t)=\Big{(}\sum_{k\in
C_{\ell,d}}p_{\ell,n,k}(t)\Big{)}\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)}:=\phi_{\ell,n}(t)\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)},$
where $\phi_{\ell,n}$ is a polynomial having coordinate degree at most
$2\ell$. The fact that each zero of $\mathrm{sinc}(\pi z)$ has multiplicity
one implies that the zero set of $\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell}(t)}$
(which is entire) is
$(\mathbb{Z}\setminus\\{-\ell,\cdots,\ell\\})^{d}\subset\mathbb{C}$. Using
that $\mathrm{SINC}\pi(x-y)$ is the reproducing kernel for
$PW_{[-\pi,\pi]^{d}}$, we see that
$G_{\ell,n}(t_{m})=\delta_{nm},$
for all $n,m\in C_{\ell,d}$. This yields that
$1=\phi_{\ell,n}(t_{n})\Big{(}\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)}\Big{)}\Big{|}_{t_{n}}.$
This shows $\phi_{\ell,n}(t_{n})\neq 0$ and $\frac{\mathrm{SINC}(\pi
t_{n})}{Q_{d,\ell}(t_{n})}\neq 0$ for $n\in C_{\ell,d}$, and that
$t_{n}\notin(\mathbb{Z}\setminus\\{-\ell,\cdots,\ell\\})^{d}$ (statement 1) of
Lemma 4.3).
For $n,m\in C_{\ell,d},n\neq m,$
$0=G_{\ell,n}(t_{m})=\phi_{\ell,n}(t_{m})\Big{(}\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)}\Big{)}\Big{|}_{t_{m}},$
We conclude that
$\phi_{\ell,n}(t_{m})=\left\\{\begin{array}[]{lr}\frac{Q_{d,\ell}(t_{n})}{\mathrm{SINC}\pi
t_{n}}\neq 0,&n=m\\\ 0,&n\neq m\end{array}\right.$
for $n,m\in C_{\ell,d}$ From this, the “existence” part of statement 2) in
Lemma 4.3 readily follows. Restated, the evaluation map taking the space of
all polynomials of coordinate degree at most $2\ell$ to
$\mathbb{R}^{(2\ell+1)^{d}}$ is onto. These spaces have the same dimension,
hence the evaluation map is a bijection, which completes the proof of
statement 2).
Suppose that 1) and 2) hold. For $n\in C_{\ell,d}$, let $p_{\ell,n}$ be the
unique polynomial of coordinate degree at most $2\ell$ such that
$p_{\ell,n}(t_{m})=\delta_{nm}$ for $m\in C_{\ell,d}$. Define
(10) $\Phi_{\ell,n}(t)=\frac{Q_{d,\ell}(t_{n})\mathrm{SINC}\pi
t}{Q_{d,\ell}(t)\mathrm{SINC}\pi t_{n}}p_{\ell,n}(t).$
Partial fraction decomposition can be used to show that $\Phi_{\ell,n}\in
PW_{[-\pi,\pi]^{d}}$. For $n,m\in C_{\ell,d}$, we therefore have
$\delta_{n,m}=\langle\Phi_{\ell,n}(\cdot),\mathrm{SINC}\pi((\cdot)-t_{m})\rangle=\langle\mathcal{F}^{-1}(\Phi_{\ell,n}),f_{m}\rangle.$
Let $L_{\ell}$ be defined as before. Let
$f=\sum_{n\in\mathbb{Z}^{d}}c_{n}e_{n}$ such that $L_{\ell}(f)=0$, then
$0=\sum_{n\in C_{\ell,d}}c_{n}f_{n}+\sum_{n\notin C_{\ell,d}}c_{n}e_{n}.$
If, for each $n\in C_{\ell,d}$ we integrate the above equation against
$\mathcal{F}^{-1}(\Phi_{\ell,n})$, we see that $c_{k}=0$ for $k\in
C_{\ell,d}$, so that $c_{k}=0$ for all $k\in\mathbb{Z}^{d}$. $L_{\ell}$ is one
to one, so by Lemma 3.5, it is an onto isomorphism from
$L_{2}([-\pi,\pi]^{d})$ to itself. ∎
###### Proof of (a) and (b) of Theorem 4.2.
Lemmas 3.6 and 4.3 imply the existence of a unique sequence of polynomials
satisfying statements (a) and (b) of Theorem 4.2, namely,
$\Psi_{\ell}(t)=\sum_{n\in C_{\ell,d}}f(t_{n})p_{\ell,n}(t),$
where $p_{\ell,n}$ is defined as in the proof of Lemma 4.3.
∎
It remains to show that this sequence of polynomials satisfies the statement
(c) of Theorem 4.2.
###### Proposition 4.4.
The following are equivalent.
1) $L$ is uniformly invertible with respect to the projections
$(P_{\ell})_{\ell\in\mathbb{N}}$.
2) For all $x\in H$,
$\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}(I-P_{\ell})x=0$.
###### Proof of Proposition 4.4.
It is clear that 1) implies 2). For the other direction, note that the
equality $L_{\ell}^{*}=P_{\ell}L^{*}+I-P_{\ell}$ implies that
(11) $I=(L_{\ell}^{*})^{-1}P_{\ell}L^{*}+(L_{\ell}^{*})^{-1}(I-P_{\ell}).$
This implies that $(L_{\ell}^{*})^{-1}P_{\ell}$ is pointwise bounded. Together
with the assumption in 2), this implies $(L_{\ell}^{*})^{-1}$ is pointwise
bounded, hence norm bounded by the uniform boundedness principle. This yields
uniform invertibility of $L$. ∎
###### Lemma 4.5.
The following are equivalent:
1) For all $g\in L_{2}([-\pi,\pi]^{d})$, we have
(12) $g=\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}P_{\ell}L^{*}g.$
2) $L$ is uniformly invertible.
###### Proof of Lemma 4.5.
Recall equation (11) and apply Proposition 4.4. ∎
###### Proposition 4.6.
Statement (c) of Theorem 4.2 is true iff
(13) $\displaystyle 0=\lim_{\ell\rightarrow\infty}\sum_{n\in
C_{\ell,d}}|f(t_{n})|^{2}\bigg{[}1-\frac{\mathrm{SINC}\pi
t_{n}}{Q_{d,\ell}(t_{n})}\bigg{]}^{2}:=\lim_{\ell\rightarrow\infty}S_{\ell,d},\quad
f\in PW_{[-\pi,\pi]^{d}}.$
###### Proof of Proposition 4.6.
Note that $Le_{n}=f_{n}$ implies that $f_{n}^{*}=(L^{*})^{-1}e_{n}.$
Similarly, $f_{\ell,n}^{*}=(L_{\ell}^{*})^{-1}e_{n}$. Given $f\in
PW_{[-\pi,\pi]^{d}}$, let $g=\mathcal{F}^{-1}(f)$. Equation (12) shows:
$\displaystyle\mathcal{F}^{-1}(f)$ $\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}\sum_{n\in
C_{\ell,d}}\langle L^{*}g,e_{n}\rangle
e_{n}=\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}\sum_{n\in
C_{\ell,d}}\langle g,f_{n}\rangle e_{n}$ $\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\sum_{n\in C_{\ell,d}}\langle
g,f_{n}\rangle f_{\ell,n}^{*}=\lim_{\ell\rightarrow\infty}\sum_{n\in
C_{\ell,d}}f(t_{n})f_{\ell,n}^{*}.$
Passing to the Fourier transform, we have
(14) $f=\lim_{\ell\rightarrow\infty}\sum_{n\in
C_{\ell,d}}f(t_{n})\mathcal{F}(f_{\ell,n}^{*}),\quad f\in
PW_{[-\pi,\pi]^{d}},$
where convergence in both $L_{2}$ and uniform. The values of a function in
$PW_{[-\pi,\pi]^{d}}$ on the set $(t_{n})_{n\in\mathbb{Z}^{d}}$ uniquely
determine the function. This and equation (10) show that
$\mathcal{F}(f_{\ell,n}^{*})(t)=G_{\ell,n}(t)=\frac{Q_{d,\ell}(t_{n})\mathrm{SINC}\pi
t}{Q_{d,\ell}(t)\mathrm{SINC}\pi t_{n}}p_{\ell,n}(t),\quad n\in C_{\ell,d}.$
This implies that
$\displaystyle\Psi_{\ell}(t)\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell}(t)}$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{n\in
C_{\ell,d}}f(t_{n})p_{\ell,n}(t)\Big{)}\frac{\mathrm{SINC}(\pi
t)}{Q_{d,\ell}(t)}$ $\displaystyle=$ $\displaystyle\sum_{n\in
C_{\ell,d}}f(t_{n})\frac{\mathrm{SINC}\pi
t_{n}}{Q_{d,\ell}(t_{n})}\mathcal{F}(f_{\ell,n}^{*})(t).$
Combined with equation (14), we see that statement (c) of Theorem 4.2 holds
iff
$\displaystyle 0=\lim_{\ell\rightarrow\infty}\sum_{n\in
C_{\ell,d}}f(t_{n})\bigg{[}1-\frac{\mathrm{SINC}\pi
t_{n}}{Q_{d,\ell}(t_{n})}\bigg{]}\mathcal{F}(f_{\ell,n}^{*}),\quad f\in
PW_{[-\pi,\pi]^{d}},$
where the limit is in the $L_{2}$ sense. Passing to the inverse Fourier
transform, statement (c) holds iff
(15) $\displaystyle
0=\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}\bigg{(}\sum_{n\in
C_{\ell,d}}f(t_{n})\bigg{[}1-\frac{\mathrm{SINC}\pi
t_{n}}{Q_{d,\ell}(t_{n})}\bigg{]}e_{n}\bigg{)},\quad f\in
PW_{[-\pi,\pi]^{d}}.$
In addition to having uniformly norm bounded inverses, $L_{\ell}$ is pointwise
bounded, so there exists $m,M>0$ such that for all $\ell\in\mathbb{N}$, $f\in
PW_{[-\pi,\pi]^{d}}$, $m\|f\|\leq\|(L_{\ell}^{*})^{-1}f\|\leq M\|f\|$. This,
combined with equation (15), proves the proposition. ∎
###### Proposition 4.7.
The following are true:
1)
$\sup_{x\in\mathbb{R}}\sup_{\ell\in\mathbb{N}}\Big{|}\frac{\mathrm{sinc}(\pi
x)}{Q_{1,\ell}(x)}\Big{|}=1.$
2) Define
$\Delta_{\ell,d}=\Big{\\{}n\in\mathbb{Z}^{d}\big{|}\Big{\|}\frac{t_{n}}{\ell+1}\Big{\|}_{\infty}<\frac{1}{\ell^{2/3}}\Big{\\}}$
for $\ell\in\mathbb{N}$, then
(16) $\displaystyle 0\leq 1-\frac{\mathrm{SINC}(\pi
t_{n})}{Q_{d,\ell}(t_{n})}<1-e^{\frac{-d(\ell+2)}{\ell^{4/3}-1}},\quad
n\in\Delta_{\ell,d}.$
###### Proof of Proposition 4.7.
For 1), the identity
$\mathrm{sinc}(\pi t)=\prod_{k=1}^{\infty}\Big{(}1-\frac{t^{2}}{k^{2}}\Big{)}$
implies
(17) $\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}=\prod_{k=\ell+1}^{\infty}\Big{(}1-\frac{t^{2}}{k^{2}}\Big{)},$
where convergence is uniform on compact subsets of $\mathbb{C}$. Fix
$\ell\in\mathbb{N}$. If $t\in[0,\ell+1]$, then $\Big{|}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{|}\leq 1.$ Note that
$|Q_{1,\ell}(t)|=\prod_{k=1}^{\ell}\Big{(}\frac{t^{2}}{k^{2}}-1\Big{)}$ is
increasing on $(\ell+1,\infty)$. If $t\in(\ell+1,\infty)$, then
$\Big{|}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{|}=\Big{|}\frac{\sin(\pi t)}{\pi
tQ_{1,\ell}(t)}\Big{|}<\frac{1}{\pi(\ell+1)|Q_{1,\ell}(\ell+1)|}.$
Computation yields
$|Q_{1,\ell}(\ell+1)|=\frac{(2\ell+1)!}{\ell!(\ell+1)!},$
so
$\Big{|}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{|}<\frac{(\ell!)^{2}}{\pi(2\ell+1)!}<1.$
Observing that $\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}$ is even proves 1).
For 2), let $t\in\mathbb{R}$ such that
$\Big{|}\frac{t}{\ell+1}\Big{|}<\frac{1}{\ell^{2/3}}$, then
$0<\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}$, and
$\displaystyle-\log\Big{(}\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}\Big{)}$
$\displaystyle=$
$\displaystyle-\sum_{k=\ell+1}^{\infty}\log\Big{(}1-\frac{t^{2}}{k^{2}}\Big{)}=\sum_{k=\ell+1}^{\infty}\sum_{j=1}^{\infty}\frac{t^{2}}{jk^{2j}}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\frac{1}{j}\Big{(}\sum_{k=\ell+1}^{\infty}\frac{1}{k^{2j}}\Big{)}t^{2j}.$
Basic calculus shows that
$\sum_{k=\ell+1}^{\infty}\frac{1}{k^{2j}}<\frac{1}{(\ell+1)^{2j}}+\frac{1}{(2j-1)(\ell+1)^{2j-1}}.$
Equality (4) implies
$\displaystyle-\log\Big{(}\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}\Big{)}$
$\displaystyle<$
$\displaystyle\sum_{j=1}^{\infty}\frac{1}{j}\Big{(}\frac{t}{\ell+1}\Big{)}^{2j}+(\ell+1)\sum_{j=1}^{\infty}\frac{1}{j(2j-1)}\Big{(}\frac{t}{\ell+1}\Big{)}^{2j}$
$\displaystyle<$
$\displaystyle(\ell+2)\sum_{j=1}^{\infty}\Big{(}\frac{t}{\ell+1}\Big{)}^{2j}<\frac{\ell+2}{\ell^{4/3}-1}.$
If $n\in\Delta_{\ell,d}$, then for each $1\leq k\leq d$,
$\Big{|}\frac{t_{n}(k)}{\ell+1}\Big{|}>\frac{\ell+2}{\ell^{4/3}-1}$, so that
$\log\Big{(}\frac{\mathrm{SINC}(\pi
t_{n})}{Q_{d,\ell}(t_{n})}\Big{)}=\sum_{k=1}^{d}\log\Big{(}\frac{\mathrm{sinc}(\pi
t_{n}(k))}{Q_{1,\ell}(t_{n}(k))}\Big{)}>-\frac{d(\ell+2)}{\ell^{4/3}-1}.$
Statement 2) of Proposition 4.7 follows readily. ∎
###### Proof of statement (c) in Theorem 4.2.
Proposition 4.7 gives the following:
(20) $\displaystyle S_{\ell,d}$ $\displaystyle\leq$
$\displaystyle\Big{(}\sum_{n\in\Delta_{\ell,d}}+\sum_{n\in\mathbb{Z}^{d}\setminus\Delta_{\ell,d}}\Big{)}|f(t_{n})|^{2}\bigg{[}1-\frac{\mathrm{SINC}\pi
t_{n}}{Q_{d,\ell}(t_{n})}\bigg{]}^{2}$ $\displaystyle\leq$
$\displaystyle\bigg{(}1-e^{\frac{-d(\ell+2)}{\ell^{4/3}-1}}\bigg{)}^{2}\sum_{n\in\mathbb{Z}^{d}}|f(t_{n})|^{2}+\sum_{n:\
\frac{\ell+1}{\ell^{2/3}}\leq\|t_{n}\|_{\infty}}4|f(t_{n})|^{2}.$
Now $(f(t_{n}))_{n\in\mathbb{Z}^{d}}\in\ell_{2}(\mathbb{Z}^{d})$ implies that
$\lim_{\ell\rightarrow\infty}S_{\ell,d}=0$, so that by Proposition 4.6,
statement (c) in Theorem 4.2 is true. ∎
## 5\. The second main result
Theorem 4.2 can be simplified. The function
$t\mapsto\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell}(t)}$
becomes more computationally complex for large values of $\ell$. If, at the
cost of global $L_{2}$ and uniform convergence, we adopt an approximation
(21) $\mathrm{SINC}(\pi t)\simeq
Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}},$
we bypass this difficulty as the exponent of the above quantity is simply a
rational function of $\ell>0$. This is precisely quantified in Theorem 5.1,
which is the second main result of this paper.
###### Theorem 5.1.
Let $(t_{n})_{\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ be a sequence such that
the associated exponential functions are a uniformly invertible Riesz basis
for $L_{2}([-\pi,\pi]^{d})$. For $N\in\\{0,1,2,\ldots\\}$, $A>0$, let
$E_{\ell,N,A}=[-A(\ell+1/2)^{\frac{2N+1}{2N+2}},A(\ell+1/2)^{\frac{2N+1}{2N+2}}]$.
Let $f\in PW_{[-\pi,\pi]^{d}}$ where $(\Psi_{\ell})_{\ell}$ is the sequence of
interpolating polynomials from Theorem 4.2, and let $N\geq 0$. Let $f\in
PW_{[-\pi,\pi]^{d}}$, and define
$I_{f,\ell}(t)=\Psi_{\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}},$
then
(22)
$\lim_{\ell\rightarrow\infty}\Big{\|}f(t)-I_{f,\ell}(t)\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}=0,$
and
(23)
$\lim_{\ell\rightarrow\infty}\Big{\|}f(t)-I_{f,\ell}(t)\Big{\|}_{L_{\infty}((E_{\ell,N,A})^{d})}=0.$
If $N=0$ in Theorem 5.1, we have the following extension of Corollary 1.2 to
arbitrary multivariate bandlimited functions (at the expense of introducing
uniform invertibility):
###### Corollary 5.2.
For all $f\in PW_{[-\pi,\pi]^{d}}$, we have
(24)
$\lim_{\ell\rightarrow\infty}\Big{\|}f(t)-\Psi_{\ell}(t)\Big{\|}_{L_{2}([-A(\ell+1/2)^{1/2},A(\ell+1/2)^{1/2}]^{d})}=0,$
and
(25)
$\lim_{\ell\rightarrow\infty}\Big{\|}f(t)-\Psi_{\ell}(t)\Big{\|}_{L_{\infty}([-A(\ell+1/2)^{1/2},A(\ell+1/2)^{1/2}]^{d})}=0.$
It is evident that if $(t_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ is
any subset such that the associated sequence of exponentials
$(f_{n})_{n\in\mathbb{Z}^{d}}$ is a Riesz basis for $L_{2}([-\pi,\pi]^{d})$,
then the map $f\mapsto(f(t_{n}))_{n\in\mathbb{Z}^{d}}$ is a bijection from
$PW_{[-\pi,\pi]^{d}}$ to $\ell_{2}(\mathbb{Z}^{d})$. This allows for a nice
interpretation of Corollary 5.2. Given a sequence
$(t_{n})_{n\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ (subject to the hypotheses
of Theorem 5.1), and sampled data
$\big{(}(t_{n},c_{n})\big{)}_{n\in\mathbb{Z}^{d}}$ where
$(c_{n})_{n\in\mathbb{Z}^{d}}\in\ell_{2}(\mathbb{Z}^{d})$, then a unique
sequence of Lagrangian polynomial interpolants exists, and in the appropriate
limit, converges to the unique band-limited interpolant of the same data.
When $N=1$, we have a sampling theorem with a Gaussian multiplier:
$f(t)\simeq\Psi_{\ell}(t)e^{-\frac{\|t\|_{2}^{2}}{(\ell+1/2)}},\quad f\in
PW_{[-\pi,\pi]^{d}}.$
Compare Theorem 5.1 with Theorem 2.6 in [2], which is a multivariate sampling
theorem with a Gaussian multipler with global $L_{2}$ and uniform convergence.
Also compare Theorem 5.1 with Theorem 2.1 in [11], which, when $d=1$ and the
data nodes are equally spaced, gives another recovery formula involving a
Gaussian mulitplier in the context of over-sampling.
The proof of Theorem 5.1 relies on two lemmas:
###### Lemma 5.3.
Let $d\in\mathbb{N}$, $N\in\\{0,1,2,\ldots\\}$, and $A>0$. There exists $M>0$
such that for sufficiently large $\ell$, and any $t\in(E_{\ell,N,A})^{d}$, we
have
$\displaystyle\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi
t)\Big{|}$ $\displaystyle\quad\leq
M(\ell+1/2)^{-\frac{1}{N+1}}|\mathrm{SINC}(\pi t)|.$
###### Lemma 5.4.
For all $f\in PW_{[-\pi,\pi]^{d}}$ and $N\in\\{0,1,2,\ldots\\}$, we have
$\lim_{\ell\rightarrow\infty}\sup_{t\in(E_{\ell,N,A})^{d}}\Big{|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}f(t)\Big{|}=0.$
The proof of Lemma 5.3 relies on the following proposition.
###### Proposition 5.5.
If $f:(0,\infty)\rightarrow(0,\infty)$ is convex, decreasing, differentiable,
and integrable away from $0$, then
$\frac{1}{4}f^{\prime}(\ell+1/2)\leq\sum_{k=\ell+1}^{\infty}f(k)-\int_{\ell+1/2}^{\infty}f(x)dx\leq
0.$
###### Proof of Proposition 5.5.
The proof follows naturally from geometric considerations. ∎
###### Proof of Lemma 5.3.
Letting $|t|<\ell+1/2$ and recalling equation (4), we see that
(26) $\displaystyle-\log\Big{(}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{)}-\sum_{k=1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}$
$\displaystyle\quad=\sum_{k=1}^{\infty}\Big{[}\sum_{j=\ell+1}^{\infty}\frac{1}{j^{2k}}-\frac{1}{(2k-1)(\ell+1/2)^{2k-1}}\Big{]}\frac{t^{2k}}{k}.$
Applying Proposition 5.5 to the function $f(t)=\frac{1}{t^{2k}}$ when $k\geq
1$, we obtain
$\frac{-k}{2(\ell+1/2)^{2k+1}}\leq\sum_{j=\ell+1}^{\infty}\frac{1}{j^{2k}}-\frac{1}{(2k-1)(\ell+1/2)^{2k-1}}\leq
0.$
Equation (26) becomes
$\displaystyle\frac{-1}{2(\ell+1/2)}\sum_{k=1}^{\infty}\Big{(}\frac{t}{\ell+1/2}\Big{)}^{2k}\leq-\log\Big{(}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{)}-\sum_{k=1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}\leq
0.$
Restated,
(27)
$\displaystyle-\frac{1}{2(\ell+1/2)}\frac{\Big{(}\frac{t}{\ell+1/2}\Big{)}^{2}}{1-\Big{(}\frac{t}{\ell+1/2}\Big{)}^{2}}+\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}$
$\displaystyle\quad\leq-\log\Big{(}\frac{\mathrm{sinc}(\pi
t)}{Q_{1,\ell}(t)}\Big{)}-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}$
$\displaystyle\quad\leq\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}.$
Exponentiating,
(28) $\displaystyle
e^{\bigg{(}-\frac{1}{2(\ell+1/2)}\frac{\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}{1-\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}\bigg{)}}e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}$
$\displaystyle\quad\leq\frac{Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{sinc}(\pi
t)}\leq
e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}.$
Let $\ell$ be chosen large enough so that
$A(\ell+1/2)^{\frac{2N+1}{2N+2}}<\ell+1/2$. If $\ell$ is large enough, then
for any $t\in E_{\ell N,A}$, $t=c(\ell+1/2)^{\frac{2N+1}{2N+2}}$ for some
$c\in[-A,A]$. For such $t$, inequality (28) implies
$\displaystyle
e^{\bigg{(}-\frac{1}{2(\ell+1/2)^{\frac{N+2}{N+1}}}\frac{c^{2}}{1-c^{2}(\ell+1/2))^{\frac{-1}{N+1}}}\bigg{)}}e^{\sum_{k=N+1}^{\infty}\frac{c^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}$
$\displaystyle\quad\leq\frac{Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{sinc}(\pi
t)}\leq
e^{\sum_{k=N+1}^{\infty}\frac{c^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}.$
If $t\in(E_{\ell,N,A})^{d}$, then $t=c(\ell+1/2)^{\frac{2N+1}{2N+2}}$ for some
$c\in[-A,A]^{d}$. For any such $t$, we have
(29) $\displaystyle
e^{\bigg{(}-\frac{d}{2(\ell+1/2)^{\frac{N+2}{N+1}}}\frac{A^{2}}{1-A^{2}(\ell+1/2))^{\frac{-1}{N+1}}}\bigg{)}}e^{\sum_{k=N+1}^{\infty}\frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}$
$\displaystyle\quad\leq\frac{Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi
t)}\leq
e^{\sum_{k=N+1}^{\infty}\frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}.$
On one hand,
(30) $\displaystyle
e^{\sum_{k=N+1}^{\infty}\frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}\leq
e^{\Big{(}\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}+O\big{(}(\ell+1/2)^{\frac{-1}{N+1}}\big{)}\Big{)}}$
where the “big O” constant is independent of $c\in[-A,A]^{d}$. On the other
hand,
(31) $\displaystyle e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}\leq
e^{\sum_{k=N+1}^{\infty}\frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big{(}1-\frac{k}{N+1}\big{)}}}.$
Inequality (29) yields
(32)
$\displaystyle\bigg{(}e^{-\frac{d}{2(\ell+1/2)^{\frac{N+2}{N+1}}}\frac{A^{2}}{1-A^{2}(\ell+1/2))^{\frac{-1}{N+1}}}}-1\bigg{)}e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}$
$\displaystyle\quad\leq\frac{Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi
t)}-e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}$ $\displaystyle\quad\leq
e^{\frac{dA^{2(N+1)}}{(N+1)(2N+1)}}\Big{(}e^{O\bigg{(}\frac{1}{(\ell+1/2)^{\frac{1}{N+1}}}\bigg{)}}-1\Big{)}.$
The left most side of inequality (32) is of the order
$O((\ell+1/2)^{-\frac{N+2}{N+1}})$, and the right most side of inequality (32)
is of the order $O((\ell+1/2)^{-\frac{1}{N+1}})$, where the “big O” constants
are independent of $c\in[-A,A]^{d}$. The lemma follows readily. ∎
###### Proof of Lemma 5.4.
Equivalently, we need to show
$\lim_{\ell\rightarrow\infty}\sup_{c\in[-A,A]^{d}}\Big{|}\Big{(}e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}f\big{(}c(\ell+1/2)^{\frac{2N+1}{2N+2}}\big{)}\Big{|}=0.$
Suppose the contrary. Let $c_{\ell}\in[-A,A]^{d}$ be a value that maximizes
the $\ell$-th term in the above limit. There exists
$(\ell_{k})_{k\in\mathbb{N}}$, and $\epsilon>0$ such that for all
$k\in\mathbb{N}$,
$\displaystyle\epsilon$ $\displaystyle\leq$
$\displaystyle\sup_{c\in[-A,A]^{d}}\Big{|}\Big{(}e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}f\big{(}c(\ell_{k}+1/2)^{\frac{2N+1}{2N+2}}\big{)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{(}e^{\frac{dA^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}\big{|}f\big{(}c_{\ell_{k}}(\ell_{k}+1/2)^{\frac{2N+1}{2N+2}}\big{)}\big{|},$
so that the sequence
$\big{(}f\big{(}c_{\ell_{k}}(\ell_{k}+1/2)^{\frac{2N+1}{2N+2}}\big{)}\big{)}_{k\in\mathbb{N}}$
is bounded away from $0$. This implies there exists $\delta>0$ such that
$\big{\|}c_{\ell_{k}}(\ell_{k}+1/2)^{\frac{2N+1}{2N+2}}\big{\|}_{2(N+1)}\leq\delta$
for $k\in\mathbb{N}$, that is,
$\|c_{\ell_{k}}\|_{2(N+1)}\leq\delta(\ell_{k}+1/2)^{-\frac{2N+1}{2N+2}}$. This
forces
$\displaystyle\epsilon$ $\displaystyle\leq$
$\displaystyle\sup_{c\in[-A,A]^{d}}\Big{|}\Big{(}e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}f\big{(}c(\ell_{k}+1/2)^{\frac{2N+1}{2N+2}}\big{)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{(}e^{\frac{\delta^{2(N+1)}}{(\ell_{k}+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\|f\|_{\infty}.$
The last term in the above inequality has limit $0$ as
$\ell\rightarrow\infty$, which is a contradiction. ∎
Now we can prove Theorem 5.1.
###### Proof of Theorem 5.1.
If $f\in PW_{[-\pi,\pi]^{d}}$, Theorem 4.2 states that
$f(t)=\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi t)}+\xi_{\ell}(t)$
where $\xi_{\ell}\rightarrow 0$ on $\mathbb{R}^{d}$ both in $L_{2}$ and
$L_{\infty}$ senses. By Lemma 5.3, we have
(33)
$\displaystyle\sup_{t\in(E_{\ell,N,A})^{d}}\Big{|}\Psi_{\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{|}$ $\displaystyle\quad\quad\leq
M(\ell+1/2)^{-\frac{1}{N+1}}\sup_{t\in(E_{\ell,N,A})^{d}}(|f(t)|-|\xi_{\ell}(t)|),$
the right side of which has zero limit. Also,
(34)
$\displaystyle\sup_{t\in(E_{\ell,N,A})^{d}}\quad\Big{|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{|}$
$\displaystyle\quad\leq\sup_{t\in(E_{\ell,N,A})^{d}}\Big{|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}f(t)\Big{|}+$
$\displaystyle\quad\quad\Big{(}e^{\frac{dA^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}\sup_{t\in(E_{\ell,N,A})^{d}}|\xi_{\ell}(t)|,$
whose right hand side, by Lemma 5.4, also has zero limit. Combining
inequalities (33) and (34), we obtain
$\lim_{\ell\rightarrow\infty}\Big{\|}\Psi_{\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{\|}_{L_{\infty}((E_{\ell,N,A})^{d})}=0.$
Equation (23) follows by a final application of Theorem 4.2.
Now we prove equation (22). Lemma 5.3 and Theorem 4.2 imply
(35)
$\displaystyle\Big{\|}\Psi_{\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}$ $\displaystyle\quad\leq
M(\ell+1/2)^{-\frac{1}{N+1}}\|f+\xi_{\ell}\|_{L_{2}((E_{\ell,N,A})^{d})},$
the right hand side of which has zero limit. Also,
(36)
$\displaystyle\Big{\|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}$
$\displaystyle\quad\leq\Big{\|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}f(t)\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}+$
$\displaystyle\quad\quad\Big{\|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\xi_{\ell}(t)\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}.$
The second term in the right hand side of inequality (36) is bounded from
above by
$\Big{(}e^{\frac{dA^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}\|\xi_{\ell}\|_{L_{2}((E_{\ell,N,A})^{d})},$
which has zero limit. The integrand of the first term in the right hand side
of inequality (36) (as a function over $\mathbb{R}^{d}$), converges uniformly
to zero by Lemma 5.4, and is bounded from above by
$\Big{(}e^{\frac{dA^{2(N+1)}}{(N+1)(2N+1)}}-1\Big{)}|f(t)|^{2}\in
L_{1}(\mathbb{R}^{d}),$
so this term has zero limit by the Dominated Convergence Theorem. Combining
equations (35) and (36) yields
$\lim_{\ell\rightarrow\infty}\Big{\|}\Psi_{\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-\frac{\Psi_{\ell}(t)}{Q_{d,\ell}(t)}\mathrm{SINC(\pi
t)}\Big{\|}_{L_{2}((E_{\ell,N,A})^{d})}=0.$
Equation (22) follows by a final application of Theorem 4.2. ∎
The optimal growth for any $(E_{\ell,N,k})_{\ell}$ such that Theorem 5.1 holds
is not known, but an upper bound for the rate is established in the Appendix.
## 6\. Examples of uniformly invertible exponential Riesz Bases
Theorems 4.2 and 5.1 both require uniform invertibility of
$(f_{n})_{n\in\mathbb{Z}^{d}}$. Fortunately, there are significant classes of
exponential Riesz bases $(f_{n})_{n\in\mathbb{Z}^{d}}$ which have this
property. Consider the exponential Riesz bases described given by Theorem 6.1
(Corollary 6.1 in [1]) and Theorem 6.2 (Theorem 1.3 in [12]).
###### Theorem 6.1.
Let $(t_{k})_{k\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ such that
$\sup_{n\in\mathbb{Z}^{d}}\Arrowvert
n-t_{n}\Arrowvert_{\infty}=L<\frac{\ln(2)}{\pi d},$
then the sequence $(f_{k})_{k\in\mathbb{Z}^{d}}$ defined by
$f_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,t_{k}\rangle}$ is a Riesz basis
for $L_{2}([-\pi,\pi]^{d})$.
###### Theorem 6.2.
For $d\geq 1$, define
$D_{d}(x):=\Big{(}1-\cos\pi x+\sin\pi x+\frac{\sin\pi x}{\pi
x}\Big{)}^{d}-\Big{(}\frac{\sin\pi x}{\pi x}\Big{)}^{d},$
and let $x_{d}$ be the unique number such that $0<x_{d}\leq 1/4$ and
$D_{d}(x_{d})=1$. Let $(t_{k})_{k\in\mathbb{Z}^{d}}\subset\mathbb{R}^{d}$ such
that
$\sup_{n\in\mathbb{Z}^{d}}\|n-t_{n}\|_{\infty}<x_{d},$
then the sequence $(f_{k})_{k\in\mathbb{Z}^{d}}$ defined by
$f_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle x,t_{k}\rangle}$ is a Riesz basis
for $L_{2}([-\pi,\pi]^{d})$.
It is worth noting that when $d=1$, Theorem 6.2 reduces to the classical
Kadec’s 1/4 Theorem, first proven in [3]. A proof of Kadec’s 1/4 Theorem can
also be found in [13, Theorem 14, page 36]. The proofs of the above theorems
show that the map $Le_{n}=f_{n}$ satisfies $\|I-L\|=\delta<1$, from which we
see that $L$ is invertible. Uniform invertibility is readily verified, or can
be seen as a consequence of the following more general proposition.
###### Proposition 6.3.
Let $L:H\rightarrow H$ be a uniformly invertible operator with respect to
$(P_{\ell})_{\ell\in\mathbb{N}}$, where
$\limsup_{\ell\rightarrow\infty}\|(P_{\ell}LP_{\ell})^{-1}\|=M<\infty.$
If $A$ is an operator such that
(37) $\|L-A\|=\frac{\gamma}{M}$
for some $\gamma<1$, then there exists $N\in\mathbb{N}$ such that $A$ is
uniformly invertible with respect to $(P_{\ell})_{\ell\geq N}$.
###### Proof of Proposition 6.3.
Using uniform invertiblity of $L$, and noting that
$(L_{\ell}^{*})^{-1}-(L^{*})^{-1}=(L_{\ell}^{*})^{-1}(L^{*}-L_{\ell}^{*})(L^{*})^{-1}$
and $\lim_{\ell\rightarrow\infty}L_{\ell}^{*}x=L^{*}x$ for all $x\in H$, we
see that $\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}x=(L^{*})^{-1}x$ for
all $x\in H$. Equation (6) implies that
$\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}x-(P_{\ell}L^{*}P_{\ell})^{-1}x=0$
for all $x\in H$. Together we have
$\lim_{\ell\rightarrow\infty}(P_{\ell}L^{*}P_{\ell})^{-1}x=(L^{*})^{-1}x,\quad
x\in H.$
General principles imply
$\|L^{-1}\|\leq\liminf_{\ell\rightarrow\infty}\|(P_{\ell}LP_{\ell})^{-1}\|\leq
M.$
This yields $\|L-A\|\leq\frac{\gamma}{\|L^{-1}\|}$, so that $A$ is invertible
by usual Neumann series manipulation. Equation (37) yields that
$\|P_{\ell}LP_{\ell}-P_{\ell}AP_{\ell}\|\leq\frac{\gamma}{M}$, where the norm
is now the standard matrix norm on the set of matrices of dimension
$\mathrm{dim}(\mathrm{ran}{P_{\ell}})$. This implies
$\|P_{\ell}-(P_{\ell}LP_{\ell})^{-1}(P_{\ell}AP_{\ell})\|\leq\frac{\gamma}{M}\|(P_{\ell}LP_{\ell})^{-1}\|,\quad\ell\in\mathbb{N}.$
Choose $N$ large enough so that
$\|(P_{\ell}LP_{\ell})^{-1}\|\leq\frac{\gamma+1}{2\gamma}M$ when $\ell\geq N$.
This yields
$\|P_{\ell}-(P_{\ell}LP_{\ell})^{-1}(P_{\ell}AP_{\ell})\|\leq\frac{\gamma+1}{2},\quad\ell\geq
N.$
Standard manipulation shows that $P_{\ell}AP_{\ell}$ is invertible for
$\ell\geq N$, and
$\sup_{\ell\geq
N}\|(P_{\ell}AP_{\ell})^{-1}\|\leq\frac{\gamma+1}{\gamma(1-\gamma)}M.$
∎
Note that in the previous proof, if $M$ is redefined to be
$\sup_{\ell\in\mathbb{N}}\|(P_{\ell}LP_{\ell})^{-1}\|$, then $A$ is uniformly
invertible with respect to $(P_{\ell})_{\ell\in\mathbb{N}}$.
The following proposition shows that compact perturbations (of arbitrary
norm), of a uniformly invertible operator also gives a uniformly invertible
operator.
###### Proposition 6.4.
Let $L:H\rightarrow H$ be uniformly invertible with respect to
$(P_{\ell})_{\ell\in\mathbb{N}}$. If $\Delta:H\rightarrow H$ is compact such
that $\tilde{L}=L+\Delta$ is an onto isomorphism, then there exists $N$ such
that $\tilde{L}$ is uniformly invertible with respect to $(P_{\ell})_{\ell\geq
N}$.
###### Proof of Proposition 6.4.
From the definition of $L_{\ell}$, we have
$I=(I-P_{\ell})L_{\ell}^{-1}+LP_{\ell}L_{\ell}^{-1}$, so that
$L^{-1}(P_{\ell}-I)L_{\ell}^{-1}=P_{\ell}L_{\ell}^{-1}-L^{-1}.$
This implies
(38)
$(L_{\ell}^{*})^{-1}P_{\ell}-(L^{*})^{-1}=(L_{\ell}^{*})^{-1}(P_{\ell}-I)(L^{*})^{-1}.$
As $\ell\rightarrow\infty$, the right hand side of equation (38) has 0 limit
pointwise. Combined with the compactness of $\Delta$, we obtain
$\lim_{\ell\rightarrow\infty}(L_{\ell}^{*})^{-1}P_{\ell}\Delta=(L^{*})^{-1}\Delta$
where limit is in the operator norm topology. This yields
(39) $\lim_{\ell\rightarrow\infty}I+\Delta P_{\ell}L_{\ell}^{-1}=I+\Delta
L^{-1}=(L+\Delta)L^{-1},$
where the limit is also in the operator norm topology. The right had side of
equation (39) is invertible, so there exists $N$ such that $\ell\geq N$
implies $(I+\Delta P_{\ell}L_{\ell}^{-1})^{-1}$ exists, and
(40) $\sup_{\ell\geq N}\|(I+\Delta P_{\ell}L_{\ell}^{-1})^{-1}\|<\infty.$
From the definition of $\tilde{L}_{\ell}$, we obtain
$\displaystyle\tilde{L}_{\ell}=L_{\ell}+\Delta P_{\ell}=(I+\Delta
P_{\ell}L_{\ell}^{-1})L_{\ell}.$
When $\ell\geq N$, we have
$\tilde{L}_{\ell}^{-1}=L_{\ell}^{-1}(I+\Delta P_{\ell}L_{\ell}^{-1})^{-1},$
and equation (40) implies
(41) $\sup_{\ell\geq N}\|\tilde{L}_{\ell}^{-1}\|\leq\sup_{\ell\geq
N}\|L_{\ell}^{-1}\|\sup_{\ell\geq N}\|(I+\Delta
P_{\ell}L_{\ell}^{-1})^{-1}\|<\infty.$
This completes the proof. ∎
The following lemma holds:
###### Lemma 6.5.
Choose $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$(h_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),t_{k}\rangle}\big{)}_{k\in\mathbb{N}}$
satisfies
$\Big{\|}\sum_{k=1}^{n}a_{k}h_{k}\Big{\|}_{L_{2}[-\pi,\pi]^{d}}\leq
B\Big{(}\sum_{k=1}^{n}|a_{k}|^{2}\Big{)}^{1/2},\quad\mathrm{for\
all}\quad(c_{k})_{k=1}^{n}\subset\mathbb{C}.$
If $(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$, and
$(f_{k})_{k\in\mathbb{N}}:=\big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),\tau_{k}\rangle}\big{)}_{k\in\mathbb{N}}$,
then for all $r,s\geq 1$ and any finite sequence $(a_{k})_{k}$, we have
$\Bigg{\Arrowvert}\sum_{k=r}^{s}a_{k}(h_{k}-f_{k})\Bigg{\Arrowvert}_{L_{2}[-\pi,\pi]^{d}}\\\
{}\leq B\Big{(}e^{\pi d\big{(}{\sup\atop{r\leq k\leq
s}}\Arrowvert\tau_{k}-t_{k}\Arrowvert_{\infty}\big{)}}-1\Big{)}\Big{(}\sum_{k=r}^{s}|a_{k}|^{2}\Big{)}^{\frac{1}{2}}.$
This lemma is a slight generalization of Lemma 5.3, found in [1] using simple
estimates. Lemma 6.5 is proven similarly. A consequence of Lemma 6.5 is the
following corollary.
###### Corollary 6.6.
Let $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$ such that
$(f_{k})_{k\in\mathbb{N}}$ (the usual exponential sequence defined in terms of
$(t_{n})_{n}$) is a Riesz basis for $L_{2}([-\pi,\pi]^{d})$. Let
$(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$, and define
$(g_{k})_{k\in\mathbb{N}}$ by $g_{k}(x)=\frac{1}{(2\pi)^{d/2}}e^{i\langle
x,\tau_{k}\rangle}$. Let $(b_{k})_{k\in\mathbb{Z}}$ be an orthonormal basis
for $L_{2}([-\pi,\pi]^{d})$. If
$\lim_{k\rightarrow\infty}\|t_{k}-\tau_{k}\|_{\infty}=0,$
then the operator defined by $b_{k}\mapsto f_{k}-g_{k}$ is compact.
The proof of Corollary 6.6 is similar to the proof of Corollary 5.5 in [1], so
it is omitted.
###### Corollary 6.7.
Let $(t_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$. Let
$(f_{k})_{k\in\mathbb{N}}$ be a Riesz basis for $L_{2}([-\pi,\pi]^{d})$ which
is uniformly invertible with respect to the projections
$(P_{\ell})_{\ell\in\mathbb{N}}$ defined at the beginning of section 4. If
$(\tau_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{d}$, and
$(g_{k})_{k\in\mathbb{N}}$ are as in Corollary (6.6), and additionally,
$(g_{k})_{k\in\mathbb{N}}$ is a Riesz basis for $L_{2}([-\pi,\pi]^{d})$, then
$(g_{k})_{k\in\mathbb{N}}$ is uniformly invertible with respect to a sequence
of projections $(P_{\ell})_{\ell\geq N}$ for some $N>0$.
###### Proof of Corollary 6.7.
Apply Corollaries 6.4 and 6.6. ∎
This corollary relates to Theorem 4.2 in the following way. Let
$(g_{k})_{k\in\mathbb{N}}$ is as in the preceding theorem. Usage of Corollary
6.4 in the proof of Corollary 6.7 does not ensure that low order polynomial
interpolants will exist; however, they will existence for sufficiently large
$\ell$. Simple examples show that in Corollary 6.7, the additional assumption
that $(g_{k})_{k\in\mathbb{N}}$ is a Riesz basis for $L_{2}([-\pi,\pi]^{d})$
cannot be dropped when $d\geq 2$. Example: The standard exponential
orthonormal basis $(e_{n})_{n\in\mathbb{Z}^{d}}$ is of course uniformly
invertible, but the set
$\Big{(}\frac{1}{(2\pi)^{d/2}}e^{i\langle(\cdot),(1,1/2,0,\cdots,0)\rangle}\Big{)}\cup(e_{n})_{n\neq
0}$
is not a Riesz basis. However, this condition can be dropped when $d=1.$ This
follows from Corollary 6.6 and the following theorem.
###### Theorem 6.8.
Let $(t_{n})_{n\in\mathbb{Z}}\ \subset\mathbb{R}$ be a sequence such that
$(f_{n})_{n\in\mathbb{Z}}$ (defined as before) is a Riesz basis for
$L_{2}[-\pi,\pi]$. If $(\tau_{n})_{n\in\mathbb{Z}}\subset\mathbb{R}$ is a
sequence of distinct points such that
$\lim_{|n|\rightarrow\infty}|t_{n}-\tau_{n}|=0,$
then $(g_{n})_{n\in\mathbb{Z}}$ (defined as before) is a Riesz basis for
$L_{2}[-\pi,\pi].$
The proof of Theorem 6.8 relies on Lemma 6.9 below, which originally appears
as Lemma 3.1 in [7]. The proof of Lemma 6.9 found in [7] itself relies on a
citation, so for the sake of completeness it is presented here with a self-
contained proof.
###### Lemma 6.9.
Let $(f_{n})_{n\in\mathbb{Z}}$ be an exponential Riesz basis for
$L_{2}[-\pi,\pi]$. If $(g_{n})_{|n|\leq\ell}$ is a sequence of complex
exponentials such that
$(g_{n})_{|n|\leq\ell}\cup(f_{n})_{|n|>{\ell_{0}}}$
consists of distinct functions, then this set is a Riesz basis for
$L_{2}[-\pi,\pi]$.
###### Proof of Lemma 6.9.
If we can prove the case when $\ell=0$, the general result follows
inductively. Let
$f_{n}(\cdot)=\frac{1}{\sqrt{2\pi}}e^{i\langle\cdot,t_{n}\rangle}$ for $n\neq
0$, and $g_{0}(\cdot)=\frac{1}{\sqrt{2\pi}}e^{i\langle\cdot,\tau_{0}\rangle}$
where $\tau_{0}\in\mathbb{R}$ and $\tau_{0}\neq t_{n}$ for $n\neq 0$. Let
$(e_{n})_{n\in\mathbb{Z}}$ be an orthonormal basis for $L_{2}[-\pi,\pi]$.
Lemma 3.5 shows that $(g_{0})\cup(f_{n})_{n\neq 0}$ is a Riesz basis if and
only if the map defined by
$e_{0}\mapsto g_{0},\quad e_{k}\mapsto f_{k},\ \mathrm{for}\ k\neq 0$
is one to one. This is readily seen to be equivalent to $\langle
g_{0},f_{1}^{*}\rangle\neq 0$, or by passing to the Fourier transform, to
$G_{0}(\tau_{0})\neq 0$, (recall that $G_{0}=\mathcal{F}(f_{0}^{*})$). If we
can show that the only zeros of $G_{0}$ in $\mathbb{R}$ are $(t_{n})_{n\neq
0}$, we are done.
Suppose there exists $\lambda\in\mathbb{R}$, $\lambda\notin(t_{n})_{n\neq 0}$.
Such that $G_{0}(\lambda)=0$ with multiplicity $m$. Define the entire function
$H(t)=\frac{(t_{0}-\lambda)^{m}}{(t-\lambda)^{m}}G_{0}(t).$
Note that $H|_{\mathbb{R}}\in L_{2}(\mathbb{R})$, and $H$ is of exponential
type $\pi$, so $H\in PW_{[-\pi,\pi]}$ by the Theorem 2.4. The expansion
$H(t)=\sum_{n\in\mathbb{Z}}H(t_{n})G_{n}(t),$
combined with $H(t_{n})=\delta_{n,0}$, shows that $H(t)=G_{0}(t)$ for all
$t\in\mathbb{R}$, an immediate contradiction. We conclude that
$G_{0}(\lambda)\neq 0$. ∎
###### Proof of Theorem 6.8.
Define $Le_{n}=f_{n}$ and $\tilde{L}e_{n}=g_{n}$. By Corollary 6.6,
$\tilde{L}$ is bounded linear and $\tilde{L}=L+\Delta$ for some compact
operator $\Delta$. Define the operator
$R_{\ell}e_{n}=\left\\{\begin{array}[]{lr}f_{n},&|n|\leq\ell\\\
g_{n}&|n|>\ell\end{array}\right..$
Rewritten, we have
$R_{\ell}=LP_{\ell}+(L+\Delta)(I-P_{\ell})=L+\Delta(I-P_{\ell}).$
Compactness of $\Delta$ implies that $\lim_{\ell\rightarrow\infty}R_{\ell}=L$
in the operator norm topology. We conclude that $R_{\ell_{0}}$ is an onto
isomorphism for some $\ell_{0}$ sufficently large; that is, the set
(42) $(f_{n})_{|n|\leq\ell_{0}}\cup(g_{n})_{|n|>{\ell_{0}}}$
is a Riesz basis for $L_{2}[-\pi,\pi]$. If we apply Lemma 6.9, by replacing
$(f_{n})_{|n|\leq\ell_{0}}$ with $(g_{n})_{|n|\leq\ell_{0}}$ in expression
(42), we have that $(g_{n})_{n\in\mathbb{Z}}$ is a Riesz basis for
$L_{2}[-\pi,\pi].$ ∎
## 7\. Appendix: Comments regarding the optimality of Theorem 5.1
In the statement of Theorem 5.1, it is not apparent whether or not
$(E_{\ell,N,k})_{\ell}$ can be replaced with a more rapidly growing sequence
of intervals; however, Proposition 7.1 shows that if $f(t)=\mathrm{SINC}(\pi
t)$, equations (23) and (22) can hold for a sequence of intervals
$(E_{\ell,N})_{\ell}$ which grow faster than $(E_{\ell,N,A})_{\ell}$.
Propositions 7.6 and 7.8 show that growth bounds of the intervals in
Proposition 7.1 are optimal for the conclusion of said proposition to hold.
Thus, the bounds in Proposition 7.1 provide upper bounds for the growth of any
sequence $(E_{\ell,N,A})_{\ell}$ such that either equation (23) or equation
(22) hold for general multivariate bandlimited functions.
###### Proposition 7.1.
Define
$\displaystyle C_{\ell,N}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{1}{4}(2N+1)^{2}(\ell+1/2)^{2N+1}\log(\ell+1/2)\Big{)}^{\frac{1}{2(N+1)}},\
\mathrm{and}$ $\displaystyle D_{\ell,N}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{1}{2}(2N+1)^{2}(\ell+1/2)^{2N+1}\log(\ell+1/2)\Big{)}^{\frac{1}{2(N+1)}},$
then
(43) $\lim_{\ell\rightarrow\infty}\Big{\|}\mathrm{SINC}(\pi
t)-I_{\mathrm{SINC}\pi(\cdot),\ell}(t)\Big{\|}_{L_{2}([-C_{\ell,N},C_{\ell,N}]^{d})}=0,$
and
(44) $\lim_{\ell\rightarrow\infty}\Big{\|}\mathrm{SINC}(\pi
t)-I_{\mathrm{SINC}\pi(\cdot),\ell}(t)\Big{\|}_{L_{\infty}([-D_{\ell,N},D_{\ell,N}]^{d})}=0.$
The proof of equation (43) requires the following two propositions.
###### Proposition 7.2.
(45)
$\lim_{\ell\rightarrow\infty}\Big{\|}\Big{(}e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\mathrm{SINC}(\pi
t)\Big{\|}_{L_{2}([-C_{\ell,N},C_{\ell,N}]^{d})}=0.$
###### Proof of Proposition 7.2.
Let $t=\alpha C_{\ell,N}$ where $\alpha\in[-1,1]^{d}$. Noting that
$e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}=\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}},$
the quantity in equation (45) becomes
$\displaystyle\bigg{(}\int_{[-C_{\ell,N},C_{\ell,N}]^{d}}\bigg{|}\Big{(}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}-1\Big{)}\mathrm{SINC}(\pi
t)\bigg{|}^{2}dt\bigg{)}^{1/2}$
$\displaystyle\quad\leq\frac{1}{C_{\ell,N}^{d/2}}\bigg{(}\int_{[-1,1]^{d}}\bigg{|}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}-1\bigg{|}^{2}d\alpha\bigg{)}^{1/2}$
$\displaystyle\quad\leq\frac{2^{d/2}\Big{(}\ell+\frac{1}{2}\Big{)}^{d\frac{2N+1}{4(N+1)}}+2^{d/2}}{\big{(}\frac{1}{4}(2N+1)^{2}\big{)}^{\frac{d}{4(N+1)}}\big{(}\log(\ell+1/2)\big{)}^{\frac{d}{4(N+1)}}(\ell+1/2)^{d\frac{2N+1}{4(N+1)}}}.$
The last term in the above inequality has limit $0$ as
$\ell\rightarrow\infty$. This proves the proposition. ∎
###### Proposition 7.3.
$\displaystyle\lim_{\ell\rightarrow\infty}\Big{\|}Q_{d,\ell}(t)e^{-\\!\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\\!\\!-\\!e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi
t)\Big{\|}_{L_{2}([-C_{\ell,N},C_{\ell,N}]^{d})}\\!=\\!0.$
###### Proof of Proposition 7.3.
If $t\in\mathbb{R}^{d}$ and $\|t\|_{\infty}<\ell+1/2$, then equation (28)
implies
(46)
$\displaystyle\bigg{(}e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\bigg{)}\prod_{k=1}^{d}e^{\bigg{(}-\frac{1}{2(\ell+1/2)}\frac{\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}{1-\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}\bigg{)}}$
$\displaystyle\quad\leq\frac{Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi
t)}\leq
e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}.$
Let $t\in[-C_{\ell,N},C_{\ell,N}]^{d}$ where $t=\alpha C_{\ell,N}$,
$\alpha\in[-1,1]$. Consider the right hand side of inequality (46) for such
$t$.
(47) $\displaystyle
e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\leq\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}e^{(\ell+1/2)O\Big{(}\big{\|}\frac{t}{\ell+1/2}\big{\|}_{2(N+2)}^{2(N+2)}\Big{)}}$
$\displaystyle\quad\leq\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}e^{M(\ell+1/2)^{-\frac{1}{N+1}}(\log(\ell+1/2))^{\frac{N+2}{N+1}}\|\alpha\|_{2(N+2)}^{2(N+2)}}.$
for some constant $M$. Noting that
$\frac{t^{2}}{(\ell+1/2)^{3}}=\frac{\|\alpha\|_{2}^{2}\big{(}\frac{1}{4}(2N+1)^{2}\big{)}^{\frac{1}{N+1}}\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{N+1}}}{(\ell+1/2)^{\frac{N+2}{N+1}}},$
we can bound the left hand side of inequality (46) from below as follows:
(48) $\displaystyle
e^{\Big{(}-m\frac{\|\alpha\|_{2}^{2}\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{N+1}}}{(\ell+1/2)^{\frac{N+2}{N+1}}}\Big{)}}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}$
$\displaystyle\quad\leq\bigg{(}e^{\sum_{k=N+1}^{\infty}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\bigg{)}\prod_{k=1}^{d}e^{\bigg{(}-\frac{1}{2(\ell+1/2)}\frac{\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}{1-\big{(}\frac{t}{\ell+1/2}\big{)}^{2}}\bigg{)}},$
where $m>0$ is chosen independently of $\ell$. Relations (46) through (48)
imply
$\displaystyle\bigg{(}e^{\Big{(}-m\frac{(\log(\ell+1/2))^{\frac{1}{N+1}}\|\alpha\|_{2}^{2}}{(\ell+1/2)^{\frac{N+2}{N+1}}}\Big{)}}-1\bigg{)}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}|\mathrm{SINC}(\pi
t)|$
$\displaystyle\quad\leq\bigg{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi
t)\bigg{|}$
$\displaystyle\quad\leq\bigg{(}e^{\Big{(}M\frac{(\log(\ell+1/2))^{\frac{N+2}{N+1}}\|\alpha\|_{2(N+2)}^{2(N+2)}}{(\ell+1/2)^{\frac{1}{N+1}}}\Big{)}}-1\bigg{)}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}|\mathrm{SINC}(\pi
t)|.$
Further simplification implies (for appropriate constants $C$, $C^{\prime}$,
and $C^{\prime\prime}$) that
$\displaystyle\Big{\|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{\|t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi
t)\Big{\|}_{L_{2}([-C_{\ell,N},C_{\ell,N}]^{d})}$ $\displaystyle\quad\leq
C\frac{(\log(\ell+1/2))^{\frac{N+2}{N+1}}}{(\ell+1/2)^{\frac{1}{N+1}}}\Bigg{(}\int_{[-C_{\ell,N},C_{\ell,N}]^{d}}\Bigg{|}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}\|\alpha\|_{2}^{2}\mathrm{SINC}(\pi
t)\Bigg{|}^{2}dt\Bigg{)}^{1/2}$
$\displaystyle\quad=C^{\prime}\frac{(\log(\ell+1/2))^{\frac{N+2}{N+1}}}{(\ell+1/2)^{\frac{1}{N+1}}}\Bigg{(}\int_{[-1,1]^{d}}\frac{\Bigg{|}\Big{(}\ell+\frac{1}{2}\Big{)}^{\frac{2N+1}{4(N+1)}\|\alpha\|_{2(N+1)}^{2(N+1)}}\|\alpha\|_{2}^{2}\mathrm{SINC}(\pi
t)\Bigg{|}^{2}}{(\log(\ell+1/2))^{\frac{d}{2(N+1)}}\big{(}\ell+\frac{1}{2}\big{)}^{\frac{2N+1}{2(N+1)}d}}d\alpha\Bigg{)}^{1/2}$
$\displaystyle\quad\leq
C^{\prime\prime}\frac{(\log(\ell+1/2))^{\frac{N+2}{N+1}}}{(\ell+1/2)^{\frac{1}{N+1}}(\log(\ell+1/2))^{\frac{d}{4(N+1)}}},$
after the change in variable $t=\alpha C_{\ell,N}$ and simple estimates. This
proves the proposition. ∎
###### Proof of equation (43).
Equation (43) follows immediately from Propositions 7.2 and 7.3. ∎
The proof of equation (44) requires the following two propositions.
###### Proposition 7.4.
(49)
$\lim_{\ell\rightarrow\infty}\Big{\|}\Big{(}e^{\frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big{)}\mathrm{sinc}(\pi
t)\Big{\|}_{L_{\infty}[-D_{\ell,N},D_{\ell,N}]}=0.$
###### Proof of Proposition 7.4.
Let $t\in[-D_{\ell,N},D_{\ell,N}]$, then $t=\alpha D_{\ell,N}$ for
$\alpha\in[-1,1]$. Simplification shows that equation (49) holds if
(50)
$\lim_{\ell\rightarrow\infty}\sup_{\alpha\in[0,1]}\bigg{|}\frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{2N+1}{2(N+1)}}}\bigg{|}=0.$
Note that for large $\ell$,
(51)
$\sup_{\alpha\in[1/2,1]}\Bigg{|}\frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{2N+1}{2(N+1)}}}\Bigg{|}\leq\frac{2}{\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{2(N+1)}}}.$
Let $0<\alpha\leq 1/2$. The Mean Value Theorem implies
(52)
$\bigg{|}\frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha}\bigg{|}\leq(2N+1)(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}\alpha^{2N+1}\log(\ell+1/2).$
This yields
$\displaystyle\sup_{\alpha\in[0,1/2]}\bigg{|}\frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha\big{(}\log(\ell+1/2)\big{)}^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{2N+1}{2(N+1)}}}\bigg{|}\leq
M\frac{\big{(}\log(\ell+1/2)\big{)}^{\frac{2N+1}{2(N+1)}}}{(\ell+1/2)^{\frac{2N+1}{2(N+1)}(1-(1/2)^{2(N+1)})}}$
for some constant $M$. Combined with inequality (51), we have equation (50),
which proves the proposition. ∎
###### Proposition 7.5.
$\lim_{\ell\rightarrow\infty}\Big{\|}Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{sinc}(\pi
t)\Big{\|}_{L_{\infty}[-D_{\ell,N},D_{\ell,N}]}=0.$
###### Proof of Proposition 7.5.
Let $t\in[-C_{\ell,N},C_{\ell,N}]$ where $t=\alpha C_{\ell,N}$,
$\alpha\in[-1,1]$. Proceeding in the same manner as in the proof of
Proposition 7.3, we see (for appropriate constants $C$ and $C^{\prime}$) that
$\displaystyle\bigg{|}Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}-e^{\frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{sinc}(\pi
t)\bigg{|}_{L_{\infty}([-C_{\ell,N},C_{\ell,N}])}$
$\displaystyle\quad\leq\frac{C(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}\alpha^{2}(\log(\ell+1/2))^{\frac{N+2}{N+1}}|\sin(\pi
t)|}{\alpha(\ell+1/2)^{\frac{1}{N+1}}(\log(\ell+1/2))^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{2N+1}{2(N+1)}}}$
$\displaystyle\quad\leq\frac{C^{\prime}(\log(\ell+1/2))^{\frac{2N+3}{2(N+1)}}}{(\ell+1/2)^{\frac{1}{N+1}}}.$
This proves the proposition. ∎
###### Proof of equation (44).
The previous two propositions prove equation when $d=1$. The multidimensional
case follows inductively. ∎
###### Proposition 7.6.
Let $N\geq 0$. If $(M_{\ell,N})_{\ell}$ is a sequence of positive numbers such
that (43) holds when $(C_{\ell,N})_{\ell}$ is replaced by
$(M_{\ell,N})_{\ell}$, then
(53) $\limsup_{\ell\rightarrow\infty}\frac{M_{\ell,N}}{C_{\ell,N}}\leq 1.$
The proof of Proposition 7.6 requires the following simple estimate:
###### Proposition 7.7.
Let $a>1/2$, $\epsilon>0$, $0<\omega<1$, then
$\int_{a}^{(1+\epsilon)a}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx>\frac{\epsilon}{2a^{\omega}(1+\epsilon)^{\omega}}-\frac{a}{2(a-1/2)^{1+\omega}}.$
###### Proof of Proposition 7.7.
Let $b=(1+\epsilon)a$. We have
$\int_{a}^{b}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx+\int_{a}^{b}\frac{\cos^{2}\pi
x}{x^{1+\omega}}dx=\frac{1}{\omega}\Big{(}\frac{1}{a^{\omega}}-\frac{1}{b^{\omega}}\Big{)}$
and
$\int_{a}^{b}\frac{\cos^{2}\pi
x}{x^{1+\omega}}dx=\int_{a-1/2}^{b-1/2}\frac{\sin^{2}\pi
x}{(x+1/2)^{1+\omega}}dx<\int_{a-1/2}^{b-1/2}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx.$
This yields
$2\int_{a}^{b}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx-\int_{b-1/2}^{b}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx+\int_{a-1/2}^{a}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx>\frac{1}{\omega}\Big{(}\frac{1}{a^{\omega}}-\frac{1}{b^{\omega}}\Big{)},$
so that
$\int_{a}^{b}\frac{\sin^{2}\pi
x}{x^{1+\omega}}dx>\frac{1}{2\omega}\Big{(}\frac{1}{a^{\omega}}-\frac{1}{b^{\omega}}\Big{)}-\frac{1}{2(a-1/2)^{1+\omega}}.$
Noting that
$\frac{1}{2\omega}\Big{(}\frac{1}{a^{\omega}}-\frac{1}{b^{\omega}}\Big{)}=\frac{\epsilon}{2\omega
a^{\omega}(1+\epsilon)^{\omega}}\frac{(1+\epsilon)^{\omega}-1}{\epsilon}>\frac{\epsilon}{2a^{\omega}(1+\epsilon)^{\omega}}$
proves the proposition. ∎
###### Proof of Proposition 7.6.
Fix $N\geq 0$, and define $c=\frac{2N+1}{2N+4}+\delta/2$ where $0<\delta$ is
small enough so that $c<1/2$. Define
$A_{\ell}=(c(N+1)(2N+1)\log(\ell+1/2))^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{-1}{2(N+1)}}$
and
$\epsilon_{\ell}=(\ell+1/2)^{1-2c}A_{\ell}.$
Note that $\lim_{\ell\rightarrow\infty}\epsilon_{\ell}=0.$ Let
$t\in[A_{\ell}(\ell+1/2),(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)],$ then
$t=\alpha(\ell+1/2)$ for some
$\alpha\in[A_{\ell},(1+\epsilon_{\ell})A_{\ell}]$. For large $\ell$, note that
inequality (28) implies
(54)
$\displaystyle\frac{1}{2\pi}e^{\frac{(\ell+1/2)\alpha^{2(N+1)}}{(N+1)(2N+1)}}\frac{|\sin\pi\alpha(\ell+1/2)|}{\alpha(\ell+1/2)}\leq\Big{|}Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}.$
Moving to the multivariate case, if
$t\in[A_{\ell}(\ell+1/2),(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)]^{d}$, then
$t=\alpha(\ell+1/2)$ for some
$\alpha\in[A_{\ell},(1+\epsilon_{\ell})A_{\ell}]^{d}$. This yields
$\displaystyle\prod_{i=1}^{d}\frac{1}{2\pi\alpha_{i}^{c}}\frac{|\sin\pi\alpha_{i}(\ell+1/2)|}{(\alpha_{i}(\ell+1/2))^{1-c}}\leq\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}.$
For sufficiently large $\ell$, we can conclude
$\displaystyle\bigg{[}\frac{1}{9\pi^{2}A_{\ell}^{2c}}\int_{A_{\ell}(\ell+1/2)}^{(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)}\frac{\sin^{2}\pi
x}{x^{2-2c}}dx\bigg{]}^{d}$
$\displaystyle\quad\leq\int_{[A_{\ell}(\ell+1/2),(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)]^{d}}\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}^{2}dt.$
Applying Proposition 7.7 for $a=A_{\ell}(\ell+1/2)$,
$\epsilon=\epsilon_{\ell},$ and $\omega=1-2c$, and using the definition of
$\epsilon_{\ell}$, we obtain
$\displaystyle\bigg{[}\frac{1}{9\pi^{2}}\Big{[}\frac{1}{2(1+\epsilon_{\ell})^{1-2c}}-\frac{1}{2A_{\ell}^{2c}(A_{\ell}(\ell+1/2)-1)^{2-2c}}\Big{]}\bigg{]}^{d}$
$\displaystyle\quad\leq\int_{[A_{\ell}(\ell+1/2),(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)]^{d}}\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}^{2}dt.$
The first term in the brackets in the previous equation has limit $1/2$, while
the second term has limit $0$. We conclude there exists a constant $\beta>0$
such that
(55)
$\beta\leq\int_{[A_{\ell}(\ell+1/2),(1+\epsilon_{\ell})A_{\ell}(\ell+1/2)]^{d}}\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}^{2}dt,\quad\ell>0.$
If $M_{\ell,N}\geq(\ell+1/2)(1+\epsilon_{\ell})A_{\ell}$ for infinitely many
$\ell$, there exists a subsequence $(\ell_{k})_{k\in\mathbb{N}}$ such that (in
particular),
$\displaystyle\lim_{\ell_{k}\rightarrow\infty}\\!\Big{\|}\mathrm{SINC}(\pi
t)-Q_{d,\ell_{k}}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell_{k}+1/2)^{2k-1}}}\Big{\|}_{L_{2}([A_{\ell_{k}}(\ell_{k}+1/2)),A_{\ell_{k}}(\ell_{k}+1/2)(1+\epsilon_{\ell_{k}})]^{d})}\\!\\!=\\!0.$
This contradicts inequality (55). This yields that for sufficiently large
$\ell$,
$\displaystyle M_{\ell,N}$ $\displaystyle<$
$\displaystyle(\ell+1/2)(1+\epsilon_{\ell})A_{\ell}$ $\displaystyle=$
$\displaystyle(1+\epsilon_{\ell})\Big{(}\Big{(}\frac{2N+1}{4N+4}+\delta/2\Big{)}(N+1)(2N+1)(\ell+1/2)^{2N+1}\log(\ell+1/2)\Big{)}^{\frac{1}{2(N+1)}}.$
Note that since $\epsilon_{\ell}\rightarrow 0$, for large $\ell$, the quantity
$(1+\epsilon_{\ell})\Big{(}\frac{2N+1}{4N+4}+\delta/2\Big{)}^{\frac{1}{2(N+1)}}$
is less than, (and bounded away from) the quantity
$\Big{(}\frac{2N+1}{4N+4}+\delta\Big{)}^{\frac{1}{2(N+1)}}$. We conclude that
for any $\delta>0$, there exists $\ell_{N,\delta}$ such that
$\sup_{\ell>\ell_{N,\delta}}\frac{M_{\ell,N}}{((N+1)(2N+1)\log(\ell+1/2))^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{2N+1}{2(N+1)}}}<\Big{(}\frac{2N+1}{4N+4}+\delta\Big{)}^{\frac{1}{2(N+1)}}.$
Proposition 7.6 follows. ∎
###### Proposition 7.8.
Let $N\geq 0$. If $(M_{\ell,N})_{\ell}$ is a sequence of positive numbers such
that equation (44) holds when $(D_{\ell,N})_{\ell}$ is replaced by
$(M_{\ell,N})_{\ell}$, then
(56) $\limsup_{\ell\rightarrow\infty}\frac{M_{\ell,N}}{D_{\ell,N}}\leq 1.$
The proof of Proposition 7.8 requires the following fact:
###### Proposition 7.9.
Let $0<\epsilon\leq 1$. If $A>0$, there exists $t\in[A,A+\epsilon]$ such that
$|\sin(\pi t)|\geq|\sin(\pi\epsilon/2)|$.
###### Proof of Proposition 7.9.
The proof is clear from geometric considerations. ∎
###### Proof of Proposition 7.8.
Let $N\geq 0$. Choose $\delta>0$ such that $c:=\frac{2N+1}{2N+2}+\delta/2<1$.
Define
$A_{\ell}=(c(N+1)(2N+1)\log(\ell+1/2))^{\frac{1}{2(N+1)}}(\ell+1/2)^{\frac{-1}{2(N+1)}}$
and
$\epsilon_{\ell}=A_{\ell}(\ell+1/2)^{1-c}.$
Note that $\lim_{\ell\rightarrow\infty}\epsilon_{\ell}=0$. Let
$t\in[A_{\ell}(\ell+1/2),A_{\ell}(\ell+1/2)+\epsilon_{\ell}].$ Proceeding as
before, for sufficiently large $\ell$, we have
$\frac{1}{2\pi}e^{\Big{(}\frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1))}\Big{)}}\frac{|\sin(\pi
t)|}{t}\leq\Big{|}Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}.$
Now for all $t\in[A_{\ell}(\ell+1/2),A_{\ell}(\ell+1/2)+\epsilon_{\ell}]$,
$\frac{1}{2\pi}\frac{(\ell+1/2)^{c}}{A_{\ell}(\ell+1/2)+\epsilon_{\ell}}|\sin(\pi
t)|\leq\Big{|}Q_{1,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}.$
In the multivariate case, if
$t\in[A_{\ell}(\ell+1/2),A_{\ell}(\ell+1/2)+\epsilon_{\ell}]^{d}$, we obtain
$\frac{1}{(2\pi)^{d}}\frac{(\ell+1/2)^{cd}}{(A_{\ell}(\ell+1/2)+\epsilon_{\ell})^{d}}\prod_{i=1}^{d}|\sin(\pi
t_{i})|\leq\Big{|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{|}.$
For large $\ell$, an application of Proposition 7.9 yields
$\frac{1}{(3\pi)^{d}}\frac{|\sin(\pi\epsilon_{\ell}/2)|^{d}}{A_{\ell}^{d}(\ell+1/2)^{(1-c)d}}\leq\Big{\|}Q_{d,\ell}(t)e^{-\sum_{k=1}^{N}\frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big{\|}_{L_{\infty}([A_{\ell}(\ell+1/2),A_{\ell}(\ell+1/2)+\epsilon_{\ell}]^{d})}.$
By the definition of $\epsilon_{\ell}$, the right hand side of the above
equation tends to a positive constant. The remainder of the proof is almost
identical to that of Proposition 7.6. ∎
The following is trivially deduced from Propositions 7.6 and 7.8: Fix $N>0$.
If $(E_{\ell,N})_{\ell}$ is a sequence of intervals such that either equation
(23) or equation (22) holds for all $f\in PW_{[-\pi,\pi]^{d}}$, then
$\max_{x\in(E_{\ell,N})^{d}}\|x\|_{\infty}=o\Big{(}\max_{x\in(E_{\ell,N+1,A})^{d}}\|x\|_{\infty}\Big{)}.$
## References
* [1] B. A. Bailey, Sampling and recovery of multidimensional bandlimited functions via frames, J. Math. Anal. Appl., (2010), 374-388.
* [2] B. A. Bailey, Th. Schlumprecht, N. Sivakumar, Nonuniform sampling and recovery of multidimensional bandlimited functions by Gaussian radial basis functions, J. Fourier Anal. Appl., (2010), Preprint.
* [3] M. I. Kadec, The exact value of the Paley-Wiener constant (Russian), Dokl. Akad. Nauk SSSR, (1964), 1253-1254.
* [4] B. Ja. Levin, Interpolation of entire functions of exponential type (Russian), Mat. Fiz. i Funkcional. Anal., (1969), 136-146.
* [5] N. Levinson, Gap and Density Theorems, American Mathematical Society, (1940).
* [6] Yu. Lyubarski and K. Seip, Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s $(A_{p})$ condition, Rev. Mat. Iberoamericana, (1997), 361-376.
* [7] H. Pak and C. Shin, Perturbation of nonharmonic Fourier series and nonuniform sampling theorem, Bull. Korean Math. Soc., (2007), 351-358.
* [8] R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society, (1944).
* [9] B.S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Acad. Nauk, (1979), 37-40.
* [10] W. Rudin, Real and complex analysis (3rd ed.), New York: McGraw-Hill, (1987).
* [11] G. Schmeisser, F. Stenger, Sinc approximation with a Gaussian multiplier, Sampl. Theory Sample Image Process., (2007), 199-221.
* [12] W. Sun and X. Zhou, On the stability of multivariate trigonometric systems, J. Math. Anal. Appl., (1999), 159-167.
* [13] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, (2001).
|
arxiv-papers
| 2010-09-10T16:37:45 |
2024-09-04T02:49:12.806807
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. A. Bailey",
"submitter": "Benjamin Bailey Mr.",
"url": "https://arxiv.org/abs/1009.2047"
}
|
1009.2123
|
# Detection of Anomalous Reactor Activity Using Antineutrino Count Rate
Evolution Over the Course of a Reactor Cycle
Vera Bulaevskaya, Adam Bernstein
###### Abstract
This paper analyzes the sensitivity of antineutrino count rate measurements to
changes in the fissile content of civil power reactors. Such measurements may
be useful in IAEA reactor safeguards applications. We introduce a hypothesis
testing procedure to identify statistically significant differences between
the antineutrino count rate evolution of a standard ’baseline’ fuel cycle and
that of an anomalous cycle, in which plutonium is removed and replaced with an
equivalent fissile worth of uranium. The test would allow an inspector to
detect anomalous reactor activity, or to positively confirm that the reactor
is operating in a manner consistent with its declared fuel inventory and power
level. We show that with a reasonable choice of detector parameters, the test
can detect replacement of 73 kg of plutonium in 90 days with 95% probability,
while controlling the false positive rate at 5%. We show that some improvement
on this level of sensitivity may be expected by various means, including use
of the method in conjunction with existing reactor safeguards methods. We also
identify a necessary and sufficient daily antineutrino count rate to achieve
the quoted sensitivity, and list examples of detectors in which such rates
have been attained.
###### Index Terms:
Safeguards applications, neutrinos, nuclear monitoring, nuclear power plants,
nuclear power simulation, particle detectors
## I Introduction
The International Atomic Energy Agency (IAEA) nuclear safeguards regime is
designed to detect diversion of fissile material from civil nuclear fuel cycle
facilities to weapons programs [1]. In previous work, we predicted [2] and
demonstrated [3],[4],[5] that cubic meter scale antineutrino detectors,
operating at a distance of tens of meters from a 1 gigawatt electric (GWe)
pressurized water reactor (PWR), can directly detect changes in operational
status, power levels, and fissile inventory of the reactor core. Similar
results were achieved earlier by a Russian group [6]. These metrics are all of
potential use for the IAEA reactor safeguards regime.
In this paper, we demonstrate a possible methodology for using antineutrino
detection in a safeguards context. We introduce a hypothesis testing procedure
to identify statistically significant differences between the antineutrino
count rate evolution of a standard ’baseline’ fuel cycle and that of an
anomalous cycle in which 73 kg of plutonium has been removed and replaced with
the equivalent fissile worth of uranium. (This quantity of plutonium
represents the removal and replacement of ten partially burnt assemblies with
ten fresh fuel assemblies.) The test would allow an inspector to detect
anomalous reactor activity, or to positively confirm that the reactor is
operating in a manner consistent with its declared fuel inventory and power
level. We show that with a reasonable choice of detector parameters, the test
can detect the net loss from the core of 73 kg of plutonium in 90 days with
95% probability, while controlling the false positive rate at 5%.
The purpose of the study is to explore this possible alternative method of
reactor safeguards, by quantifying the sensitivity of an antineutrino count
rate measurement to anomalous changes in fissile content. In describing our
example, we avoid the standard IAEA term ’diversion’, since we do not
explicitly specify the fate of the removed plutonium. In particular, we are
not asserting that the removal of plutonium in this example could not be
uncovered by existing IAEA safeguards methodologies.
One of IAEA’s inspection goals is to be able to detect diversion of 8 kg 1118
kg is designated by the IAEA as a ’significant quantity’ (SQ) of plutonium.
The IAEA definition of a significant quantity is ’the approximate quantity of
nuclear material in respect of which, taking into account any conversion
process involved, the possibility of manufacturing a nuclear explosive device
cannot be excluded’ [7]. of plutonium from a civil nuclear facility in a
90-day period [8]. Our current sensitivity to anomalous reactor operation
caused by removal of plutonium is at the level of several significant
quantities. Enhancements to the detector, including the capability to measure
the antineutrino energy spectrum, may allow for detection of even smaller
changes in the reactor’s fissile content. While the demonstrated sensitivity
is not to actual diversion but to anomalous reactor operations, we expect that
this method can be used in conjunction with existing IAEA safeguards
methodologies to achieve IAEA SQ goals for diverted material. We note that
other IAEA surveillance and accountancy measurement devices do not in
isolation reach the SQ goals, but are used as part of a comprehensive
accountancy strategy. Examples include Cherenkov light monitors in spent fuel
cooling ponds, which are not sensitive at the SQ level, but which provide
continuity of knowledge and confirm the presence of large numbers of
radioactive spent fuel assemblies.
We begin by briefly describing the relationship between the antineutrino count
rate and the reactor fissile inventory, and contrast our method for anomaly
detection with current IAEA reactor safeguards practice. Next, we describe the
test procedure and its inputs, including the fuel loadings of the baseline and
anomalous scenario cycles. We then examine the statistical power of the
procedure to distinguish between the two cycles and thereby identify an
anomaly in reactor operations. We include the effects of counting statistics,
a fixed systematic bias in detector response, deliberate malfeasance on the
part of the reactor operator, the starting point and duration of data
acquisition, and simulation errors. We also establish a range of acceptable
detector masses, intrinsic efficiencies and standoff distances that would
permit discovery of the anomaly in our example. We conclude by summarizing the
potential impact of this approach on current IAEA safeguards and useful next
steps.
In this paper, we assume that the background count rates are negligible
relative to those produced by the antineutrino signal. This assumption is
based on high signal-to-background ratios achieved in several past
experiments, discussed in Section VI. If these ratios are not achieved, the
results described here do not apply, and an additional analysis would be
required to account for the effects of higher background rates.
## II Current IAEA Reactor Safeguards and Antineutrino-Based Safeguards
Currently, the IAEA uses nuclear material accountancy, as well as containment
and surveillance (CS) techniques to verify the quantities of fuel used in and
discharged from reactors. Nuclear material accountancy refers to a
quantitative and independent check of fuel inventories, performed by the
Agency. At reactors, the predominant material accountancy method is item
accountancy, or counting of items (fresh and spent fuel assemblies and rods)
considered to contain fixed and known quantities of fissile material. The
presence and integrity of radioactive spent fuel assemblies and rods in
cooling ponds at the reactor is also checked by Cherenkov light measurements
and other methods. CS techniques, such as videocameras and seals on the
reactor head, are also used [7].
By contrast, antineutrino-based safeguards offer a form of near-real-time and
nondestructive bulk accountancy. In contrast to item accountancy, bulk
accountancy methods provide estimates of the total fissile mass without
relying on assumptions about the mass contents of premeasured items. Examples
include coincidence neutron counting, mass spectroscopy and chemical analyses.
As such, antineutrino based methods are complementary to the existing
safeguards regime, since they provide independent quantitative information
about fissile material inventories as long as the reactor is operational.
Among other uses, this information can provide independent confirmation that
the fuel inventory at beginning and throughout the reactor cycle is consistent
with operator declarations. In principle, the inventory estimate so derived
can also be used to check for shipper/receiver differences, both for fresh
fuel taken in by the operator and for spent fuel sent to downstream
reprocessing or storage facilities.
While the measurement capability appears promising, its actual import for IAEA
safeguards is beyond the scope of this paper. As an example of the
complications that arise, we note that for existing power reactors, the
antineutrino-based inventory estimates would have to be reconciled with and
integrated into the full accounting of all materials at the reactor site,
including that in spent fuel cooling ponds. For such sites, with decades of
accumulated and largely unassayed fuel, containing many tens of tons of
fissile material, such accounting may prove impractical. For this reason, we
recommend that a more detailed analysis of the capability be conducted by
safeguards experts, both for existing and future reactor safeguards regimes.
## III Modeling the Antineutrino Count Rate for Safeguards Applications
A change in fissile mass content in a reactor core - such as that occurring
when uranium is consumed and plutonium produced in the course of a reactor
fuel cycle - creates a measurable systematic shift in the antineutrino count
rate (and energy spectrum). In previous work [5], we have shown that the
antineutrino count rate is reduced by about 10$\%$ relative to its value at
the beginning of the cycle over the course of a typical 1.5 year pressurized
water reactor (PWR) fuel cycle. This reduction occurs even when (as is
typical) the reactor maintains constant power throughout the cycle; therefore,
monitoring the antineutrino count rate provides information about core fissile
inventory evolution that is not accessible through a measurement of the
reactor power alone.
In a safeguards context, the measured antineutrino count rate evolution would
be compared to a predicted count rate evolution assuming normal conditions
(i.e., no removal of plutonium) over some portion or all of the fuel cycle.
The predicted evolution under normal operating conditions will be referred to
as the “baseline scenario” for the remainder of this paper. The prediction is
obtained from a reactor simulation code, such as ORIGEN [9], which takes as
inputs the operator-declared thermal power and initial fissile isotopic
masses, as well as other reactor parameters, and returns fission rates for
each isotope. The individual fission rates are then converted into a predicted
emitted antineutrino flux using standard analytical formulae. The emitted
antineutrino flux is finally converted to a measured antineutrino count rate,
using a detector response function derived from experiment and modeling.
In the present work, we simulate both the baseline and anomalous antineutrino
count rates over the course of the fuel cycle for use in our hypothesis test.
We use an ORIGEN simulation of the core of Unit 2 of the San Onofre Nuclear
Generating Station (SONGS), originally published in [10]. The detector
response function was derived from the SONGS1 experiment [2], for which the
antineutrino signal was approximately 360 counts per day at beginning of cycle
after subtraction of reactor-off backgrounds.
Following [6], we describe the PWR core antineutrino count rate evolution
$N_{\bar{\nu}}(t)$ at time $t$ in the fuel cycle as a product of two time-
dependent factors:
$N_{\bar{\nu}}(t)=P_{th}(t)\cdot\gamma(1+k(t)).$ (1)
$P_{th}(t)$ is the reactor thermal power. The term $(1+k(t))$ depends on the
changing fissile isotopic content of the core, embodied in the parameter
$k(t)$. $\gamma$ is a constant related to the detector mass, efficiency, and
standoff distance. This parametrization highlights the direct dependence of
the count rate on the thermal power, an important consideration we return to
in Section V-D.
For the PWR core being considered here, (1) is well approximated by a
quadratic function of time:
$N_{\bar{\nu}}(t)=\beta_{0}+\beta_{1}t+\beta_{2}t^{2}.$ (2)
The quadratic model in (2) is valid for PWRs loaded with typical Low Enriched
Uranium (LEU) fuel. Other fuel loadings and reactor types can result in an
antineutrino count rate evolution that is substantially different in form from
(2).
The coefficients $\beta_{0}$, $\beta_{1}$ and $\beta_{2}$ in (2) can be used
to detect a departure from the baseline scenario. The measured antineutrino
count rate evolution can be used to estimate the coefficients, which can then
be compared to those predicted for the baseline scenario. A statistically
significant difference in at least one of the estimated coefficients from its
baseline counterpart could indicate a departure of the observed evolution from
that of the baseline scenario.
## IV Testing For Anomalous Activity
Following the model in (2), the true baseline evolution of antineutrino count
rate as a function of time $t$ in the fuel cycle is given by
$N^{(B)}_{\bar{\nu}}(t)=\beta^{(B)}_{0}+\beta^{(B)}_{1}t+\beta^{(B)}_{2}t^{2}$
(3)
(The superscript “B” in the above equation and for the remainder of the paper
indicates “baseline”). As discussed earlier, the true baseline evolution is
obtained from a reactor simulation. To account for simulation error, we modify
the model in (3) by representing the baseline count rate at time $t$ as
Gaussian with the mean equal to the simulation value and the standard
deviation equal to 1% of the simulation value, i.e.,
$N^{(B)}_{\bar{\nu}}(t)\sim\ \mbox{Gaussian}(\mu(t),0.01\mu(t)).$ (4)
$\mu(t)$ is the baseline evolution antineutrino count rate value at time $t$
from the simulation and can be modeled as
$\mu(t)=\beta^{(B)}_{0}+\beta^{(B)}_{1}t+\beta^{(B)}_{2}t^{2}.$ (5)
One percent random error is typical for these and other ORIGEN simulations
[10],[11]. (Systematic shifts of the predicted and measured response are
treated separately in Section V-C.)
Let $\\{N^{(M)}_{\bar{\nu}}(t)\\}$ denote the measured count rate evolution
(the superscript “M” indicates “measured”) which is to be tested against the
baseline scenario evolution. Since the measurements follow Poisson statistics,
$N^{(M)}_{\bar{\nu}}(t)\sim\
\mbox{Poisson}(\beta^{(M)}_{0}+\beta^{(M)}_{1}t+\beta^{(M)}_{2}t^{2}).$ (6)
To determine whether the measured antineutrino count rate evolution deviates
significantly from that of the baseline, we can compare the coefficient
$\beta^{(B)}_{i}$ in (5) to its counterpart $\beta^{(M)}_{i}$ in (6) for each
$i=0,1,2$. This requires us to estimate each of these coefficients.
One way to do this is to perform the least squares (LS) regression of both the
modeled baseline count rates $N^{(B)}_{\bar{\nu}}(t)$ and the measured count
rates $N^{(M)}_{\bar{\nu}}(t)$ on $t$ and $t^{2}$. LS regression is best
suited to the case of Gaussian noise with constant variance [12]. In our case,
the baseline count rates do in fact have Gaussian noise by construction, and
high Poisson statistics make the noise in the measured count rates
approximately Gaussian. Moreover, as noted in Section III, the change in the
count rate variance over the course of the cycle for a standard PWR is about
10%. Thus, LS regression should produce statistically near-optimal coefficient
estimates in this context (if necessary, weighted least squares regression
could be used to alleviate the issue of non-constant variance).
Of greater concern for the regression analysis is that $t$ and $t^{2}$ are
highly correlated, which can lead to very unstable coefficient estimates. A
common way to overcome this problem is to perform regression on deviations
from the sample mean of times $(t-\bar{t})$ and deviations from the mean
squared $(t-\bar{t})^{2}$ because the correlation between these two terms is
substantially lower than that between $t$ and $t^{2}$ [12]. Therefore, we
reparameterize the model for the measured count rates $N^{(M)}_{\bar{\nu}}(t)$
in (6) as follows:
$N^{(M)}_{\bar{\nu}}(t)\sim\
\mbox{Poisson}(\gamma^{(M)}_{0}+\gamma^{(M)}_{1}(t-\bar{t})+\gamma^{(M)}_{2}(t-\bar{t})^{2}).$
(7)
We must also reparametrize the model in (5). The baseline count rate
$N^{(B)}_{\bar{\nu}}(t)$ still follows (4), but the baseline mean function
$\mu(t)$ is now given by
$\mu(t)=\gamma^{(B)}_{0}+\gamma^{(B)}_{1}(t-\bar{t})+\gamma^{(B)}_{2}(t-\bar{t})^{2}.$
(8)
Each coefficient $\gamma^{(M)}_{i}$ in (7) can then be compared to its
counterpart $\gamma^{(B)}_{i}$ in (8) for $i=0,1,2$ by testing the following
pairs of hypotheses:
$\displaystyle H^{(0)}_{o}:\gamma^{(M)}_{0}=\gamma^{(B)}_{0}\ $ versus
$\displaystyle H^{(0)}_{a}:\gamma^{(M)}_{0}\neq\gamma^{(B)}_{0}$
$\displaystyle H^{(1)}_{o}:\gamma^{(M)}_{1}=\gamma^{(B)}_{1}\ $ versus
$\displaystyle H^{(1)}_{a}:\gamma^{(M)}_{1}\neq\gamma^{(B)}_{1}$ (9)
$\displaystyle H^{(2)}_{o}:\gamma^{(M)}_{2}=\gamma^{(B)}_{2}\ $ versus
$\displaystyle H^{(2)}_{a}:\gamma^{(M)}_{2}\neq\gamma^{(B)}_{2}$
The test procedure then consists of the following steps:
1. 1.
Generate $\\{N^{(B)}_{\bar{\nu}}(t)\\}$ according to (4).
2. 2.
Obtain coefficient estimates $\hat{\gamma}^{(B)}_{0}$,
$\hat{\gamma}^{(B)}_{1}$ and $\hat{\gamma}^{(B)}_{2}$, and their standard
errors $se(\hat{\gamma}^{(B)}_{0})$, $se(\hat{\gamma}^{(B)}_{1})$ and
$se(\hat{\gamma}^{(B)}_{2})$ from the least squares regression of generated
count rates $\\{N^{(B)}_{\bar{\nu}}(t)\\}$ on time deviations $(t-\bar{t})$
and time deviations squared $(t-\bar{t})^{2}$ (where $\bar{t}$ is the sample
average of all the time values $t$).
3. 3.
Similarly, obtain coefficient estimates $\hat{\gamma}^{(M)}_{0}$,
$\hat{\gamma}^{(M)}_{1}$ and $\hat{\gamma}^{(M)}_{2}$, and their standard
errors $se(\hat{\gamma}^{(M)}_{0})$, $se(\hat{\gamma}^{(M)}_{1})$ and
$se(\hat{\gamma}^{(M)}_{2})$ from the least squares regression of measured
count rates $\\{N^{(M)}_{\bar{\nu}}(t)\\}$ on $(t-\bar{t})$ and
$(t-\bar{t})^{2}$.
4. 4.
Obtain test statistics
$s_{i}=\frac{\hat{\gamma}^{(M)}_{i}-\hat{\gamma}^{(B)}_{i}}{\sqrt{se^{2}(\hat{\gamma}^{(M)}_{i})+se^{2}(\hat{\gamma}^{(B)}_{i})}}$
(10)
for $i=0,1,2$ and their corresponding $p$-values, given by
$p_{i}=2\cdot P(S>|s_{i}|)$ (11)
where $S$ has a Student’s $t$ distribution with $2\cdot(n-3)$ degrees of
freedom with $n$ equal to the number of count rate measurements.
5. 5.
Determine the acceptable false positive (FP) rate (see Section V) and apply
the false discovery rate (FDR) procedure, described in [13]222As described in
more detail in [13], the FDR procedure controls the error rate of testing
multiple hypotheses., to determine whether to reject each of the $H^{(i)}_{o}$
in favor of $H^{(i)}_{a}$ in (IV). If at least one of the null hypotheses is
rejected, conclude that the measured evolution deviates significantly from
that of the baseline. Otherwise, conclude that the measured evolution does not
significantly deviate from that of the baseline.
## V Test Performance
The test can produce two types of errors: it could find a significant
difference from the baseline in at least one coefficient when the evolution
was in fact produced by a baseline scenario (a false positive, or FP, result),
or it could miss a significant difference in all coefficients when the
evolution was different from the baseline (a false negative, or FN, result).
The complement of the FN rate is the true positive (TP) rate. The TP rate is
defined as the probability of finding a significant difference in at least one
of the coefficients from its baseline counterpart when the evolution in
question is in fact different from that of the baseline. A good test has a low
FP rate and a high TP rate. There is a trade-off between these two quantities:
all else being equal, increasing the TP rate of the test comes at a price of a
higher FP rate. A Receiver Operating Characteristic (ROC) curve for a
particular test procedure shows the former as a function of the latter, thus
allowing one to determine the minimum FP rate that yields the desired TP rate.
### V-A ROC Curve Simulation
To estimate the ROC curve of the test, we carried out a simulation (not to be
confused with the reactor simulation) that estimates the TP rate of the test
for a given FP rate. This simulation was performed for a scenario in which ten
once-burned assemblies with the highest plutonium content are removed and
replaced with 3.91% enriched fresh fuel. This represents the removal of 73 kg
of 239Pu from the core. Complete fissile inventories at beginning of cycle for
the baseline and anomalous scenario are shown in Table I.
TABLE I: The initial inventories of the main fissioning isotopes for the baseline and anomalous scenarios. The final column is the difference in fissile content between the two scenarios. A negative (positive) value indicates that the isotope was removed (added) in the anomalous scenario. Isotope | Baseline mass | Anomalous scenario | Mass difference
---|---|---|---
| (kg) | mass (kg) | (kg)
235U | 2834 | 2849 | 15
238U | 82912 | 83351 | 439
239Pu | 225 | 152 | -73
241Pu | 21 | 12 | -9
Fig. 1 shows the antineutrino count rate evolutions predicted by the ORIGEN
simulation for the baseline scenario (solid green) and the anomalous scenario
(red). (The shifted baseline evolution, shown in dashed green, is discussed in
Section V-D).
Figure 1: Baseline (solid green), shifted baseline (dashed green) and
anomalous (red) scenario evolutions of daily antineutrino count rates versus
time (in days), as simulated in ORIGEN.
A given point on a ROC curve is obtained as follows. One hundred thousand
pairs of anomalous and baseline evolutions are generated, with the former from
a Poisson distribution with the coefficients $\gamma^{(M)}_{i}$, $i=0,1,2$,
obtained from the ORIGEN reactor simulation for the given scenario and time
period, and the latter from a Gaussian distribution according to (4). The test
procedure introduced in Section IV is then applied at the given FP rate (the
$x$ coordinate of the point on the ROC curve) to each pair of evolutions. We
then estimate the TP rate (the $y$ coordinate of the point on the ROC curve)
with the fraction of the 100,000 evolution pairs for which at least one of the
null hypotheses in (IV) is rejected. This is repeated for a sequence of FP
rate values from 0 to 1, thus producing a curve. The large number of generated
evolutions ensures that every TP rate estimate is within 1% of the relevant
true TP rate.
To verify that the nominal FP rate of our test procedure corresponds to its
actual FP rate, we also generated 100,000 baseline evolutions from a Poisson
distribution with the coefficients $\gamma^{(B)}_{i}$, $i=0,1,2$, obtained
from the ORIGEN reactor simulation, for the given time period. We estimated
the actual FP rates with the fractions of these evolutions for which at least
one of the null hypotheses in (IV) was rejected and found them to be very
close to the nominal FP rates.
While the performance of the test will depend on the specific scenario, the
present example allows us to identify several important factors that influence
our ability to detect any anomalous reactor operation. In the following
sections we assess the impact on our test performance of finite counting
statistics, systematic error in the detector response, operator malfeasance,
and the starting point and duration of data acquisition within the cycle.
### V-B Effect of Counting Statistics
For the evolutions shown in Fig. 1, antineutrino count rates range from
approximately 375 per day at the beginning of cycle to approximately 335 per
day at the end of cycle. As discussed in Section VI, easily achievable
increases in the combined detector mass and efficiency can lead to a five-fold
improvement in counting statistics. We considered the impact of these changes
on the test performance, simply by increasing the count rate used in our test
by a factor of 5.
Fig. 2 shows that this dramatically improves the performance of the test. The
ROC curve for high count rates collected over the first 90 days in the cycle,
shown in purple, is up to six times higher than the ROC curve for the low
count rates for the same time period, shown in orange. For example, at the FP
rate of 5%, the high count TP rate of the former is 95%, while the low count
TP rate is 34%. This strong effect was observed for other data acquisition
periods. These results, as well as those discussed in Sections V-D and V-E,
are summarized in Table II. For the particular scenario considered here, we
verified that a minimum five-fold improvement in counting statistics is
necessary in order to achieve a 95% TP rate at the 5% FP rate. This was
accomplished by progressively increasing the count rate in the testing
procedure until the 95% TP% / 5% FP standard was achieved.
TABLE II: True positive rates (in %) at the false positive rate of 5% for the various factors considered here. See text for an explanation of each factor. | 2000 counts | 375 counts
---|---|---
Dura- | per day | per day
tion | Un- | Shifted Due To | Un-
(days) | shifted | Mal- | Detector Bias | shifted
| | fesance | Uncorr. | Corrected |
first 30 | 58 | | | |
first 90 | 95 | 23 | 0.4 | 12 | 34
first 180 | | | | 62 |
first 250 | 99 | 56 | | 96 |
last 90 | | 32 | | |
last 250 | | 73 | | |
500 | 99.99 | 99 | | |
Figure 2: ROC curves for the test using days 0–90, assuming low counts
(orange) and high counts (purple). The dotted vertical line corresponds to the
FP rate of 5%, while the dashed and dotted horizontal line corresponds to the
TP rate of 95%.
### V-C Effect of a Systematic Shift in Detector Response
Systematic shifts in the detector response could cause upward or downward
shifts in the measured antineutrino count rate. In that case, even if fuel has
not been removed, the detector measurements may deviate significantly from the
predicted baseline evolution. In this section we analyze the consequences of
such shifts for the hypothesis testing procedure.
The absolute count rate of reactor antineutrinos has been measured with 3$\%$
systematic uncertainty [14]. Antineutrino count rate measurements made
relative to an initial value have a considerably smaller systematic error, of
less than 1$\%$ [15], since fixed systematic errors present in the absolute
measurement are cancelled by subtraction. As we will show, a hypothesis test
that uses antineutrino count rate trajectories made relative to a premeasured
value are much less sensitive to systematic detector shifts than a test on
data not referred to an initial value.
In an actual safeguards deployment, a detector bias would become evident by a
comparison of measured and predicted antineutrino counts integrated over a few
weeks. For example, with measured antineutrino count rates of 2000 counts per
day, 20 days of data acquisition would suffice to reduce the statistical error
to 0.5%, small enough to measure a few percent difference between predicted
and actual rates. In the context of the hypothesis test considered here, such
a shift can be mistakenly interpreted as evidence for anomalous reactor
operations, or correctly as a previously undiscovered systematic shift in the
detector response, not attributable to the anomaly.
We examined the impact of a systematic shift incorrectly interpreted as
evidence for anomalous reactor operations. We adjusted both the baseline and
anomalous measured count rate evolutions by 1%. (We report only a downward
shift result, which is conservatively worse than the impact of the upward
shift for the scenarios considered here). A 1% absolute systematic error is
smaller than that typically obtained in reactor antineutrino experiments, but
is already large enough to illustrate the strong impact of detector bias. Fig.
3 shows the resulting shifted baseline and anomalous count rate evolutions, as
well as the original unshifted evolutions. As can be seen from this plot, the
shifted baseline evolution is now further from the reference (original)
baseline than the shifted anomalous evolution. As a result, the performance of
a test deteriorates dramatically. At 5% FP rate, the TP rate is 0.4%, compared
to 95% in the absence of a detector bias. The test attains the desired 95% TP
rate only at the FP rate of practically 100%. Thus, even a small bias in the
detector response severely weakens the statistical power of the hypothesis
test if an absolute comparison of count rate trajectories is made.
The negative impact on the test of an absolute systematic shift in detector
response can be mitigated in two ways: either by using relative count rate
data, referred to a corrected value measured at startup, or by comparison with
a template from a previous cycle known by other means to be standard.
For the first case, we investigated the TP rate of the hypothesis test
assuming the measured antineutrino count rates are corrected by the difference
between the predicted and measured values averaged over the first 20 days of
the cycle. Thus, agreement of predicted and measured count rates at beginning
of cycle is enforced before the testing procedure is applied. (This is
equivalent to making an initial assumption that no anomaly is present. When
the shifted antineutrino counts are corrected in this way, the TP rate is 12%
at 5% FP rate with 90 days of data acquisition, which is a significant
improvement over 0.4% TP rate reported above in the case of shifted
measurements not referred to an initial value (referred to as “uncorrected” in
Table II). When the acquisition period is increased to 180 days, this
correction yields a TP rate of 62% at the 5% FP rate. For 250 days, the TP
rate is 96%, which is only slightly below the rate in the absence of a shift
for the same acquisition period.
While a measurement relative to startup improves the power of the test, the
most favorable approach is to use a measured template for the antineutrino
count rate, derived from a previous cycle known to be standard by other means.
By definition, this removes any systematic detector bias, since the relation
between the baseline fuel evolution and the measured antineutrino count rate
has been empirically established. This case reverts to our earlier result for
high statistics acquisition - 95% TP rate and a 5% FP rate with 90 days of
data acquisition. The approach of using a predefined template from a previous
and well known fuel cycle has a further advantage that it no longer depends on
a reactor simulation and its associated errors. This appears to be the most
effective method for identifying anomalous fuel loadings, so long as
systematic errors in antineutrino detector predicted and measured response
remain at the level of a few percent.
Figure 3: Baseline (solid green), anomalous (solid red), shifted baseline
(dashed green) and shifted anomalous (dashed red) scenario evolutions of
antineutrino daily count rates versus time (in days), as simulated in ORIGEN.
Figure 4: ROC curves for the test applied to the original unshifted (purple)
and detector-shifted (orange) evolutions. The time period shown here is days
0-90 in the fuel cycle. The dotted vertical line corresponds to the FP rate of
5%, while the dashed and dotted horizontal line corresponds to the TP rate of
95%.
### V-D Effect of Operator Malfeasance
Equation (1) shows that both thermal power and fissile isotopic content can be
altered to change the antineutrino count rate. Thus, in an attempt to conceal
the removal of plutonium in the present example, the reactor operator could
report a higher thermal power value than the true operating power. This input
information would cause the simulation to incorrectly predict a systematic
upward shift in the baseline evolution.
To assess the impact of a misreported power history, we considered the effect
of a 1% upward systematic shift of the baseline evolution that was originally
obtained from the ORIGEN simulation (solid green curve in Fig. 1). Fig. 1
shows the resulting shifted baseline evolution (dashed green curve). As can be
seen from the plot, this evolution is much less distinguishable from the
anomalous evolution than the true baseline evolution, so that this shift can
be expected to deteriorate the test’s performance.
Fig. 5 confirms this loss of sensitivity. Both ROC curves shown in this plot
were obtained from the test using count rate data for days 0–90 in the cycle,
assuming high counting statistics. For this particular time period, the TP
rate for the test applied to the shifted baseline was as low as one-ninth of
that observed using the original baseline. For example, at the FP rate of 5%,
the TP rate of the former is 95%, while that of the latter is 23%. In Section
VII, we discuss operational and experimental means to address the problem of
deliberate misreporting.
It should be noted that longer duration of data acquisition reduces the impact
of malfeasance. As Table II shows, with high count rate data, the TP rates at
5% FP rate are respectively 56% and 99.99% for 250 and 500 days of data
acquisition. The complete ROC curves for the various acquisition times in the
case of the shifted baseline are shown in Fig. 7. Hence, even in the presence
of malfeasance, the anomaly can be detected with high sensitivity if one
acquires antineutrino data over the entire cycle.
Figure 5: ROC curves for the test applied to the original unshifted (purple)
and operator-shifted (orange) baseline evolutions. The time period shown here
is days 0–90 in the fuel cycle. The dotted vertical line corresponds to the FP
rate of 5%, while the dashed and dotted horizontal line corresponds to the TP
rate of 95%.
### V-E Effect of the Starting Point and Duration of the Data Acquisition
Period
Naturally, the estimates of the evolution coefficients
$\hat{\gamma}^{(M)}_{i}$ and the test performance both improve as data are
acquired for longer periods. In our ROC curve simulation, we considered the
following four durations: days 0–500 (roughly full cycle length), days 0–250
(half cycle length), days 0–90 and days 0–30 in the cycle. Fig. 6 shows the
ROC curves for these four duration periods, assuming high count rates. At the
FP rate of 5%, the TP rate is 99.99% for 500 days versus 99%, 95% and 58% for
250, 90 and 30 days, respectively.
Moreover, as Fig. 1 reveals, when the baseline is shifted due to incorrect
input information, in addition to the duration of data acquisition, the
location of the time window in the cycle during which the data are acquired
will also affect the performance of the test. For example, the shifted
baseline evolution is less distinguishable from the anomalous evolution in the
first 250 days of the cycle than in the last 250 days. The same is true when
comparing the first 90 days to the last 90 days of the cycle. Therefore, we
also compared the performance of the test for the shifted baseline using high
count rate data from the first 90, last 90 (days 411-500), first 250, last 250
(days 251–500), and all 500 days of the cycle.
Fig. 7 shows the ROC curves for these five periods for the case of the shifted
baseline. As was noted earlier, as the number of days goes down, the test
performance degrades. Moreover, the test applied to the count rate data for
the last 250 days performs better than for the first 250 days because the
shifted baseline and the anomalous evolutions are further apart at later times
in the fuel cycle. The same is true when comparing the performance for the
first 90 days to the last 90 days. However, the test is less sensitive to the
starting point than to the duration of the data acquisition period.
These various effects are summarized in Table II. The effect of duration and
period was very similar for the low count rates, so these results are not
included in the table.
Figure 6: ROC curves for the test using high count rates acquired over the
full cycle, or 500 days (turquoise), days 0–250 of the cycle (blue), days 0–90
of the cycle (orange), and days 0–30 of the cycle (green). The dotted vertical
line corresponds to the FP rate of 5%, while the dashed and dotted horizontal
line corresponds to the TP rate of 95%. Figure 7: ROC curves for the test
applied to the operator-shifted baseline and high count rates acquired over
the full cycle, or 500 days (turquoise), first 250 days of the cycle (blue),
last 250 days of the cycle (red), first 90 days of the cycle (orange), and
last 90 days of the cycle (green). The dotted vertical line corresponds to the
FP rate of 5%, while the dashed and dotted horizontal line corresponds to the
TP rate of 95%.
## VI Impact on Detector Design and Operation
The test performance described above can be used to guide the design of future
safeguards antineutrino detectors. For a given anomalous scenario and desired
true and false positive rate, a minimum antineutrino count rate requirement
can be established. Within practical limits set by the reactor site, detector
cost and complexity, a desired event rate may be achieved by adjusting the
detector standoff distance, size or intrinsic efficiency.
As discussed earlier, the antineutrino rate in the SONGS1 experiment [2] was
approximately 360 counts per day at beginning of cycle after subtraction of
reactor-off backgrounds. According to the ROC curve in Fig. 2, this
antineutrino count rate gives a 34% TP rate for a 5% FP rate with a 90-day
acquisition period. We assume that an acceptable test for IAEA safeguards or a
similar monitoring regime will require at least 95% TP rate at the 5% FP rate.
In the previous sections, we have shown that for the anomalous scenario we
considered, a 2000 count per day net antineutrino event rate is necessary and
sufficient to achieve this TP/FP rate combination.
The SONGS1 detector was located 24.5 meters from the reactor core, with a 0.48
ton target mass, and 11% intrinsic detection efficiency [5]. An increase in
event rate compared to SONGS1 could be accomplished by a combination of
reduced standoff distance, increased detector target mass and/or increased
intrinsic detection efficiency. For example, at 24.5 meter standoff, a one ton
detector with 30% intrinsic efficiency, or a two ton detector with 15%
intrinsic efficiency would reach the 2000 count rate level and thus, the
desired 95%/5% TP/FP rates. Alternatively, a one ton, 11% efficient detector
at 15 meter standoff would reach the same TP/FP rate combination.
As shown in Table III, previous antineutrino detectors had masses and
efficiencies required to achieve the desired TP/FP rate performance. The
series of deployments at the Rovno reactor complex in the Ukraine is of
particular interest since the efficiencies are high, while the overburden and
other conditions are similar to those that would be encountered in many
reactors under the IAEA safeguards. By contrast, the high efficiency of the
CHOOZ detector reflects the state of the art for this class of detectors, but
is achieved in part through significantly greater overburden and reduced
ambient radioactivity compared to the other experiments, so such a device is
unlikely to be practical in a safeguards context.
TABLE III: Power, mass, standoff distance, efficiency, and signal-to-background ratios of some previous antineutrino experiments. Experiment | Power | Mass | Distance | Efficiency | Signal/Bkgd
---|---|---|---|---|---
| (GW) | (ton) | (m) | (%) | Cts/Day
Rovno 1 [6] | 1.375 | 0̃.5 | 18 | 20 | 909/149
Rovno 2 [16] | 1.375 | 0̃.2 | 18 | 30 | 267/94
CHOOZ [17] | 4.4 | 5.0 | 1000 | 69.8 | 24/1.2
Palo Verde [18] | 11.6 | 11.3 | 800 | 10 | 200/300
SONGS1 [2] | 3.4 | 0.64 | 24.5 | 11 | 564/105
Bugey [19] | 3.4 | 0.64 | 24.5 | 10 | 62/2.5
## VII Conclusions and Possible Future Work
This paper introduced a test procedure that determines whether a given
antineutrino count rate evolution significantly deviates from that of the
baseline. The procedure uses a quadratic model for the antineutrino count rate
as a function of time since the beginning of the fuel cycle. However, the
procedure can be adapted to a much wider class of models. The procedure
involves least squares estimation of the parameters in the quadratic model for
the evolution in question and a multiple hypothesis testing procedure, known
as False Discovery Rate (FDR), to determine whether at least one of the
estimated parameters is significantly different from its baseline counterpart.
The anomalous operations identified in this paper do not constitute a
diversion scenario per se, since we have not specified the ultimate fate of
the removed fuel. Instead, we have estimated the sensitivity of antineutrino
rate measurements to changes in typical civil power reactor fuel loadings. An
important future exercise, best conducted by IAEA safeguards experts, is a
fuller analysis of the reactor safeguards implications of this novel bulk
accountancy method.
While the specific performance of the test will depend on the scenario, this
work has identified the factors that most influence our ability to detect
anomalous fuel loadings generally. Among the factors that we considered,
counting statistics, the presence of detector bias, and introduction of a
systematic shift due to operator malfeasance had the most dramatic impact on
the test performance. High counting statistics collected over longer periods
of time in the absence of a deliberate shift in the baseline or detector bias
yield the best performance and attain the target 95% TP rate at the 5% FP
rate. We also found that the effect of a systematic error in detector bias
response can be substantially reduced by an initial correction of the
predicted to the observed count rates, or most effectively by an empirical
calibration of detector response using antineutrino count rate data from a
previous fuel cycle. The latter approach has the further advantage of
lessening the dependence of the method on a reactor simulation. Changes in the
starting point of data acquisition had a smaller impact on the performance.
Past experience has demonstrated that increasing the antineutrino count rate
through efficiency or mass increases is achievable, so that our target 95% TP
/ 5% FP rate combination can be attained with practical detectors. More
problematic in a safeguards context is the issue of deliberate misreporting of
power levels on the part of the operator that would undermine the statistical
power of our test. While this is a serious concern, we note that the
operator’s misreporting must be fully consistent with the antineutrino data,
which are independently acquired by and remain under the control of the
safeguards inspector. This independently acquired information places an
important additional constraint on the operator compared to current practice,
in which declarations, along with item accountancy, are the primary sources of
quantitative information about the reactor thermal power and fuel loading.
Moreover, the misrepresentation must be tuned to the particular anomalous
operational state chosen by the operator. If different amounts or types of
fissile material are removed, the hypothesis test may still detect a
significant departure from the baseline. To further examine the robustness of
this method, it is necessary to investigate a wider class of anomalous
scenarios, varying both fuel and reactor type.
As described in [20], a direct measurement of the antineutrino spectrum would
provide sufficient information to simultaneously constrain both power and
fissile isotopic content. This would severely undermine or even eliminate the
benefit to the operator of misreporting the thermal power. However, since the
antineutrino rate per energy bin will be necessarily reduced, the statistical
power of the test may be compromised, or, alternatively, a larger detector may
be required than is the case for a pure rate measurement. In future work, we
will apply a hypothesis testing procedure on a spectrally resolved
antineutrino measurement, including realistic statistical and systematic
uncertainties, to quantify any additional sensitivity inherent in the spectral
analysis.
Finally, as noted earlier, we used an ORIGEN simulation of the SONGS Unit 2
reactor core. Assemblies were assumed to have no spatial extent: the only
spatial information in our calculation was the variation in distance of each
pointlike assembly from the detector. A full three-dimensional treatment of
the assemblies would allow inclusion of effects, such as the variation of the
centroid of fission over the cycle.
## VIII Acknowledgments
We thank the DOE Office of Nonproliferation Research and Engineering for their
sustained support of this project. We also thank Nathaniel Bowden and Scott
Kiff for insightful comments on an earlier version of this manuscript.
Finally, we express our gratitude to the management and staff of the San
Onofre Nuclear Generating Station, for allowing us to deploy and take data
with our prototype safeguards antineutrino detectors.
## References
* [1] IAEA, “http://www.iaea.org/Publications/Factsheets/English/sg_overview.html.”
* [2] A. Bernstein, Y. Wang, G. Gratta, and T. West, “Nuclear reactor safeguards and monitoring with antineutrino detectors,” _J. Appl. Phys._ , vol. 91, p. 4672, 2002.
* [3] N. S. Bowden, A. Bernstein, M. Allen, J. S. Brennan, M. Cunningham, J. K. Estrada, C. M. R. Greaves, C. Hagmann, J. Lund, W. Mengesha, T. D. Weinbeck, and C. D. Winant, “Experimental results from an antineutrino detector for cooperative monitoring of nuclear reactors,” _NIM A_ , vol. 572, pp. 985–998, 2007.
* [4] A. Bernstein, N. S. Bowden, A. Misner, and T. Palmer, “Monitoring the thermal power of nuclear reactors with a prototype cubic meter antineutrino detector,” _J. Appl. Phys._ , vol. 103, p. 074905, 2008.
* [5] N. Bowden, A. Bernstein, S. Dazeley, R. Svoboda, A. Misner, and T. Palmer, “Observation of the isotopic evolution of pressurized water reactor fuel using an antineutrino detector,” _J. Appl. Phys._ , vol. 105, p. 064902, 2009\.
* [6] Y. Klimov, V. I. Kopeikin, L. A. Mika lyan, K. V. Ozerov, and V. V. Sinev, “Neutrino method remote measurement of reactor power and power output,” _Atomnaya Energiya_ , vol. 76, p. 130, 1994.
* [7] IAEA, “http://www.iaea.org/Publications/Booklets/TeamingInspectors/index.html.”
* [8] N.Harms and P.Rodriguez, “Safeguards at light-water reactors: Current practices, future directions,” _IAEA Bulletin_ , vol. 38, pp. 16–19, 1996\.
* [9] _SCALE: A Modular Code System for PerformingStandardized Computer Analyses for LicensingEvaluations_ , Version 5 ed., 2006, available from Radiation Safety Information Computational Center at ORNL as CCC-732.
* [10] A. Misner, “Simulated antineutrino signatures of nuclear reactors for nonproliferation applications,” Ph.D. dissertation, Oregon State University, 2008, http://ir.library.oregonstate.edu/jspui/handle/1957/8495.
* [11] O. Hermann, “Benchmark of scale (sas2h) isotopic predictions of depletion analyses for san onofre pwr mox fuel,” Lockheed Martin, Oak Ridge National Laboratory, Tech. Rep. ORNL/TM-1999/326, 2000.
* [12] J. Neter, R. Berger, and M. Kutner, _Applied Linear Statistical Models_. Homewood, Illinois: Irwin, 1990\.
* [13] Y. Benjamini and Y. Hochberg, “Controlling the false discovery rate: A practical and powerful approach to multiple testing,” _Journal of Royal Statistical Society, series B_ , vol. 57, pp. 289–300, 1995.
* [14] P. V. C. Bemporad G. Gratta, “Reactor-based neutrino oscillation experiments,” _Rev. Mod. Phys._ , vol. 74, pp. 297–328, 2002.
* [15] T. D. C. collaboration, “Double chooz, a search for the neutrino mixing angle theta-13,” _arXiv:hep-ex/0606025v4_ , 2006.
* [16] V. A. Korovkin, S. A. Kodanev, A. D. Yarichin, A. A. Borovoi, V. I. Kopeikin, L. A. Mikae’lyan, and V. D. Sidorenko, “Measurement of burnup of nuclear fuel in a reactor by neutrino emission,” _??_ , p. ??, 1984.
* [17] M. A. et al. (CHOOZ), “Search for neutrino oscillations on a long base-line at the chooz nuclear power station,” _Eur. Phys. J._ , vol. C27, p. 331, 2003\.
* [18] F. B. et al., “Final results from the palo verde neutrino oscillation experiment,” _Phys. Rev. D_ , vol. 64, p. 112001, 2001.
* [19] B. A. et al., “Search for neutrino oscillations at 15, 40, and 95 meters from a nuclear power reactor at bugey,” _Nucl. Phys. B_ , pp. 503–532, 1995.
* [20] P. Huber and T. Schwetz, “Precision spectroscopy with reactor antineutrinos,” _Phys. Rev. D_ , vol. 70, p. 053011, 2004.
| Vera Bulaevskaya and Adam Bernstein are with the Lawrence Livermore
National Laboratory in Livermore, CA.
---|---
|
arxiv-papers
| 2010-09-11T00:34:21 |
2024-09-04T02:49:12.817082
|
{
"license": "Public Domain",
"authors": "Vera Bulaevskaya, Adam Bernstein",
"submitter": "Adam Bernstein",
"url": "https://arxiv.org/abs/1009.2123"
}
|
1009.2171
|
# Subgroup S–commutativity degree of finite groups
Daniele Ettore Otera Département de Mathématique,Université Paris-Sud 11
Batiment 425, Faculté de Science d’Orsay
F-91405, Orsay Cedex, France daniele.otera@math-psud.fr and Francesco G.
Russo Department of Mathematics, University of Palermo
Via Archirafi 34, 90123, Palermo, Italy. francescog.russo@yahoo.com
###### Abstract.
The so–called subgroup commutativity degree $sd(G)$ of a finite group $G$ is
the number of permuting subgroups $(H,K)\in\mathrm{L}(G)\times\mathrm{L}(G)$,
where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by
$|\mathrm{L}(G)|^{2}$. It allows us to measure how $G$ is far from the
celebrated classification of quasihamiltonian groups of K. Iwasawa. Here we
generalize $sd(G)$, looking at suitable sublattices of $\mathrm{L}(G)$, and
show some new lower bounds.
###### Key words and phrases:
Subgroup commutativity degree, sublattices, abelian groups.
###### 2010 Mathematics Subject Classification:
Primary: 06B23; Secondary: 20D60.
## 1\. Introduction
All groups in the present paper are supposed to be finite. Given two subgroups
$H$ and $K$ of a group $G$, the product $HK=\\{hk\ |\ h\in H,k\in K\\}$ is not
always a subgroup of $G$. $H$ and $K$ permute if $HK=KH$, or equivalently, if
$HK$ is a subgroup of $G$. $H$ is said to be permutable (or quasinormal) in
$G$, if it permutes with every subgroup of $G$. It is possible to strengthen
this notion in various ways. $H$ is S–permutable (or _S–quasinormal_) in $G$,
if $H$ permutes with all Sylow subgroups of $G$ (for all primes in the set
$\pi(G)$ of the prime divisors of $|G|$). Historically, O. Kegel introduced
S–permutable subgroups in 1962, generalizing a well–known result of O. Ore of
1939, who proved that permutable subgroups are subnormal (see [9, 15] for
details). Roughly speaking, this notion deals with subgroups which are
permutable with maximal subgroups. Several authors investigated the topic in
the successive years and we mention [1, 2, 14, 15] for our aims.
The subgroup lattice $\mathrm{L}(G)$ of a group $G$ is the set of all
subgroups of $G$ and is a complete bounded lattice with respect to the set
inclusion, having initial element the trivial subgroup $\\{1\\}$ and final
element $G$ itself (see [8, 15]). Its binary operations $\wedge,\vee$ are
defined by $X\wedge Y=X\cap Y$, $X\vee Y=\langle X\cup Y\rangle$, for all
$X,Y\in\mathrm{L}(G)$. Furthermore, $\mathrm{L}(G)$ is modular, if all the
subgroups of $G$ satisfy the modular law. $G$ is modular, if $\mathrm{L}(G)$
is modular (see [15, Section 2.1]). This notion is important, because of the
following concept. A group $G$ is quasihamiltonian, if all its subgroups are
permutable. By a result of K. Iwasawa [15, Theorem 2.4.14], quasihamiltonian
groups are classified, but, at the same time, these groups are characterized
to be nilpotent and modular (see [15, Exercise 3, p.87]).
Now we recall some terminology from [14], which will be useful in the rest of
the paper. Any non–empty set of subgroups $\mathrm{S}(G)$ of $G$ may be always
regarded as a sublattice of $\mathrm{L}(G)$ having initial element
${\underset{S\in\mathrm{S}(G)}{\bigwedge}}S$ and final element
${\underset{S\in\mathrm{S}(G)}{\bigvee}}S$. The symbol $\mathrm{S}^{\perp}(G)$
denotes the set of all subgroups $H$ of $G$ which are permutable with all
$S\in\mathrm{S}(G)$ and it is easy to check that $\mathrm{S}^{\perp}(G)$ is a
sublattice of $\mathrm{L}(G)$ (see [14, Section 1]). There is a wide
literature when we choose $\mathrm{S}(G)$ to be equal to the sublattice
$\mathrm{M}(G)$ of all maximal subgroups of $G$, or to the sublattice
$\mathrm{sn}(G)$ of all subnormal subgroups of $G$, or also to the sublattice
$\mathrm{n}(G)$ of all normal subgroups of $G$. Consequently,
$\mathrm{L}^{\perp}(G)$ is the sublattice of all permutable subgroups of $G$,
$\mathrm{M}^{\perp}(G)$ that of the subgroups permutable with all maximal
subgroups of $G$ and so on for $\mathrm{sn}^{\perp}(G)$ and
$\mathrm{n}^{\perp}(G)=\mathrm{L}(G)$. Immediately, the role of the operator
$\perp$ appears to be very intriguing for the structure of $G$ and several
authors investigated this aspect. For instance, $G$ is quasihamiltonian if and
only if $\mathrm{L}(G)=\mathrm{L}^{\perp}(G)$.
In Section 2 we will describe a notion of probability on $\mathrm{L}(G)$,
beginning from groups in which the subgroups in $\mathrm{sn}(G)$ permutes with
those in $\mathrm{M}(G)$. The generality of the methods (we follow [3, 4, 6,
7, 10, 11, 12, 13, 17]) may be translated in terms of arbitrary sublattices,
satisfying a prescribed restriction. Section 3 shows some consequences on the
size of $|\mathrm{L}(G)|$.
## 2\. Measure theory on subgroup lattices
The following notion has analogies with [6, Definitions 2.1,3.1,4.1] and [12,
Equation 1.1] and will be treated as in [3, 4, 6, 7, 10, 11, 12, 13, 17].
###### Definition 2.1.
For a group $G$,
(2.1) $spd(G)=\frac{|\\{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)\ |\
XY=YX\\}|}{|\mathrm{sn}(G)|\ |\mathrm{M}(G)|},$
is the subgroup S–commutativity degree of $G$.
$0<spd(G)\leq 1$ denotes the probability that a randomly picked pair
$(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)$ is permuting, that is, $XY=YX$.
(2.1) may be rewritten, introducing the function
$\chi:\mathrm{sn}(G)\times\mathrm{M}(G)\rightarrow\\{0,1\\}$ defined by
(2.2) $\chi(X,Y)=\left\\{\begin{array}[]{lcl}1,&&\mathrm{if}\ XY=YX,\\\
0,&&\mathrm{if}\ XY\not=YX,\end{array}\right.$
in the following form
(2.3) $spd(G)=\frac{1}{|\mathrm{sn}(G)|\
|\mathrm{M}(G)|}{\underset{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)}{\sum}}\chi(X,Y).$
In Definition 2.1 and (2.3), we may replace
$\mathrm{sn}(G)\times\mathrm{M}(G)$ with $\mathrm{S}(G)\times\mathrm{T}(G)$,
where $\mathrm{S}(G)$ and $\mathrm{T}(G)$ are two arbitrary sublattices of
$\mathrm{L}(G)$. We have chosen $\mathrm{sn}(G)\times\mathrm{M}(G)$, because
[1, 2] describe the structure of the groups in which the subnormal subgroups
permute with all Sylow subgroups (called $PST$–groups). If $\mathrm{Syl}(G)$
is the set of all Sylow subgroups of $G$,
$\mathrm{Syl}(G)\subseteq\mathrm{M}(G)$ and this means that we have already a
classification for a group $G$ such that
$\mathrm{sn}(G)\subseteq\mathrm{Syl}(G)^{\perp}$.
(2.3) allows us to to treat the problem from the point of view of the measure
theory on groups. A computational advantage may be found in a formula for
$spd(G_{1}\times G_{2})$, where $G_{1}$ and $G_{2}$ are two given groups.
###### Corollary 2.2.
Let $G_{i}$ be a family of groups of coprime orders for $i=1,2,\ldots,k$. Then
$spd(G_{1}\times G_{2}\times\ldots\times G_{k})=spd(G_{1})\ spd(G_{2})\
\ldots\ spd(G_{k}).$
The techniques of proof are straightforward applications of (2.3) and the
details are omitted. However, it is good to note that Corollary 2.2 shows the
stability with respect to forming direct products of $spd(G)$: this fact was
proved in [3, 4, 6, 7, 10, 12, 13, 17] in different contexts. Another basic
property is to relate $spd(G)$ to quotients and subgroups of $G$.
Let $G=NH$ for a normal subgroup $N$ of $G$ and a subgroup $H$ of $G$
isomorphic to $G/N$ (briefly, $H\simeq G/N$). In general, it is easy to check
that $\mathrm{sn}(G/N)$ is lattice isomorphic to $\mathrm{sn}(H)$ (briefly,
$\mathrm{sn}(G/N)\sim\mathrm{sn}(H)$) and that
$\mathrm{M}(G/N)\sim\mathrm{M}(H)$. We will concentrate on some special
classes of groups, satisfying
(2.4)
$\Big{(}\mathrm{sn}(G/N)\times\mathrm{M}(G/N)\Big{)}\sim\Big{(}\mathrm{sn}(H)\times\mathrm{M}(H)\Big{)}\subseteq\mathrm{sn}(G)\times\mathrm{M}(G)$
(2.5)
$\mathrm{sn}(N)\times\mathrm{M}(N)\subseteq\mathrm{sn}(G)\times\mathrm{M}(G).$
(2.4)–(2.5), jointly with (2.3), allow us to conclude
(2.6)
$\sum_{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)}\chi(X,Y)\geq\sum_{(X,Y)\in\mathrm{sn}(N)\times\mathrm{M}(N)}\chi(X,Y);$
(2.7)
$\sum_{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)}\geq{\underset{(X/N,Y/N)\in\mathrm{sn}(G/N)\times\mathrm{M}(G/N)}{\sum}}\chi(X/N,Y/N)=\sum_{(Z,T)\in\mathrm{sn}(H)\times\mathrm{M}(H)}\chi(Z,T)$
and consequently
(2.8) $2|\mathrm{sn}(G)|\ |\mathrm{M}(G)|\
spd(G)\geq\sum_{(X,Y)\in\mathrm{sn}(N)\times\mathrm{M}(N)}\chi(X,Y)+\sum_{(Z,T)\in\mathrm{sn}(H)\times\mathrm{M}(H)}\chi(Z,T).$
$=|\mathrm{sn}(N)|\ |\mathrm{M}(N)|\ spd(N)+|\mathrm{sn}(G/N)|\
|\mathrm{M}(G/N)|\ spd(G/N).$
Similar techniques have been used by Tǎrnǎuceanu [17] in order to study the
subgroup commutativity degree
(2.9) $sd(G)=\frac{|\\{(X,Y)\in\mathrm{L}(G)^{2}\ |\
XY=YX\\}|}{|\mathrm{L}(G)|^{2}}=\frac{1}{|\mathrm{L}(G)|^{2}}{\underset{(X,Y)\in\mathrm{L}(G)^{2}}{\sum}}\chi(X,Y).$
[17] can be seen as a natural extension, to the context of the lattice theory,
of the concept of commutativity degree
(2.10) $d(G)=\frac{|\\{(x,y)\in G^{2}\ |\
xy=yx\\}|}{|G|^{2}}=\frac{1}{|G|^{2}}{\underset{x\in G}{\sum}}|C_{G}(x)|,$
where $C_{G}(x)=\\{g\in G\ |\ gx=xg\\}$. There are several contributions on
$d(G)$ in [3, 4, 6, 7, 10, 11, 12, 13]. The main strategy of investigation is
to begin with the case of equality at 1 and then describe the situation, when
we leave this extremal case. Upper and lower bounds will measure the distance
from known classes of groups. For instance, $d(G)=1$ if and only if $G$ is
abelian; $sd(G)=1$ if and only if $\mathrm{L}(G)=\mathrm{L}(G)^{\perp}$.
Therefore the next are mile stones for the rest of the paper.
###### Corollary 2.3.
In a group $G$ we have $spd(G)=1$ if and only if
$\mathrm{sn}(G)\subseteq\mathrm{M}^{\perp}(G)$ or
$\mathrm{M}(G)\subseteq\mathrm{sn}^{\perp}(G)$.
###### Proof.
It follows from the above considerations. ∎
###### Corollary 2.4.
If $G$ is a nilpotent group, then $spd(G)=1$.
###### Proof.
Application of Corollary 2.3, noting that
$\mathrm{M}(G)\subseteq\mathrm{n}(G)\subseteq\mathrm{sn}^{\perp}(G)$. ∎
###### Corollary 2.5.
In a group $G$ we have $\frac{|\mathrm{sn}(G)|\
|\mathrm{M}(G)|}{|\mathrm{L}(G)|^{2}}\ spd(G)\leq\ sd(G)$ and the equality
holds if and only if $\mathrm{sn}(G)=\mathrm{M}(G)=\mathrm{L}(G)$.
###### Proof.
Since $\mathrm{sn}(G)\times\mathrm{M}(G)\subseteq\mathrm{L}(G)^{2}$,
$\\{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)\ |\
XY=YX\\}\subseteq\\{(X,Y)\in\mathrm{L}(G)^{2}\ |\ XY=YX\\}$ and then
$|\mathrm{sn}(G)|\ |\mathrm{M}(G)|\
spd(G)=|\\{(X,Y)\in\mathrm{sn}(G)\times\mathrm{M}(G)\ |\ XY=YX\\}|$
$\leq|\\{(X,Y)\in\mathrm{L}(G)^{2}\ |\ XY=YX\\}|=|\mathrm{L}(G)|^{2}\ sd(G)$
from which the inequality follows. The rest is clear. ∎
Corollary 2.4 clarifies the situation for nilpotent groups. Then we proceed to
study solvable groups. Unfortunately, these cannot be described as in [17,
Proposition 2.4]: Different techniques are necessary. We recall that an
abelian group $A$ of order $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{m}^{n_{m}}$,
for suitable powers of $p_{1},p_{2},\ldots,p_{m}\in\pi(A)$, has a canonical
decomposition of the form $A\simeq A_{1}\times A_{2}\times\ldots A_{m},$ where
$n_{1},\ldots,n_{m}$ are positive integers and $A_{1},A_{2}\ldots,A_{m}$ are
the primary factors. It is well–known that
$|\mathrm{L}(A)|=|\mathrm{L}(A_{1})|\cdot|\mathrm{L}(A_{2})|\cdot\ldots\cdot|\mathrm{L}(A_{m})|.$
In case $p=p_{1}=p_{2}=\ldots=p_{m}$ [18, Proposition 3.2] shows that the
number of maximal subgroups of the elementary abelian $p$–group
$\mathbb{Z}_{p^{\alpha_{1}}}\times\mathbb{Z}_{p^{\alpha_{2}}}\times\ldots\times\mathbb{Z}_{p^{\alpha_{k}}}$
is equal to $\frac{p^{k}-1}{p-1}$, for suitable integers
$1\leq\alpha_{1}\leq\alpha_{2}\leq\ldots\leq\alpha_{k}$ and $k\geq 1$.
###### Lemma 2.6.
Let $N=\mathbb{Z}_{p^{\alpha_{1}}}\times\mathbb{Z}_{p^{\alpha_{2}}}$ be a
non–trivial normal abelian subgroup of $G$ with $0\leq\alpha_{1}+\alpha_{2}$
and $1\leq\alpha_{1}\leq\alpha_{2}$ such that $G/N$ is of prime order and
(2.4)–(2.5) are satisfied. Then
$spd(G)\geq\frac{f(p,\alpha_{1},\alpha_{2})}{2\ |\mathrm{sn}(G)|\
|\mathrm{M}(G)|},$
where
$f(p,\alpha_{1},\alpha_{2})=\frac{1}{p^{2}-2p+1}\Big{(}(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+3}+2p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}-(\alpha_{1}+\alpha_{2}-1)p^{2}-(\alpha_{1}+\alpha_{2}+11)p+(\alpha_{1}+\alpha_{2}+5)\Big{)}$
is a polynomial function depending only on $N$.
###### Proof.
We note that $G=HN$, where $G/N\simeq H$ is of prime order, so that it is
meaningful to formulate the conditions in (2.4) and (2.5), requiring that they
are satisfied. From (2.8),
(2.11) $spd(G)\geq\frac{|\mathrm{sn}(N)|\
|\mathrm{M}(N)|spd(N)+|\mathrm{sn}(G/N)|\ |\mathrm{M}(G/N)|spd(G/N)}{2\
|\mathrm{sn}(G)|\ |\mathrm{M}(G)|}$
$|\mathrm{sn}(G/N)|=|\mathrm{M}(G/N)|=2$, by a counting argument, and
$spd(N)=spd(G/N)=1$, by Corollary 2.4, then
(2.12) $=\frac{|\mathrm{sn}(N)|\ |\mathrm{M}(N)|}{2\ |\mathrm{sn}(G)|\
|\mathrm{M}(G)|}+\frac{4}{2\ |\mathrm{sn}(G)|\ |\mathrm{M}(G)|}=\frac{1}{2\
|\mathrm{sn}(G)|\ |\mathrm{M}(G)|}\left(|\mathrm{sn}(N)|\
|\mathrm{M}(N)|+4\right)$
[18, Theorem 3.3] implies
$|\mathrm{sn}(N)|=|\mathrm{L}(N)|=\frac{1}{(p-1)^{2}}[(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}-(\alpha_{1}+\alpha_{2}+3)p+(\alpha_{1}+\alpha_{2}+1)]$,
and, as noted above, $|\mathrm{M}(N)|=\frac{p^{2}-1}{p-1}=p+1$, hence
(2.13) $=\frac{1}{2\ |\mathrm{sn}(G)|\ |\mathrm{M}(G)|}\ \cdot\
\Big{(}\frac{p+1}{(p-1)^{2}}\Big{(}(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}$
$-(\alpha_{1}+\alpha_{2}+3)p+(\alpha_{1}+\alpha_{2}+1)\Big{)}+4\Big{)}$
in order to write better the above expression we introduce the coefficients
$C_{1}=\alpha_{2}-\alpha_{1}+1;\ C_{2}=\alpha_{2}-\alpha_{1}-1;\
C_{3}=\alpha_{1}+\alpha_{2}+3;C_{4}=\alpha_{1}+\alpha_{2}+1$
and then we get
$=\frac{1}{2\ |\mathrm{sn}(G)|\ |\mathrm{M}(G)|}\Big{(}\frac{1}{(p-1)^{2}}\
(C_{1}p^{\alpha_{1}+3}+(C_{1}-C_{2})p^{\alpha_{1}+2}-C_{2}p^{\alpha_{1}+1}$
$+(4-C_{3})p^{2}-(8+C_{3})p+(4+C_{4})\Big{)}.$
Developping the computations in the brackets, we get the polynomial
$f(p,\alpha_{1},\alpha_{2})$. ∎
Lemma 2.6 may be adapted to $sd(G)$ in the following way.
###### Lemma 2.7.
Let $N=\mathbb{Z}_{p^{\alpha_{1}}}\times\mathbb{Z}_{p^{\alpha_{2}}}$ be a
non–trivial normal subgroup of $G$ with $0\leq\alpha_{1}+\alpha_{2}$ and
$1\leq\alpha_{1}\leq\alpha_{2}$ such that $G/N$ is of prime order. Then
$sd(G)\geq\frac{g(p,\alpha_{1},\alpha_{2})}{2\ |\mathrm{L}(G)|^{2}},$
where
$g(p,\alpha_{1},\alpha_{2})=\frac{1}{(p-1)^{4}}\Big{(}(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}-(\alpha_{1}+\alpha_{2}+3)p+(\alpha_{1}+\alpha_{2}+1)\Big{)}^{2}+4$
is a polynomial function depending only on $N$.
###### Proof.
We note that $G=HN$, where $G/N\simeq H$ is of prime order. (2.4) is in this
case $\mathrm{L}(G/N)^{2}\sim\mathrm{L}(H)^{2}\subseteq\mathrm{L}(G)^{2}$ and
is always satisfied. Analogously, (2.5) becomes
$\mathrm{L}(N)^{2}\subseteq\mathrm{L}(G)^{2}$ and is satisfied, too. Then
(2.8) becomes
(2.14)
$sd(G)\geq\frac{|\mathrm{L}(N)|^{2}sd(N)+|\mathrm{L}(G/N)|^{2}spd(G/N)}{2\
|\mathrm{L}(G)|^{2}}$
and, from the assumptions, $|\mathrm{L}(G/N)|=2$, $sd(G/N)=sd(N)=1$ but again
[18, Theorem 3.3] implies
$|\mathrm{L}(N)|^{2}=\frac{1}{(p-1)^{4}}\Big{(}(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}-(\alpha_{1}+\alpha_{2}+3)p+(\alpha_{1}+\alpha_{2}+1)\Big{)}^{2}$.
Therefore
(2.15) $=\frac{1}{2\
|\mathrm{L}(G)|^{2}}\Big{(}\frac{1}{(p-1)^{4}}\Big{(}(\alpha_{2}-\alpha_{1}+1)p^{\alpha_{1}+2}-(\alpha_{2}-\alpha_{1}-1)p^{\alpha_{1}+1}$
$-(\alpha_{1}+\alpha_{2}+3)p+(\alpha_{1}+\alpha_{2}+1)\Big{)}^{2}+4\Big{)},$
where it appears clear what is the polynomial function
$g(p,\alpha_{1},\alpha_{2})$, which we are looking for. ∎
As usual $Fit(G)$ denotes the Fitting subgroup of $G$.
###### Theorem 2.8.
Let $G$ be a solvable group in which
$C=C_{G}(Fit(G))=\mathbb{Z}_{p^{\alpha_{1}}}\times\mathbb{Z}_{p^{\alpha_{2}}}$,
for $0\leq\alpha_{1}+\alpha_{2}$, $1\leq\alpha_{1}\leq\alpha_{2}$, $p$ a prime
and $|G:C|$ a prime.
* (i)
If (2.4)–(2.5) are satisfied, then
$spd(G)\geq\frac{f(p,\alpha_{1},\alpha_{2})}{2\ |\mathrm{sn}(G)|\
|\mathrm{M}(G)|},$ where $f(p,\alpha_{1},\alpha_{2})$ is a polynomial function
depending only on $C$.
* (ii)
$sd(G)\geq\frac{g(p,\alpha_{1},\alpha_{2})}{2\ |\mathrm{L}(G)|^{2}},$ where
$g(p,\alpha_{1},\alpha_{2})$ is a polynomial function depending only on $C$.
###### Proof.
Since $G$ is solvable, it is well–known that $C$ is an abelian normal subgroup
of $G$. Then our position is correct in assuming
$C=\mathbb{Z}_{p^{\alpha_{1}}}\times\mathbb{Z}_{p^{\alpha_{2}}}$, with
$0\leq\alpha_{1}+\alpha_{2}$, $1\leq\alpha_{1}\leq\alpha_{2}$, $p$ prime and
$G/C$ is of prime order. Now (i) is an application of Lemma 2.6 and (ii) of
Lemma 2.7. ∎
The lower bound in Lemma 2.7 for $sd(G)$ is more precise than the following
bound, which was the first to be presented in literature.
###### Corollary 2.9 (See [17], Corollary 2.6).
A group $G$ possessing a normal abelian subgroup of prime index has
$|\mathrm{L}(G)|^{2}\ sd(G)\geq|\mathrm{L}(N)|^{2}+2|\mathrm{L}(N)|+1$.
A different restriction is obtained when we multiply up (2.6)–(2.7).
###### Proposition 2.10.
Let $N$ be a normal subgroup of a group $G=NH$ satisfying (2.4) and (2.5).
Then
$spd(G)\geq\frac{1}{|\mathrm{sn}(G)|\
|\mathrm{M}(G)|}\sqrt{{\underset{(Z,T)\in\mathrm{sn}(H)\times\mathrm{M}(H)}{\underset{(X,Y)\in\mathrm{sn}(N)\times\mathrm{M}(N)}{\sum}}}\chi(X,Y)\
\chi(Z,T)}.$
###### Proof.
From (2.6)–(2.7) and the Cauchy inequality for numerical series,
$|\mathrm{sn}(G)|^{2}\ |\mathrm{M}(G)|^{2}\ spd(G)^{2}\geq\
\sum_{(X,Y)\in\mathrm{sn}(N)\times\mathrm{M}(N)}\chi(X,Y)\ \cdot\
\sum_{(Z,T)\in\mathrm{sn}(H)\times\mathrm{M}(H)}\chi(Z,T)$ (2.16)
$\geq{\underset{(Z,T)\in\mathrm{sn}(H)\times\mathrm{M}(H)}{\underset{(X,Y)\in\mathrm{sn}(N)\times\mathrm{M}(N)}{\sum}}}\chi(X,Y)\
\chi(Z,T).$
All are positive quantities and then, extracting the square root, the result
follows.
∎
The next result answers in a certain sense to [17, Problem 4.1].
###### Corollary 2.11.
Let $N$ be a normal subgroup of a group $G=NH$. Then
$sd(G)\geq\frac{1}{|\mathrm{L}(G)|^{2}}\sqrt{{\underset{(Z,T)\in\mathrm{L}(H)^{2}}{\underset{(X,Y)\in\mathrm{L}(N)^{2}}{\sum}}}\chi(X,Y)\
\chi(Z,T)}.$
###### Proof.
Firstly, we note that the corresponding versions of (2.4) and (2.5) for
$sd(G)$ are always satisfied. Then we argue as in Proposition 2.10. ∎
## 3\. Applications and final considerations
The symmetric group on 3 elements
$S_{3}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{3}=\langle a,b\ |\
a^{3}=b^{2}=1,b^{-1}ab=a^{-1}\rangle$ has $sd(S_{3})=\frac{5}{6}$ (see [17,
p.2510]), is metabelian and satisfies the description in Theorem 2.8, since it
is an example of a primitive group of affine type (see [5]). This group was
the origin of our investigation. In fact, a primitive group $P$ of affine type
is a semidirect product with normal factor $Fit(P)$. Furthermore, $Fit(P)$
turns out to be elementary abelian and $C_{P}(Fit(P))=Fit(P)$. This means that
Theorem 2.8 gives a good description for the subgroup commutativity degree and
for the subgroup S–commutativity degree of such groups. While [4, 6, 7, 11,
12, 13] show that we may classify a group, when restrictions on $d(G)$ are
given, the problem is still open for $sd(G)$ and $spd(G)$. We illustrate one
case, involving $sd(G)$. This is to justify the interest in Section 2 in the
new bounds.
###### Corollary 3.1.
A metabelian group $G$ with $|G^{\prime}|$ and $|G/G^{\prime}|$ of prime
orders is cyclic, whenever the bound in Corollary 2.9 is achieved with
$sd(G)=\frac{5}{6}$.
###### Proof.
We begin from $|\mathrm{L}(G)|^{2}\
\frac{5}{6}=|\mathrm{L}(N)|^{2}+2|\mathrm{L}(N)|+1$, which becomes
$|\mathrm{L}(G)|^{2}\ \frac{5}{6}=4+4+1=9$, then
$2\leq|\mathrm{L}(G)|=\sqrt{\frac{56}{5}}=\sqrt{11.2}<4$. This implies either
$|\mathrm{L}(G)|=2$ or $|\mathrm{L}(G)|=3$. In the first case, $G$ is cyclic
of prime order. In the second case, $G$ is lattice isomorphic to $C_{p^{2}}$
for a suitable prime $p$. In both cases $G$ is cyclic. ∎
The control of $|\mathrm{L}(G)|$ was the main ingredient in the previous
proof. Unfortunately, formulas for the growth of $\mathrm{L}(G)$ are hard to
find and [16] helps our investigations. The Möbius number of $\mathrm{L}(G)$
is a number which allows us to control the size of $|\mathrm{L}(G)|$. In case
of a symmetric group $S_{n}$, it is denoted by $\mu(1,S_{n})$ and was
conjectured to be $(-1)^{n-1}\ (|\mathrm{Aut}(S_{n})|/2)$ for all $n>1$ (see
[16, p.1]). For $n\leq 11$, this was proved by H. Pahlings. Recent progresses
are summarized below.
###### Theorem 3.2 (See [16], Theorems 1.6, 1.8, 1.10).
* (i)
Let $p$ be a prime. Then $\mu(1,S_{p})=(-1)^{p-1}\ \frac{p!}{2}$.
* (ii)
Let $n=2p$ and $p$ be an odd prime. Then
$\mu(1,S_{n})=\left\\{\begin{array}[]{lcl}-n!,&&\mathrm{if}\ n-1\ \mathrm{is\
prime\ and}\ p\equiv 3\mod 4,\\\ \frac{n!}{2},&&\mathrm{if}\ n=22,\\\
-\frac{n!}{2},&&\mathrm{otherwise.}\end{array}\right.$
* (iii)
Let $n=2^{a}$ for an integer $a\geq 1$. Then $\mu(1,S_{n})=-\frac{n!}{2}$.
Let $\mu(1,G)\in\\{\mu(1,S_{p}),\mu(1,S_{n})\\}$, being $\mu(1,S_{p})$ and
$\mu(1,S_{n})$ the values in Theorem 3.2 under the given restrictions on $n$
and $p$. In a certain sense, the following result specifies our considerations
on $S_{3}$, when we have an arbitrary primitive group of affine type for which
the subgroup lattice is growing as $S_{n}$.
###### Corollary 3.3.
Under the assumptions of Theorem 2.8, let $G$ be a solvable group such that
$|\mathrm{L}(G)|=\mu(1,G)$. Then
$sd(G)\geq\frac{g(p,\alpha_{1},\alpha_{2})}{2\ \mu(1,G)^{2}},$ where
$g(p,\alpha_{2},\alpha_{2})$ is a polynomial function depending only on
$C_{G}(Fit(G))$.
## Acknowledgement
The second author thanks D.E. Otera and the Université Paris-Sud 11 for the
hospitality during the month of May 2010, in which the significant part of the
present project has been done. We are also grateful to some colleagues, who
detected two fundamental problems in the original version of the manuscript,
allowing us to enlarge our perspectives of study in the present version.
## References
* [1] R.K. Agrawal, Finite groups whose subnormal subgroups permute with all Sylow subgroups, Proc. Amer. Math. Soc. 47 (1975), 77–83.
* [2] J.C. Beidleman and H. Heineken, Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups J. Group Theory 6 (2003), 139–158.
* [3] K. Chiti, M.R.R. Moghaddam and A.R. Salemkar, $n$-isoclinism classes and $n$\- nilpotency degree of finite groups, Algebra Coll. 12 (2005), 225–261.
* [4] A.K. Das and R.K. Nath, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo 59 (2010), 137–142.
* [5] J.D. Dixon and B. Mortimer, Permutation Groups, Springer, Berlin, 1996.
* [6] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35 (2007), 4183–4197.
* [7] A. Erfanian, R. Rezaei and F.G. Russo, Relative $n$-isoclinism classes and relative $n$-th nilpotency degree of finite groups, e-print, Cornell University Library, arXiv:1003.2306.
* [8] G. Grätzer, General Lattice Theory, 2nd edition, Birkäuser, 2003, Basel.
* [9] J. Lennox and S. Stonehewer, Permutable subgroups, Oxford University Press, Oxford, 1993.
* [10] H. Mohammadzadeh. A.R. Salemkar and H. Tavallaee, A remark on the commuting probability in finite groups, South East Bull. Math. Sci. 34 (2010), 755–763.
* [11] P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335–343.
* [12] R. Rezaei and F.G. Russo, $n$-th relative nilpotency degree and relative $n$-isoclinism classes, Carpathian J. Math., to appear.
* [13] F.G. Russo, The generalized commutativity degree in a finite group, Acta Univ. Apulensis Math. Inform 18 (2009), 161–167.
* [14] P. Schmid, Subgroups permutable wih all Sylow subgroups, J. Algebra 207 (1998), 285–293.
* [15] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994.
* [16] J. Shareshian, On the Möbius number of the subgroup lattice of the symmetric group, J. Comb. Theory Ser. A 78 (1997), 236–267.
* [17] M. Tǎrnǎuceanu, Subgroup commutativity degrees of finite groups, J. Algebra 321 (2009), 2508–2520.
* [18] M. Tǎrnǎuceanu, An arithmetic method of counting the subgroups of a finite abelian group, Bull. Math. Soc. Sci. Math. Roumanie 53 (2010), 373–386.
|
arxiv-papers
| 2010-09-11T14:17:12 |
2024-09-04T02:49:12.825211
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele Ettore Otera (Universite' Paris-Sud 11, Orsay Cedex, France)\n and Francesco G. Russo (Universita' degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1009.2171"
}
|
1009.2191
|
# Interplay between Superconductivity and Antiferromagnetism in a Multi-
layered System
H. T. Quan and Jian-Xin Zhu jxzhu@lanl.gov http://theory.lanl.gov Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
###### Abstract
Based on a microscopic model, we study the interplay between superconductivity
and antiferromagnetism in a multi-layered system, where two superconductors
are separated by an antiferromagnetic region. Within a self-consistent mean-
field theory, this system is solved numerically. We find that the
antiferromagnetism in the middle layers profoundly affects the supercurrent
flowing across the junction, while the phase difference across the junction
influences the development of antiferromagnetism in the middle layers. This
study may not only shed new light on material design and material engineering,
but also bring important insights to building Josephson-junction-based quantum
devices, such as SQUID and superconducting qubit.
###### pacs:
74.50.+r, 75.70.-i, 74.81.-g
## I Introduction
As a remarkable aspect of high-$T_{c}$ superconductivity, its unique
characteristics may result from the competition between more than one type of
order parameter. PWAnderson:97 ; PALee:06 ; SChakravarty:04 Historically, the
typical antagonistic relationship between superconductivity and magnetism has
led researchers to avoid using magnetic elements, such as iron, as potential
building blocks of superconducting (SC) material. However, it is now well
accepted that the unconventional superconductivity emerging upon doping is
closely related to the antiferromagnetism (AFM) in parent compounds JZaanen:89
; KMachida:89 ; SAKivelson:03 ; EDemler:04 . Recent research theme in
high-$T_{c}$ cuprate community centers on how to establish the connection
between antiferromagnetic (AF) and $d$-wave SC (DSC) orderings orenstein00 ,
i.e., whether they compete with each other or coexist microscopically. This
issue is a subject of current discussions. Depending on material details, some
compounds show the coexistence of DSC and AFM YSLee:99 while others exhibit
microscopic separation of these two phases SHPan:01 . Engineered heterogeneous
systems and multilayered high-$T_{c}$ cuprates offer a unique setting to study
the interplay between DSC and AFM. In these systems, the disorder effect can
be minimized significantly with atomically smooth interfaces. Experimentally,
no mixing of DSC and AFM was reported in the heterostructure artificially
grown by stacking integer number of $\mathrm{La_{1.85}Sr_{0.15}CuO_{4}}$ and
$\mathrm{La_{2}CuO_{4}}$ layers bozovic . Meanwhile microscopic evidence for
the uniform mixed phase of AFM and DSC in outer $\mathrm{CuO_{2}}$ planes was
reported mukuda on a Hg-based five-layered cuprate. Theoretically, Demler et
al. phenomenalogical have studied the proximity effect and Josephson coupling
in the SO(5) theory of high-$T_{c}$ superconductors. Depending on the
thickness, the middle antiferromagnetic region could behave like a
superconductor, metal or insulator. On the other hand, it was shown in the
perturbation theory perturbative that the spin exchange coupling in the
insulating AF layer can allow the tunneling of Cooper pairs.
In this article, we study microscopically a multi-layered system with two
superconductors separated by AF layers. Through self-consistent mean-field
theory and numerical diagonalization, we are able to solve the Hamiltonian,
which allows us to study the interplay between DSC and AFM in detail. Varying
some external parameters, such as the SC phase difference across the system
$\delta\varphi$ and Coulomb interaction $U$ in the AF layers leads to
interesting results about the interplay.
Figure 1: Schematic drawing of a multi-layered system. The middle region
consists of AF layers (in X-Y plane) and the two sides are SC layers with
$d$-wave pairing symmetry. The phases of the SC order parameter (or the
pairing potential) in the first (far left) and the last (far right) layer are
fixed at $\varphi_{L}$ and $\varphi_{R}$, respectively.
## II Model and setup
The model system under consideration is schematically presented in Fig. 1, for
which the Hamiltonian can be written as:
$\displaystyle H$ $\displaystyle=$
$\displaystyle-\sum_{i,j,\sigma}t_{i,j}c_{i,\sigma}^{\dagger}c_{j,\sigma}+\sum_{i}U_{i}(n_{i,\uparrow}-\frac{1}{2})(n_{i,\downarrow}-\frac{1}{2})$
(1) $\displaystyle-\frac{1}{2}\sum_{i,j}V_{ij}n_{i}n_{j}-\mu\sum_{i}n_{i}\;,$
where $t_{i,j}$ denotes the hopping integral between the nearest neighbor
sites. For in-plane (X-Y plane) hopping, $t_{i,j}=t$, and for inter-layer (Z
direction) hopping, $t_{i,j}=t_{\perp}$. $c_{i,\sigma}$
($c_{i,\sigma}^{\dagger}$) is the annihilation (creation) operator of
electrons on the $i$th lattice site with spin $\sigma$
($\sigma=\uparrow,\downarrow$). The quantity
$n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i,\sigma}$ is the number operator on
the $i$th lattice site. Both $U_{i}$ and $V_{ij}$ are positive. $U_{i}$
indicates the on-site repulsive Coulomb interaction, which is nonzero _only_
in the middle layers, and $V_{ij}$ describes the in-plane nearest neighbor
attractive interaction, which is nonzero _only_ in the SC layers. We are going
to fix the phase of the SC order parameter in the first (far left) and the
last (far right) layer at $\varphi_{L}$ and $\varphi_{R}$, respectively, which
can be introduced by a gauge flux.
When there is no inter-layer coupling, $t_{\perp}=0$, the system is decomposed
into individual two-dimensional (2D) systems. It is known that in a 2D SC
layer, the in-plane nearest neighbor attractive interaction $V_{ij}$ leads to
a nonzero energy gap $\Delta$ with $d$-wave symmetry, and in a 2D AF layer the
on-site repulsive interaction $U_{i}$ leads to a nonzero $(\pi,\pi)$ spin
density wave (SDW) order. In a genuine cuprate compound, there exists the
inter-layer coupling, which is usually weaker than the in-plane hopping.
Because of the inter-layer coupling, there arises interesting interplay
between SC layers and AF layers (see Fig. 1).
The strategy of solving the Hamiltonian is briefly summarized as follows: Due
to the translational symmetry in the 2D X-Y plane, the 3D problem can be
decomposed into a 2D (X-Y) plus 1D (Z) problem. In the X-Y plane,
$\vec{k}=(k_{x},k_{y})$ is a good quantum number, where
$k_{x}=\frac{1}{2}(\frac{n_{x}}{N_{x}}-\frac{n_{y}}{N_{y}})\pi$,
$k_{y}=\frac{1}{2}(\frac{n_{x}}{N_{x}}+\frac{n_{y}}{N_{y}})\pi$, and
$n_{x}=-\frac{1}{2}N_{x}$, $-\frac{1}{2}N_{x}+1$, $\cdots$,
$\frac{1}{2}N_{x}-1$, $n_{y}=-\frac{1}{2}N_{y}$, $-\frac{1}{2}N_{y}+1$,
$\cdots$, $\frac{1}{2}N_{y}-1$. In order to simplify the Hamiltonian, we adopt
the mean-field approximation, which leads to the following Bogoliubov-de
Gennes equation PGdeGennes:1965 :
$\sum_{m}\left[\begin{array}[]{cccc}\xi_{\vec{k}n}\delta_{nm}+t_{nm}&-\frac{U_{n}}{2}M_{n}\delta_{nm}&\Delta_{\vec{k},n}\delta_{nm}&0\\\
-\frac{U_{n}}{2}M_{n}\delta_{nm}&\xi_{\vec{k}+\vec{Q},n}\delta_{nm}+t_{nm}&0&\Delta_{\vec{k}+\vec{Q},n}\delta_{nm}\\\
\Delta_{\vec{k},n}^{\ast}\delta_{nm}&0&-\xi_{-\vec{k}n}\delta_{nm}-t_{nm}^{\ast}&-\frac{U_{n}}{2}M_{n}\delta_{nm}\\\
0&\Delta_{\vec{k}+\vec{Q},n}^{\ast}\delta_{nm}&-\frac{U_{n}}{2}M_{n}\delta_{nm}&-\xi_{-(\vec{k}+\vec{Q})n}\delta_{nm}-t_{nm}^{\ast}\\\
\end{array}\right]\left[\begin{array}[]{c}u_{\vec{k},m,\uparrow}^{\alpha}\\\
u_{\vec{k}+\vec{Q},m,\uparrow}^{\alpha}\\\
v_{\vec{k},m,\downarrow}^{\alpha}\\\
v_{\vec{k}+\vec{Q},m,\downarrow}^{\alpha}\end{array}\right]=E_{\alpha}\left[\begin{array}[]{c}u_{\vec{k},n,\uparrow}^{\alpha}\\\
u_{\vec{k}+\vec{Q},n,\uparrow}^{\alpha}\\\
v_{\vec{k},n,\downarrow}^{\alpha}\\\
v_{\vec{k}+\vec{Q},n,\downarrow}^{\alpha}\end{array}\right]\;,$ (2)
where $m$ and $n$ label the layer number; $\vec{k}$ and $\vec{k}+\vec{Q}$ with
$\vec{Q}=(\pi,\pi)$ denote the wave vectors in the first Brillouin zone of the
X-Y plane, $t_{mn}=t_{\perp}(\delta_{n,m+1}+\delta_{n,m-1})$ describes the
nearest neighbor inter-plane hopping, $\xi_{\vec{k}n}=-2t(\cos k_{x}+\cos
k_{y})-\mu$ is the normal metal dispersion relation, $\mu$ is the chemical
potential, the variables $U_{n}$ is equal to $U$ in the AF layers and zero
otherwise while $V_{n}$ is equal to $V$ in the SC layers and zero otherwise.
The superconducting pairing potential $\Delta_{\vec{k},n}=V_{n}\Psi_{n}(\cos
k_{x}-\cos k_{y})/2$, where $\Psi_{n}$ is the superconducting pairing
wavefunction. Obviously, the superconducting pairing potential, which is the
SC order parameter has the same phase as that of the superconducting pairing
wavefunction. $M_{n}=\left\langle n_{n,\uparrow}\right\rangle-\left\langle
n_{n,\downarrow}\right\rangle$ describes the SDW in the $n$-th layer. The
average electron number $\left\langle n_{n,\uparrow}\right\rangle$,
$\left\langle n_{n,\downarrow}\right\rangle$ and the SC pairing wavefunction
$\Psi_{n}$ can be determined self-consistently through iteration over the
following relation:
$\displaystyle\Psi_{n}=\frac{2}{N}\sum_{\vec{k},\alpha}\frac{(\cos k_{x}-\cos
k_{y})}{2}g(u,v)\tanh{\frac{\beta E_{\alpha}}{2}}\;,$ (3a)
$\displaystyle\left\langle
n_{n,\uparrow}\right\rangle=\frac{1}{N}\sum_{\vec{k},\alpha}\left|u_{\vec{k},n,\uparrow}^{\alpha}+u_{\vec{k}+\vec{Q},n,\uparrow}^{\alpha}\right|^{2}f(E_{\alpha})\;,$
(3b) $\displaystyle\left\langle
n_{n,\downarrow}\right\rangle=\frac{1}{N}\sum_{\vec{k},\alpha}\left|v_{\vec{k},n,\downarrow}^{\alpha}+v_{\vec{k}+\vec{Q},n,\downarrow}^{\alpha}\right|^{2}f(-E_{\alpha})\;,$
(3c)
where $N=N_{x}\times N_{y}$ is the sites number in a 2D plane. $\vec{k}$
samples half of the first Brillouin zone.
$g(u,v)=u_{\vec{k},n,\uparrow}^{\alpha}v_{\vec{k},n,\downarrow}^{\alpha,\ast}-u_{\vec{k}+\vec{Q},n,\uparrow}^{\alpha}v_{\vec{k}+\vec{Q},n,\downarrow}^{\alpha,\ast}$,
and $f(E_{\alpha})=1/(1+e^{\beta E_{\alpha}})$ is the Fermi distribution.
Through this simplification, the multi-layered system is then solved
numerically. We are especially interested in the interplay between the DSC and
the AFM, which is characterized by the SDW $M_{n}$ and the SC pairing
wavefunction $\Psi_{n}$. We will focus on zero temperature $T=0$. The size of
X-Y plane is $N_{x}=N_{y}=40$. The total layer number is $N_{z}=22$. The two
middle layers are AF layers ($U_{n}=U$, $V_{n}=0$, for $n=11$, $12$), and the
rest are SC layers ($U_{n}=0$, $V_{n}=V$, for $n=1$, $2$, $\cdots$, $10$, and
$n=13$, $\cdots$, $22$). For simplicity, we choose $t=1$, $t_{\perp}=0.1$,
$V=1.5$, and $\mu=0$. We will vary the control parameters, such as $U$ and
$\delta\varphi=\varphi_{L}-\varphi_{R}$, to study the competition between DSC
and AFM. When the phase difference $\delta\varphi$ is introduced, there is a
current flowing across the system and its expression is given by:
$\displaystyle I$ $\displaystyle=$ $\displaystyle
2\frac{t_{\perp}e}{N}\sum_{\vec{k},\alpha}\mathrm{Im}\biggl{\\{}\left[u_{\vec{k},n,\uparrow}^{\alpha,\ast}u_{\vec{k},n+1,\uparrow}^{\alpha}+\left(\vec{k}\leftrightarrow\vec{k}+\vec{Q}\right)\right]f_{+}$
(4)
$\displaystyle+\left[v_{\vec{k},n,\downarrow}^{\alpha}v_{\vec{k},n+1,\downarrow}^{\alpha,\ast}+v_{\vec{k}+\vec{Q},n,\downarrow}^{\alpha}v_{\vec{k}+\vec{Q},n+1,\downarrow}^{\alpha,\ast}\right]f_{-}\biggr{\\}}\;,$
where $f_{\pm}=f(\pm E_{\alpha})$.
## III Influence of antiferromagnetism on superconductivity
In the model introduced above, the change of the Coulomb interaction $U$ in
the middle layers can drive a metal-Mott insulator phase transition at a
certain value $U_{c}$. Correspondingly, the multi-layered system changes from
a superconductor/normal metal/superconductor (SNS) weak link bardeen72 to a
superconductor/insulator/superconductor (SIS) Josephson junction josephson .
The studies of SNS and SIS junctions with static potential barrier have been
well documented bardeen72 ; josephson ; MTinkham:75 . Here we start from a
microscopic model and drive the middle layers to change from one electronic
state into another by tuning the on-site Coulomb interaction $U$. We are
interested in how the SC pairing wavefunction and the current across the
junction change with the Coulomb interaction $U$. We plot the current as a
function of the phase difference $\delta\varphi$ in Fig. 2. It can be seen
that below a threshold value of $U_{c}\approx 0.8$, the current is a piecewise
periodic function. In each periodic region, it varies linearly with
$\delta\varphi$. This agrees with the result obtained in Ref. bardeen72, for
a junction with a normal metal. We also find that when the Coulomb interaction
is larger than $U_{c}$, the current as a function of $\delta\varphi$ changes
gradually with the increase of $U$. When $U$ is in the range $0.8<U<1.5$, the
current shows a shape intermediate between a piecewise linear function and a
sinusoidal function. Similar results have been observed in Refs. bardeen72 ;
phenomenalogical ; EDemler:04 when one varies the temperature or the
thickness of the middle layers instead of the the Coulomb interaction $U$.
When $U\approx 2.0$, the current is well approximated by a sinusoidal function
of $\delta\varphi$ (see Fig. 2), which is a typical feature of the dc
Josephson junction (JJ) current josephson . The magnitude of current decreases
rapidly with the increase of $U$.
Figure 2: Current across the junction as a function of the phase difference
$\delta\varphi$ of two superconductors for different $U$. Here $V=1.5$,
$t_{\perp}=0.1$, $N_{x}=N_{y}=40$, $N_{z}=22$. Up panel: $U=0$, $0.1$, $0.2$,
$\cdots$, $1.5$, and $\delta\varphi=0\sim\pi$. Down panel: $U=2.0$ and
$\delta\varphi=0\sim 2\pi$.
Figure 3: Absolute value (Up panel) and argument (Down panel) of
superconducting pairing wavefunction $\Psi_{n}$ as a function of layer number.
The phase of SC pairing wavefunction is the same as that of the pairing
wavefunction. Here $V=1.5$, $t_{\perp}=0.1$, $N_{x}=N_{y}=40$, $N_{z}=22$,
$\delta\varphi=3\pi/4$. $U$ is fixed at different values: $U=0$, $0.1$, $0.2$,
$\cdots$, $1.5$.
In order to have a better understanding of the influence of AFM on DSC, we
plot the absolute value and the phase of the SC pairing wavefunction in Fig.
3. It can be seen that when $U$ is small, there is a nonzero paring potential
in the middle layers due to the proximity effect, and the two superconductors
are weakly linked by the normal metallic layers. In this weak link regime, the
phase of the SC pairing wavefunction is almost linear in layer number $n$.
This means that when the middle layers are in the normal metallic state, it
has little influence on the SC layers on two sides. However, when $U$ is
larger than $U_{c}\approx 0.8$, SDW develops, and the middle layers become
Mott insulator. Quantum mechanically, Cooper pairs can still tunnel through
the SDW region when $U$ is not very large. We call this intermediate $U$
regime as the tunneling regime. From the weak link regime to tunneling regime,
the phase of the SC paring wavefunction changes gradually from a (nearly)
linear function to a step function of the layer number (see Fig. 3). This
corresponds to the observation that the current shape changes from being
piecewise linear to sinusoidal. Therefore, when $U$ is very large, the
current-phase relation becomes identical to the famous Josephson relation
josephson ; feynman . Here for the first time, we establish the connection
between the profile of the SC phase distribution (Fig. 3) and the current-
phase dependence (Fig. 2), which we believe is rather intriguing.
When one continues to increase $U$, the middle layers becomes more opaque and
the superconducting pairing wavefunction is completely suppressed in this
region. As a result, the tunneling of Cooer pairs will be completely quenched,
and the two comprising superconductors become decoupled.
We also would like to mention that when the middle layers are in the normal
metallic state, the current across the junction is a mesoscopic effect
JXZhu:10 . The current is proportional to the phase gradient of the SC pairing
wavefunction. That is, when one increases the number of SC layers, the current
will decrease and finally vanish. However, when the middle layer is in the AF
state, the whole system becomes a JJ. The current of a JJ is no longer a
mesoscopic effect. It is completely determined by the phase difference of the
SC pairing wavefunction on both ends of the system. Therefore, it will not
decrease with the increase of the number of the SC layers. In this sense we
say that AFM insulator layers enhance supercurrent.
## IV Influence of superconductivity on antiferromagnetism
In the above discussion, we find the profound influence of the Coulomb
interaction $U$ (and hence AFM) of the middle layers on the DSC. We now study
the back-action of DSC on AFM. We will fix the phase difference
$\delta\varphi$ at different values, and see if the SDW in the middle layers
changes with $\delta\varphi$. We plot in Fig. 4 the SDW as a function of $U$
for various values of $\delta\varphi$. Clearly, the SDW in the middle layers
is influenced by $\delta\varphi$ in certain range of $U$. When
$\delta\varphi=\pi$, the critical value $U_{c}$, after which the SDW develops,
is much smaller than in the case of $\delta\varphi=0$, though the current is
zero in both cases. This is very similar to the result in Ref.
phenomenalogical, that the middle AF layers will be influenced by the current
fed into the junction. When the Coulomb interaction is very weak, the SDW does
not develop for $\delta\varphi\neq\pi$, and the middle layers are in the
normal metallic state. In contrast, when the Coulomb interaction is very
strong, e.g. $U>1.2$, the AFM is robust against the phase difference across
the junction and strongly suppresses the pairing wavefunction in the middle
region (see Fig. 3). Only when $U$ is small, is the emergence of the SDW
sensitive to the phase difference.
Figure 4: (Color online) SDW in the middle layer ($n=11$) as a function of the
Coulomb interaction for different fixed $\delta\varphi$. Here $U=0\sim 1.5$,
$V=1.5$, $t_{\perp}=0.1$, $N_{x}=N_{y}=40$, $N_{z}=22$, $\delta\varphi=0$
(solid), $3\pi/4$ (dashed), and $\pi$ (dot dashed).
When switching off the interlayer coupling ($t_{\perp}=0$), the system is
decoupled into 2D systems. The DSC on two sides will not influence the AFM in
the middle layer. The middle layer is described by a 2D Hubbard model, in
which the SDW will develop for an infinitesimal $U$. In the present model
($t_{\perp}\neq 0$), when $\delta\varphi=0$, the SDW in the middle layers is
suppressed due to the inter-layer coupling, which weakens the perfect nesting
at $\vec{Q}=(\pi,\pi)$. The SDW will not develop until $U$ is larger than a
threshold value $U_{c}$ (see Fig. 4). Hence both $t_{\perp}$ and
$\delta\varphi$ will influence the AFM. The results in Fig. 4 indicate that
the AFM shows a crossover behavior for $\delta\varphi=\pi$ rather than a
criticality as obtained for $\delta\varphi=0$. For $\delta\varphi\in(0,\pi)$,
the SDW critical point $U_{c}$ is less than that for $\delta\varphi=0$. This
result is in agreement with that in Ref. phenomenalogical, .
In order to have a better understanding of the influence of the DSC on the AFM
in the middle layers, we plot in Fig. 5 the local density of state (LDOS), as
given by
$\displaystyle\rho_{n,\uparrow}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{\vec{k},\alpha}\left|u_{\vec{k},n,\uparrow}^{\alpha}+u_{\vec{k}+\vec{Q},n,\uparrow}^{\alpha}\right|^{2}\left[-\frac{\partial
f(\omega)}{\partial\omega}\right],$ (5a) $\displaystyle\rho_{n,\downarrow}(E)$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{\vec{k},\alpha}\left|v_{\vec{k},n,\downarrow}^{\alpha}+v_{\vec{k}+\vec{Q},n,\downarrow}^{\alpha}\right|^{2}\left[-\frac{\partial
f(\omega^{\prime})}{\partial\omega^{\prime}}\right],$ (5b)
where $\omega=E-E_{\alpha}$ and $\omega^{\prime}=E+E_{\alpha}$. From Fig. 5,
we can see a sharp intensity decay of LDOS at $E_{F}$ when $\delta\varphi=0$.
When $\delta\varphi=\pi$, the LDOS exhibits a peak structure at $E_{F}$. Since
the development of the SDW is also determined by the normal-state DOS at low
energies, it explains the different SDW critical behavior between the cases
with $\delta\varphi=\pi$ and $\delta\varphi=0$.
Figure 5: (Color online) LDOS of the middle layer ($n=11$)
$\rho_{11,\uparrow}(E)$. Here $U=0$, $V=1.5$, $t_{\perp}=0.1$,
$N_{x}=N_{y}=40$, $N_{z}=22$, $\delta\varphi=0$, and $\pi$.
## V Conclusion and discussion
We have studied the interplay between the DSC and AFM in a multi-layered
system. It is revealed that the AF layers have profound influence on the DSC.
A metal-Mott insulator phase transition makes the multi-layer system change
from a SNS to a SIS junction. We have also established the connection between
the profile of the SC phase distribution and the current-phase dependence. We
find that when the barrier of the middle layers is not high enough (metal or
semiconductor regime), the middle layers cannot maintain a step-function-like
phase of two superconductors on two sides. Accordingly, the current across the
junction will deviate from a sinusoidal function of the phase difference. When
the barrier of the middle layer is high (deep insulator regime), as predicted
by Josephson josephson the phase across the junction is a step function, and
the current is a sinusoidal function of the phase difference. Meanwhile, the
phase difference across the system will dramatically influence the development
of SDW in the middle layers. The results from our simulations may shed new
light on material engineering. The DSC and AFM can coexist in a single layer,
but they also compete with each other. Another important insight from the
present study is the microscopic model of JJ. JJ has been widely used in a lot
of quantum devices, such as in SQUID for ultra-sensitive magnetic field
measurement and superconducting qubit qubit for quantum computing. Our study
provides a microscopic theory for JJ, hence has possible applications in
building adaptive JJ based devices by engineering the middle layers of the JJ.
Finally, with the development of experimental techniques, it is now possible
to synthesize heterogeneous systems with layer by layer at atomic scale
threelayer . We expect that our results can be experimentally verified, as
some experiments on engineered material have been reported threelayer ;
bozovic ; mukuda .
Acknowledgments: We thank A. V. Balatsky, Quanxi Jia, A. J. Taylor, and S.
Trugman for useful discussions. This work was carried out under the auspices
of the National Nuclear Security Administration of the U.S. DOE at LANL under
Contract No. DE-AC52-06NA25396, the LANL LDRD-DR Project X96Y, and the U.S.
DOE Office of Science.
## References
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* (11) I. Bozovic, G. Logvenov, M. A. J. Verhoeven, P. Caputo, E. Goldobin and T. H. Geballe, Nature (London) 422, 873 (2003).
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|
arxiv-papers
| 2010-09-11T18:18:11 |
2024-09-04T02:49:12.830660
|
{
"license": "Public Domain",
"authors": "H. T. Quan and Jian-Xin Zhu",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/1009.2191"
}
|
1009.2261
|
# Recurrence relation for the $6j$-symbol of $\mathrm{su}_{q}(2)$ as a
symmetric eigenvalue problem
Igor Khavkine1 1 Institute for Theoretical Physics, Utrecht, Leuvenlaan 4,
NL-3584 CE Utrecht, The Netherlands i.khavkine@uu.nl
###### Abstract
A well known recurrence relation for the $6j$-symbol of the quantum group
$\mathrm{su}_{q}(2)$ is realized as a tridiagonal, symmetric eigenvalue
problem. This formulation can be used to implement an efficient numerical
evaluation algorithm, taking advantage of existing specialized numerical
packages. For convenience, all formulas relevant for such an implementation
are collected in the appendix. This realization is a biproduct of an
alternative proof of the recurrence relation, which generalizes a classical
($q=1$) result of Schulten and Gordon and uses the diagrammatic spin network
formalism of Temperley-Lieb recoupling theory to simplify intermediate
calculations.
###### pacs:
02.20.Uw, 03.65.Fd
††: J. Phys. A: Math. Gen.
## 1 Introduction
Quantum groups first appeared in the study of quantum integrable systems.
Since then, they have proven useful in many applications, including among
others conformal field theory, statistical mechanics, representation theory
and the theory of hypergeometric functions, along with exhibiting a rich
internal structure. Quantum groups have appeared seminally in the mathematical
physics literature in connection with topological quantum field theory [1].
More recently, the quantum group $\mathrm{su}_{q}(2)$ was used to construct
“loop” [2] and “spin foam” [3] models of quantum gravity with a positive
cosmological constant.
The $6j$-symbol (or Racah-Wigner coefficients and henceforth the $6j$ or the
$q$-$6j$ for the classical and $q$-deformed versions respectively) has been
studied extensively. It first appeared in work on $q$-hypergeometric functions
[4]. Later, it was found to play a central role in the representation theory
of $\mathrm{su}_{q}(2)$ [5]. It is known to satisfy some recurrence relations
[4, 6], including a particular linear, three term, single argument one [5, 7,
8]. This recurrence has been used to analyze the asymptotics of the classical
[9, 10] and quantum [11] $6j$.
However, this $q$-$6j$ and the need to know its numerical values for a large
number of arguments also appear in other segments of the literature [3, 12],
less aware of these properties. An explicit formula for the $q$-$6j$ is well
known and involves a number of arithmetic operations that is linear in its
arguments (see Appendix). In applications where a large number of $6j$-symbols
is needed at once, e.g., for all values of one argument with others fixed as
is the case in [3], the total number of operations becomes quadratic in the
arguments. The above mentioned recurrence relation can be used to greatly
increase the efficiency of the calculation by reducing the total operation
count to be linear in the arguments. The main goal of this paper is to show
that, moreover, this recurrence can be realized as a tridiagonal, symmetric
eigenvalue problem, a property not shared by most recurrence relations,
especially since the equivalence is established using only rational
operations.
A significant advantage of the eigenvalue formulation is the ability to make
use of readily available, robust linear algebra packages, such as LAPACK [13],
which automatically take care of the important issues of numerical accuracy
and stability. When $q=1$ or when $q$ is a primitive root of unity, the
relevant inner product becomes either positive- or negative-definite and
standard, specialized numerical methods can be exploited to increase the
efficiency of the calculation even further. A secondary goal of this paper is
to concisely collect all the relevant information needed to readily implement
such an efficient $q$-$6j$ numerical evaluation algorithm without intimate
familiarity with the literature on quantum groups or $q$-hypergeometric
functions.
Sections 2 and 3, which can be skipped by those familiar with the mathematical
literature on $\mathrm{su}_{q}(2)$ recoupling theory, introduce the basic
notions of the spin network formalism [14, 15, 16], define the Kauffman-Lins
convention for the $6j$-symbol and summarize basic diagrammatic identities
needed for Section 4, where the recurrence relation is realized as an
eigenvalue problem. This is accomplished as a byproduct of an alternative
proof of the recurrence itself that generalizes the classical argument from
the Appendix of [10]. The diagrammatic spin network formalism makes all
intermediate calculations easy to check and reproduce. The Appendix
conveniently summarizes all formulas needed for a direct computer
implementation of the recurrence-based evaluation of the $q$-$6j$, including
the connection between the Kauffman-Lins and Racah-Wigner notational
convention, which is traditionally used in physics.
## 2 Spin networks
In a variety of physical and mathematical applications, one often encounters
tensor contraction expressions of the form
$T_{j\cdots}^{lm\cdots}=A_{ij\cdots}^{kl\cdots}B_{k\cdots}^{im\cdots}\cdots
Z_{\cdots}^{\cdots},$ (1)
where $T$, $A$, $B$, …, $Z$ are invariant tensors, with each index
transforming under a representation of a group or an algebra. The application
at hand usually calls for evaluating $T$, or at least simplifying it. An
extensive literature on this subject exists for the classical group $SU(2)$ or
its Lie algebra, a subject known as _angular momentum recoupling_ [17, 18]. It
is well known that such tensor contractions can be very efficiently expressed,
manipulated, and simplified using diagrams known as _spin networks_ [14].
Extensions of these techniques [15, 16] are also known for the _quantum_ (or
_$q$ -deformed_, since they depend on an arbitrary complex number $q\neq 0$)
analogs, the quantum group $\mathrm{su}_{q}(2)$ or $U_{q}(su(2))$. The basics
of this diagrammatic formalism, as needed for the derivation of the recurrence
relation, are given in this and next sections. All relevant formulas,
including explicit spin network evaluations in terms of _quantum integers_ are
listed in the Appendix.
Single spin networks are edge-labeled graphs111Spin networks are actually
ribbon graphs, but since all diagrams in this paper are planar, the ribbon
structure can be added through blackboard framing., where each vertex has
valence either $1$ or $3$. General spin networks are formal linear
combinations of single spin networks. Edges attached to univalent vertices are
called _free_. Spin networks without free edges are called _closed_.
Conventionally, the labels are either integers (_spins_) of half-integers
(_twice-spins_), which correspond to irreducible representations of
$\mathrm{su}_{q}(2)$. Reference [15] labels all spin networks with twice-
spins. Unless otherwise indicated, all conventions in this paper follow [15].
Two spin networks may be equal even if not represented with identical labeled
graphs. A complete description of these identities are given in [15] and [16];
their study constitutes _spin network recoupling_ and is what allows us to
equate222A down-to-earth guide to this correspondence, for the classical $q=1$
case, can be found in Appendix A of [19]. Complete details with proofs can be
found in [16]. spin networks with $\mathrm{su}_{q}(2)$-invariant tensors and
their contractions.
In this correspondence, each index of a tensor, transforming under an
irreducible representation, corresponds to a spin network edge, labeled by the
same representation (free indices correspond to free edges). In particular, a
closed spin network corresponds to a complex number. Spin networks form a
graded algebra over $\mathbb{C}$ (as do tensors). The grading is given by the
number of free edges (free indices) and the product is diagrammatic
juxtaposition (tensor product).
## 3 Diagrammatic identities
The spin networks with $n$ free edges with fixed labels (_$n$ -valent_ spin
networks) form a linear space with a natural bilinear form (or inner product).
Suppose that the free edges are ordered in some canonical way, then, given two
spin networks, we can reflect one of them in a mirror and connect the free
edges in order. The value of the resulting closed spin network defines the
bilinear form, which is symmetric and non-degenerate [15]. We use the bra-ket
notation for this inner product $\langle s^{\prime}|s\rangle$, where $s$ and
$s^{\prime}$ are two spin networks. We also let $|s\rangle$ stand for $s$ and
$\langle s^{\prime}|$ for the reflection of $s^{\prime}$. The existence of an
inner product allows the following identities, whose proofs can be found in
[15]. For each identity, the corresponding well known fact of $SU(2)$
representation theory is given.
The space of $2$-valent spin networks, with ends labeled $a$ and $b$, is
$1$-dimensional if $a=b$, and $0$-dimensional otherwise. For non-trivial
dimension, the single edge gives a complete basis and therefore the _bubble
identity_ :
$\raisebox{-4.95801pt}{\leavevmode\hbox{\par\begin{picture}(654.0,293.0)(1249.0,-159.0)\put(1261.0,-16.0){\line(1,0){630.0}}
\put(1306.0,29.0){\makebox(0.0,0.0)[b]{\smash{{a}}}}
\put(1846.0,29.0){\makebox(0.0,0.0)[b]{\smash{{b}}}}
{\color[rgb]{.72,.72,.72}\put(1576.0,-16.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1576.0,-16.0){\circle{240.0}}
}\end{picture}}}~{}=\delta_{ab}\frac{\raisebox{-7.70862pt}{\leavevmode\hbox{\par\begin{picture}(477.0,380.0)(1339.0,-253.0)\put(1456.0,-136.0){\oval(210.0,210.0)[bl]}
\put(1456.0,-121.0){\oval(210.0,210.0)[tl]}
\put(1696.0,-136.0){\oval(210.0,210.0)[br]}
\put(1696.0,-121.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-241.0){\line(1,0){240.0}} \put(1456.0,-16.0){\line(1,0){240.0}}
\put(1351.0,-136.0){\line(0,1){15.0}} \put(1801.0,-136.0){\line(0,1){15.0}}
\put(1801.0,-16.0){\makebox(0.0,0.0)[lb]{\smash{{a}}}}
{\color[rgb]{.72,.72,.72}\put(1576.0,-16.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1576.0,-16.0){\circle{240.0}}
}\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}}~{}~{}\raisebox{-2.8081pt}{\leavevmode\hbox{\par\begin{picture}(474.0,225.0)(1339.0,-121.0)\put(1351.0,-16.0){\line(1,0){450.0}}
\put(1576.0,29.0){\makebox(0.0,0.0)[b]{\smash{{a}}}}
\put(1576.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{1,1,1}a}}}}}
\end{picture}}}~{}.$ (2)
This identity the diagrammatic analog of Schur’s lemma for intertwiners
between irreducible representations.
The space of $3$-valent spin networks, with ends labeled $a$, $b$ and $c$, is
also $1$-dimensional if the triangle inequalities (31) and parity constraints
(32) are satisfied, and $0$-dimensional otherwise, if $q$ is generic. When $q$
is a primitive root of unity, the dimension also vanishes whenever the further
$r$-boundedness constraint (33) is violated. In the case of nontrivial
dimension, the canonical trivalent vertex gives a complete basis and therefore
the _vertex collapse identity_ :
cab
$\displaystyle=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(605.0,608.0)(983.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
{\color[rgb]{.72,.72,.72}\put(1126.0,164.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1126.0,164.0){\circle{240.0}}
}\end{picture}}}}{\theta(a,b,c)}~{}~{}\raisebox{-13.7157pt}{\leavevmode\hbox{\par\begin{picture}(342.0,570.0)(1834.0,-751.0){\color[rgb]{0,0,0}\put(2071.0,-466.0){\oval(450.0,450.0)[tl]}
\put(2071.0,-466.0){\oval(450.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1846.0,-466.0){\line(1,0){225.0}}
}\put(2161.0,-736.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(2161.0,-286.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2161.0,-511.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}~{}~{},$ (3) $\displaystyle\theta(a,b,c)$
$\displaystyle=\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}.$ (4)
The normalization of the vertex, the value of the $\theta$-network, is
evaluated in Equation (30). This identity is the diagrammatic analog of the
uniqueness (up to normalization) of the Clebsch-Gordan intertwiner.
Now, consider the space of $4$-valent networks with free edges labeled $a$,
$b$, $c$ and $d$. There are two natural bases, the vertical $\langle l|$ and
the horizontal $|\bar{j}\rangle$:
$\raisebox{-18.93237pt}{\leavevmode\hbox{\par\begin{picture}(300.0,735.0)(1426.0,-166.0){\color[rgb]{0,0,0}\put(1576.0,164.0){\oval(180.0,450.0)[tl]}
\put(1576.0,164.0){\oval(180.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1666.0,389.0){\oval(180.0,180.0)[tl]}
\put(1666.0,389.0){\oval(180.0,180.0)[bl]}
}{\color[rgb]{0,0,0}\put(1666.0,-61.0){\oval(180.0,180.0)[tl]}
\put(1666.0,-61.0){\oval(180.0,180.0)[bl]}
}\put(1711.0,299.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1711.0,29.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1711.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1711.0,-151.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1441.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}\quad,\quad\raisebox{-18.93237pt}{\leavevmode\hbox{\par\begin{picture}(345.0,735.0)(1741.0,-166.0){\color[rgb]{0,0,0}\put(1801.0,164.0){\oval(180.0,270.0)[br]}
\put(1801.0,164.0){\oval(180.0,270.0)[tr]}
}{\color[rgb]{0,0,0}\put(1801.0,164.0){\oval(450.0,630.0)[br]}
\put(1801.0,164.0){\oval(450.0,630.0)[tr]}
}{\color[rgb]{0,0,0}\put(1891.0,164.0){\line(1,0){135.0}}
}\put(1756.0,-151.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1756.0,29.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1756.0,299.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1756.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2071.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}~{}.$ (5)
The admissible ranges for $j$ and $l$, the dimension $n$ of this space, and
the conditions on $(a,b,c,d)$ under which $n>0$ are given by Equations (41)
through (52). The transition matrix between the two bases is given by the so-
called $\operatorname{Tet}$-network:
$\operatorname{Tet}(a,b,c,d;j,l)=\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}=\langle\bar{j}|l\rangle.$ (6)
The coefficients expressing the vertical basis in terms of the horizontal one
define the _$6j$ -symbol_, which can be expressed in terms of the
$\operatorname{Tet}$-network:
$\displaystyle|l\rangle$ $\displaystyle=\sum_{j}\begin{Bmatrix}{a}&{b}&{j}\\\
{c}&{d}&{l}\end{Bmatrix}_{KL}|\bar{j}\rangle,$ (7)
$\displaystyle\begin{Bmatrix}{a}&{b}&{j}\\\ {c}&{d}&{l}\end{Bmatrix}_{KL}$
$\displaystyle=\frac{\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
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}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}}{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\end{picture}}}}~{}\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}.$ (8)
Note the subscript $KL$ for Kauffman-Lins, since this $6j$-symbol is defined
with respect to the conventions of [15]. The relation to the classical Racah-
Wigner $6j$-symbol used in the physics literature is given explicitly in
Equation 53.
## 4 Recurrence relation for the $\operatorname{Tet}$-network
The identities given in the previous section allow an alternative, elementary
derivation of the three-term recurrence relation for the
$\operatorname{Tet}$-network, distinct from the standard one. The standard
derivation is given in [7] and another is possible using the general theory of
recurrences for $q$-hypergeometric functions [6], but neither directly yields
the symmetric eigenvalue problem form.
It is easy to check, using the bubble identity, that both the vertical and
horizontal bases are orthogonal and that they are normalized as
$\displaystyle\langle\bar{j}|\bar{j}\rangle$
$\displaystyle=\raisebox{-19.78601pt}{\leavevmode\hbox{\par\begin{picture}(750.0,762.0)(1426.0,-163.0){\color[rgb]{0,0,0}\put(1981.0,164.0){\line(1,0){135.0}}
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}{\color[rgb]{0,0,0}\put(1711.0,74.0){\oval(450.0,450.0)[bl]}
\put(1711.0,254.0){\oval(450.0,450.0)[tl]}
\put(1891.0,74.0){\oval(450.0,450.0)[br]}
\put(1891.0,254.0){\oval(450.0,450.0)[tr]}
\put(1711.0,-151.0){\line(1,0){180.0}} \put(1711.0,479.0){\line(1,0){180.0}}
\put(1486.0,74.0){\line(0,1){180.0}} \put(2116.0,74.0){\line(0,1){180.0}}
}{\color[rgb]{0,0,0}\put(1756.0,164.0){\oval(270.0,270.0)[bl]}
\put(1756.0,164.0){\oval(270.0,270.0)[tl]}
\put(1846.0,164.0){\oval(270.0,270.0)[br]}
\put(1846.0,164.0){\oval(270.0,270.0)[tr]} \put(1756.0,29.0){\line(1,0){90.0}}
\put(1756.0,299.0){\line(1,0){90.0}} \put(1621.0,164.0){\line(0,1){0.0}}
\put(1981.0,164.0){\line(0,1){0.0}}
}\put(2161.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1801.0,524.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1801.0,344.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1801.0,74.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1801.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1441.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\end{picture}}}~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}},$ (9) $\displaystyle\langle l|l\rangle$
$\displaystyle=\raisebox{-19.78601pt}{\leavevmode\hbox{\par\begin{picture}(660.0,762.0)(1201.0,-163.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\oval(180.0,450.0)[tl]}
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}{\color[rgb]{0,0,0}\put(1711.0,164.0){\oval(180.0,450.0)[br]}
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\put(1456.0,-76.0){\oval(210.0,210.0)[tl]}
\put(1606.0,-46.0){\oval(210.0,210.0)[br]}
\put(1606.0,-76.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-151.0){\line(1,0){150.0}} \put(1456.0,29.0){\line(1,0){150.0}}
}{\color[rgb]{0,0,0}\put(1456.0,404.0){\oval(210.0,210.0)[bl]}
\put(1456.0,374.0){\oval(210.0,210.0)[tl]}
\put(1606.0,404.0){\oval(210.0,210.0)[br]}
\put(1606.0,374.0){\oval(210.0,210.0)[tr]}
\put(1456.0,299.0){\line(1,0){150.0}} \put(1456.0,479.0){\line(1,0){150.0}}
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\put(1846.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1531.0,344.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1531.0,524.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}~{}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}}.$ (10)
Curiously, when these normalizations are fully expanded using formulas from
the Appendix, they take the form $(-)^{\sigma}P/Q$, where $P$ and $Q$ are
products of positive quantum integers. In both cases, $\sigma=(a+b+c+d)/2$, is
an integer independent of $j$ or $l$. When $q=1$ or when $q$ is a primitive
root of unity, positive quantum integers are positive real numbers. Hence the
above inner product is real and either positive- or negative-definite. On the
other hand, for arbitrary complex $q$, the normalizations (9) and (10) can be
essentially arbitrary complex numbers.
If we can find a linear operator $L$ that is diagonal in one basis, but not in
the other, then we can obtain $\langle\bar{j}|l\rangle$ as matrix elements of
the diagonalizing transformation. Furthermore, if the non-diagonal form of $L$
is tridiagonal, then the linear equations defining $\langle\bar{j}|l\rangle$
reduce to a three-term recurrence relation.
We can construct such an operator by generalizing the argument for the
classical case, found in the Appendix of [10]. For brevity of notation, we
introduce a special modified version of the trivalent vertex:
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}\put(2071.0,-466.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1531.0,-466.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2881.0,-466.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2341.0,-466.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2656.0,-286.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}2}}}}}
\put(2611.0,-646.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$[a]$}}}}}
\put(2206.0,-421.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$=$}}}}}
\end{picture}}}.$ (11)
The unlabeled edge implicitly carries twice-spin $2$ and the bold dot
indicates the multiplicative factor of $[a]$. Using it, we can define a
symmetric operator $L$. Its diagrammatic representation and its matrix
elements are given below.
The operator $L$ is diagonal in the $|l\rangle$-basis and its matrix elements
$L_{ll^{\prime}}=\langle l|L|l^{\prime}\rangle$ are
$\displaystyle L_{ll^{\prime}}$
$\displaystyle=\raisebox{-18.36328pt}{\leavevmode\hbox{\par\begin{picture}(660.0,717.0)(1201.0,-163.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\oval(180.0,450.0)[tl]}
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}{\color[rgb]{0,0,0}\put(1711.0,164.0){\oval(180.0,450.0)[br]}
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\put(1456.0,-76.0){\oval(210.0,210.0)[tl]}
\put(1606.0,-46.0){\oval(210.0,210.0)[br]}
\put(1606.0,-76.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-151.0){\line(1,0){150.0}} \put(1456.0,29.0){\line(1,0){150.0}}
}{\color[rgb]{0,0,0}\put(1456.0,404.0){\oval(210.0,210.0)[bl]}
\put(1456.0,374.0){\oval(210.0,210.0)[tl]}
\put(1606.0,404.0){\oval(210.0,210.0)[br]}
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\put(1666.0,209.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1666.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}~{}=\frac{[a][b]}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
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\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
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}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}~{}\delta_{ll^{\prime}}$ (12)
$\displaystyle=\lambda(a,b,l)\langle l|l^{\prime}\rangle,$ (13)
with
$\lambda(a,b,l)=\frac{\left[\frac{a-b+l}{2}\right]\left[\frac{-a+b+l}{2}\right]-\left[\frac{a+b-l}{2}\right]\left[\frac{a+b+l}{2}+2\right]}{[2]},$
(14)
where we have evaluated $\operatorname{Tet}(a,a,b,b;l,2)$ as
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}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}}{[a][b]}\lambda(a,b,l).$ (15)
This result may be obtained directly from Equation (34), where the sum reduces
to two terms, or from more fundamental considerations [20]. In the limit,
$q\to 1$, the eigenvalues simplify to
$\lambda(a,b,l)=\frac{1}{4}[l(l+2)-a(a+2)-b(b+2)]$, which shows that the
operator $L$ is closely related to the “square of angular momentum” in quantum
mechanics, which was used to obtain the classical version of this recurrence
relation [10].
On the other hand, in the $|\bar{j}\rangle$ basis, the operator $L$ is not
diagonal and the matrix elements
$\bar{L}_{jj^{\prime}}=\langle\bar{j}|L|\bar{j}^{\prime}\rangle$, making use
of the vertex collapse identity, are
$\bar{L}_{jj^{\prime}}=\raisebox{-18.36328pt}{\leavevmode\hbox{\par\begin{picture}(750.0,717.0)(1426.0,-163.0){\color[rgb]{0,0,0}\put(1801.0,479.0){\circle*{46.0}}
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}{\color[rgb]{0,0,0}\put(1981.0,164.0){\line(1,0){135.0}}
}{\color[rgb]{0,0,0}\put(1486.0,164.0){\line(1,0){135.0}}
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\put(1711.0,254.0){\oval(450.0,450.0)[tl]}
\put(1891.0,74.0){\oval(450.0,450.0)[br]}
\put(1891.0,254.0){\oval(450.0,450.0)[tr]}
\put(1711.0,-151.0){\line(1,0){180.0}} \put(1711.0,479.0){\line(1,0){180.0}}
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\put(1756.0,164.0){\oval(270.0,270.0)[tl]}
\put(1846.0,164.0){\oval(270.0,270.0)[br]}
\put(1846.0,164.0){\oval(270.0,270.0)[tr]} \put(1756.0,29.0){\line(1,0){90.0}}
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\put(1981.0,164.0){\line(0,1){0.0}}
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\put(1981.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1621.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
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\put(2161.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
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\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
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}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}},$ (16)
with the special case $\bar{L}_{00}=0$. Fortunately, though
$\bar{L}_{jj^{\prime}}$ is not diagonal, it is tridiagonal. This property is a
consequence of the conditions enforced at the central vertex in both
$\operatorname{Tet}$-networks above: the triangle inequality,
$|j-j^{\prime}|\leq 2$, and the parity constraint, which forces admissible
values of $j$ to change by $2$. If these conditions are violated, the matrix
element $\bar{L}_{jj^{\prime}}$ vanishes.
The diagonal elements $\bar{L}_{jj}$ can be evaluated using (15). For the off-
diagonal elements $\bar{L}_{j+2,j}=\bar{L}_{j,{j+2}}$ we also need
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\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}=\frac{1}{[a]}\left[\frac{a+d-j}{2}\right]~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}},$ (17)
which can be obtained in the same way as (15). Finally, we need the identities
$\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}=-\frac{[j+2]}{[2][j]}~{}\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}\quad\text{and}\quad\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
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}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\end{picture}}}=\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\end{picture}}}\phantom{j+{}}.$ (18)
The $|j\rangle$-basis matrix elements can now be expressed as (again, recall
the special case $\bar{L}_{00}=0$)
$\displaystyle\bar{L}_{jj}$ $\displaystyle=-\langle
j|j\rangle\frac{[2]\lambda(a,j,d)\lambda(b,j,c)}{[j][j+2]},$ (19)
$\displaystyle\bar{L}_{j,j+2}$ $\displaystyle=\langle
j{+}2|j{+}2\rangle\left[\frac{a+d-j}{2}\right]\left[\frac{b+c-j}{2}\right].$
(20)
The transition matrix elements $\langle\bar{j}|l\rangle$ can now be obtained
by solving an eigenvalue problem in the $|\bar{j}\rangle$-basis:
$\displaystyle\langle\bar{j}|L-\lambda_{l}|l\rangle$
$\displaystyle=\sum_{j^{\prime}}\frac{\langle\bar{j}|L-\lambda_{l}|\bar{j}^{\prime}\rangle}{\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle}\langle\bar{j}^{\prime}|l\rangle,$
(21) $\displaystyle 0$
$\displaystyle=\sum_{j^{\prime}}\left(\frac{\bar{L}_{jj^{\prime}}}{\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle}-\lambda_{l}\delta_{jj^{\prime}}\right)\langle\bar{j}^{\prime}|l\rangle,$
(22) $\displaystyle 0$
$\displaystyle=\sum_{j^{\prime}}\left(\bar{L}_{jj^{\prime}}-\lambda_{l}\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle\delta_{jj^{\prime}}\right)\frac{\langle\bar{j}^{\prime}|l\rangle}{\langle\bar{j}^{\prime}|j^{\prime}\rangle},$
(23)
where $\lambda_{l}=\lambda(a,b,l)$. Since $\bar{L}_{jj^{\prime}}$ is
tridiagonal, we obtain a three-term recurrence relation for the
$\langle\bar{j}|l\rangle$ transition coefficients. Expanding the expression
for $\bar{L}_{jj^{\prime}}$, we find the following general form of the
recurrence relation:
$\frac{\bar{L}_{j,j-2}}{\langle\overline{j-2}|\overline{j-2}\rangle}\langle\overline{j-2}|l\rangle+\left(\frac{\bar{L}_{jj}}{\langle\bar{j}|\bar{j}\rangle}-\lambda_{l}\right)\langle\bar{j}|l\rangle+\frac{\bar{L}_{j,j+2}}{\langle\overline{j+2}|\overline{j+2}\rangle}\langle\overline{j+2}|l\rangle=0,$
(24)
with the provision that $\bar{L}_{jj^{\prime}}$ vanishes whenever either of
the indices fall outside the admissible range or $j=j^{\prime}=0$. Finally,
the transition coefficients are uniquely determined (up to sign) by requiring
the normalization condition
$\sum_{j}\frac{\langle
l|\bar{j}\rangle\langle\bar{j}|l\rangle}{\langle\bar{j}|\bar{j}\rangle}=\langle
l|l\rangle.$ (25)
Practically, it is more convenient to recover the correct normalization for
all $j$ and fixed $l$, or vice versa, by requiring $\langle\bar{j}|l\rangle$
to agree with (34) for $j=\underline{j}$, cf. (41), where the sum reduces to a
single term.
Once the $\operatorname{Tet}$-network has been evaluated recursively, the
$6j$-symbol can be obtained from Equation (7). Alternatively, a linear, three-
term recurrence relation directly for the $6j$-symbol follows from (24) and
the linear, two-term recurrence relations for the bubble and
$\theta$-networks, obvious from (29) and (30). However, because of the
additional normalization factors in Equation (7), this direct recurrence
relation cannot be cast in the form of a symmetric eigenvalue problem like
(23) using rational operations alone.
The author thanks Dan Christensen for many helpful discussions as well as
Nicolai Reshetikhin and an anonymous referee for pointing out previous work on
the above recurrence relation. In the course of this work, the author was
supported by Postgraduate (PGS) and Postdoctoral (PDF) Fellowships from the
Natural Science and Engineering Research Council (NSERC) of Canada.
## Appendix A Formulas
For a complex number $q\neq 0$ and an integer $n$ the corresponding _quantum
integer_ is defined as
$[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}.$ (26)
In the limit $q\to 1$, we recover the regular integers, $[n]\to n$. When
$q=\exp(i\pi/r)$, for some integer $r>1$, it is a _primitive root of unity_
and the definition reduces to
$[n]=\frac{\sin(n\pi/r)}{\sin(\pi/r)},$ (27)
This expression is clearly real and positive in the range $0<n<r$. _Quantum
factorials_ are direct analogs of classical factorials:
$[0]!=1,\quad[n]!=[1][2]\cdots[n].$ (28)
Next, we give the evaluations of some spin networks needed in the paper. They
are reproduced from Ch. 9 of [15]. The _bubble diagram_ evaluates to
$\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}=(-)^{j}[j+1]$ (29)
whenever it is non-vanishing. For generic $q$, it vanishes if $j<0$ and if $q$
is a primitive root of unity then it also vanishes when $j>r-2$. The _$\theta$
-network_ evaluates to
$\theta(a,b,c)=\frac{(-)^{s}[s+1]![s-a]![s-b]![s-c]!}{[a]![b]![c]!},$ (30)
with $s=(a+b+c)/2$, whenever the twice-spins $(a,b,c)$ are admissible and
vanishes otherwise. Admissibility consists of the following criteria (besides
the obvious $a,b,c\geq 0$):
triangle inequalities $\displaystyle\left\\{\begin{aligned} a&\leq b+c\\\
b&\leq c+a\\\ c&\leq a+b\end{aligned}\right.,$ (31) parity $\displaystyle
a+b+c\equiv 0\pmod{2}.$ (32) When $q$ is a primitive root of unity, further
constraints needs to be satisfied: $r$-boundedness
$\displaystyle\left\\{\begin{aligned} a,b,c&\leq r-2\\\ a+b+c&\leq
2r-4\end{aligned}\right..$ (33)
The _tetrahedral-_ or _$\operatorname{Tet}$ -network_ evaluates to
$\operatorname{Tet}(a,b,c,d;j,l)=\frac{\mathcal{I}!}{\mathcal{E}!}\sum_{S}\frac{(-)^{S}[S+1]!}{\prod_{\imath}[S-a_{\imath}]!\prod_{\jmath}[b_{\jmath}-S]!},$
(34)
where the summation is over the range $m\leq S\leq M$ and
$\displaystyle\mathcal{I}!$
$\displaystyle=\prod_{\imath,\jmath}[b_{\jmath}-a_{\imath}]!,$
$\displaystyle\mathcal{E}!$ $\displaystyle=[a]![b]![c]![d]![j]![l]!,$ (35)
$\displaystyle a_{1}$ $\displaystyle=(a+d+j)/2,$ $\displaystyle b_{1}$
$\displaystyle=(b+d+j+l)/2,$ (36) $\displaystyle a_{2}$
$\displaystyle=(b+c+j)/2,$ $\displaystyle b_{2}$ $\displaystyle=(a+c+j+l)/2,$
(37) $\displaystyle a_{3}$ $\displaystyle=(a+b+l)/2,$ $\displaystyle b_{3}$
$\displaystyle=(a+b+c+d)/2,$ (38) $\displaystyle a_{4}$
$\displaystyle=(c+d+l)/2,$ $\displaystyle m$
$\displaystyle=\max\\{a_{\imath}\\},$ (39) $\displaystyle M$
$\displaystyle=\min\\{b_{\jmath}\\}.$ (40)
The indices $\imath$ and $\jmath$ fully span the defined ranges. Each of the
triples of twice-spins $(a,b,l)$, $(c,d,l)$, $(a,d,j)$ and $(c,b,j)$ must be
admissible, otherwise the $\operatorname{Tet}$-network vanishes. Then, due to
parity constraints, $a_{\imath}$, $b_{\jmath}$, $m$, $M$, and $S$ are always
integers. If the twice-spins $(a,b,c,d)$ are fixed, the admissibility
conditions for generic $q$ enforce the ranges of $\underline{j}\leq
j\leq\overline{j}$ and $\underline{l}\leq l\leq\overline{l}$ to
$\displaystyle\underline{j}$ $\displaystyle=\max\\{|a-d|,|b-c|\\},$
$\displaystyle\overline{j}$ $\displaystyle=\min\\{a+d,b+c\\},$ (41)
$\displaystyle\underline{l}$ $\displaystyle=\max\\{|a-b|,|c-d|\\},$
$\displaystyle\overline{l}$ $\displaystyle=\min\\{a+b,c+d\\},$ (42)
with
$\displaystyle j$ $\displaystyle\equiv a+b\equiv c+d\pmod{2},$ (43)
$\displaystyle l$ $\displaystyle\equiv a+d\equiv b+c\pmod{2}.$ (44)
The number of admissible values is the same for $j$ and $l$ and is equal to
$n=\max\\{0,\bar{n}\\}$, where
$\displaystyle\bar{n}$ $\displaystyle=\min\\{m,s-M\\}+1,$ $\displaystyle m$
$\displaystyle=\min\\{a,b,c,d\\},$ (45) $\displaystyle s$
$\displaystyle=(a+b+c+d)/2,$ $\displaystyle M$
$\displaystyle=\max\\{a,b,c,d\\}.$ (46)
This number $n$ is also the dimension of the space of $4$-valent spin networks
with fixed twice-spins $(a,b,c,d)$ labeling the free edges. This dimension is
non-vanishing, $n>0$, precisely when the twice-spins satisfy the conditions
$\displaystyle a+b+c+d$ $\displaystyle\leq 2\max\\{a,b,c,d\\},$ (47)
$\displaystyle a+b+c+d$ $\displaystyle\equiv 0\pmod{2}.$ (48)
When $q$ is a primitive root of unity, the admissible ranges shrink to
$\underline{l}\leq j\leq\overline{j}_{r}$ and $\underline{l}\leq
l\leq\overline{l}_{r}$, where
$\displaystyle\overline{j}_{r}$
$\displaystyle=\min\left\\{\overline{j},r-2,2r-4-\max\\{a+d,b+c\\}\right\\},$
(49) $\displaystyle\overline{l}_{r}$
$\displaystyle=\min\left\\{\overline{l},r-2,2r-4-\max\\{a+b,c+d\\}\right\\}.$
(50)
The number of admissible values in each range is thus restricted to
$n=\max\\{0,\bar{n}_{r}\\}$, where
$\bar{n}_{r}=\min\left\\{\bar{n},r-1-\max\\{M,s-m\\}\right\\}.$ (51)
The condition $n>0$ requires (47), (48) and
$a+b+c+d\leq 2\min\\{a,b,c,d\\}+2r-4.$ (52)
The above admissibility criteria are well known. However, the consequent
explicit expressions for the constraints on $(a,b,c,d)$, the bounds on $j$ and
$l$, and the dimension $n$ are not easily found in the literature.
In the classical $q=1$ case, the Kauffman-Lins version of the $6j$-symbol (7)
differs from the Racah-Wigner convention used in the physics literature, which
preserves the symmetries of the underlying $\operatorname{Tet}$-network. The
two $6j$-symbols are related through the formula
$\begin{Bmatrix}{j_{1}/2}&{j_{2}/2}&{j_{3}/2}\\\
{J_{1}/2}&{J_{2}/2}&{J_{3}/2}\end{Bmatrix}_{RW}=\frac{\operatorname{Tet}(J_{1},J_{2},j_{1},j_{2};J_{3},j_{3})}{\sqrt{|\theta(J_{1},J_{2},j_{3})\theta(j_{1},j_{2},j_{3})\theta(J_{1},j_{2},J_{3})\theta(J_{2},j_{1},J_{3})}|},$
(53)
which can be obtained by comparing the explicit expressions (34) and (6.3.7)
of [17]. Note that the argument of the absolute value under the square root
has sign $(-)^{j_{3}-J_{3}}$.
## References
## References
* [1] Edward Witten. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121(3):351–399, 1989.
* [2] Lee Smolin. Quantum gravity with a positive cosmological constant. 2002\.
* [3] Igor Khavkine and J. Daniel Christensen. $q$-deformed spin foam models of quantum gravity. Classical and Quantum Gravity, 24(13):3271–3290, 2007.
* [4] Richard Askey and James Wilson. A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or $6-j$ Symbols. SIAM Journal on Mathematical Analysis, 10(5):1008, 1979.
* [5] A. N. Kirillov and N. Yu Reshetikhin. Representations of the algebra $U_{q}(sl(2))$, $q$-orthogonal polynomials and invariants of links. In Victor G. Kac, editor, Infinite-dimensional Lie Algebras and Groups, volume 7 of Advanced Series in Mathematical Physics, pages 285–339, Singapore, 1989. World Scientific.
* [6] I. I. Kachurik and A. U. Klimyk. General recurrence relations for clebsch-gordan coefficients of the quantum algebra $U_{q}(\mathrm{su}_{2})$. Journal of Physics A: Mathematical and General, 24(17):4009–4015, 1991.
* [7] Shun’ya Mizoguchi and Tsukasa Tada. 3-Dimensional Gravity and the Turaev-Viro Invariant. Progress of Theoretical Physics Supplement, 110:207–227, 1992.
* [8] Shun’ya Mizoguchi and Tsukasa Tada. Three-dimensional gravity from the Turaev-Viro invariant. Physical Review Letters, 68(12):1795–1798, 1992.
* [9] G. Ponzano and T. Regge. Semiclassical limit of racah coefficients. In F. Bloch, S. G. Cohen, A. De-Shalit, S. Sambursky, and I. Talmi, editors, Spectroscopic and group theoretical methods in physics, pages 1–98, Amsterdam, 1968. North-Holland.
* [10] Klaus Schulten and Roy G. Gordon. Exact recursive evaluation of 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta. Journal of Mathematical Physics, 16(10):1961–1970, 1975.
* [11] Yuka Taylor and Christopher Woodward. 6j symbols for u_q(sl_2) and non-Euclidean tetrahedra. Selecta Mathematica (New Series), 11(3):539–571, 2006.
* [12] P. P. Raychev, R. P. Roussev, and Yu F. Smirnov. The quantum algebra SUq(2) and rotational spectra of deformed nuclei. Journal of Physics G: Nuclear and Particle Physics, 16(8):L137, 1990\.
* [13] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users’ Guide (Software, Environments and Tools). Society for Industrial Mathematics, 3 edition, 1987.
* [14] R. Penrose. Applications of negative dimension tensors. In D. J. A. Welsh, editor, Combinatorial Mathematics and its Applications, pages 221–244. Academic Press, 1971.
* [15] Louis H. Kauffman and Sóstenes Lins. Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, volume 134 of Annals of Mathematics Studies. Princeton University Press, 1994.
* [16] J. Scott Carter, Daniel E. Flath, and Masahico Saito. The Classical and Quantum 6j-symbols, volume 43 of Mathematical Notes. Princeton University Press, 1995.
* [17] A. R. Edmonds. Angular Momentum in Quantum Mechanics. Princeton Landmarks in Physics. Princeton University Press, 1996.
* [18] A. P. Yutsis, J. B. Levinson, and V. V. Vanagas. Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Scientific Translation, 1962.
* [19] J. Wade Cherrington, J. Daniel Christensen, and Igor Khavkine. Dual computations of non-abelian yang-mills theories on the lattice. Physical Review D, 76(9):094503, 2007.
* [20] G. Massbaum and P. Vogel. 3-valent graphs and the kauffman bracket. Pacific Journal of Mathematics, 164(2):361–381, 1994.
|
arxiv-papers
| 2010-09-12T19:33:16 |
2024-09-04T02:49:12.837558
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Igor Khavkine",
"submitter": "Igor Khavkine",
"url": "https://arxiv.org/abs/1009.2261"
}
|
1009.2322
|
# Deterministic Online Call Control in Cellular Networks and Triangle-Free
Cellular Networks
Joseph Wun-Tat Chan College of International Education, Hong Kong Baptist
University, Hong Kong, cswtchan@gmail.com Francis Y.L. Chin Department of
Computer Science, The University of Hong Kong, Hong Kong, chin@cs.hku.hk,
Research supported by HK RGC grant HKU-7117/09E and the William M.W. Mong
Engineering Research Fund Xin Han School of Software, Dalian University of
Technology, China, hanxin.mail@gmail.com. Partially supported by Start-up
Funding (1600-893335) provided by DUT, China Ka-Cheong Lam College of Computer
Science, Zhejiang University, China, pandaman@163.com Hing-Fung Ting
Department of Computer Science, The University of Hong Kong, Hong Kong,
hfting@cs.hku.hk, Research supported by HK RGC grant HKU-7171/08E Yong Zhang
Department of Computer Science, The University of Hong Kong, Hong Kong,
yzhang@cs.hku.hk
###### Abstract
Wireless Communication Networks based on Frequency Division Multiplexing (FDM
in short) plays an important role in the field of communications, in which
each request can be satisfied by assigning a frequency. To avoid interference,
each assigned frequency must be different to the neighboring assigned
frequencies. Since frequency is a scarce resource, the main problem in
wireless networks is how to fully utilize the given bandwidth of frequencies.
In this paper, we consider the online call control problem. Given a fixed
bandwidth of frequencies and a sequence of communication requests arrive over
time, each request must be either satisfied immediately after its arrival by
assigning an available frequency, or rejected. The objective of call control
problem is to maximize the number of accepted requests. We study the
asymptotic performance of this problem, i.e., the number of requests in the
sequence and the bandwidth of frequencies are very large. In this paper, we
give a 7/3-competitive algorithm for call control problem in cellular network,
improving the previous 2.5-competitive result. Moreover, we investigate the
triangle-free cellular network, propose a 9/4-competitive algorithm and prove
that the lower bound of competitive ratio is at least 5/3.
Keywords: Online algorithms, Call control problem, Cellular networks,
Triangle-free cellular network
## 1 Introduction
Frequency Division Multiplexing (FDM in short) is commonly used in wireless
communications. To implement FDM, the wireless network is partitioned into
small regions (cell) and each cell is equipped with a base station. When a
call request arrives at a cell, the base station in this cell will assign a
frequency to this request, and the call is established via this frequency.
Since frequency is a scarce resource, to satisfy the requests from many users,
a straightforward idea is reusing the same frequency for different call
requests. But if two calls which are close to each other are using the same
frequency, interference will happen to violate the quality of communications.
Thus, to avoid interference, the same frequency cannot be assigned to two
different calls with distance close to each other. In general, the same
frequency cannot be assigned to two calls in the same cell or neighboring
cells.
There are two research directions on the fully utilization of the frequencies.
One is frequency assignment problem, and the other is call control problem. In
frequency assignment problem, each call request must be accepted, and the
objective is to minimize the number of frequencies to satisfy all requests. In
call control problem, the bandwidth of frequency is fixed, thus, when the
number of call requests in a cell or in some neighboring cells is larger than
the total bandwidth, the request sequence cannot be totally accepted, i.e.,
some requests would be rejected. The objective of call control problem is to
accept the requests as many as possible.
Problem Statement:
In this paper, we consider the online version of call control problem. There
are $\omega$ frequencies available in the wireless networks. A sequence
$\sigma$ of call requests arrives over time, where
$\sigma=\\{r_{1},r_{2},...,r_{t},...\\}$, $r_{t}$ denotes the $t$-th call
request and also represent the cell where the $t$-th request arrives. When a
request arrives at a cell, the system must either choose a frequency to
satisfy this request without interference with other assigned frequencies in
this cell and its neighboring cells, or reject this request. When handling a
request, the system does not know any information about future call requests.
We assume that when a frequency is assigned to a call, this call will never
terminate and the frequency cannot be changed. The objective of this problem
is to maximize the number of accepted requests.
We focus on the call control problem in cellular networks and triangle-free
cellular networks. In the cellular networks, each cell is a hexagonal region
and has six neighbors, as shown in Figure 1(a). The cellular network is widely
used in wireless communication networks. A network is triangle-free if there
are no 3-cliques in the network, i.e., there are no three mutually-adjacent
cells. An example of a triangle-free cellular network is shown in Fig. 1(b).
(a) cellular network
(b) triangle-free cellular network
Figure 1: An example of the cellular network and triangle-free cellular
network
Performance Measure:
To measure the performance of online algorithms, we use the competitive ratio
to compare the performance between the online algorithm and the optimal
offline algorithm, which knows the whole request sequence in advance. In call
control problem, the output is the number of accepted requests. For a request
sequence $\sigma$, let $A(\sigma)$ and $O(\sigma)$ denote the number of
accepted request of an online algorithm $A$ and the optimal offline algorithm
$O$, respectively. We focus on the asymptotic performance for the call control
problem, i.e., the number of requests and the number of frequencies are very
large positive integers. The asymptotic competitive ratio for an online
algorithm $A$ is
$R_{A}^{\infty}=\limsup_{n\rightarrow\infty}\max_{\sigma}\\{\frac{O(\sigma)}{A(\sigma)}|O(\sigma)=n\\}.$
Related Works:
How to fully utilize the frequencies to satisfy the communication requests is
a very fundamental problem in theoretical computer science and wireless
communications. Both the frequency assignment problem and the call control
problem are well studied during these years. From the description of these two
problems, we know that the call control problem is the dual problem of the
frequency assignment problem.
The offline version of the frequency assignment problem in cellular networks
was proved to be NP-hard by McDiarmid and Reed [6], and two 4/3-approximation
algorithms were given in [6, 7]. In the online frequency assignment problem,
when a call request arrives, the network must immediately assign a frequency
to this call without any interference. There are mainly three strategies:
Fixed Allocation [5], Greedy Assignment [1], and Hybrid Assignment [3]. If the
duration of each call is infinity and the assigned frequency cannot be
changed, the hybrid algorithm gave the best result for online frequency
assignment, i.e., a 2-competitive algorithm for the absolute performance and a
1.9126-competitive algorithm for the asymptotic performance. When the
background network is triangle-free, a 2-local 5/4-competitive algorithm was
given in [9], an inductive proof for the 7/6 ratio was reported in [4], where
$k$-local means when assigning a frequency, the base station only knows the
information of its neighboring cells within distance $k$. In [11], a 1-local
4/3-competitive algorithm was given.
For the call control problem, the offline version is NP-hard too [6]. To
handle such problem, greedy strategy is always the first try, when a call
request arrives, the network choose the minimal available frequency to serve
this request, if any frequency is interfere with some neighboring assigned
frequency, the request will be rejected. Pantziou et al. [8] analyzed the
performance of the greedy strategy, proved that the asymptotic competitive
ratio of the greedy strategy is equal to the maximal degree of the network.
Caragiannis et al. [1] gave a randomized algorithm for the call control
problem in cellular networks, the asymptotic competitive ratio of their
algorithm is 2.651. Later, the performance of the randomized algorithms was
improved to 16/7 by the same authors [2], they also proved the lower bound of
the asymptotic competitive ratio for the randomized algorithm is at least 2.
Very recently, a deterministic algorithm with asymptotic competitive ratio 2.5
was given in [10], and the lower bound of the asymptotic competitive ratio for
the deterministic algorithm was proved to be 2.
Our Contributions:
In this paper, we consider the deterministic algorithms for the online call
control problem in cellular networks and triangle-free cellular networks. In
cellular network, we give a 7/3-competitive algorithm, improving the previous
2.5-competitive result. In triangle-free network, we propose a 9/4-competitive
algorithm, moreover, we show that the lower bound of the competitive ratio in
triangle-free network is at least 5/3.
## 2 Call Control in Cellular Networks
The idea of our algorithm for call control problem in cellular networks is
similar to the algorithm in [10]. By using a totally different analysis, the
performance of our algorithm is better. Moreover, our algorithm is best
possible among algorithms using this kind of idea.
Cellular networks are 3 colorable, each cell can be associated with a color
from $\\{R,G,B\\}$ and any two neighboring cells are with different colors.
Partition the frequencies into four sets, $F_{R}$, $F_{B}$, $F_{G}$, and
$F_{S}$, where $F_{X}$ ($X\in\\{R,G,B\\}$) can be only used in cells with
color $X$ and $F_{S}$ can be used in any cell. Since we consider the
asymptotic performance of the call control problem, we may regard the number
$\omega$ of frequencies in the system is a multiple of 7. Divide the the
frequencies into four disjoint sets as follows:
$\begin{array}[]{ll}F_{R}=&\\{1,...,2\omega/7\\},\\\
F_{G}=&\\{2\omega/7+1,...,4\omega/7\\},\\\
F_{B}=&\\{4\omega/7+1,...,6\omega/7\\},and\\\
F_{S}=&\\{6\omega/7+1,...,\omega\\}\end{array}$
Obviously, the ratio between the number of frequencies in $F_{R}$, $F_{G}$,
$F_{B}$, and $F_{S}$ is $2:2:2:1$.
Now we describe our algorithm CACO as follows:
Algorithm 1 CACO: When a request arrives at a cell $C$ with color
$c\in\\{R,G,B\\}$
1:if $F_{c}$ is not totally used up then
2: assign the minimal available frequency from $F_{c}$ to satisfy this
request.
3:else if $F_{S}$ is not totally used up in cell $C$ and its neighboring cells
then
4: assign the minimal available frequency from $F_{S}$ to satisfy this
request.
5:else
6: reject this request.
7:end if
The high level idea to show the performance of our algorithm CACO is to prove
that the ratio between the total number of accepted requests by CACO and the
total number of satisfied requests by the optimal offline algorithm is at
least 3/7. To prove this, we analyze the number of satisfied requests in each
cell and its neighboring cells, then compare the number with the optimum
value.
Let $R_{i}$ be the number of the requests arrived in cell $C_{i}$. Let $O_{i}$
be the number of requests accepted by the optimal offline algorithm in cell
$C_{i}$. $\sum_{i}O_{i}$ is the total number of accepted request by the
optimal offline algorithm. Let $A_{i}$ be the number of requests accepted by
our online algorithm CACO in cell $C_{i}$. $\sum_{i}A_{i}$ is the total number
of accepted request by CACO. Let $A_{x}(C_{i})$ be the the number of requests
accepted by CACO in cell $C_{i}$ by assigning frequencies from frequency set
$F_{x}$. It can be seen that
$A_{i}=A_{R}(C_{i})+A_{G}(C_{i})+A_{B}(C_{i})+A_{S}(C_{i})$. If $C_{i}$ is
colored with $x\in\\{R,G,B\\}$, then $A_{i}=A_{x}(C_{i})+A_{S}(C_{i})$.
###### Fact 1.
For each cell $C_{i}$, $O_{i}\leq R_{i}$, $A_{i}\leq R_{i}$, and $A_{i}\geq
2\omega/7$ when $R_{i}\geq 2\omega/7$.
According to the number of satisfied requests by the optimal offline
algorithm, we classify the cells into two types: cell $C_{i}$ is $safe$ if
$O_{i}\leq 2\omega/3$, and $dangerous$ otherwise.
###### Lemma 2.
Suppose cell $C_{i}$ is with color $x$, if $C_{i}$ is safe, then $A_{i}\geq
3O_{i}/7$
###### Proof.
Consider the following two cases.
* •
$R_{i}\leq 2\omega/7$
According to CACO, all requests in this cell must be satisfied when $R_{i}\leq
2\omega/7$, thus, $A_{i}=R_{i}$. Since $R_{i}\geq O_{i}$, we have $A_{i}\geq
3O_{i}/7$.
* •
$R_{i}>2\omega/7$
In this case, CACO will accept at least $2\omega/7$ requests by assigning
frequencies from $F_{x}$, thus, $A_{i}\geq 2\omega/7$. Since $C_{i}$ is safe,
$O_{i}\leq 2\omega/3$, therefore, we have $A_{i}\geq 3O_{i}/7$.
Combining the above two cases, this lemma is true. ∎
###### Fact 3.
A safe cell has at most 3 dangerous neighboring cells. All neighboring cells
around a dangerous cell are safe.
###### Proof.
This fact can be proved by contradiction. If a safe cell $C$ has more than 3
dangerous neighboring cells, since $C$ has 6 neighboring cells, there must
exist two dangerous cells which are neighbors. From the definition of
dangerous cell, the total number of accepted request in these two dangerous
neighboring cells is strictly more than $\omega$, contradiction!
Similarly, if a dangerous cell $C^{\prime}$ is a neighboring cell of another
dangerous cell $C$, the total number of accepted request in $C$ and
$C^{\prime}$ is strictly more than $\omega$. Contradiction! ∎
According to the algorithm CACO, when a request cannot be satisfied in a cell
$C$ with color $c$, all frequencies in $F_{c}$ must be used in $C$, and all
frequencies in $F_{S}$ must be used in $C$ and its six neighbors. Thus, we
have the following fact:
###### Fact 4.
If cell $C$ cannot satisfy a request according to the algorithm CACO, then
$A_{S}(C)+\sum_{k}A_{S}(C_{k})\geq\omega/7$, where $C_{k}$ represents the
neighboring cell of $C$.
To compare the number of satisfied requests by CACO in each cell with the
optimal offline solution, we define $B_{i}$ as follows, where $C_{k}$
represents the neighboring cell of $C_{i}$.
$B_{i}=\left\\{\begin{array}[]{ll}3O_{i}/7&\textrm{if $C_{i}$ is safe}\\\
A_{i}+\sum_{k}(A_{k}-3O_{k}/7)/3&\textrm{if $C_{i}$ is dangerous.}\\\
\end{array}\right.$
###### Lemma 5.
$\sum_{i}B_{i}\leq\sum_{i}A_{i}$.
###### Proof.
Suppose $C_{k}$ is a safe cell. According to Lemma 2, we have $A_{k}\geq
3O_{k}/7$. From Fact 3, we know that there are at most three dangerous
neighbors around $C_{k}$, thus, after counting $B_{k}=3O_{k}/7$ frequencies in
$C_{k}$, the remaining $A_{k}-3O_{k}/7$ frequencies can compensate the
frequencies in its dangerous neighbors, and each dangerous cell receives
$(A_{k}-3O_{k}/7)/3$ frequencies. From the definition of $B_{i}$, we can see
that $\sum_{i}B_{i}\leq\sum_{i}A_{i}$. ∎
###### Theorem 6.
The asymptotic competitive ratio of algorithm CACO is at most 7/3.
###### Proof.
From the definition of $O_{i}$ and $B_{i}$, we can say that $O_{i}/B_{i}\leq
7/3$ for any cell $C_{i}$ leads to the correctness of this theorem. That is
because
$\frac{\sum_{i}O_{i}}{\sum_{i}A_{i}}\leq\frac{\sum_{i}O_{i}}{\sum_{i}B_{i}}\leq\max_{i}\frac{O_{i}}{B_{i}}.$
If the cell $C_{i}$ is safe, i.e., $O_{i}\leq 2\omega/3$, we have
$O_{i}/B_{i}=7/3$.
If the cell $C_{i}$ is dangerous, i.e., $O_{i}>2\omega/3$, since $R_{i}\geq
O_{i}>2\omega/3>3\omega/7$, that means the number of requests $R_{i}$ in this
cell is larger than $A_{i}$. Thus, some requests are rejected, and this cell
cannot accept any further requests.
* •
If the number of accepted requests in any neighbor of $C_{i}$ is no more than
$2\omega/7$, that means all frequencies in $F_{S}$ are assigned to requests in
cell $C_{i}$. Thus, $A_{i}=3\omega/7$. In this case, we have
$O_{i}/B_{i}=O_{i}/(A_{i}+(\sum_{k}(A_{k}-3O_{k}/7))/3)\leq
O_{i}/A_{i}\leq\omega/A_{i}=7/3.$
* •
Otherwise, suppose there are $m$ neighbors of $C_{i}$ in which the number of
accepted requests are more than $2\omega/7$. Let $\hat{O_{i}}$ denote the
average number of the optimum value of accepted requests in these $m$
neighboring cells around $C_{i}$.
$\displaystyle B_{i}=A_{i}+(\sum_{k}(A_{k}-3O_{k}/7))/3$ $\displaystyle=$
$\displaystyle 2\omega/7+A_{S}(C_{i})+(\sum_{k}(A_{k}-3O_{k}/7))/3$
$\displaystyle\geq$ $\displaystyle 2\omega/7+A_{S}(C_{i})+(m\times
2\omega/7+\sum_{\begin{subarray}{c}\textrm{for the neighbors}\\\ \textrm{with
$A_{k}>2\omega/7$}\end{subarray}}A_{S}(C_{k})-m\times 3\hat{O_{i}}/7)/3$
$\displaystyle\geq$ $\displaystyle 2\omega/7+(m\times
2\omega/7+\sum_{\begin{subarray}{c}\textrm{for the neighbors}\\\ \textrm{with
$A_{k}>2\omega/7$}\end{subarray}}A_{S}(C_{k})+A_{S}(C_{i})-m\times
3\hat{O_{i}}/7)/3$ $\displaystyle\geq$ $\displaystyle 2\omega/7+(m\times
2\omega/7+\omega/7-m\times 3\hat{O_{i}}/7)/3$ $\displaystyle\geq$
$\displaystyle 2\omega/7+(2\omega/7+\omega/7-3\hat{O_{i}}/7)/3$
$\displaystyle(\textrm{that is because for any neighbor with }A_{k}>2w/7,$
$\displaystyle O_{k}\leq(\omega-
O_{i})\leq\omega/3,\textrm{thus},\hat{O_{i}}\leq\omega/3\textrm{ and
}2\omega/7-3\hat{O_{i}}/7\geq 0.)$ $\displaystyle\geq$ $\displaystyle
2\omega/7+(3\omega/7-3(\omega-O_{i})/7)/3$ $\displaystyle(\textrm{since
}O_{k}\leq\omega-O_{i}\textrm{, we have }\hat{O_{i}}\leq\omega-O_{i})$
$\displaystyle=$ $\displaystyle 2\omega/7+O_{i}/7$
Thus, $O_{i}/B_{i}\leq O_{i}/(2w/7+O_{i}/7)\leq 7/3$.
From the above analysis, we can say that the asymptotic competitive ratio of
the algorithm CACO is at most 7/3. ∎
In this kind of algorithms, the frequencies are partitioned into $F_{R}$,
$F_{G}$, $F_{B}$ and $F_{S}$, when a request arrives at a cell with color $c$,
first choose the frequency from the set $F_{c}$, then from $F_{S}$ if no
interference appear. The performances are different w.r.t. the ratio between
$|F_{R}|$ ($|F_{G}|$, $|F_{B}|$) and $|F_{S}|$. Note that from symmetry, the
size of $F_{R}$, $F_{G}$ and $F_{B}$ should be same. Now we show that CACO is
best possible among such kind of algorithms. Suppose the ratio between
$|F_{R}|$ and $|F_{S}|$ is $x:y$. Consider the configuration shown in Figure
2. In the first step, $\omega$ requests arrive at the center cell $C$ with
color $c$, the algorithm will use up all frequencies in $F_{c}$ and $F_{S}$,
in this case, the ratio of accepted requests by the optimal offline algorithm
and the online algorithm is $(3x+y)/(x+y)$ since the optimal algorithm will
accept all these requests. In the second step, $\omega$ requests arrive at
$C_{1}$, $C_{2}$ and $C_{3}$ with the same color $c^{\prime}$. The online
algorithm can only accept $x\omega/(3x+y)$ requests in each $C_{i}$ ($1\leq
i\leq 3$) since the frequencies in $F_{S}$ are all used in $C$. In this case,
the ratio between the optimal offline algorithm and the online algorithm is
$3(3x+y)/(4x+y)$ since the optimal algorithm will accept all $\omega$ requests
in $C_{i}$ ($1\leq i\leq 3$) and reject all requests in $C$. Balancing these
two ratios, we have $x:y=2:1$. From the description of the above two steps,
the lower bound of competitive ratio for this kind of algorithm is $7/3$.
Figure 2: Algorithm CACO is best possible among this kind of algorithms
## 3 Call Control in Triangle-Free Cellular Networks
The call control problem in cellular network is hard. But for some various
graph classes, this problem may have a better performance. For example, in
linear network, an optimal online algorithm with competitive ratio 3/2 can be
achieved [10]. An interesting induced network, triangle-free cellular network,
has been studied for many problems including frequency assignment problem[4,
9, 11].
For a cell $C_{i}$ in triangle-free cellular networks, there are only two
possible configurations for its neighboring cells, which are shown in Fig. 3.
If $C_{i}$ has 3 neighbors, the neighboring vertices are of the same color. On
the other hand, if the neighbors are of different colors, $C_{i}$ has 2
neighbors. There exists a simple structure in triangle-free cellular network,
i.e., a cell has only one neighbor, this structure can be regarded as the case
in Fig. 3(b).
(a) Structure A: neighbors with the same base color
(b) Structure B: neighbors with different base colors
Figure 3: Structure of neighboring cells
For the three base colors $R$, $G$ and $B$, we define a cyclic order among
them as $R\rightarrow G$, $G\rightarrow B$ and $B\rightarrow R$. Partition the
frequency set $\\{1,...,\omega\\}$ into three disjoint sets:
$F_{R}=\\{1,...,\omega/3\\},\emph{
}F_{G}=\\{\omega/3+1,...,2\omega/3\\},\emph{
}F_{B}=\\{2\omega/3+1,...,\omega\\}$
To be precisely, assigning frequencies from a set must in order of bottom-to-
top (assigning frequencies from the lower number to the higher number) or top-
to-bottom (assigning frequencies from the higher number to the lower number).
Now we describe our algorithm for call control problem in triangle-free
cellular networks.
Algorithm CACO2: Handling arrival requests in a cell $C$ with color
$X\in\\{R,G,B\\}$
1. 1.
If cell $C$ has no neighbors, just assign frequencies from 1 to $\omega$.
2. 2.
If cell $C$ has neighboring structure $A$ (Fig. 3(a)), let $Y$ be the base
color of $C$’s neighbors and $Z$ be the other third color. Assign frequency in
cell $C$ as follows if no interference appear:
1. (a)
Assign frequencies from $F_{X}$ in bottom-to-top order.
2. (b)
If all frequencies in $F_{X}$ are used up, assign frequencies from $F_{Z}$ in
bottom-to-top order if $X\rightarrow Y$; and in top-to-bottom order otherwise.
Such assignment guarantees that if $C$ uses the frequency from $F_{Z}$ after
using up all frequencies from $F_{X}$, and its neighboring cell $C^{\prime}$
also uses the frequency from $F_{Z}$ after using up the frequencies from
$F_{Y}$, $C$ and $C^{\prime}$ must assign frequency from $F_{Z}$ in different
order no matter what the neighbor configuration of $C^{\prime}$ is. (This can
be verified by checking this case (case 2) and the next case (case 3) of
CACO2.)
3. 3.
If cell $C$ has neighboring configuration $B$ (Fig. 3(b)), let $Y$ and $Z$ be
the base colors of its two neighbors, respectively. Without loss of
generality, assume that $X\rightarrow Y$. Assign frequency in cell $C$ as
follows if no interference appear:
1. (a)
Assign frequencies from $F_{X}$ in bottom-to-top order.
2. (b)
If all frequencies in $F_{X}$ are used up, assign frequencies from $F_{Y}$ in
top-to-bottom order.
###### Theorem 7.
The competitive ratio of CACO2 is at most 9/4.
###### Proof.
For a given request sequence, let $O_{i}$ and $A_{i}$ be the numbers of
accepted requests in cell $C_{i}$ by the optimal offline algorithm and online
algorithm CACO2, respectively. This theorem holds if
$\sum_{i}O_{i}/\sum_{i}A_{i}\leq 9/4$. Similar to the analysis for CACO,
define $B_{i}$ as the amortized number of accepted requests in cell $C_{i}$.
Again, our target is to prove that $\sum_{i}B_{i}\leq\sum_{i}A_{i}$ and
$O_{i}/B_{i}\leq 9/4$ in any cell $C_{i}$. W.l.o.g., let $X$, $Y$ and $Z$
denote the three colors in the network.
Intuitively, we may set $B_{i}=4O_{i}/9$ if $A_{i}\geq 4O_{i}/9$, and the
remaining uncounted frequencies can be used to compensate the number of
accepted frequencies in its neighboring cells. Next, we describe how to
partition the remaining uncounted frequencies according to cell $C_{i}$’s
neighboring configuration. Let $H_{ij}$ be the number of frequencies used in
$C_{i}$ and compensates the number of frequencies in $C_{j}$.
1. 1.
The neighboring configuration of $C_{i}$ is $A$ (Fig. 3(a)), the uncounted
number of frequencies is $A_{i}-4O_{i}/9$, evenly distribute this number to
its three neighboring cells, i.e., each neighbors $C_{j}$ of $C_{i}$ receives
$H_{ij}=(A_{i}-4O_{i}/9)/3$.
2. 2.
The neighboring configuration of $C_{i}$ is $B$ (Fig. 3(b)). Assume that the
color of $C_{i}$ is $X$, the colors of its neighboring cells are $Y$ (cell
$C_{j}$) and $Z$ (cell $C_{k}$) respectively. W.l.o.g., assume that
$X\rightarrow Y$, $Y\rightarrow Z$ and $Z\rightarrow X$.
* •
If $A_{i}>\omega/3$,
In this case, the requests in cell $C_{i}$ will use some frequencies from the
top part of $F_{Y}$.
* –
If $A_{j}<4O_{j}/9$,
there exist rejected request in $C_{j}$, thus, $A_{i}+A_{j}=2\omega/3$. The
remaining uncounted number of frequencies in $C_{i}$ is partitioned into
$(4O_{j}/9-A_{j})$ and $\omega/9$. The former part $(4O_{j}/9-A_{j})$
compensates the number in $C_{j}$ (i.e., $H_{ij}=4O_{j}/9-A_{j}$), and the
latter part $\omega/9$ compensates the number in $C_{k}$ (i.e.,
$H_{ik}=\omega/9$) if $A_{k}<4O_{k}/9$. This compensation is justified since
$4O_{i}/9+(4O_{j}/9-A_{j})+\omega/9=4(O_{i}+O_{j})/9-A_{j}+\omega/9\leq
5\omega/9-A_{j}<A_{i}$.
* –
If $A_{j}\geq 4O_{j}/9$,
in this case, no compensation is needed in $C_{j}$. Let
$H_{ik}=A_{i}-4O_{i}/9$ if $A_{k}<4O_{k}/9$.
* •
If $A_{i}\leq\omega/3$,
In this case, all frequencies used in $C_{i}$ are from $F_{X}$, and some
frequencies used in $C_{k}$ may from $F_{X}$ too. If $A_{k}<4O_{k}/9$, all
remaining uncounted number $A_{i}-4O_{i}/9$ compensates the number in $C_{k}$,
i.e., $H_{ik}=A_{i}-4O_{i}/9$. No extra number of frequencies compensates the
number of frequencies in $C_{j}$, i.e., $H_{ij}=0$.
We define $B_{i}$ as follows, where $H_{ji}$ is the compensation from its
neighbor $C_{j}$.
$B_{i}=\left\\{\begin{array}[]{ll}4O_{i}/9&\textrm{if $A_{i}\geq 4O_{i}/9$}\\\
A_{i}+\sum_{j}H_{ji}&\textrm{if $A_{i}<4O_{i}/9$},\\\ \end{array}\right.$
From previous description, we have $4O_{i}/9+\sum_{j}H_{ij}\leq A_{i}$ if
$A_{i}\geq 4O_{i}/9$, thus,
$\displaystyle\sum_{i}B_{i}\emph{ }\hbox
to0.0pt{$\displaystyle=\sum_{A_{i}\geq
4O_{i}/9}4O_{i}/9+\sum_{A_{i}<4O_{i}/9}(A_{i}+\sum_{\textrm{$C_{i}$ and
$C_{j}$ are neighbors}}H_{ji})$\hss}$ $\displaystyle=$
$\displaystyle\sum_{A_{i}\geq 4O_{i}/9}(4O_{i}/9+\sum_{\textrm{$C_{i}$ and
$C_{j}$ are neighbors}}H_{ij})+\sum_{A_{i}<4O_{i}/9}A_{i}$ $\displaystyle\leq$
$\displaystyle\sum_{A_{i}\geq 4O_{i}/9}A_{i}+\sum_{A_{i}<4O_{i}/9}A_{i}$
$\displaystyle=$ $\displaystyle\sum_{i}A_{i}$
Now we analyze the relationship between $B_{i}$ and $O_{i}$ for any cell
$C_{i}$. Assuming that the color of $C_{i}$ is $X$.
1. 1.
If $A_{i}\geq 4O_{i}/9$, $B_{i}=4O_{i}/9$.
2. 2.
If $A_{i}<4O_{i}/9$,
1. (a)
If $A_{i}<\omega/3$
Since $A_{i}<4O_{i}/9$, there must exist some rejected requests in $C_{i}$.
Some frequencies in $F_{X}$ are used in one of $C_{i}$’s neighbor $C_{j}$.
According to the algorithm, the neighboring structure of $C_{j}$ is $B$ (Fig.
3(b)), and $A_{i}+A_{j}=2\omega/3$. We say that $A_{j}\geq 4O_{j}/9$.
Otherwise, $A_{i}+A_{j}<4O_{i}/9+4O_{j}/9=4(O_{i}+O_{j})/9\leq 4\omega/9$,
contradiction!
In this case, $H_{ji}=4O_{i}/9-A_{i}$, thus,
$B_{i}=A_{i}+\sum_{\textrm{$C_{k}$ and $C_{i}$ are neighbors}}H_{ki}\geq
A_{i}+H_{ji}=4O_{i}/9.$
2. (b)
If $A_{i}\geq\omega/3$ and $C_{i}$ has two neighbors $C_{j}$ with color $Y$
and $C_{k}$ with color $Z$ as shown in Fig. 3(b).
W.l.o.g., assume that $X\rightarrow Y$, $Y\rightarrow Z$ and $Z\rightarrow X$.
According to the algorithm, after using up the frequencies in $F_{X}$, $C_{i}$
will use some frequencies from $F_{Y}$ until interference appear, thus,
$A_{i}+A_{j}\geq 2\omega/3$. We claim that $A_{j}>4O_{j}/9$. That is because
$O_{j}\leq\omega-O_{i}<\omega-9A_{i}/4\leq\omega-9\omega/12=\omega/4$,
$A_{i}<4O_{i}/9\leq 4\omega/9$, and $A_{i}+A_{j}\geq 2\omega/3$. Thus, we have
$A_{j}\geq 2\omega/9>4O_{j}/9$.
1. i.
If the neighboring configuration around $C_{j}$ is $A$ (Fig. 3(a)),
$H_{ji}=(A_{j}-4O_{j}/9)/3$, and
$\displaystyle B_{i}$ $\displaystyle\geq$ $\displaystyle A_{i}+H_{ji}$
$\displaystyle=$ $\displaystyle A_{i}+(A_{j}-4O_{j}/9)/3$ $\displaystyle=$
$\displaystyle 2A_{i}/3+(A_{i}+A_{j})/3-4O_{j}/27$ $\displaystyle\geq$
$\displaystyle 4\omega/9-4O_{j}/27$ $\displaystyle\geq$ $\displaystyle
4O_{i}/9$
2. ii.
If the neighboring configuration around $C_{j}$ is $B$ (Fig. 3(b)),
* •
If $A_{j}\leq\omega/3$, we have $H_{ji}=A_{j}-4O_{j}/9$. Thus,
$B_{i}\geq A_{i}+H_{ji}=A_{i}+A_{j}-4O_{j}/9\geq 2\omega/3-4O_{j}/9\geq
4O_{i}/9.$
* •
If $A_{j}>\omega/3$, according to the description of the compensation,
$H_{ji}=\omega/9$ or $H_{ji}=A_{j}-4O_{j}/9$. In the former case,
$B_{i}\geq A_{i}+H_{ji}=A_{i}+\omega/9\geq\omega/3+\omega/9=4\omega/9\geq
4O_{i}/9.$
In the latter case,
$B_{i}\geq A_{i}+H_{ji}=A_{i}+A_{j}-4O_{j}/9\geq 2\omega/3-4O_{j}/9\geq
4O_{i}/9.$
3. (c)
If $A_{i}\geq\omega/3$ and the neighbors of $C_{i}$ are of the same color
(Fig. 3(a)),
Assume that the color of its neighboring cell is $Y$. According to the
algorithm, after using up the frequencies from $F_{X}$, $C_{i}$ will use some
frequencies from $F_{Z}$ to satisfy some requests. Since $C_{i}$ rejects some
requests, we have either $A_{i}=2\omega/3$, or $A_{i}+A_{j}=\omega$ for some
neighboring cell $C_{j}$ of $C_{i}$, which is because $C_{i}$ and $C_{j}$
assign frequencies from $F_{Z}$ in different order, and $C_{j}$ will use the
frequency from $F_{Z}$ after using up the frequency from $F_{Y}$. Since
$A_{i}<4O_{i}/9\leq 4\omega/9$, the first case does not happen.
We claim that $A_{j}\geq 4O_{j}/9$, which is because $A_{j}=\omega-
A_{i}>\omega-4O_{i}/9\geq 5\omega/9>4O_{j}/9$.
* •
If the neighboring configuration of $C_{j}$ is $A$ (Fig. 3(a)),
$H_{ji}=(A_{j}-4O_{j}/9)/3$. Thus,
$\displaystyle B_{i}$ $\displaystyle\geq$ $\displaystyle A_{i}+H_{ji}$
$\displaystyle=$ $\displaystyle A_{i}+(A_{j}-4O_{j}/9)/3$ $\displaystyle=$
$\displaystyle 2A_{i}/3+(A_{i}+A_{j})/3-4O_{j}/27$ $\displaystyle\geq$
$\displaystyle 5\omega/9-4O_{j}/27$ $\displaystyle>$ $\displaystyle 4O_{i}/9$
* •
If the neighboring configuration of $C_{j}$ is $B$ (Fig. 3(b)),
$H_{ji}=\omega/9$ or $A_{j}-4O_{j}/9$. In the former case,
$B_{i}\geq A_{i}+H_{ji}=A_{i}+\omega/9\geq 4\omega/9\geq 4O_{i}/9.$
In the latter case,
$B_{i}\geq A_{i}+H_{ji}=A_{i}+A_{j}-4O_{j}/9=\omega-4O_{j}/9\geq 4O_{i}/9.$
Combine all above cases, we have $O_{i}/B_{i}\leq 9/4$ in each cell $C_{i}$.
Since $\sum_{i}B_{i}\leq\sum_{i}A_{i}$, we have
$\sum_{i}O_{i}/\sum_{i}A_{i}\leq 9/4$. ∎
Next, we prove that the lower bound of the competitive ratio for call control
problem in triangle-free cellular networks is at least 5/3.
###### Theorem 8.
The competitive ratio for call control problem in triangle-free cellular
network is at least 5/3.
###### Proof.
We prove the lower bound by using an adversary who sends requests according to
the assignment of the online algorithm.
Figure 4: lower bound of competitive ratio is at least 5/3
Consider the configuration shown in Figure 4.
In the first step, the adversary sends $\omega$ requests in the center cell
$C$. Suppose the online algorithm accepts $x$ requests. If $x\leq 3\omega/5$,
the adversary stop sending request. In this case, the optimal offline
algorithm can accept all these $\omega$ requests, thus, the ratio is at least
$5/3$.
If $x>3\omega/5$, the adversary then sends $\omega$ requests in each cell of
$C_{1}$, $C_{2}$ and $C_{3}$. To avoid interference, the online algorithm
accepts at most $\omega-x$ requests in each cell, and the total number of
accepted requests is $x+3(\omega-x)=3\omega-2x$. In this case, the optimal
offline algorithm will accept $3\omega$ requests, i.e., reject all requests in
the center cell $C$. Thus, the ratio in this case is $3\omega/(3\omega-2x)$.
Since $x>3\omega/5$, this value is at least $5/3$.
Combine the above two cases, the competitive ratio for call control problem in
triangle-free cellular network is at least 5/3. ∎
## 4 Concluding Remarks
We have studied online call control problem in wireless communication networks
and presented online algorithms in cellular networks and triangle-free
cellular networks. In cellular networks, we derived an upper bound of 7/3,
while in triangle-free cellular networks, the upper bound and lower bound we
achieved in this paper are 9/4 and 5/3, respectively. These bounds surpass the
previous best known results. The gap between the upper and lower bound is
quite big in both cases, and closing the gaps are very interesting problems
for future research.
## References
* [1] Ioannis Caragiannis, Christos Kaklamanis, and Evi Papaioannou. Efficient on-line frequency allocation and call control in cellular networks. Theory Comput. Syst., 35(5):521-543, 2002. A preliminary version of the paper is in SPAA 2000.
* [2] Ioannis Caragiannis, Christos Kaklamanis, and Evi Papaioannou. Competitive Algorithms and Lower Bounds for On-Line Randomized Call Control in Cellular Networks. Networks 52(4): 235-251, 2008. Preliminary versions are in WAOA 03 and EUROPAR 05.
* [3] Wun-Tat Chan, Francis Y.L. Chin, Deshi Ye and Yong Zhang. Online Frequency Allocation in Cellular Networks. In Proc. of the 19th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2007), pp. 241-249.
* [4] Fr$\acute{\textmd{e}}$d$\acute{\textmd{e}}$ric Havet. Channel assignment and multicoloring of the induced subgraphs of the triangular lattice. Discrete Math. 233, 219-231 (2001).
* [5] V. H. MacDonald. Advanced mobile phone service: The cellular concept. Bell Systems Technical Journal, 58(1):15-41, 1979.
* [6] Colin McDiarmid and Bruce Reed. Channel assignment and weighted coloring. Networks, 36(2):114-117, 2000.
* [7] Lata Narayanan, and Sunil Shende. Static frequency assignment in cellular networks. Algorithmica, 29(3):396-409, 2001.
* [8] Grammati E. Pantziou, George P. Pentaris, and Paul G. Spirakis Competitive Call Control in Mobile Networks. Theory of Computing Systems, 35(6): 625-639, 2002.
* [9] Petra $\check{\textmd{S}}$parl, and Janez $\check{\textmd{Z}}$erovnik. 2-local 5/4-competitive algorithm for multicoloring triangle-free hexagonal graphs. Inf. Process. Lett. 90, 239-246 (2004)
* [10] Deshi Ye, Xin Han, and Guochuan Zhang. Deterministic On-line Call Control in Cellular Networks. Theor. Comput. Sci. 411(31-33): 2871-2877 (2010)
* [11] Yong Zhang, Francis Y.L. Chin, and Hong Zhu. A 1-Local Asymptotic 13/9-Competitive Algorithm for Multicoloring Hexagonal Graphs. Algorithmica (2009) 54:557-567.
|
arxiv-papers
| 2010-09-13T08:36:19 |
2024-09-04T02:49:12.844819
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joseph Wun-Tat Chan, Francis Y.L. Chin, Xin Han, Ka-Cheong Lam,\n Hing-Fung Ting, and Yong Zhang",
"submitter": "Yong Zhang",
"url": "https://arxiv.org/abs/1009.2322"
}
|
1009.2375
|
# Multidimensional Kruskal–Katona theorem222The paper is in public domain, and
is not protected by copyright. The paper is available at arXiv:1009.2375.
Boris Bukh111B.Bukh@dpmms.cam.ac.uk. Centre for Mathematical Sciences,
Cambridge CB3 0WB, England and Churchill College, Cambridge CB3 0DS, England.
###### Abstract
We present a generalization of a version of the Kruskal–Katona theorem due to
Lovász. A shadow of a $d$-tuple $(S_{1},\dotsc,S_{d})\in\binom{X}{r}^{d}$
consists of $d$-tuples
$(S_{1}^{\prime},\dotsc,S_{d}^{\prime})\in\binom{X}{r-1}^{d}$ obtained by
removing one element from each of the $S_{i}$. We show that if a family
$\mathcal{F}\subset\binom{X}{r}^{d}$ has size
$\lvert\mathcal{F}\rvert=\binom{x}{r}^{d}$ for a real number $x\geq r$, then
the shadow of $\mathcal{F}$ has size at least $\binom{x}{r-1}^{d}$.
## Introduction
An $r$-uniform set family $\mathcal{F}$ is simply a collection of $r$-element
sets. The shadow of $\mathcal{F}$, denoted $\partial\mathcal{F}$, consists of
all $(r-1)$-element sets that can be obtained by removing an element from a
set in $\mathcal{F}$. If $(X,<)$ is an ordered set, then $A\subset X$ is
colexicographically smaller than $B\subset X$ if the largest element of
$(A\cup B)\setminus(A\cap B)$ lies in $B$.
The Kruskal–Katona theorem [Kru63, Kat68] is a classic result in combinatorics
that states that $\lvert\partial\mathcal{F}\rvert\geq\lvert\partial
F_{0}\rvert$, where $\mathcal{F}_{0}$ is the initial segment of length
$\lvert\mathcal{F}\rvert$ in colexicographic order on $r$-tuples of some
ordered set. Moreover equality is achieved only if $\mathcal{F}$ is an initial
segment of such a colexicographic order. As the quantitative form of the
Kruskal–Katona theorem is unwieldy, in applications one usually uses the
weaker form due to Lovász [Lov79, Ex. 13.31(b)]: if
$\lvert\mathcal{F}\rvert=\binom{x}{r}$ for some real number333For real $x$ and
integer $r$ the binomial coefficient $\binom{x}{r}$ is defined by
$x(x-1)\dotsb(x-r+1)/r!$. $x\geq r$, then $\lvert\partial
F\rvert\geq\binom{x}{r-1}$.
In this paper we present a generalization of Lovász’s theorem to
multidimensional $r$-uniform families. A _$d$ -dimensional $r$-uniform family_
is a collection of $d$-tuples of $r$-element sets. In other words, if we
denote by $\binom{X}{r}$ the family of all $r$-element subsets of $X$, then a
$d$-dimensional $r$-uniform family is a subset of $\binom{X}{r}^{d}$. A
_shadow_ of such a family $\mathcal{F}\subset\binom{X}{r}^{d}$ is defined to
be
$\partial\mathcal{F}\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\\{(S_{1}\setminus\\{x_{i}\\},\dotsc,S_{d}\setminus\\{x_{d}\\}):(S_{1},\dotsc,S_{d})\in\mathcal{F},\text{and
}x_{i}\in S_{i}\text{ for }i=1,\dotsc,d\\}.$
The special case $d=1$ of the following theorem is Lovász’s result.
###### Theorem 1.
Suppose $\mathcal{F}\subset\binom{X}{r}^{d}$ is a $d$-dimensional $r$-uniform
family of size
$\lvert\mathcal{F}\rvert=\binom{x}{r}^{d},$
where $x\geq r$ is a real number. Then
$\lvert\partial\mathcal{F}\rvert\geq\binom{x}{r-1}^{d}.$
Moreover, equality holds only if $\mathcal{F}$ is of the form
$\binom{Y_{1}}{r}\times\dotsb\times\binom{Y_{d}}{r}$ for some sets
$Y_{1},\dotsc,Y_{d}\subset X$.
The rest of the paper contains the proof of this result.
## Proof
For simplicity of notation we shall assume that the ground set is
$[n]\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\\{1,2,\dotsc,n\\}$, with
the ordering on it being the standard ordering of the integers. This incurs no
loss of generality.
A $k$-dimensional _section_ of a $d$-dimensional family
$\mathcal{F}\subset\binom{X}{r}^{d}$ is the subfamily of $\mathcal{F}$
obtained by fixing $d-k$ coordinates. For example, for any $(d-k)$-tuple
$S=(S_{1},\dotsc,S_{d-k})\in\binom{X}{r}^{d-k}$ the family
$\mathcal{F}_{S}\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\bigl{\\{}(S_{d-k+1},\dotsc,S_{d})\in\binom{X}{r}^{k}:(S_{1},\dotsc,S_{d})\in\mathcal{F}\bigr{\\}}$
is a $k$-section of $\mathcal{F}$. In general, any $d-k$ coordinates might be
fixed, not necessarily the first $d-k$.
We say that a family $\mathcal{F}\subset\binom{X}{r}^{d}$ is _monotone_ if
every $1$-dimensional section is an initial segment in the colexicographic
order.
###### Lemma 2 (Proof deferred to p. Deferred lemmas).
For every family $\mathcal{F}\subset\binom{[n]}{r}^{d}$ there is a monotone
family $\mathcal{F}_{0}\subset\binom{[n]}{r}^{d}$ of the same size as
$\mathcal{F}$, and such that
$\lvert\partial\mathcal{F}_{0}\rvert\leq\lvert\partial\mathcal{F}\rvert$.
By the Lemma 2 it suffices to restrict the attention to monotone families. The
shadows of monotone families are most easily described using the
colexicographic ordering. This will permit us to establish a correspondence
between $d$-dimensional monotone families and subsets of $\mathbb{N}^{d}$. Let
$\mathbb{N}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\\{1,2,\dotsc\\}$ be
the set of positive integers, and partially order $\mathbb{N}^{d}$ by
$(x_{1},\dotsc,x_{d})\leq(y_{1},\dotsc,y_{d})\text{ whenever }x_{i}\leq
y_{i}\text{ for every }i=1,\dotsc,d.$ (1)
A set $L\subset\mathbb{N}^{d}$ is said to be _monotone_ if whenever
$x=(x_{1},\dotsc,x_{d})\in L$, then $L$ contains all the elements smaller than
$x$.
If $S\in\binom{[n]}{r}$ is the $i$’th in the colexicographic ordering on
$\binom{[n]}{r}$, then we put $\operatorname{ind}_{r}(S)=i$. A tuple
$S=(S_{1},\dots,S_{d})\in\binom{[n]}{r}^{d}$ is mapped to
$\operatorname{ind}_{r}(S)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}(\operatorname{ind}_{r}(S_{1}),\dotsc,\operatorname{ind}_{r}(S_{d}))$.
In this manner every $\mathcal{F}\subset\binom{[n]}{r}^{d}$ is associated with
its image $\operatorname{ind}_{r}(\mathcal{F})\subset\mathbb{N}^{d}$. An
_extreme point_ of a monotone set $L\subset\mathbb{N}^{d}$ is a point $x\in L$
such that no point in $L$ is larger than $x$. The set of extreme points of $L$
will be denoted $\operatorname{extr}L$. The _monotone closure_ of a set
$L\subset\mathbb{N}^{d}$ is the set
$\operatorname{mclos}(L)=\\{x\in\mathbb{N}^{d}:x\leq y\text{ for some }y\in
L\\}$. It is clear that $L=\operatorname{mclos}\operatorname{extr}L$ for any
finite set $L$.
For an integer $m\geq 1$ let $KK_{r}(m)$ be the size of a shadow of the
initial segment of length $m$ in colexicographic order of $\binom{[n]}{r}$.
The Kruskal–Katona theorem states that if $\mathcal{F}\subset\binom{[n]}{r}$,
then $\lvert\partial\mathcal{F}\rvert\geq KK_{r}(\lvert\mathcal{F}\rvert)$. We
extend the definition of $KK_{r}$ to
$KK_{r}\colon\mathbb{N}^{d}\to\mathbb{N}^{d}$ by
$KK_{r}(a_{1},\dotsc,a_{d})\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}(KK_{r}(a_{1}),\dotsc,KK_{r}(a_{d}))$.
###### Lemma 3 (Proof deferred to p. Deferred lemmas).
Let $\mathcal{F}\subset\binom{[n]}{r}^{d}$ be a monotone family. Then its
shadow $\partial\mathcal{F}$ is also a monotone family, and
$\operatorname{extr}\operatorname{ind}_{r-1}(\partial\mathcal{F})=KK_{r}(\operatorname{extr}\operatorname{ind}_{r}(\mathcal{F})).$
The preceding lemma permits us to forget about shadows of set families, and
instead think about images of monotone sets under $KK_{r}$. However, as
$KK_{r}$ is quite an erratic function, our next step is to replace it by a
smoother function. For an integer $r\geq 2$ put
$LL_{r}\Bigl{(}\binom{x}{r}\Bigr{)}=\binom{x}{r-1}\qquad\text{ if }x\geq r.$
(2)
Since $\binom{x}{r}$ is an increasing function of $x$ for $x\geq r-1$, the
function $LL_{r}$ is well-defined on $[1,\infty)$. We would like to extend
$LL_{r}$ to $[0,1)$ while maintaining the inequality $LL_{r}\leq KK_{r}$.
Furthermore, as it will become clear below, it will be essential for $LL_{r}$
to be increasing, concave and to satisfy
$x\frac{f^{\prime}(x)}{f(x)}<y\frac{f^{\prime}(y)}{f(y)}\qquad\text{when
}x>y.$ (3)
Any extension of $LL_{r}$ to $[0,\infty)$ satisfying these conditions is
equally good for us. For example, one permissible extension is
$LL_{r}(x)=r\left(x+\frac{1}{\sum_{i=1}^{r}1/i}(x-x^{2})\right)\qquad\text{ if
}0\leq x\leq 1.$ (4)
###### Lemma 4 (Proof deferred to p. Deferred lemmas).
The function $LL_{r}$ defined by (2) and (4) is a continuously differentiable
function that is strictly increasing, concave, and satisfies (3).
Extend $LL_{r}$ to $LL_{r}\colon\mathbb{R}_{+}^{d}\to\mathbb{R}_{+}^{d}$ by
$LL_{r}(x_{1},\dotsc,x_{d})\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}(LL_{r}(x_{1}),\dotsc,LL_{r}(x_{d}))$. Put
$\mathbb{R}_{+}=[0,\infty)$. Partially order $\mathbb{R}_{+}^{d}$ according to
(1), and extend the definitions of the terms “monotone” and “extreme point” in
the obvious way. We associate to every monotone set $L\subset\mathbb{N}^{d}$
the set $M\subset\mathbb{R}_{+}^{d}$ given by $M=L+[-1,0]^{d}$. Geometrically,
$M$ is the set obtaining by filling in the square lattice boxes indexed by
$L$. The volume of $M$ is equal to the number of points in $L$. The set $M$ so
obtained is monotone. Since $LL_{r}(0)=0$ and $LL_{r}\leq KK_{r}$, Lemma 3
implies that if $\lvert\partial\mathcal{F}\rvert\leq t$ for some family
$\mathcal{F}\subset\binom{[n]}{r}^{d}$, then there is a closed monotone set
$M\subset\mathbb{R}_{+}^{d}$ with
$\operatorname{vol}(M)=\lvert\mathcal{F}\rvert$ for which
$\operatorname{vol}(LL_{r}(M))\leq t$. The Theorem 1 thus follows from the
following claim.
###### Claim 5.
Suppose $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuously
differentiable, strictly increasing, concave function satisfying (3) and
$f(0)=0$. Define $f\colon\mathbb{R}_{+}^{d}\to\mathbb{R}_{+}^{d}$ by
$f(x_{1},\dotsc,x_{d})=(f(x_{1}),\dotsc,f(x_{d}))$. Then for every closed
monotone set $M\subset\mathbb{R}_{+}^{d}$ we have
$\operatorname{vol}(f(M))\geq\operatorname{vol}(f(M_{0}))$
where $M_{0}=[0,\sqrt[d]{\operatorname{vol}(M)}]^{d}$ is the cube of the same
volume as $M$, and one of whose vertices is at the origin. Furthermore
equality holds only if $M=M_{0}$.
To prove the claim we shall first establish it in the dimension $d=2$, and use
that to deduce the general case. Indeed, assume that the two-dimensional case
is known, $d\geq 3$, and $M$ is not a cube. Pick any $2$-dimensional
coordinate plane $P$. On each $2$-dimensional section of $M$ by a plane
parallel to $P$, replace the section of $M$ by a square of the same area as
the area of that section. The operation yields a monotone set, and by the case
$d=2$ of the claim, it reduces the volume of $f(M)$ unless every section of
$M$ is a square. Therefore, the only minimizer of $\operatorname{vol}(f(M))$
is the cube $[0,\sqrt[d]{\operatorname{vol}(M)}]^{d}$.
$\Delta X$$\Delta Y$
$\Delta X$$\Delta Y$
Figure 1: The area-reducing transformation for an elongated rectangle (left),
and for a general monotone set (right).
So assume $d=2$. To see where the condition (3) comes from consider the case
where $M$ is a rectangle, i.e. a set of the form $M=[0,X]\times[0,Y]$, with
say $X>Y$. In that case, if we are to move a small amount of mass from the
shorter side to the longer one, to obtain a less elongated rectangle
$M^{*}=[0,X-\Delta X]\times[0,Y+\Delta Y]$, then (3) is exactly what is
necessary to conclude that
$\operatorname{area}(f(M^{*}))<\operatorname{area}(f(M))$.
The situation when $M$ is not a rectangle is to our advantage because $f$ is
concave and we place the mass farther from the origin than in the case when
$M$ is a rectangle. The only complication is that we need to introduce
continuous time to avoid technicalities arising from discrete time increments.
Since $M$ is monotone there is a decreasing function
$g_{\infty}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ so that
$M=\\{(x,y)\in\mathbb{R}_{+}^{2}:y\leq g_{\infty}(x)\\}$. Since $M$ is closed,
$g_{\infty}$ is left-continuous. Define
$g_{t}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ by
$g_{t}(x)=\begin{cases}g_{\infty}(x)+\frac{1}{t}\int_{[t,\infty)}g_{\infty}(y)\,dy&\text{if
}x\leq t,\\\ 0&\text{if }x>t.\end{cases}$
Let $M_{t}=\\{(x,y)\in\mathbb{R}_{+}^{2}:y\leq g_{t}(x)\\}$. Then
$\operatorname{area}(M_{t})=\operatorname{area}(M)$. Differentiating
$\operatorname{area}(f(M_{t}))=\int_{[0,t]}f(g_{t}(x))f^{\prime}(x)\,dx,$
we obtain
$\displaystyle\frac{\partial\operatorname{area}(f(M_{t}))}{\partial t}$
$\displaystyle=f(g_{t}(t))f^{\prime}(t)+\int_{[0,t]}f^{\prime}(g_{t}(x))\frac{\partial
g_{t}}{\partial t}(x)f^{\prime}(x)\,dx$ $\displaystyle\geq
f(g_{t}(t))f^{\prime}(t)+f^{\prime}(g_{t}(t))\int_{[0,t]}\frac{\partial
g_{t}}{\partial t}(x)f^{\prime}(x)\,dx$
$\displaystyle=f(g_{t}(t))f^{\prime}(t)+f^{\prime}(g_{t}(t))\left(\frac{\partial
g_{t}}{\partial t}(t)f(t)-\int_{[0,t]}f(x)\frac{\partial^{2}g_{t}(x)}{\partial
x\partial t}\,dx\right)$
$\displaystyle=f(g_{t}(t))f^{\prime}(t)+f^{\prime}(g_{t}(t))\frac{\partial
g_{t}}{\partial t}(t)f(t),$
where the inequality holds since $f$ is concave, and $(\partial g_{t}/\partial
t)f^{\prime}$ is negative (see Figure 1 for a geometric illustration of the
inequality). Since $\partial g_{t}/\partial t\geq-g_{t}(t)/t$, from (3) it
follows that $\operatorname{area}(f(M_{t}))$ is an increasing function of $t$
as long as $g_{t}(t)<t$.
Let $T=\sqrt{\operatorname{area}(M)}$. Since $\operatorname{area}(M_{t})\geq
tg_{t}(t)$, it follows that $g_{t}<t$ for every $t>T$. Thus
$\operatorname{area}(f(M_{T}))\leq\operatorname{area}(f(M))$, with equality
only if $M\subset[0,T]\times\mathbb{R}_{+}$. Since $g_{T}(x)\leq
g_{\infty}(x)+\operatorname{area}(M)/T$ it follows that if
$M\subset\mathbb{R}_{+}\times[0,Y]$, then
$M_{T}\subset[0,T]\times[0,Y+\operatorname{area}(M)/T]=[0,T]\times[0,Y+T]$.
Reversing the roles of $x$ and $y$ axes, and applying the argument to $M_{T}$,
it follows that for every closed monotone set $M\subset\mathbb{R}_{+}^{2}$
there is a compact monotone set $M^{\prime}\subset[0,2T]\times[0,T]$ for which
$\operatorname{area}(f(M^{\prime}))\leq\operatorname{area}(f(M))$ with
equality holding only for $M=[0,T]^{2}$. Since the space of compact monotone
subsets of $[0,2T]\times[0,T]$ endowed with Hausdorff distance is a compact
space, and $\operatorname{area}(f(\cdot))$ is a continuous function on the
space, it follows that $[0,T]^{2}$ is a unique set minimizing this function.
This completes the proof of the Claim 5 in the case $d=2$.
## Deferred lemmas
###### Proof of Lemma 2.
For the duration of this proof define the _weight_ of
$\mathcal{F}\subset\binom{[n]}{r}^{d}$ to be
$\sum_{S\in\mathcal{F}}\lVert\operatorname{ind}_{r}(\mathcal{F})\rVert_{1}$,
where $\lVert(m_{1},\dotsc,m_{d})\rVert_{1}=m_{1}+\dotsb+m_{d}$. We may assume
that $\mathcal{F}$ has smallest weight among families of size
$\lvert\mathcal{F}\rvert$ and whose shadow does not exceed
$\lvert\partial\mathcal{F}\rvert$.
Suppose some $1$-dimensional section of $\mathcal{F}$ is not an initial
segment of the colexicographic order. Without loss of generality we may assume
that the section is of the form $\mathcal{F}_{S}$ for some $S$. Define a
compression operator $\Delta\colon 2^{\binom{[n]}{r}}\to 2^{\binom{[n]}{r}}$
which takes $\mathcal{F}\subset\binom{[n]}{r}$ to the initial segment of
$\binom{[n]}{r}$ in the colexicographic order. One can write $\mathcal{F}$ as
a disjoint union of its $1$-dimensional sections as
$\mathcal{F}=\bigcup_{S\in\binom{[n]}{r}^{d-1}}\\{S\\}\times\mathcal{F}_{S}.$
Define
$\mathcal{F}^{\prime}=\bigcup_{S\in\binom{[n]}{r}^{d-1}}\\{S\\}\times\Delta\mathcal{F}_{S}.$
We claim that
$\lvert\partial\mathcal{F}^{\prime}\rvert\leq\lvert\partial\mathcal{F}\rvert$.
Indeed, let $S^{\prime}\in\binom{[n]}{r-1}^{d-1}$ be arbitrary, and consider
the section $(\partial\mathcal{F}^{\prime})_{S^{\prime}}$. The section has at
least $t$ elements if and only if there is a $S\in\binom{[n]}{r}^{d-1}$ such
that $S^{\prime}\in\partial S$ and $KK_{r}(\lvert\mathcal{F}_{S}\rvert)\geq
t$. Hence, if $\lvert(\partial\mathcal{F}^{\prime})_{S^{\prime}}\rvert\geq t$,
then by the classical Kruskal–Katona inequality
$\lvert(\partial\mathcal{F})_{S^{\prime}}\rvert\geq t$. Since the inequality
holds for every $S^{\prime}$, it follows that
$\lvert\partial\mathcal{F}\rvert=\sum_{S^{\prime}\in\binom{[n]}{r-1}^{d-1}}\left\lvert(\partial\mathcal{F})_{S^{\prime}}\right\rvert\geq\sum_{S^{\prime}\in\binom{[n]}{r-1}^{d-1}}\left\lvert(\partial\mathcal{F}^{\prime})_{S^{\prime}}\right\rvert=\lvert\partial\mathcal{F}^{\prime}\rvert$
Since the weight of $\mathcal{F}^{\prime}$ is less than that of $\mathcal{F}$,
this contradicts the choice of $\mathcal{F}$. ∎
###### Proof of Lemma 3.
First we establish that $\partial\mathcal{F}$ is monotone. Suppose
$S=(S_{1},S_{2},\dotsc,S_{d})\in\partial\mathcal{F}$ and $S_{1}^{\prime}$
precedes $S_{1}$ in colexicographic order. There is an
$\bar{S}=(\bar{S}_{1},\dotsc,\bar{S}_{d})\in\mathcal{F}$ so that
$S\in\partial\bar{S}$. Since the shadow of an initial segment of
colexicographic order is an initial segment of colexicographic order, there is
an $\bar{S}_{1}^{\prime}\in\binom{[n]}{r}$ so that $\bar{S}_{1}^{\prime}$
precedes $\bar{S}_{1}$ in the order, and
$S_{1}^{\prime}\in\partial\bar{S}_{1}^{\prime}$. Thus
$(S_{1}^{\prime},S_{2},\dotsc,S_{d})\in\partial(\bar{S}_{1}^{\prime},S_{2},\dotsc,S_{d})\subset\partial\mathcal{F}$.
This shows that the $1$-dimensional section
$(\partial\mathcal{F})_{S_{2},\dotsc,S_{d}}$ of $\partial\mathcal{F}$ is
monotone. Since ordering of coordinates is arbitrary, it follows that every
$1$-dimensional section of $\mathcal{F}$ is monotone, i.e. $\mathcal{F}$ is
monotone.
From the definition of $KK_{r}$ it follows that
$\max\operatorname{ind}_{r-1}(\partial\mathcal{F}_{0})=KK_{r}(\lvert
F\rvert_{0})$ whenever $\mathcal{F}_{0}$ is the initial segment of
$\binom{[n]}{r}$ in the colexicographic order. The second claim of the Lemma
is then again a consequence of the fact that an image of an initial segment of
colexicographical order on $\binom{[n]}{r}$ is an initial segment on
$\binom{[n]}{r-1}$. ∎
###### Proof of Lemma 4.
It is clear that the function defined by (2) is a continuous monotone
increasing function. The concavity of $LL_{r}$ on $(1,\infty)$ follows from a
simple derivative calculation: Indeed, for $x\geq r$
$\displaystyle\frac{d}{dx}LL_{r}\Bigl{(}\binom{x}{r}\Bigr{)}$
$\displaystyle=\frac{d}{dx}\binom{x}{r-1},$ $\displaystyle
LL_{r}^{\prime}\Bigl{(}\binom{x}{r}\Bigr{)}\binom{x}{r}\left(\frac{1}{x}+\dotsb+\frac{1}{x-r+1}\right)$
$\displaystyle=\binom{x}{r-1}\left(\frac{1}{x}+\dotsb+\frac{1}{x-r+2}\right),$
$\displaystyle 1/LL_{r}^{\prime}\Bigl{(}\binom{x}{r}\Bigr{)}$
$\displaystyle=\frac{x-r+1}{r}\frac{\frac{1}{x}+\dotsb+\frac{1}{x-r+1}}{\frac{1}{x}+\dotsb+\frac{1}{x-r+2}},$
$\displaystyle 1/LL_{r}^{\prime}\Bigl{(}\binom{x}{r}\Bigr{)}$
$\displaystyle=\frac{1}{r}\left(x-r+1+\frac{1}{\frac{1}{x}+\dotsb+\frac{1}{x-r+2}}\right),$
from which it is clear that $LL_{r}^{\prime}$ is decreasing on $(1,\infty)$.
Moreover this expression for $LL_{r}^{\prime}$ and
$LL\Bigl{(}\binom{x}{r}\Bigr{)}/\binom{x}{r}=\binom{x}{r-1}/\binom{x}{r}=r/(x-r+1)$
imply that
$\frac{LL_{r}\Bigl{(}\binom{x}{r}\Bigr{)}}{\binom{x}{r}LL_{r}^{\prime}\Bigl{(}\binom{x}{r}\Bigr{)}}=1+\frac{1}{(x-r+1)\left(\frac{1}{x}+\dotsb+\frac{1}{x-r+2}\right)}.$
Since $(x-r+1)/(x-t)$ is a decreasing function of $x$ for every $t<r-1$, it
follows that
$\frac{LL_{r}\Bigl{(}\binom{x}{r}\Bigr{)}}{\binom{x}{r}LL_{r}^{\prime}\Bigl{(}\binom{x}{r}\Bigr{)}}$
is increasing, i.e. $LL_{r}$ satisfies (3) on $(1,\infty)$.
Since $x-x^{2}$ is concave, the function given by (4) is concave on $[0,1)$.
For brevity of notation put $\epsilon\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\left(\sum_{i=1}^{r}1/i\right)^{-1}$. Monotonicity of $LL_{r}$ on
$[0,1)$ follows from $\epsilon<1$. Furthermore, for $x\in[0,1)$ we have
$x\frac{LL_{r}^{\prime}(x)}{LL_{r}(x)}=x\frac{1+\epsilon(1-2x)}{x+\epsilon(x-x^{2})}=2-\frac{1+\epsilon}{1+\epsilon(1-x)},$
from which we see that $LL_{r}$ satisfies (3) on $[0,1)$. Finally, it is easy
to check that at $x=1$ the function $LL_{r}(x)$ is continuous and the left and
right derivatives agree. ∎
## Concluding remarks
For us the original motivation for the study of shadows of $d$-dimensional
families was in their application to convexity spaces, and Eckhoff’s
conjecture[Buk]. For that application Theorem 1 sufficed. However, it would be
interesting to find the sharp multidimensional generalization of
Kruskal–Katona theorem.
It is worth noting that the argument given in this paper is largely
insensitive to the poset structure of $2^{X}$. The only input it uses is the
one-dimensional Kruskal–Katona theorem. First, Lemma 2 is a direct consequence
of the fact that the Kruskal–Katona theorem equality is attained only for an
initial segment of a certain linear order. Secondly, a weaker quantitative
form of the Kruskal–Katona theorem is used in Lemma 4 to construct a
continuous function to which Claim 5 applies.
Acknowledgement. I am thankful to Peter Keevash for a helpful discussion. I
thank the referee for many improvements in presentation.
## References
* [Buk] Boris Bukh. Radon partitions in convexity spaces. Discrete Comput. Geom., accepted. arXiv:1009.2384.
* [Kat68] G. Katona. A theorem of finite sets. In Theory of graphs (Proc. Colloq., Tihany, 1966), pages 187–207. Academic Press, New York, 1968.
* [Kru63] Joseph B. Kruskal. The number of simplices in a complex. In Mathematical optimization techniques, pages 251–278. Univ. of California Press, Berkeley, Calif., 1963.
* [Lov79] L. Lovász. Combinatorial problems and exercises. North-Holland Publishing Co., Amsterdam, 1979.
|
arxiv-papers
| 2010-09-13T12:58:17 |
2024-09-04T02:49:12.851197
|
{
"license": "Public Domain",
"authors": "Boris Bukh",
"submitter": "Boris Bukh",
"url": "https://arxiv.org/abs/1009.2375"
}
|
1009.2384
|
# Radon partitions in convexity spaces222The paper is in public domain, and is
not protected by copyright.
Boris Bukh111B.Bukh@dpmms.cam.ac.uk. Centre for Mathematical Sciences,
Cambridge CB3 0WB, England and Churchill College, Cambridge CB3 0DS, England.
###### Abstract
Tverberg’s theorem asserts that every $(k-1)(d+1)+1$ points in
$\mathbb{R}^{d}$ can be partitioned into $k$ parts, so that the convex hulls
of the parts have a common intersection. Calder and Eckhoff asked whether
there is a purely combinatorial deduction of Tverberg’s theorem from the
special case $k=2$. We dash the hopes of a purely combinatorial deduction, but
show that the case $k=2$ does imply that every set of $O(k^{2}\log^{2}k)$
points admits a Tverberg partition into $k$ parts.
## Introduction
Radon’s lemma [Rad21] states that every set $P$ of $d+2$ points in
$\mathbb{R}^{d}$ can be partitioned into two classes $P=P_{1}\cup P_{2}$ so
that the convex hulls of $P_{1}$ and $P_{2}$ intersect. Birch [Bir59] (for
$d=2$) and Tverberg [Tve66] (for general $d$) extended Radon’s theorem to the
analogous statement for partitions of a set into more than two parts: For a
set $P\subset\mathbb{R}^{d}$ of $\lvert P\rvert\geq(k-1)(d+1)+1$ points there
is a partition $P=P_{1}\cup\dotsb\cup P_{k}$ into $k$ parts, such that the
intersection of the convex hulls
$\operatorname{conv}P_{1}\cap\dotsb\cap\operatorname{conv}P_{k}$ is non-empty.
The bound of $(k-1)(d+1)+1$ is sharp, as witnessed by any set of points in
sufficiently general position.
Calder [Cal71] conjectured and Eckhoff [Eck79] speculated that Tverberg’s
theorem is a consequence of Radon’s theorem in the context of abstract
convexity spaces. The conjecture, which we now present, is commonly referred
as “Eckhoff’s conjecture”, and we will maintain this tradition to avoid
additional confusion. If true, the conjecture would have provided a purely
combinatorial proof of Tverberg’s theorem. However, we will show that the
conjecture is false.
A _convexity space_ on the ground set $X$ is a family $\mathcal{F}\subset
2^{X}$ of subsets of $X$, called _convex sets_ , such as both $\emptyset$ and
$X$ are convex, and intersection of any collection of convex sets is convex.
For example, the familiar convex sets in $\mathbb{R}^{d}$ form a convexity
space on $\mathbb{R}^{d}$. Among the other examples are axis-parallel boxes in
$\mathbb{R}^{d}$, finite subsets on any ground set, closed sets in any
topological space (see the book [vdV93] for a through overview of convexity
spaces). If the ground set $X$ in the convexity space $(X,\mathcal{F})$ is
clear from the context, we will speak simply of a convexity space
$\mathcal{F}$. The _convex hull_ of a set $P\subset X$, denoted
$\operatorname{conv}P$, is the intersection of all the convex sets containing
$P$. We write $\operatorname{conv}_{\mathcal{F}}P$ if the convexity space is
not clear from the context. The $k$-th _Radon number_ of $(X,\mathcal{F})$ is
the minimum natural number $r_{k}$, if it exists, so that every set $P\subset
X$ of at least $r_{k}$ points admits a partition $P=P_{1}\cup\dotsb\cup P_{k}$
into $k$ parts whose convex hulls have an element in common. It is not hard to
show333According to [Eck00] it was first shown by R.E.Jamison (1976). The
first published proofs appear to be in [DRS81] and [JW81]. that if $r_{2}$ is
finite, then so is $r_{k}$. Eckhoff’s conjecture states that
$r_{k}\leq(k-1)(r_{2}-1)+1$ in every convexity space. The conjecture has been
proved for $r_{2}=3$ by Jamison [JW81], and for convexity space with at most
$2r_{2}$ points by Sierksma and Boland [SB83]. In section 4 we reproduce a
version of Jamison’s proof.
The best bounds on $r_{k}$ are
$\displaystyle r_{k_{1}k_{2}}$ $\displaystyle\leq r_{k_{1}}r_{k_{2}}\qquad$
$\displaystyle\text{(due to Jamison
\cite[cite]{[\@@bibref{}{jamison_r3}{}{}]})},$ $\displaystyle r_{2k+1}$
$\displaystyle\leq(r_{2}-1)(r_{k+1}-1)+r_{k}+1\qquad$ $\displaystyle\text{(due
to Eckhoff \cite[cite]{[\@@bibref{}{eckhoff_survey}{}{}]})}.$
In particular,
$r_{k}\leq k^{\lceil\log_{2}r_{2}\rceil}.$ (1)
The following result improves on (1).
###### Theorem 1.
Let $(X,\mathcal{F})$ be a convexity space, and assume that $r_{2}$ is finite.
Then
$r_{k}\leq c(r_{2})k^{2}\log^{2}k,$
where $c(r_{2})$ is a constant that depends only on $r_{2}$.
Though this bound is not far from Eckhoff’s conjecture, the conjecture itself
is false.
###### Theorem 2.
For each $k\geq 3$ there is a convexity space $(X,\mathcal{F})$ such that
$r_{2}=4$, but $r_{k}\geq 3(k-1)+2$.
Despite the failure of Eckhoff’s conjecture, we have been unable to rule out
that the convexity spaces with finite $r_{2}$ might behave similarly to
Euclidean spaces. It is conceivable that $r_{k}$ is bounded by a linear
function of $k$ for each $r_{2}$. Moreover, it is possible that other results
from combinatorial convexity extend to such spaces. For instance, Radon proved
the lemma now bearing his name to give an alternative proof of Helly’s theorem
that if in some family of convex sets in $\mathbb{R}^{d}$ every $d+1$ sets
intersect, then all of them do. One of the easy but startling consequences of
Helly’s theorem is the centrepoint theorem. The centrepoint theorem asserts
that for every finite set $P\subset\mathbb{R}^{d}$ there is a point
$p\in\mathbb{R}^{d}$ (the “centrepoint”) such that every convex set containing
more than $\frac{d}{d+1}\lvert P\rvert$ points of $P$ also contains $p$. Both
the deduction of Helly’s theorem from Radon’s theorem, and the deduction of
centrepoint theorem from Helly’s theorem remain valid in the context of the
convexity spaces with finite $r_{2}$. This prompts the following question:
###### Question 3 (Weak epsilon-nets).
Suppose $(X,\mathcal{F})$ is a convexity space with finite $r_{2}$. Let
$\varepsilon>0$ be given. Let $P\subset X$ be a set of points in the space. Is
there a set $N$ of $\lvert N\rvert\leq f(\epsilon,r_{2})$ points such that
every convex set $S$ containing more than $\varepsilon\lvert P\rvert$ points
of $P$ also contains at least one point of $N$?
The set $N$ as in the question above is called a _weak $\varepsilon$-net_
(with respect to convex sets) for $P$. In $\mathbb{R}^{d}$ it is known that
there are weak $\varepsilon$-nets of size only
$(1/\varepsilon)^{d}\log^{c_{d}}(1/\varepsilon)$. The discussion above shows
that the answer to the question is positive if $\epsilon>1-1/(r_{2}-1)$. It is
unclear whether the weak $\varepsilon$-nets of size $f(\varepsilon,r_{2})$
exist for any $\epsilon<1-1/(r_{2}-1)$.
Bárány [Bár82] showed that if $P$ is an $n$-point set in $\mathbb{R}^{d}$,
then there is a point $p$ in $c_{d}\binom{n}{d+1}$ of all the $\binom{n}{d+1}$
simplices spanned by $P$, where $c_{d}$ is a positive constant that depends
only on $d$. In $\mathbb{R}^{1}$, it is immediate that $c_{1}=1/2$ is
admissible, and is best possible. The situation for convexity spaces with
bounded $r_{2}$ is again unclear, except if $r_{2}=3$:
###### Proposition 4 (Selection theorem).
Let $(X,\mathcal{F})$ be a space with $r_{2}=3$. Let $P\subset X$ be point
set. Then there is a point $p\in X$ that is contained in at least
$\frac{1}{3}\binom{n}{2}+O(n)$ of all the sets $\operatorname{conv}\\{x,y\\}$.
###### Question 5.
Does the preceding proposition hold with $1/2$ in place of $1/3$?
The standard greedy argument of Alon, Bárány, Füredi, Kleitman [ABFK92,
Section 8] shows that the selection theorem implies an affirmative answer to
Question 3. In particular, it gives $f(\epsilon,3)\leq
O\bigl{(}(1/\epsilon)^{2}\bigr{)}$, which is probably not sharp.
The rest of the paper is organized as follows. In section 1 we introduce our
only technical tool, the nerves of convex sets. In lemma 7 we will show that
the nerves encode all the information about the convexity space that we need.
In section 2 we present a counterexample to Eckhoff’s conjecture. It is then
followed in section 3 by the proof of Theorem 1. We conclude the paper with a
short discussion of convexity spaces with $r_{2}=3$.
## 1 Nerves
Let $P$ be a set of points in a some convexity space. We associate to $P$ a
collection $\boldsymbol{\mathcal{N}}(P)$ of subsets of $2^{P}$. A family
$\mathcal{F}\subset 2^{P}$ belongs to $\boldsymbol{\mathcal{N}}(P)$ if and
only if the intersection $\bigcap_{S\in\mathcal{F}}\operatorname{conv}S$ is
non-empty. In the conventional terminology one would say that the collection
$\boldsymbol{\mathcal{N}}(P)$ is the nerve of the family of convex sets
$\\{\operatorname{conv}S:S\subset P\\}$. Since we will not use the nerves of
any other families of sets, in this paper we abuse the language and say that
$\boldsymbol{\mathcal{N}}(P)$ is the nerve of $P$.
###### Proposition 6.
If $\boldsymbol{\mathcal{N}}=\boldsymbol{\mathcal{N}}(P)$, then
$\boldsymbol{\mathcal{N}}$ satisfies the following properties:
1. (N1)
$\boldsymbol{\mathcal{N}}$ is a downset: if
$\mathcal{F}\in\boldsymbol{\mathcal{N}}$ and
$\mathcal{F}^{\prime}\subset\mathcal{F}$, then
$\mathcal{F}^{\prime}\in\boldsymbol{\mathcal{N}}$.
2. (N2)
If $\mathcal{F}$ is in $\boldsymbol{\mathcal{N}}$, then so is
$\hat{\mathcal{F}}\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\\{S^{\prime}:S^{\prime}\supset S\in\mathcal{F}\\}$.
3. (N3)
For every $p\in P$ the family
$\mathcal{F}(p)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\\{S:p\in S\\}$
is in $\boldsymbol{\mathcal{N}}$.
4. (N4)
The set $P$ can be partitioned into $k$ parts $P=P_{1}\cup\dotsb\cup P_{k}$ so
that
$(\operatorname{conv}P_{1})\cap\dotsb\cap(\operatorname{conv}P_{k})\neq\emptyset$
if and only if there is a family
$\\{P_{1},\dotsc,P_{k}\\}\in\boldsymbol{\mathcal{N}}$ consisting of $k$
disjoint sets.
5. (N5)
If $r_{t}$ exists, then for every set of $r_{t}$ families
$\boldsymbol{\mathcal{F}}=\\{\mathcal{F}_{1},\dotsc,\mathcal{F}_{r_{t}}\\}\subset\boldsymbol{\mathcal{N}}$
there is a partition
$\boldsymbol{\mathcal{F}}=\boldsymbol{\mathcal{F}}_{1}\cup\dotsb\cup\boldsymbol{\mathcal{F}}_{t}$
of $\boldsymbol{\mathcal{F}}$ into $t$ parts so that
$(\bigcap\boldsymbol{\mathcal{F}}_{1})\cup\dotsb\cup(\bigcap\boldsymbol{\mathcal{F}}_{t})\in\boldsymbol{\mathcal{N}}$.
###### Proof.
The first properties four properties are immediate from the definition of
$\boldsymbol{\mathcal{N}}(P)$.
The final property is easy too: Suppose
$\boldsymbol{\mathcal{F}}=\\{\mathcal{F}_{1},\dotsc,\mathcal{F}_{r_{t}}\\}$ is
given. Let $q_{i}$ be any point in
$\bigcap_{S\in\mathcal{F}_{i}}\operatorname{conv}S$. The set
$Q=\\{q_{1},\dotsc,q_{r_{t}}\\}$ of $r_{t}$ points can be partition into $t$
parts $Q=Q_{1}\cup\dotsb\cup Q_{t}$ so that
$(\operatorname{conv}Q_{1})\cap\dotsb\cap(\operatorname{conv}Q_{t})$ is non-
empty, thus containing some point $p$. The partition $Q=Q_{1}\cup\dotsb\cup
Q_{t}$ naturally induces the partition
$\boldsymbol{\mathcal{F}}=\boldsymbol{\mathcal{F}}_{1}\cup\dotsb\cup\boldsymbol{\mathcal{F}}_{t}$.
It is easy to see that the point $q$ belongs to
$\bigcap_{S\in\cap\boldsymbol{\mathcal{F}}_{i}}\operatorname{conv}S$ for each
$i=1,\dotsc,t$. ∎
Thanks to the following lemma we can avoid the convexity spaces in the rest of
the paper, and work exclusively with nerves.
###### Lemma 7.
Let $P$ be a set, and let $\boldsymbol{\mathcal{N}}$ be a collection of
subsets of $2^{P}$ that satisfies the first three properties in the
Proposition 6. Then there are a ground set $X\supset P$ and a convexity space
on $X$ so that $\boldsymbol{\mathcal{N}}(P)=\boldsymbol{\mathcal{N}}$.
###### Proof.
For an arbitrary family $\mathcal{F}$ let
$C(\mathcal{F})=\\{\mathcal{F}^{\prime}\in\boldsymbol{\mathcal{N}}:\mathcal{F}\subset\mathcal{F}^{\prime}\\}$,
and denote by $\mathcal{C}$ the family of all the sets of the form
$C(\mathcal{F})$. Put $X=\boldsymbol{\mathcal{N}}$. We claim that
$\mathcal{C}$ forms a desired convexity space on $X$. It is clear that
$\emptyset,X\in\mathcal{C}$. Since $C(\mathcal{F}_{1})\cap
C(\mathcal{F}_{2})=C(\mathcal{F}_{1}\cup\mathcal{F}_{2})$, and similarly for
intersections of more than two sets, the collection $\mathcal{C}$ indeed forms
a convexity space on $X$. Define $\phi\colon P\to X$ by
$\phi(p)=\mathcal{F}(p)$. The map $\phi$ is well-defined by property (N3), and
provides the embedding of $P$ into $X$. We need to check that
$\boldsymbol{\mathcal{N}}(\phi(P))=\phi(\boldsymbol{\mathcal{N}})$
Since $\mathcal{F}(p)\in C(\mathcal{F})$ if and only if
$\mathcal{F}\subset\mathcal{F}(p)$, it follows that
$\\{\mathcal{F}(p_{1}),\dots,\mathcal{F}(p_{t})\\}\subset C(\mathcal{F})$
precisely when $\mathcal{F}\subset\bigcap\mathcal{F}(p_{i})$. Hence, if
$P^{\prime}\subset P$, then
$\displaystyle\operatorname{conv}_{\mathcal{C}}(\phi(P^{\prime}))=\bigcap_{\phi(P^{\prime})\subset
C(\mathcal{F})}C(\mathcal{F})=\bigcap_{\mathcal{F}\subset\bigcap_{p\in
P^{\prime}}\mathcal{F}(p)}C(\mathcal{F})=C\left(\bigcap_{p\in
P^{\prime}}\mathcal{F}(p)\right).$
Hence, $\mathcal{F}\in\operatorname{conv}_{\mathcal{C}}(\phi(P^{\prime}))$ if
and only if $\\{P:P^{\prime}\subset P\\}\subset\mathcal{F}$. The intersection
$\bigcap_{P\in\mathcal{F}^{\prime}}\operatorname{conv}_{\mathcal{C}}(\phi(P))$
is non-empty if and only if there is an
$\mathcal{F}\in\boldsymbol{\mathcal{N}}$ so that
$\hat{\mathcal{F}}^{\prime}\subset\mathcal{F}$. Thus by the properties (N1)
and (N2)
$\bigcap_{P\in\mathcal{F}^{\prime}}\operatorname{conv}_{\mathcal{C}}(\phi(P))\neq\emptyset\iff
F^{\prime}\in\boldsymbol{\mathcal{N}}.$
Therefore
$\boldsymbol{\mathcal{N}}_{\mathcal{C}}(\phi(P))=\phi(\boldsymbol{\mathcal{N}})$
as claimed. ∎
## 2 Counterexample to Eckhoff’s conjecture
###### Proof of Theorem 2.
We shall use the Lemma 7 to construct the requisite convexity space. Let
$P=[3(k-1)+1]$. Consider the three kinds of families:
$\displaystyle A[x]$
$\displaystyle=\bigl{\\{}\\{x\\}\bigr{\\}}\cup\binom{P}{4},$ $\displaystyle
B[xy:zw]$
$\displaystyle=\bigl{\\{}\\{x,y\\},\\{z,w\\}\bigr{\\}}\cup\bigl{\\{}S\\!\in\\!\binom{P}{3}:\\{x,y,z,w\\}\cap
S\neq\emptyset\bigr{\\}}\cup\binom{P}{4},\text{ distinct }x,y,z,w$
$\displaystyle C[xy]$
$\displaystyle=\bigl{\\{}\\{x,y\\}\bigr{\\}}\cup\binom{P}{3},\qquad x,y\text{
are distinct}.$
Here $x,y,z,w$ are elements of $P=[3(k-1)+1]$. Let $\hat{A[x]},\hat{B}[xy:zw]$
and $\hat{C}[xy]$ be as in Proposition 6 property (N2). Let
$\boldsymbol{\mathcal{N}}$ consist of all the families,
$\hat{A[x]},\hat{B}[xy:zw]$ and $\hat{C}[xy]$ and all their subfamilies. Let
$\boldsymbol{\mathcal{N}}_{0}$ consist only of families $\hat{A}[x]$,
$\hat{B}[xy:zw]$ and $\hat{C}[xy]$. As $\boldsymbol{\mathcal{N}}$
automatically satisfies properties (N1) and (N2) in Proposition 6 and
$\mathcal{F}(p)\subset\hat{A}[p]$, by Lemma 7 it is a nerve of some convexity
space. As $k\geq 3$, no family of the form $A[x],B[xy:zw]$ or $C[xy]$ contains
$t$ disjoint sets. From that it follows that none of
$\hat{A}[x],\hat{B}[xy:zw]$ or $\hat{C}[xy]$ contain $k$ disjoint sets either,
and same holds for every family in $\boldsymbol{\mathcal{N}}$. Therefore, to
establish the theorem it remains to verify the property (N5) with $r_{2}=4$.
As $\hat{A}$-, $\hat{B}$\- and $\hat{C}$-families are the maximal families in
$\boldsymbol{\mathcal{N}}$, it suffices to show that whenever
$\boldsymbol{\mathcal{F}}=\\{\mathcal{F}_{1},\dotsc,\mathcal{F}_{4}\\}$ is a
collections of four families in $\boldsymbol{\mathcal{N}}_{0}$, then there is
a partition
$\boldsymbol{\mathcal{F}}=\boldsymbol{\mathcal{F}}_{1}\cup\boldsymbol{\mathcal{F}}_{2}$
so that
$(\bigcap\boldsymbol{\mathcal{F}}_{1})\cup(\bigcap\boldsymbol{\mathcal{F}}_{2})$
is contained in some $\mathcal{F}\in\boldsymbol{\mathcal{N}}_{0}$.
To every family $\mathcal{F}$ we associate a subset
$e(\mathcal{F})=\mathcal{F}\cap\binom{P}{2}.$
That is
$\displaystyle e(\hat{A}[x])$ $\displaystyle=\bigl{\\{}\\{x,y\\}:y\in
P\setminus\\{x\\}\bigr{\\}}$ $\displaystyle e(\hat{B}[xy:zw])$
$\displaystyle=\bigl{\\{}\\{x,y\\},\\{z,w\\}\bigr{\\}},$ $\displaystyle
e(\hat{C}[xy])$ $\displaystyle=\bigl{\\{}\\{x,y\\}\bigr{\\}}.$
Note that $e(\mathcal{F}_{1}\cap\mathcal{F}_{2})=e(\mathcal{F}_{1})\cap
e(\mathcal{F}_{2})$. It is convenient think of $e(\mathcal{F})$ as an edge of
a hypergraph on the ground set $\binom{P}{2}$.
Note that if $\mathcal{F}_{1},\mathcal{F}_{2}\in\boldsymbol{\mathcal{N}}_{0}$
are two distinct families, then $\mathcal{F}_{1}\cap\mathcal{F}_{2}$ is
contained in a $\hat{C}$-set. Moreover, if $e(\mathcal{F}_{1})\cap
e(\mathcal{F}_{2})=\emptyset$, then $\mathcal{F}_{1}\cap\mathcal{F}_{2}$ in
contained in $\binom{P}{3}$.
Suppose $\mathcal{F}_{1},\dotsc,\mathcal{F}_{4}$ are four families in
$\boldsymbol{\mathcal{N}}_{0}$. If $e(\mathcal{F}_{1})\cap
e(\mathcal{F}_{2})=\emptyset$, then
$\mathcal{F}_{1}\cap\mathcal{F}_{2}\subset\binom{P}{3}$ and
$\mathcal{F}_{3}\cap\mathcal{F}_{4}\subset\hat{C}[xy]$ for some $x,y$. Hence
$(\mathcal{F}_{1}\cap\mathcal{F}_{2})\cup(\mathcal{F}_{3}\cap\mathcal{F}_{4})\subset\binom{P}{3}\cup\hat{C}[xy]=\hat{C}[xy]$.
We may thus assume that $e(F_{1})\cap e(F_{2})$ is non-empty, and similarly
for other pairs of sets $\mathcal{F}_{1},\dotsc,\mathcal{F}_{4}$.
There are five cases according to the number of $\hat{A}$-families among the
four families $\mathcal{F}_{1},\dotsc\mathcal{F}_{4}$.
There no $\hat{A}$-families: Since every two families meet, and
$e(\mathcal{F}_{1}),\dotsc,e(\mathcal{F}_{4})$ contain $1$ or $2$ vertices
each, it follows that $e(\mathcal{F}_{1}),\dotsc,e(\mathcal{F}_{4})$ must have
a common vertex, say $\\{1,2\\}\in\binom{P}{2}$. Then
$(\mathcal{F}_{1}\cap\mathcal{F}_{2})\cup(\mathcal{F}_{3}\cap\mathcal{F}_{4})\subset\hat{C}[12]$.
There is a single $\hat{A}$-family $\mathcal{F}_{1}$: As $e(\mathcal{F}_{2})$,
$e(\mathcal{F}_{3})$ and $e(\mathcal{F}_{4})$ pairwise meet, they either have
a vertex in common, or $\mathcal{F}_{2},\mathcal{F}_{3},\mathcal{F}_{4}$ are
$\hat{B}$-families, and
$e(\mathcal{F}_{2}),e(\mathcal{F}_{3}),e(\mathcal{F}_{4})$ form a triangle.
However, they cannot form the triangle because $e(\mathcal{F}_{1})$ would not
meet each of $e(\mathcal{F}_{2})$, $e(\mathcal{F}_{3})$ and
$e(\mathcal{F}_{4})$. Thus, $e(\mathcal{F}_{1})\cap\dotsb\cap
e(\mathcal{F}_{4})$ is non-empty, and equals to say
$\\{1,2\\}\in\binom{P}{2}$. Then
$(\mathcal{F}_{1}\cap\mathcal{F}_{2})\cup(\mathcal{F}_{3}\cap\mathcal{F}_{4})\subset\hat{C}[12]$.
There are two $\hat{A}$-families $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$: The
intersection $e(\mathcal{F}_{3})\cap e(\mathcal{F}_{4})$ contains just one
element, say $\\{x,y\\}$. If $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are just
$\hat{A}[x]$ and $\hat{A}[y]$, then
$(\mathcal{F}_{1}\cap\mathcal{F}_{2})\cup(\mathcal{F}_{3}\cap\mathcal{F}_{4})\subset\hat{C}[xy]$.
If $\mathcal{F}_{1}=\hat{A}[z]$ and $z\not\in\\{x,y\\}$, then it necessarily
follows that $e(\mathcal{F}_{3})=\bigl{\\{}\\{x,y\\},\\{z,w_{3}\\}\bigr{\\}}$
and $e(\mathcal{F}_{4})=\bigl{\\{}\\{x,y\\},\\{z,w_{4}\\}\bigr{\\}}$ for some
$w_{3}$ and $w_{4}$. Thus $\mathcal{F}_{2}$ is either $\hat{A}[x]$ or
$\hat{A}[y]$. In either case $(\mathcal{F}_{1}\cap
F_{3})\cup(\mathcal{F}_{2}\cap\mathcal{F}_{4})\subset\hat{B}[xy:zw_{3}]$.
There are three $\hat{A}$-families $\mathcal{F}_{1}$, $\mathcal{F}_{2}$ and
$\mathcal{F}_{3}$: As $e(\mathcal{F}_{4})$ has to meet all of
$e(\mathcal{F}_{1})$, $e(F_{2})$, $e(F_{3})$, it must be that
$\mathcal{F}_{4}$ is a $\hat{B}$-family, implying that
$(\mathcal{F}_{1}\cap\mathcal{F}_{2}\cap\mathcal{F}_{3})\cup\mathcal{F}_{4}=\mathcal{F}_{4}$.
All four families are $\hat{A}$-families: Say, $\mathcal{F}_{1}=\hat{A}[x]$,
$\mathcal{F}_{2}=\hat{A}[y]$, $\mathcal{F}_{3}=\hat{A}[z]$ and
$\mathcal{F}_{4}=\hat{A}[w]$. In that case
$(\mathcal{F}_{1}\cap\mathcal{F}_{2})\cup(F_{3}\cap\mathcal{F}_{4})\subset\hat{B}[xy:zw]$.
∎
## 3 Upper bound on Radon numbers
The main ingredient in the proof of theorem 1 is a version of Kruskal–Katona
theorem from [Buk10]. A _$d$ -dimensional $r$-uniform family_ is a $d$-tuple
of $r$-element sets. In other words, if we denote by $\binom{X}{r}$ the family
of all $r$-element subsets of $X$, then $d$-dimensional $r$-uniform family is
a subset of $\binom{X}{r}^{d}$. A _shadow_ of such a family
$\mathcal{F}\subset\binom{X}{r}^{d}$ is defined to be
$\partial\mathcal{F}\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\bigl{\\{}(S_{1}\setminus\\{x_{i}\\},\dotsc,S_{d}\setminus\\{x_{d}\\}):(S_{1},\dotsc,S_{d})\in\mathcal{F},\text{and
}x_{i}\in S_{i}\text{ for }i=1,\dotsc,d\bigr{\\}}.$
###### Lemma 8 (Theorem 1 of [Buk10]).
Suppose $\mathcal{F}\subset\binom{X}{r}^{d}$ is a $d$-dimensional $r$-uniform
family of size
$\lvert\mathcal{F}\rvert=\binom{x}{r}^{d},$
where $x\geq r$ is a real number. Then
$\lvert\partial\mathcal{F}\rvert\geq\binom{x}{r-1}^{d}.$
In addition to multidimensional Kruskal–Katona theorem, we shall need four
lemmas. The first two lemmas are a bound on Turán numbers of hypergraphs and a
bound on the independence numbers of graphs in which every subgraph have a
large independence number.
###### Lemma 9 ([dC83]).
If $H$ is an $s$-uniform hypergraph on $n$ vertices with fewer than
$\binom{l-1}{s-1}^{-1}\frac{n-l+1}{n-s+1}\binom{\lvert H\rvert}{s}$ edges,
then $H$ contains an independent set on $l$ vertices.
###### Lemma 10 (Special case of Theorem 2.1 from [AS07]).
Let $t<s\leq 2s-3$, and let $G$ be a graph on $n$ vertices. Suppose that every
set of size $s$ contains an independent set of size $t$. Then $G$ contains an
independent set of size $n-s+1$.
Our third lemma is purely computational. We say that a tuple
$(S_{1},\dotsc,S_{d})\in\binom{P}{a}^{d}$ is _$r$ -good_ if there are $r$
pairwise disjoint sets $S_{i_{1}},\dotsc,S_{i_{r}}$ among $S$’s.
###### Lemma 11.
Let $P$ be a finite set. There are at most
$C(d)(a^{2}/\lvert P\rvert)^{d-r+1}\binom{\lvert P\rvert}{a}^{d}$
$r$-bad tuples in $\binom{P}{a}^{d}$, where $C(d)$ is a constant that depends
only on $d$.
###### Proof.
For $S=(S_{1},\dots,S_{d})\in\binom{P}{a}^{d}$ let $G[S]$ be a graph on
$\\{1,\dotsc,d\\}$ with $ij$ forming an edge if $S_{i}\cap
S_{j}\neq\emptyset$. A tuple $S$ is $r$-bad if and only if the independence
number of $G[S]$ is less than $r$. Suppose that the largest forest in $G[S]$
has $m$ edges, then by contracting these edges we obtain an independent set of
size $d-m$. Thus if a tuple $S$ is $r$-bad, then $G[S]$ contains a forest $F$
with $d-r+1$ edges. We say that the forest $F$ _witnesses_ that $S$ is
$r$-bad.
Fix a forest $F$. We shall bound the number of $r$-bad tuples $S$ for which
$F$ is a witness that $S$ is $r$-bad. Let $v_{1},\dotsc,v_{d}$ be a
relabelling of $\\{1,\dotsc,d\\}$ so that in $F$ the vertex $v_{i}$ is
adjacent to at most one vertex $v_{j}$ with $j<i$. Pick $S_{1},\dotsc,S_{d}$
uniformly at random from $\binom{P}{a}$. If $v_{i}$ is adjacent to some
$v_{j}$ with $j<i$ let $E_{i}$ be the event that $S_{i}\cap
S_{j}\neq\emptyset$. If $v_{j}$ is adjacent to none $v_{j}$ with $j<i$ let
$E_{i}$ be the event that holds with probability $1$. Then
$\displaystyle\Pr[F\text{ is a witness that }S\text{ is
$r$-bad}]=\prod_{i=1}^{d}\Pr[E_{i}|E_{1},\dotsc,E_{i-1}]=\prod_{i=1}^{d}\Pr[E_{i}]\leq(a^{2}/\lvert
P\rvert)^{d-r+1}.$
As the number of forests on $d$ vertices depends only on $d$, the lemma
follows by the union bound. ∎
Finally, the third lemma that we need is a restatement of Jamison’s upper
bound $r_{2^{t}}\leq r_{2}^{t}$ in terms of nerves. We include the proof for
completeness.
###### Lemma 12.
Suppose $P$ a set in a convexity space, and
$\boldsymbol{\mathcal{N}}=\boldsymbol{\mathcal{N}}(P)$ is its nerve. Then for
every set $P^{\prime}\subset P$ of size $\lvert P^{\prime}\rvert=r_{2}^{t}$
there is a family $\mathcal{F}\in\boldsymbol{\mathcal{N}}$ containing $2^{t}$
disjoint subsets of $P^{\prime}$.
###### Proof.
The proof is by induction on $t$. The base case $t=0$ is trivial. Suppose
$t\geq 1$. Let $P^{\prime}=P_{1}^{\prime}\cup\dotsb\cup P_{r_{2}}^{\prime}$ be
a partition of $P^{\prime}$ into sets of size $r_{2}^{t-1}$. By the induction
hypothesis, there are families $\mathcal{F}_{1},\dotsc,\mathcal{F}_{r_{2}}$
such that each $\mathcal{F}_{i}$ contains $2^{t-1}$ disjoint subsets of
$P_{i}^{\prime}$. Let these subsets be $R_{i,1},\dotsc,R_{i,2^{t-1}}$. By
property (N5) of the proposition 6, there is a a set $I\subset[r_{k}]$ so that
$\bigcap_{i\in I}\mathcal{F}_{i}\cup\bigcap_{i\not\in
I}\mathcal{F}_{i}\in\boldsymbol{\mathcal{N}}.$
By property (N2) the intersection $R_{j}=\bigcap_{i\in I}R_{i,j}$ is in
$\bigcap_{i\in I}\hat{\mathcal{F}}_{i}$ for each $j=1,\dotsc,2^{t-1}$. The
sets $R_{j}$ are $2^{t-1}$ disjoint subsets of $\bigcup_{i\in
I}P^{\prime}_{i}$. Similarly one obtains $2^{t-1}$ disjoint subsets of
$\bigcup_{i\not\in I}P^{\prime}_{i}$, for the total of $2^{t}$ disjoint
subsets of $P^{\prime}$. ∎
###### Proof of theorem 1.
It suffices to show that for every nerve $\boldsymbol{\mathcal{N}}$ on $\lvert
P\rvert=k^{2}\log^{2}k$ points there are $k$ disjoint sets
$S_{1},\dotsc,S_{k}\subset P$ and a family $\mathcal{F}$ that contains all of
these sets.
For brevity we shall write $r=r_{2}$ and $t=1+\lceil\log_{2}r\rceil$. Define a
$(2r-3)$-dimensional family $\mathcal{T}\subset\binom{P}{r^{t}}^{2r-3}$ as
follows: A tuple $(S_{1},\dotsc,S_{2r-3})\in\binom{P}{r^{t}}^{2r-3}$ is in
$\mathcal{T}$ if there is a family $\mathcal{F}\in\boldsymbol{\mathcal{N}}$
such that $\\{S_{1},\dotsc,S_{2r-3}\\}\subset\mathcal{F}$. Let $P_{0}\subset
P$ be any $(2r-3)r^{t}$-element subset of $P$. Let $P^{\prime}\subset P_{0}$
be an arbitrary $r^{t}$-element subset of $P_{0}$. By the preceding lemma
there is a family $\mathcal{F}$ that contains $2^{t}$ disjoint subsets of
$P^{\prime}$. Since $2r-3\leq 2^{t}$, by property (N2) it follows that
$\mathcal{F}$ contains $2r-3$ disjoint subsets of size $r^{t}$ each that
partition $P_{0}$. In other words, $P_{0}$ gives rise to at least one tuple in
$\mathcal{T}$. Since $P_{0}$ is an arbitrary $(2r-3)r^{t}$-element subset of
$P$, we conclude that
$\lvert\mathcal{T}\rvert\geq\binom{\lvert P\rvert}{(2r-3)r^{t}}\geq
c_{1}(r)\lvert P\rvert^{(2r-3)r^{t}}\geq\binom{\lvert
P\rvert}{r^{t}}^{2r-3}-\binom{(1-c_{2}(r))\lvert P\rvert}{r^{t}}^{2r-3}$
for some positive constants $c_{1}(r),c_{2}(r)$ that depend only on $r$.
Let $m=\lceil\log k/c_{2}(r)\rceil$. Define a $(2r-3)$-dimensional family
$\mathcal{T}^{\prime}\subset\binom{P}{mr^{t}}^{2r-3}$ in the same way as
$\mathcal{T}$ was defined: namely, $S\in\mathcal{T}^{\prime}$ if there is an
$\mathcal{F}\in\boldsymbol{\mathcal{N}}$ such that $S\subset\mathcal{F}$. Note
that the property (N2) implies that if $S\in\binom{P}{mr^{t}}^{2r-3}$ is not
in $\mathcal{T}^{\prime}$, then neither is any family obtained from $S$ by
removing some elements from each set in $S$. Lemma 8 applied to the complement
of $\mathcal{T}^{\prime}$ yields
$\lvert\mathcal{T}^{\prime}\rvert\geq\binom{\lvert
P\rvert}{mr^{t+1}}^{2r-3}-\binom{(1-c_{2}(r))\lvert P\rvert}{mr^{t}}^{2r-3}.$
Let $H\subset\binom{\binom{P}{mr^{t}}}{2r-3}$ be a $(2r-3)$-uniform hypergraph
on $\binom{P}{mr^{t}}$ with edges
$\\{S_{1},\dotsc,S_{2r-3}\\}\in
H\iff(S_{1},\dotsc,S_{2r-3})\in\mathcal{T}^{\prime}\text{ and
}(S_{1},\dotsc,S_{2r-3})\text{ is $r$-good}.$
By Lemma 11, it follows that
$\displaystyle\lvert H\rvert$
$\displaystyle\geq\frac{1}{(2r-3)!}\left(\lvert\mathcal{T}^{\prime}\rvert-
c_{3}(r)(m^{2}r^{2t}/\lvert P\rvert)^{r-2}\binom{\lvert
P\rvert}{mr^{t}}^{2r-3}\right)$ $\displaystyle\geq\binom{\binom{\lvert
P\rvert}{mr^{t}}}{2r-3}\left(1-(1-c_{2}(r))^{(2r-3)mr^{t}}-c_{4}(r)(m^{2}/\lvert
P\rvert)^{r-2}\right)$
Since $m>\log k/c_{2}(r)$, and $\lvert
P\rvert\geq(9c_{4}(r))^{1/(r-2)}m^{2}k^{2}$ it follows that the density of $H$
is
$\lvert H\rvert/\binom{\binom{\lvert P\rvert}{mr^{t}}}{2r-3}\geq
1-k^{(2r-3)r^{t}}-(3k)^{-(2r-4)}\geq 1-(2k)^{-(2r-4)}$
for $k$ large enough.
By Lemma 9 the hypergraph $H$ contains a clique on $2k$ vertices. Let
$S_{1},\dotsc,S_{2k}\in\binom{P}{mr^{t}}$ be the vertices of this clique.
Since edges of $H$ are $r$-good among every $2r-3$ of these $2k$ sets there
are $r$ that are pairwise disjoint. Thus, by Lemma 10 there are $k$ of them,
say $S_{1},\dotsc,S_{k}$, that are pairwise disjoint.
We claim that for every $I\subset[k]$ there is a family
$\mathcal{F}_{I}\in\boldsymbol{\mathcal{N}}$ that contains $S_{i}$ for every
$i\in I$. The proof is by induction on $\lvert I\rvert$ starting with $\lvert
I\rvert=2r-3$. If $\lvert I\rvert=2r-3$, then the claim holds because
$\\{S_{i}:i\in I\\}$ is an edge in $H$. Suppose $\lvert I\rvert>2r-3$. Pick
any $r$ distinct $\lvert I\rvert-1$-element subsets $I_{1},\dotsc,I_{r}$ of
$I$. Then by by property (N5) applied to families
$\mathcal{F}_{I_{1}},\dotsc,\mathcal{F}_{I_{r}}$ it follows that there is a
$J\subset[r]$ so that $\mathcal{F}=(\bigcap_{j\in
J}\mathcal{F}_{I_{j}})\cup(\bigcap_{j\not\in
J}\mathcal{F}_{I_{j}})\in\boldsymbol{\mathcal{N}}$. Since the family
$\mathcal{F}$ contain $\mathcal{F}_{i}$ for every $i\in I$, we may put
$\mathcal{F}_{I}=\mathcal{F}$.
Finally, the family $\mathcal{F}_{[k]}$ contains $k$ disjoint sets
$S_{1},\dotsc,S_{k}$, as required. ∎
## 4 Convexity spaces with $r_{2}=3$
The space with $r_{2}=3$ are especially nice because of the following lemma,
which is implicit in [JW81].
###### Lemma 13.
Let $P$ be a set in a convexity space with $r_{2}=3$, and let
$\boldsymbol{\mathcal{N}}=\boldsymbol{\mathcal{N}}(P)$ be its nerve. Then
there is a family $\mathcal{F}_{p}\in\boldsymbol{\mathcal{N}}$ for each $p\in
P$, and these families satisfy
1. (J1)
$\\{p\\}\in\mathcal{F}_{p}$.
2. (J2)
If $p,q,r$ are any three points of $P$, then either
$\\{p,q\\}\in\mathcal{F}_{r}$ or $\\{p,r\\}\in F_{q}$ or
$\\{q,r\\}\in\mathcal{F}_{p}$.
3. (J3)
If $\\{q,r\\}\in\mathcal{F}_{p}$ and $\\{r,s\\}\in\mathcal{F}_{q}$, then
$\\{r,s\\}\in\mathcal{F}_{p}$.
###### Proof.
Let $\mathcal{F}_{p}$ be a maximal family containing $\\{p\\}$. Then the other
conditions follow from the property (N5) applied to the triple of families
$\mathcal{F}_{p},\mathcal{F}_{q},\mathcal{F}_{r}$. ∎
###### Proof of Proposition 4.
Let $I=\\{(p,q,r):p\in\operatorname{conv}\\{q,r\\}\\}$. Since there are
$\binom{n}{3}$ triples $\\{p,q,r\\}$, each of which contributes at least at
least one element $I$, the proposition follows by the pigeonhole principle. ∎
Since Jamison’s proof of Eckhoff’s conjecture is especially short in the
language of nerves, we include it:
###### Theorem 14.
If $r_{2}=3$, then $r_{k}\leq 2(k-1)+1$.
###### Proof.
Suppose $\lvert P\rvert=2(k-1)+1$. We shall show that one of $\mathcal{F}_{p}$
contains $k$ pairwise disjoint sets. We claim that there is a pair of elements
$p,q\in P$ so that $\\{p,q\\}\in\mathcal{F}_{r}$ for every $r\neq p,q$.
Indeed, it is true if $\lvert P\rvert\leq 3$. If $\lvert P\rvert\geq 4$, and
$s$ is any element of $\mathcal{F}_{p}$, then by induction there is a $p,q\in
P\setminus\\{s\\}$ so that $\\{p,q\\}\in\mathcal{F}_{r}$ for every $r\neq
p,q,s$. If in addition $\\{p,q\\}\in\mathcal{F}_{s}$, then we are done.
Otherwise by property (J2) either $\\{p,s\\}\in\mathcal{F}_{q}$ or
$\\{q,s\\}\in\mathcal{F}_{p}$. Say $\\{p,s\\}\in\mathcal{F}_{q}$. Then by
property (J3) applied to $\\{p,q\\}\in\mathcal{F}_{r}$ and either
$\\{p,s\\}\in\mathcal{F}_{q}$ we conclude that $\\{p,s\\}$ is in every
$\mathcal{F}_{r}$, $r\neq p,s$. The claim is proved.
Let $p,q$ be a pair of element so that $\\{p,q\\}\in\mathcal{F}_{r}$ for
$r\neq p,q$. By the induction hypothesis applied to $P\setminus\\{p,q\\}$
there is $r\in\\{p,q\\}$ so that $\mathcal{F}_{r}$ contains $k-1$ disjoint
sets that are also disjoint from $\\{p,q\\}$. Together with $\\{p,q\\}$ these
form a desired family of disjoint sets. ∎
## References
* [ABFK92] Noga Alon, Imre Bárány, Zoltán Füredi, and Daniel J. Kleitman. Point selections and weak $\epsilon$-nets for convex hulls. Combin. Probab. Comput., 1(3):189–200, 1992. http://www.tau.ac.il/~nogaa/PDFS/abfk3.pdf.
* [AS07] Noga Alon and Benny Sudakov. On graphs with subgraphs having large independence numbers. J. Graph Theory, 56(2):149–157, 2007. http://www.math.ucla.edu/~bsudakov/erdos-hajnal.pdf.
* [Bár82] Imre Bárány. A generalization of Carathéodory’s theorem. Discrete Math., 40(2-3):141–152, 1982.
* [Bir59] B. J. Birch. On $3N$ points in a plane. Proc. Cambridge Philos. Soc., 55:289–293, 1959.
* [Buk10] Boris Bukh. Multidimensional Kruskal–Katona theorem. arXiv, Sep 2010.
* [Cal71] J. R. Calder. Some elementary properties of interval convexities. J. London Math. Soc. (2), 3:422–428, 1971.
* [dC83] D. de Caen. Extension of a theorem of Moon and Moser on complete subgraphs. Ars Combin., 16:5–10, 1983.
* [DRS81] Jean-Paul Doignon, John R. Reay, and Gerard Sierksma. A Tverberg-type generalization of the Helly number of a convexity space. J. Geom., 16(2):117–125, 1981.
* [Eck79] Jürgen Eckhoff. Radon’s theorem revisited. In Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978), pages 164–185. Birkhäuser, Basel, 1979.
* [Eck00] Jürgen Eckhoff. The partition conjecture. Discrete Math., 221(1-3):61–78, 2000. Selected papers in honor of Ludwig Danzer.
* [JW81] Robert E. Jamison-Waldner. Partition numbers for trees and ordered sets. Pacific J. Math., 96(1):115–140, 1981. http://projecteuclid.org/getRecord?id=euclid.pjm/1102734951.
* [Rad21] Johann Radon. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann., 83(1-2):113–115, 1921.
* [SB83] Gerard Sierksma and Jan Ch. Boland. On Eckhoff’s conjecture for Radon numbers; or how far the proof is still away. J. Geom., 20(2):116–121, 1983.
* [Tve66] H. Tverberg. A generalization of Radon’s theorem. J. London Math. Soc., 41:123–128, 1966.
* [vdV93] M. L. J. van de Vel. Theory of convex structures, volume 50 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1993.
|
arxiv-papers
| 2010-09-13T13:13:54 |
2024-09-04T02:49:12.856719
|
{
"license": "Public Domain",
"authors": "Boris Bukh",
"submitter": "Boris Bukh",
"url": "https://arxiv.org/abs/1009.2384"
}
|
1009.2419
|
# How reliable are Finite-Size Lyapunov Exponents for the assessment of ocean
dynamics?
Ismael Hernández-Carrasco Cristóbal López Emilio Hernández-García IFISC,
Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), 07122
Palma de Mallorca, Spain Antonio Turiel Institut de Ciències del Mar, CSIC,
Passeig Marítim de la Barceloneta 37-49, 08003 Barcelona, Spain
###### Abstract
Much of atmospheric and oceanic transport is associated with coherent
structures. Lagrangian methods are emerging as optimal tools for their
identification and analysis. An important Lagrangian technique which is
starting to be widely used in oceanography is that of Finite-Size Lyapunov
Exponents (FSLEs). Despite this growing relevance there are still many open
questions concerning the reliability of the FSLEs in order to analyse the
ocean dynamics. In particular, it is still unclear how robust they are when
confronted with real data. In this paper we analyze the effect on this
Lagrangian technique of the two most important effects when facing real data,
namely noise and dynamics of unsolved scales. Our results, using as a
benchmarch data from a primitive numerical model of the Mediterranean Sea,
show that even when some dynamics is missed the FSLEs results still give an
accurate picture of the oceanic transport properties.
Lagrangian viewpoint Griffa.1996 ; Buffoni.1997 ; Iudicone.2002 ; Mancho.2006
; Molcard.2006 . Lagrangian diagnostics exploit the spatio-temporal
variability of the velocity field by following fluid particle trajectories, in
contrast with Eulerian diagnostics, which analyze only frozen snapshots of
data. Among Lagrangian techniques the most used ones involve the computation
of local Lyapunov exponents (LLE) which measure the relative dispersion of
transported particles Artale.1997 ; Aurell.1997 ; Haller.1998 ; Haller.2000 ;
Boffetta.2001 ; Iudicone.2002 . LLEs give information on dispersion processes
but also, and even more importantly, can be used to detect and visualize
Lagrangian Coherent Structures (LCSs) in the flow like vortices, barriers to
transport, fronts, etc.Haller.2000b ; Joseph.2002 ; Koh.2002 ; Lapeyre.2002 ;
Beron-Vera.2008 ; Chaos.2010 .
The standard definition of Lyapunov exponents Ott.1993 involves a double
limit, in which infinitely-long times and infinitesimal initial separations
are taken. These limits can not be practically implemented when dealing with
realistic flows of geophysical origin. Over real data, LLEs are defined by
relaxing some of the limit procedures. In finite-time Lyapunov exponents
(FTLE) Ott.1993 ; Lapeyre.2002 trajectory separations are computed starting
still at infinitesimal distance, but only for a finite time. In the case of
finite-size Lyapunov exponents (FSLE), the key tool used in our work
Aurell.1997 ; Boffetta.2001 , one computes the time which is taken for two
trajectories, initially separated by a finite distance, to reach a larger
final finite distance.
FSLEs are attracting the attention of the oceanographic community Artale.1997
; Iudicone.2002 ; dOvidio.2004 ; Molcard.2006 ; Garcia-Olivares.2007 ;
Haza.2008 ; dOvidio.2009 ; TewKai.2009 ; Poje.2010 . The main reasons for this
interest, within the framework of their ability for studying LCSs and
dispersion processes, are the following: a) they identify and display the
dynamical structures in the flow that strongly organizes fluid motion (the
above mentioned LCSs, which are defined as ridges of the FSLEs fields); b)
they are relatively easy to compute; c) they provide extra information on
characteristics time-scales for the dynamics; and d) they are able to reveal
oceanic structures below the nominal resolution of the velocity field being
analyzed. In addition, FSLEs are specially suited to analyze transport in
closed areas Artale.1997 .
Despite the growing number of applications of FSLEs, a rigorous analysis of
many of their properties is still lacking. There are two main concerns before
applying FSLE to real data, namely the effect of noise and the role of
observation scale. Concerning noise, real data are discretized and noisy, and
this can affect the reliability of FSLE-based diagnostics (recent related
studies for FTLE can be consulted in Chaos.2010 ). Concerning scale
properties, FSLEs can be obtained over a grid finer than that of the data.
This enables to study submesoscale processes under the typical mesoscales
(below $10$ kilometers) that nowadays provide, for example, altimetry data
dOvidio.2009 ; TewKai.2009 . But then the question is if finer-grid LCSs are
meaningful or just an artifact. On the other side, we can have access to a
limited-resolution velocity field, and then ask ourselves if any refinement in
the velocity grid (by improved data acquisition, for instance) is going to
modify our previous assessment of LCS at the rougher scale. The main objective
of this paper is to address these questions, in particular with reference to
their potential applications into ocean dynamics. A related study of the
sensitivity of relative dispersion of particles statistics, when the spatial
resolution of the velocity field changes, has appeared recently Poje.2010 .
The benchmark for the study of FSLEs properties used in this work is a the
two-dimensional velocity field of the marine surface obtained from a numerical
model of the Mediterranean Sea. The first half of the paper is devoted to
study what we have just called scale properties of the FSLE field. By changing
gradually the resolution of the grid on which they are computed, we will show
that they have typical multifractal properties. This means, in particular,
that FSLEs obtained for a finer resolution than that provided by the data
provides non-artificial information. Subsequently, we will consider a somewhat
opposite case, i.e., what happens to the FSLEs if the velocity-data grid is
changed. Their robustness under data-resolution transformations will be
discussed. The second half of the work will analyze the effect of noise.
Again, two different scenarios are considered: a) uncertainties in the
velocity data, and b) noise in the particle trajectories. FSLEs will be shown
to be robust against these two sources of error, and the reasons for this will
be discussed.
## I Description of the data
We analyze a velocity dataset generated with the DieCast (Dietrich for Center
Air Sea Technology) numerical ocean model adapted to the Mediterranean Sea
Fernandez.2005 . The dataset has been already used in previous Lagrangian
studies dOvidio.2004 ; Schneider.2005a ; Mancho.2008 . DieCast is a primitive-
equation, z-level, finite difference ocean model which uses the hydrostatic,
incompressible and rigid lid approximations. At each grid point, horizontal
resolution is the same in both the longitudinal, $\phi$, and latitudinal,
$\lambda$, directions, with resolutions $\Delta\phi=1/8^{o}$ and
$\Delta\lambda=\Delta\phi\cos\lambda$. Vertical resolution is variable with 30
layers. Annual climatologic forcing is used, so that it is enough to keep a
temporal resolution of one day. We will use velocity data corresponding to the
second layer, which has its center at a depth of 16 meters. This sub-surface
layer is representative of the marine surface circulation and is not directly
driven by wind. We have recorded daily velocities for five years, and
concentrate our work in the area of the Balearic Sea. In Figure 1 we show a
snapshot of the velocity field from the DieCAST model.
Figure 1: Snapshot of the surface velocity field of the Balearic Sea
corresponding to day $640$ in the DieCAST simulation.
## II Definition and implementation of FSLEs
FSLEs provide a measure of dispersion as a function of the spatial resolution,
serving to isolate the different regimes corresponding to different length
scales of oceanic flows, as well as identifying the LCSs present in the data.
To calculate the FSLEs we have to know the trajectories of fluid particles,
which are computed by integrating the equations of motion for which we need
the velocity data $(u,v)$. FSLE are computed from $\tau$, the time required
for two particles of fluid (one of them placed at $(x,y)$) to separate from an
initial (at time $t$) distance of $\delta_{0}$ to a final distance of
$\delta_{f}$ , as:
$\Lambda(x,y,t,\delta_{0},\delta_{f})\;=\;\frac{1}{\tau}\log\frac{\delta_{f}}{\delta_{0}}.$
(1)
In principle obtaining $\Lambda(x,y,t,\delta_{0},\delta_{f})$ would imply to
consider all the trajectories starting from points at distance $\delta_{0}$
from our basis point; in practice, when confronted with regular, discretized
grids, only the four closest neighbors are considered. It is convenient to
choose $\delta_{0}$ to be the intergrid spacing among the points on which the
FSLEs will be computed, i.e., it is the resolution of the “FSLE grid”. The
details of the calculation of the FSLEs are in Appendix A.
## III Effect of sampling scale on FSLEs
Notice that we distinguish two types of grids, namely the FSLE grid (in the
following “F-grid”) and that where the velocity field is given (called
velocity grid or “V-grid”). These two grids need not to coincide. We explore
first the effect of changing the resolution of FSLE grid.
In Figure 2 we show an example of the FSLEs derived using four different
F-grids with increasing resolutions. Visually, as the resolution of the F-grid
is increased the structures already observed in the coarser version are kept,
just increasing their detail level. In addition, when resolution is increased,
new less active structures are appearing in areas previously regarded as
almost inactive. Taking a F-grid finer than the associated V-grid would make
no sense if FSLE was an Eulerian measure obtained from single snapshots of the
velocity field. But FSLE is a Lagrangian measure, i.e. they are computed using
trajectories which integrate information on the history of the velocity field.
This allows capturing the effects of the large scales on scales smaller than
the V-grid. This does not mean that we reconstruct all the effects taking
place at the smaller scales, but only the ones that have been originated by
the relatively large velocity features which are resolved on the V-grid.
Figure 2: Snapshots of FSLEs backward in time starting from day $640$ at
different F-grid resolutions: a) $\delta_{0}=1/8^{\circ}$, b)
$\delta_{0}=1/16^{\circ}$, c) $\delta_{0}=1/32^{\circ}$, d)
$\delta_{0}=1/64^{\circ}$. In all of them we take $\delta_{f}=1^{\circ}$. The
values in the color bar have units of $day^{-1}$.
The main structures of the flow, which are essentially filaments, become finer
as resolution is increased, behaving much like the geometrical persistence of
a fractal interface Falconer.1990 . The question naturally rises about the
possible multifractal character of FSLEs. Multifractality is a property
characteristic to turbulent flows, and it is associated to the development of
a complex hierarchy accross which energy is transmitted and utterly dissipated
Frisch.1995 ; Turiel.2008 . The study of the scaling properties of the
distribution of FSLEs at the different resolution scales reveal that they are
multifractal (see Appendix B). This implies that changes in scale are
accounted for by a well-defined transformation, namely a cascade
multiplicative process Frisch.1995 ; Turiel.2008 . It also implies that
information is hierarchized Turiel.2002 and so what is obtained first, at the
coarsest scales, is the most relevant information. Due to multifractality,
small-scale structures, as unveiled by the FSLEs, with typical sizes smaller
than that of the velocity resolution, are determined by the larger ones and
the multi-scale invariance properties. Thus, no artificiality is induced by
this calculation and the robustness of FSLE analysis under changes in scale is
confirmed.
A different question concerns the robustness of FSLEs when the V-grid is
changed. For instance, when a diagnosis is obtained with a low-resolution
velocity field, is this diagnosis compatible with a later improved observation
of the velocity? The answer is yes. In Figure 3 we show the FSLEs obtained at
a fixed F-grid resolution of $1/8^{\circ}$ for varying velocity resolutions
(namely, $1/8^{\circ}$, $1/4^{\circ}$ and $1/2^{\circ}$). The change of
resolution of the velocity grid is performed as indicated in Appendix C. We
observe that the global features observed with the coarser resolution V-grid
are kept when this is refined. Obviously, as the velocity field is described
with enhanced resolution new details (with short-range effect on the flow
structure and so no contradicting the large-scale picture) become apparent in
the FSLEs. However, the effect of refining the V-grid is not only introducing
new small-scale structures: there is a consistent increase in the values of
the FSLEs as the V-grid is refined. The histograms of the FSLEs,
$\Lambda_{f}$, for a given velocity resolution conditioned by the FSLEs,
$\Lambda_{c}$, obtained from a coarser velocity field are shown in Figure 4.
The modal line (the line of maximum conditioned probability) is close to a
straight line. The best linear regression fits are
$\Lambda_{1/8^{\circ}}=1.08\Lambda_{1/4^{\circ}}+0.05$ (correlation
coefficient, $\rho=0.69$) and
$\Lambda_{1/4^{\circ}}=0.99\Lambda_{1/2^{\circ}}+0.04$ ($\rho=0.69$).
According to these results, we can approximate the finer FSLEs $\Lambda_{f}$
in terms of the coarser FSLEs $\Lambda_{c}$ as:
$\Lambda_{f}(\vec{x})\;=\;\Lambda_{c}(\vec{x})\>+\>\Delta\Lambda_{fc},$ (2)
where the quantity $\Delta\Lambda_{fc}$ determines the contribution to the
FSLE by the small-scale variations in velocity not accounted for by the lower
resolution version of this field. It is hence independent of $\Lambda_{c}$, so
the intercept of the vertical axis with the linear regression equals the mean
of this quantity.
Figure 3: Snapshots of FSLEs backward in time starting from day $640$ at
different initial V-grid resolutions: a)$\Delta_{0}=1/8^{\circ}$, b)
$\Delta_{0}=1/4^{\circ}$, c) $\Delta_{0}=1/2^{\circ}$. In all of them we take
the same F-grid resolution of $\delta_{0}=1/8^{\circ}$, and
$\delta_{f}=1^{\circ}$. The color bar has units of day-1.
A linear dependence of $\Lambda_{f}$ with $\Lambda_{c}$ when scale is changed
implies that FSLEs follow a multiplicative cascade Turiel.2006 ; Turiel.2008 ,
an essential ingredient in multifractal systems which gives further
confirmation to our previous results. We have hence shown that a) the
dependence of FSLEs on both types of scale parameters reveals a multifractal
structure; b) what is diagnosed at the coarser scales is still valid when
scale is refined (although as V-grid resolution is increased the reference
level of FSLEs is increased by a constant).
Figure 4: Probability distributions (coded as grey levels) of FSLEs derived at
a coarse velocity resolution ($\Lambda_{c}$, vertical axis) conditioned by a
finer velocity grid ($\Lambda_{f}$, horizontal axis). The range of values of
both axes run linearly from $0$ to $0.5$ day-1. To obtain a statistics large
enough, we have considered $30$ FSLEs snapshots, starting form $t=640$ up to
$t=1075$ days, in steps of $15$ days. The brightest color (pure white)
corresponds to the maximum probability at each column; the darkest color (pure
black) corresponds to zero. Left panel: FLSEs derived from $1/4^{\circ}$
velocities conditioned by FSLEs at original $1/8^{\circ}$ velocities. Right
panel: FSLEs from $1/2^{\circ}$ velocities conditioned by FSLEs from
$1/4^{\circ}$ velocities.
## IV Effect of noise on FSLEs
We compute the FSLEs after applying a random perturbation to all components of
the velocity field. The velocity is changed from $(u,v)$ to
$(u^{\prime},v^{\prime})$, with
$u^{\prime}(x,t)=u(\textbf{x},t)(1+\alpha\eta_{x}(\textbf{x},t))$ and
$v^{\prime}(\textbf{x},t)=v(\textbf{x},t)(1+\alpha\eta_{y}(\textbf{x},t))$.
$\\{\eta_{x}(\textbf{x},t),\eta_{y}(\textbf{x},t)\\}$ are sets of Gaussian
random numbers of zero mean and unit variance. $\alpha$ measures the relative
size of the perturbation (it gives the ratio of the mean amplitude of noise
with respect to mean amplitude of the velocity). We introduce three different
kinds of error: uncorrelated noise, i.e. different and uncorrelated values of
$\\{\eta_{x}(\textbf{x},t),\eta_{y}(\textbf{x},t)\\}$ for each x and $t$;
correlated in time and uncorrelated in space (uncorrelated for different x but
the same values at given x for different $t$); and correlated in space and
uncorrelated in time (uncorrelated values for different $t$, but the same
values for different x at fixed $t$). Note that the perturbation is
proportional to the original velocity. Figure 5 shows a snapshot of FSLEs for
the velocity field perturbed by uncorrelated noise of a relative size
$\alpha=10$, i.e., noise is $10$ times larger than the amplitude of the
initial velocity field. The computed Lagrangian structures look rather the
same, despite the large size of the perturbation introduced.
Figure 5: Snapshots of FSLEs calculated backwards in time starting from day
$600$ at fixed spatial resolution ($\delta_{0}=1/64^{\circ}$), and at
different $\alpha$: a)$\alpha=0$, b) $\alpha=10$. In both of them we take
$\delta_{f}=1^{\circ}$. The color bar has units of $day^{-1}$. Initial
conditions for which the separation $\delta_{f}$ has not been reached after
$600$ days are assigned a value $\Lambda=0$.
In order to quantify the influence of the velocity perturbation in the FSLE
calculation we compute the relative error (RE) of perturbed FSLEs with respect
to unperturbed FSLEs, $<\epsilon(t)>$, at a given instant of time, and then
averaging in time (we have $M=100$ snapshots $:t=t_{1},...,t_{M}$) as:
$\epsilon(t_{i})=\sqrt{\frac{1}{N}\sum_{\textbf{x}}\dfrac{|\Lambda^{\alpha}(\textbf{x},t_{i})-\Lambda(\textbf{x},t_{i})|^{2}}{|\Lambda(\textbf{x},t_{i})|^{2}}},\
\ \ \hfill<\epsilon(t)>\equiv\frac{1}{M}\sum_{i=1}^{M}\epsilon(t_{i})\ .$ (3)
$\Lambda(\textbf{x},t_{i})$ and $\Lambda^{\alpha}(\textbf{x},t_{i})$ are the
FSLEs fields without and with inclusion of the perturbation in the velocity
data, respectively. The sum over x runs over the $N=2679$ spatial points.
Figure 6 shows the average RE as a function of $\alpha$. It must be remarked
that the RE has always small values: even for $\alpha=10$ the RE remains
smaller than $0.23$ for the three kinds of noise. To get an idea of how
relevant these quantities are, we have computed the RE of shuffled FSLEs
(permuting locations at random) with respect to the original ones, and
obtained a value of $1.143$. FSLEs are thus robust against uncorrelated noise;
the reason is the averaging effect produced when computing them by integrating
over trajectories which extend in time and space, that tends to cancellate
random, uncorrelated errors.
Figure 6: Relative error $<\epsilon(t)>$ of the FSLE fields for different
perturbation intensity $\alpha$ in the velocity data. Solid line is for
uncorrelated noise in space and time, dashed-dotted line is for uncorrelated
noise in time and correlated in space, and dotted line is for uncorrelated
noise in space and correlated in time. $<\epsilon(t)>$ is obtained by
averaging the RE in $100$ snapshots (see Eq. (3)). The error bar is the
statistical error of the temporal average $<\epsilon(t)>$. Left: spatial
resolution $\delta_{0}=1/8^{\circ}$. Right: spatial resolution
$\delta_{0}=1/64^{\circ}$. In all calculations we take $\delta_{f}=1^{\circ}$
Now we proceed by adding noise to the particle trajectories. This is a
simplified way of including unresolved small scales in the Lagrangian
computations Griffa.1996 . To be precise we solve numerically the system of
Equations (13) and (14) (see Appendix D), where a stochastic term with a
Gaussian random number and an effective eddy-diffusion, $D$, has been added.
For the diffusivity we use Okubo’s empirical formula Okubo.1971 , which
relates the effective eddy-diffusion, $D$ in $m^{2}/s$, with the spatial
scale, $l$ in meters: $D(l)=2.055\ 10^{-4}\ l^{1.15}$. If we take $l=12\ km$,
which is the approximate length corresponding to the $1/8^{\circ}$ DieCAST
resolution at Mediterranean latitudes, we obtain $D\sim 10\ m^{2}s^{-1}\equiv
D_{0}$.
In Figure 7 we show particle trajectories without (top panel) and with (bottom
panel) eddy diffusion. As expected diffusion introduces small scale
irregularities on the trajectories, and also substantial dispersion at large
scales. In Figure 8, FSLEs with $\delta_{0}=1/64^{\circ}$ $D=0~{}m^{2}s^{-1}$
and $D=0.9~{}m^{2}s^{-1}$ (obtained for this scale by Okubo’s formula for
spatial resolution) are shown. We can see that the main mesoscale structures
are maintained, but small-scale filamental structures are lost since filaments
become blurred. This is somehow expected a priory because diffusion introduces
a new length scale $l_{D}$ proportional to $\sqrt{D}$. A pointwise comparison
of noiseless and noise-affected FSLEs makes no sense, since the noise-induced
blurring disperses FSLEs values specially at places with low values, see
Figure 8. But for Lagrangian diagnostics high FSLE values are much more
relevant, so we compute the error restricted to the places where $\Lambda>0.2$
for $D=0$. Left panel of Figure 9 shows the RE respect to the $D=0$ case
(applying Equation (3)) for different values of $D$. The RE monotonously
increases with $D$, but remains smaller than $0.6$ for the largest value of
$D$ considered. Again, to get an idea of how relevant these relative errors
are, we have to compare them with the RE of shuffled FSLEs, of value $1.143$.
Figure 7: Trajectories of five particles without diffusion (top) and with
diffusion (bottom). The difference in the initial positions of all five
particles is about $0.06^{\circ}$, and we use these initial conditions in both
computations. The trajectories were computed for $50$ days of integration. We
used the eddy-diffusion $D_{0}\sim 10m^{2}s^{-1}$ assigned by the Okubo
formula to the resolution of the DieCAST model at Mediterranean latitudes.
Figure 8: FSLEs computed backwards from day $500$ at the same spatial
resolution ($\delta_{0}=1/64^{\circ}$), and for different eddy-diffusion
values: a)$D=0~{}m^{2}s^{-1}$ b) $D=0.9~{}m^{2}s^{-1}$. We take
$\delta_{f}=1^{\circ}$. The color bar has units of $day^{-1}$. Initial
conditions for which the separation $\delta_{f}$ has not been reached after
$500$ days are assigned a value $\Lambda=0$.
As a matter of fact, diffusion introduces an effective observation scale, and
one should not go beyond that limit to obtain senseful results; this is
illustrated in the right panel of Figure 9. As shown in the figure, for fixed
eddy diffusivity (in the case of the figure, $D_{0}=10m^{2}s^{-1}$), when
$\delta_{0}$ becomes greater the error diminishes. Hence, a fixed diffusion
will eventually be negligible at a scale large enough, in this way determining
an observation scale. A completely different situation is given when the eddy-
diffusion depends on the scale according to Okubo’s formula (in our case, at
$\delta_{0}=1/16^{\circ}$ $D=4,5m^{2}s^{-1}$; at $\delta_{0}=1/32^{\circ}$
$D=2m^{2}s^{-1}$; and at $\delta_{0}=1/64^{\circ}$, $D=0.9m^{2}s^{-1}$). Now
$<\epsilon(t)>$ takes a constant value close to $0.45$, meaning that Okubo’s
diffusion behaves the same at all scales. This is expected since Okubo’s law
is based on the hypothesis that unsolved scales act as diffusers, like in
turbulence Frisch.1995 . Our result is consistent with the ideas behind
Okubo’s hypothesis.
Figure 9: Left: Relative error $<\epsilon(t)>$ of the FSLE at the different
values of $D$ in the particle trajectories, with respect to the $D=0$ case.
Spatial resolution is $\delta_{0}=1/8^{\circ}$, and $\delta_{f}=1^{\circ}$.
$<\epsilon(t)>$ is obtained by temporally averaging the relative errors in
$100$ snapshots. The (small) error bars indicate the statistical error in the
$<\epsilon(t)>$ average. Right: Dotted line is the RE $<\epsilon(t)>$ of the
FSLE at different spatial resolution $\delta_{0}$ and at one assigned eddy-
diffusion $D$ for every spatial resolution in the particle trajectories with
respect to the $D=0$ case. Solid line is the RE $<\epsilon(t)>$ of the FSLE at
different spatial resolution $\delta_{0}$, and at the same eddy-diffusion
$D_{0}=10m^{2}s^{-1}$ in the particle trajectories with respect to the $D=0$
case. Dashed-dotted line is the RE of shuffled FSLE with respect to the
original case ($D=0$) at different spatial resolution. $<\epsilon(t)>$ by
temporally averaging the RE in $100$ snapshots. The (small) error bar
indicates the statistical error in the $<\epsilon(t)>$ average. In all of them
we take $\delta_{f}=1^{\circ}$.
We can characterize the effective diffusion scale from the properties of the
histograms. In Figure 10 we present the histograms of FSLEs at different
F-grid resolutions, and including or not eddy diffusion which takes always the
same value $D_{0}=10m^{2}s^{-1}$. For $\delta_{0}=1/8^{\circ}$ the histograms
with and without diffusion are almost coincident. This is due to the fact that
the value of diffusion we are using is the one corresponding, by the Okubo’s
formula, to $1/8^{\circ}$. i.e., we are parameterizing turbulence below
$1/8^{\circ}$, and this has no effects on the FSLE computations if the minimum
scale considered is also $1/8$. However, this behavior is different for
smaller $\delta_{0}$ (always keeping $D_{0}=10m^{2}s^{-1}$). The histograms
for $\delta_{0}=1/16,1/64^{\circ}$, with and without diffusion, are clearly
different, those including diffusion becoming closer to the histogram for
$\delta_{0}=1/8^{\circ}$.
Figure 10: Comparison between probability density function for the FSLEs at
different resolutions with different values of eddy-diffusion, and without
diffusion. It is obtained from the temporal average ($30$ snapshots) of
histograms. Solid line for $\delta_{0}=1/8^{\circ}$ with diffusion, dotted
$\delta_{0}=1/8^{\circ}$ without diffusion, dashed $\delta_{0}=1/16^{\circ}$
without diffusion, dashed-dotted line for $\delta_{0}=1/64^{\circ}$ with
diffusion, and circle-line for $\delta_{0}=1/64^{\circ}$ without diffusion.
## V Conclusions
In this paper, we have analyzed the sensibility of FSLE-based analyses for the
diagnostic of Lagrangian properties of the ocean (most notably, horizontal
mixing and dispersion). Our sensibility tests include the two most important
effects when facing real data, namely dynamics of unsolved scales and of
noise. Our results show that even if some dynamics are missed (because of lack
of sampling or inaccuracy of any kind in the measurements) FLSEs results would
still give an accurate picture of Lagrangian properties, valid for the solved
scales. This does not mean that scale and/or noise leave FSLEs unaffected, but
the way in which they modify this Lagrangian diagnostics can be properly
accounted.
Lagrangian methods provide answers to problems which have a deep impact on
risk management (e.g. control of pollutant dispersion) as well as on ecosystem
analysis (e.g. tracking nutrient mixing and transport, identifying the role of
horizontal mixing in primary productivity). They utterly will give hints about
energy exchanges in the upper ocean and will help in understanding processes
driving global change in the oceans. The use of Lagrangian techniques for the
assessment of the transport and mixing properties of the ocean has grown in
importance in the latest years, with increasing efforts devoted to the
implementation of appropriate techniques but few studies on the validity of
the results when real data, affected by realistic constraints, are used. Our
work will serve to unify and interpret the analyses provided by Lagrangian
methods when real data are processed.
###### Acknowledgements.
This work is a contribution to OCEANTECH project (CSIC PIF-2006). IH-C, CL and
EH-G acknowledge support from project FISICOS (FIS2007-60327) of MEC and
FEDER.
## Appendix A Calculation of FSLE.
It is natural to choose the initial points $(x,y)$ on the nodes of a grid with
lattice spacing coincident with the initial separation of fluid particles
$\delta_{0}$. In this way one obtains maps of values of $\Lambda$ at a spatial
resolution that will coincide with $\delta_{0}$. To compute $\Lambda$ we need
to know the trajectories of the particles. The equations of motion that
describe the horizontal evolution of particle trajectories in the velocity
field are:
$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle\frac{u(\phi,\lambda,t)}{R\cos{\lambda}},$ (4)
$\displaystyle\frac{d\lambda}{dt}$ $\displaystyle=$
$\displaystyle\frac{v(\phi,\lambda,t)}{R},$ (5)
where $u$ and $v$ represent the zonal and meridional components of the surface
velocity field coming from the simulations described above, $R$ is the radius
of the Earth (6400 km in our computations). Numerically we proceed integrating
Eqs. (4) and (5) using a standard fourth-order Runge-Kutta scheme with an
integration time step $dt$ = 6 hours. Spatiotemporal interpolation of the
velocity data is achieved by bilinear interpolation. Since this technique
requires an equally spaced grid and this is not the case for the spherical
coordinates $(\phi,\lambda)$, for which the grid is not uniformly spaced in
the latitude coordinate, we first transform it to a new system $(\phi,\mu)$
where the grid turns out to be uniformly sampled Mancho.2008 . The latitude
$\lambda$ is related to the new coordinate $\mu$ according to:
$\mu\;=\;\log|\sec\lambda\>+\>\tan\lambda|$ (6)
Using the new coordinate variables the equations of motion become:
$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle\frac{u(\phi,\mu,t)}{R\cos{\lambda(\mu)}}$ (7)
$\displaystyle\frac{d\mu}{dt}$ $\displaystyle=$
$\displaystyle\frac{v(\phi,\mu,t)}{R\cos{\lambda(\mu)}},$ (8)
and one can convert the $\mu$ values back to latitude $\lambda$ by inverting
Eq. (6): $\lambda=\pi/2-2\arctan e^{-\mu}$. Once we integrate the equations of
motion, Eqs. (7) and (8), we compute the FSLEs applying Eq. (1) for the points
of a lattice with spacing $\delta_{0}$. We will only compute FSLEs integrating
backwards-in-time the particle trajectories, since LCSs associated to this has
a direct physical interpretation, but all our results are similar for forward-
in-time dynamics.
## Appendix B Multifractal character of FSLE.
To study if FSLEs behave like multifractal systems we have computed their
probability density distribution, $P(\delta_{0},\Lambda)$, for different
resolutions $\delta_{0}$. It must follow that for any resolution scale
$\delta_{0}$ of FSLE grid we should observe:
$P(\delta_{0},\Lambda)\propto\delta_{0}^{d-D(\Lambda)},$ (9)
Notice that as $d-D(\Lambda)$ is a positive quantity, $P(\delta_{0},\Lambda)$
becomes smaller as $\delta_{0}$ is reduced. In fact, a characteristic
signature of multifractal scaling is having a scale-dependent histogram which
becomes more strongly peaked as the resolution scale becomes smaller. We know
that there is a FSLE for each domain point, so we can normalize the FSLE
distribution by its maximum (that will be attained for a value $\Lambda_{c}$)
and then retrieve the associated singularity spectrum according to the
following expression:
$D(\Lambda)\;=\;d-\frac{\log{\frac{P(\delta_{0},\Lambda)}{P(\delta_{0},\Lambda_{c})}}}{\log{\delta_{0}}}.$
(10)
In Figure 11, top, we show the histograms (averaged over $30$ snapshots
distributed among $15$ months) normalized to have the same unitary are. One
can see that when the resolution gets finer the $P(\delta_{0},\Lambda)$
narrows, and the peak height increases, in agreement with Equation (9).
According to the definition of Microcanonical Multifractals Isern.2007 ;
Turiel.2008 , the system will be multifractal if the curves $D(\Lambda)$
estimated at different resolutions $\delta_{0}$ using Equation (10) are all
equal. This can be observed in Figure 11, bottom. The collapse of the four
curves is not perfect due to the lost of translational invariance produced by
the small size of the domain -the Balearic basin- that we analyze. Thus, the
interfaces of constant $\Lambda$ values build an approximate multifractal
hierarchy and generalized scale invariance is present in the FSLE field.
Figure 11: Top: Comparison of the probability density functions
$P(\delta_{0},\Lambda)$ for the FSLEs at different resolutions. It is obtained
from the temporal average ($30$ snapshots) of histograms. Dotted line is for
$\delta_{0}=1/8^{\circ}$, dashed-dotted line $\delta_{0}=1/16^{\circ}$, dashed
$\delta_{0}=1/32^{\circ}$, and solid line for $\delta_{0}=1/64^{\circ}$.
Bottom: $D(\Lambda)$ for different values of $\delta_{0}$. Dotted for
$\delta_{0}=1/8^{\circ}$, dashed-dotted line $\delta_{0}=1/16^{\circ}$, dashed
$\delta_{0}=1/32^{\circ}$, and solid line line for $\delta_{0}=1/64^{\circ}$.
## Appendix C Change of resolution of the velocity grid.
In order to reduce the resolution of a given velocity grid, the easiest way
would be by subsampling the existing grid points or by block-averaging the
values of the velocity and assign the result to the central grid point.
However, in the case of a complex boundary such as the Mediterranean coast
such a strategy is strongly inconvenient, as the coarsening of the grid would
imply to change land circulation barriers (islands, straits). The
disappearance of a land barrier or the creation of a new one as a consequence
of the coarsening would imply a dramatic change in the value of FSLEs at all
points affected by the modified circulation; if the circulation patterns are
rather complex almost every point could be affected. We have thus preferred to
smooth the velocity with a convolution kernel weighted with a local
normalization factor, and keeping the original resolution for the data so land
barriers are equally well described than in the original data.
We define the coarsening kernel of scale factor $s$, $\kappa_{s}$, as:
$\kappa_{s}(x,y)\;=\;e^{-\frac{x^{2}+y^{2}}{2s^{2}}}$ (11)
We disregard the introduction of a normalization factor at this point since we
will need to normalize locally later. The coarsened version of the velocity
vector would hence be given by the convolution of this kernel with the
velocity, denoted by $\kappa_{s}\star\vec{v}$. A coarsening convolution kernel
turns out to be convenient with almost horizontal fields, as the derivatives
commute with the convolution operator so if $\nabla\vec{v}\approx 0$ hence
$\kappa_{s}\star\nabla\vec{v}\approx 0$. However, this coarsening scheme needs
to be improved. By convention, we take the velocity $\vec{v}$ as zero over
land points. For that reason, a simple convolution does not produce a correct
coarsened version of the velocity because points close to land would
experience a loss of energy. The easiest way to correct this is to normalize
by the weight of the sea points. Let us first define the sea mask $M(x,y)$ as
$1$ over the sea and $0$ over the land. The normalization weight is given by
$\kappa_{s}\star M$. For points very far from the land, this weight is just
the normalization of $\kappa_{s}$. For points surrounded by land points the
weight takes the contributions from sea points only. We thus define the
coarsened version by a scale factor $s$ of the velocity, $\vec{v}_{s}$, as:
$\vec{v}_{s}\;=\;\frac{\kappa_{s}\star\vec{v}}{\kappa_{s}\star M}$ (12)
In Figure 12 we show two examples of coarsened velocities. We can see that
typical circulation patterns are coarsened as $s$ increases, while land
obstacles are preserved. In fact, if $\Delta_{0}$ is the velocity resolution
scale, the effective resolution scale of $\vec{v}_{s}$ is $s\Delta_{0}$ (the
nominal resolution, on the contrary, is the original one, $\Delta_{0}$).
Figure 12: Top: Velocity field coarsened by a scale factor $s=2$, for a
equivalent resolution $\delta_{0}=1/4^{\circ}$. Bottom: Velocity field
coarsened by a scale factor $s=4$, for a equivalent resolution
$\delta_{0}=1/2^{\circ}$.
## Appendix D Introducing noise in the particle’s trajectories.
In order to introduce noise in the particle’s trajectories we resolve the
following system of stochastic equations:
$\displaystyle\frac{d\phi}{dt}=\frac{u(\phi,\lambda,t)}{R\cos(\lambda)}+\frac{\sqrt{2D}}{R\cos(\lambda)}\xi_{1}(t),$
(13)
$\displaystyle\frac{d\lambda}{dt}=\frac{v(\phi,\lambda,t)}{R}+\frac{\sqrt{2D}\xi_{2}(t)}{R}.$
(14)
$\xi_{i}(t)\ \ i=1,2$ are the components of a two-dimensional Gaussian white
noise with zero mean and correlations
$<\xi_{i}(t)\xi_{j}(t^{\prime})>=\delta_{ij}\delta(t-t^{\prime})$.Eqs. (13 and
14) use a simple white noise added to the trajectories. A more realistic
representation of small-scale Lagrangian dispersion in turbulent fields
requires using other kinds of correlated noises Griffa.1996 but, as we are
interested in examining influences of the missing scales, it is convenient to
use white noise, since this would represent the extreme case of very irregular
trajectories which gives an upper bound to the effects of more realistic
smoother small scales. Thus the tests presented here are similar to the ones
considered before (perturbation of the velocity) when adding uncorrelated
perturbations to the velocity, but here the perturbation acts at arbitrarily
small scales, as appropriate for a turbulent field, instead of being smooth
below a cutoff scale, as appropriate for modelling observational errors.
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|
arxiv-papers
| 2010-09-13T15:26:33 |
2024-09-04T02:49:12.864400
|
{
"license": "Public Domain",
"authors": "Ismael Hern\\'andez-Carrasco, Emilio Hern\\'andez-Garc\\'ia, Crist\\'obal\n L\\'opez, Antonio Turiel",
"submitter": "Ismael Hernandez-Carrasco",
"url": "https://arxiv.org/abs/1009.2419"
}
|
1009.2422
|
# Baryon Fields with $U_{L}(3)\times U_{R}(3)$ Chiral Symmetry III:
Interactions with Chiral $({\bf 3},\overline{\bf 3})\oplus(\overline{\bf
3},{\bf 3})$ Spinless Mesons
Hua-Xing Chen1 hxchen@rcnp.osaka-u.ac.jp V. Dmitrašinović2 dmitrasin@ipb.ac.rs
Atsushi Hosaka3 hosaka@rcnp.osaka-u.ac.jp 1Department of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China
2 Institute of Physics, Belgrade University, Pregrevica 118, Zemun, P.O.Box
57, 11080 Beograd, Serbia
3 Research Center for Nuclear Physics, Osaka University, Ibaraki 567–0047,
Japan
###### Abstract
Three-quark nucleon interpolating fields in QCD have well-defined
$SU_{L}(3)\times SU_{R}(3)$ and $U_{A}(1)$ chiral transformation properties,
viz. $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$, $[({\bf 3},\overline{{\bf
3}})\oplus(\overline{{\bf 3}},{\bf 3})]$, $[({\bf 8},{\bf 1})\oplus({\bf
1},{\bf 8})]$ and their “mirror” images, Ref. Chen:2008qv . It has been shown
(phenomenologically) in Ref. Chen:2009sf that mixing of the $[({\bf 6},{\bf
3})\oplus({\bf 3},{\bf 6})]$ chiral multiplet with one ordinary (“naive”) and
one “mirror” field belonging to the $[({\bf 3},\overline{{\bf
3}})\oplus(\overline{{\bf 3}},{\bf 3})]$, $[({\bf 8},{\bf 1})\oplus({\bf
1},{\bf 8})]$ multiplets can be used to fit the values of the isovector
($g_{A}^{(3)}$) and the flavor-singlet (isoscalar) axial coupling
($g_{A}^{(0)}$) of the nucleon and then predict the axial $F$ and $D$
coefficients, or vice versa, in reasonable agreement with experiment. In an
attempt to derive such mixing from an effective Lagrangian, we construct all
$SU_{L}(3)\times SU_{R}(3)$ chirally invariant non-derivative one-meson-baryon
interactions and then calculate the mixing angles in terms of baryons’ masses.
It turns out that there are (strong) selection rules: for example, there is
only one non-derivative chirally symmetric interaction between $J=\frac{1}{2}$
fields belonging to the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ and the
$[({\bf 3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$ chiral
multiplets, that is also $U_{A}(1)$ symmetric. We also study the chiral
interactions of the $[({\bf 3},\overline{{\bf 3}})\oplus(\overline{{\bf
3}},{\bf 3})]$ and $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$ nucleon
fields. Again, there are selection rules that allow only one off-diagonal non-
derivative chiral $SU_{L}(3)\times SU_{R}(3)$ interaction of this type, that
also explicitly breaks the $U_{A}(1)$ symmetry. We use this interaction to
calculate the corresponding mixing angles in terms of baryon masses and fit
two lowest lying observed nucleon (resonance) masses, thus predicting the
third $(J=\frac{1}{2},I=\frac{3}{2})$ $\Delta$ resonance, as well as one or
two flavor-singlet $\Lambda$ hyperon(s), depending on the type of mixing. The
effective chiral Lagrangians derived here may be applied to high density
matter calculations.
baryon, chiral symmetry, axial current, $F$/$D$ values
###### pacs:
14.20.-c, 11.30.Rd, 11.40.Dw
## I Introduction
Axial current “coupling constants” of the baryon flavor octet are well known,
see Ref. Yamanishi:2007zza . The zeroth (time-like) components of these axial
currents are generators of the $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry
that is one of the fundamental symmetries of QCD. The general flavor
$SU_{F}(3)$ symmetric form of the nucleon axial current contains two free
parameters, called $F$ and $D$ couplings, that are empirically determined as
$F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, see Ref. Yamanishi:2007zza .
Another, perhaps separate, yet equally important piece of information is the
flavor-singlet axial coupling $g_{A}^{(0)}=0.33\pm 0.08$ of the nucleon
Bass:2007zzb ,Vogelsang:2007zza .
Recent studies Dmitrasinovic:2009vp ; Chen:2009sf point towards baryon chiral
mixing (of $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ with the $[({\bf
3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$, $[({\bf 8},{\bf
1})\oplus({\bf 1},{\bf 8})]$ chiral multiplets 111These multiplets are not
limited to three-quark interpolators: for a discussion of the validity of our
assumptions, see Sect. III.3.) as a possible mechanism underlying the baryons’
axial couplings. This finding is in line with the old current algebra results
of Gerstein and Lee Gerstein:1966zz and of Harari Harari:1966yq ;
Harari:1966jz , updated to include recently measured values of $F$ and $D$
couplings, Ref. Yamanishi:2007zza , and extended to include the flavor-singlet
coupling $g_{A}^{(0)}$ of the nucleon, which was not considered in the
mid-1960’s at all, presumably due to the lack of data. Our own starting point
was the study of the QCD interpolating fields’ chiral properties Nagata:2007di
,Nagata:2008zzc ,Chen:2008qv .
The next step is to try and reproduce this phenomenological mixing starting
from a chiral effective model interaction, rather than per fiat. As the first
step in that direction we must look for a dynamical source of mixing. One such
mechanism is the simplest chirally symmetric non-derivative
one-$(\sigma,\pi)$-meson interaction Lagrangian; non-derivative because that
induces baryon masses via the $\sigma$-baryon coupling.
We construct all $SU_{L}(3)\times SU_{R}(3)$ chirally invariant non-derivative
one-meson-baryon interactions and then use them to calculate the mixing angles
in terms of baryons’ masses. It turns out that there are severe chiral
selection rules at work here. For example, we show that only the mirror field
$[(\overline{{\bf 3}},{\bf 3})\oplus({\bf 3},\overline{{\bf 3}})]$ can be
coupled to the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ baryon chiral
multiplet by non-derivative terms; whereas the ordinary (“naive”) multiplet
$[({\bf 3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$ requires
one (or generally an odd number of) derivative(s). Moreover, this interaction
also conserves the $U_{A}(1)$ symmetry. This is interesting, as the mixing
with a mirror baryon field of this type seems preferable from the point of
view of the two-flavor phenomenological study, Ref. Dmitrasinovic:2009vp .
We note that all, but one of the $SU_{L}(3)\times SU_{R}(3)$ symmetric
interactions, viz. the $[({\bf 3},\overline{{\bf 3}})\oplus(\overline{{\bf
3}},{\bf 3})]--[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$, also conserve the
$U_{A}(1)$ symmetry. This means that explicit $U_{A}(1)$ symmetry breaking may
occur in baryons only in so far as the $SU_{L}(3)\times SU_{R}(3)$ symmetry is
explicitly broken, with the exception mentioned above. This is in stark
contrast with the $SU_{L}(2)\times SU_{R}(2)$ case, where all of the
interaction terms have both the $U_{A}(1)$ symmetry-conserving and the
$U_{A}(1)$ symmetry-breaking version Dmitrasinovic:2009vp ;
Dmitrasinovic:2009vy . In this sense, the three-flavor chiral symmetry is more
restrictive and consequently more instructive than the two-flavor one.
The conventional models of (linearly realized) chiral $SU_{L}(3)\times
SU_{R}(3)$ symmetry, Refs. Hara:1965 ; Lee:1968 ; Bardeen:1969ra ; Christos2 ;
Zheng:1992mn ; Papazoglou:1997uw , on the other hand appear to fix the $F$ and
$D$ parameters at either ($F$=0,$D$=1), which case goes by the name of $[({\bf
3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$, or at
($F$=1,$D$=0), which case goes by the name of $[({\bf 8},{\bf 1})\oplus({\bf
1},{\bf 8})]$ chiral representation. Both of these chiral representations
suffer from the shortcoming that $F$+$D$=1$\neq g_{A}^{(3)}=$1.267 without
derivative couplings. But, even with derivative interactions, one cannot
change the value of the vanishing coupling, i.e. of $F$=0, in $[({\bf
3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$, or of $D$=0, in
$[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$. Rather, one can only renormalize
the non-vanishing coupling to 1.267. This is perhaps the most troublesome
problem of the linear realization chiral $SU_{L}(3)\times SU_{R}(3)$ symmetric
Lagrangians as it has far-reaching consequences for the kaon and hyperon
interactions, hyper-nuclear physics and nuclear astrophysics of collapsed
stars Papazoglou:1998vr ; Beckmann:2001bu .
Another, perhaps equally important and difficult problem is that of the
flavor-singlet axial coupling of the nucleon Bass:2007zzb ,Vogelsang:2007zza .
This is widely thought of as being disconnected from the $F$,$D$ problem, but
we have already shown, see Refs. Dmitrasinovic:2009vp ; Chen:2009sf , that the
chiral mixing of three-quark interpolating fields casts some new light on this
problem. Namely, the flavor-singlet axial coupling turns out to be
$g_{A}^{(0)}=(3F-D)$, i.e., a function of the flavor SU(3) octet $(F,D)$
coefficients and thus proportional to the eighth flavor component of the
$SU(3)$ symmetric axial coupling $g_{A}^{(8)}=\frac{1}{\sqrt{3}}(3F-D)$, so
long as one mixes only three-quark interpolating fields. In other words, the
ratio of these two measured quantities is fixed at ${\sqrt{3}}$ in the three-
quark assumption, so one must go beyond this approximation in order to break
the deadlock.
Even though an awareness of this mixing has been around for more than 40 years
Hara:1965 ; Lee:1968 ; Bardeen:1969ra ; Weinberg:1969hw , the $SU_{L}(3)\times
SU_{R}(3)$ chiral interactions necessary to describe such chiral mixing(s)
have not been considered in print 222D. Jido and A. Ohnishi have shown us the
results of some of their unpublished studies Jido09 ; Ohnishi09 . Some
$SU_{L}(2)\times SU_{R}(2)$ results can be found in Refs. Dmitrasinovic:2009vp
; Dmitrasinovic:2009vy and some limited $SU_{L}(3)\times SU_{R}(3)$ results
can be found in Refs. Christos2 ,Zheng:1992mn ., let alone derived. The
present paper serves to provide a dynamical model of chiral mixing that is the
“best” approximation to the phenomenological solution of both the $(F,D)$ and
the flavor-singlet axial coupling problems, assuming only three-quark baryon
interpolating fields. We found two simple solutions/fits 333which does not
preclude existence of more complicated solutions.: one that conserves the
$U_{A}(1)$ symmetry and another one that does not. This goes to show that the
“QCD $U_{A}(1)$ anomaly” may, but need not be the underlying source of the
“nucleon spin problem” Bass:2007zzb ,Vogelsang:2007zza , as was once widely
thought Zheng:1991pk . In all likelihood the $U_{A}(1)$ anomaly provides only
a (relatively) small part of the solution, the largest part coming from the
chiral structure (“mixing”) of the nucleon.
One immediate application of our results ought to be in high density matter
calculations, where only one baryon chiral multiplet ($[({\bf
3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$) and its
interaction with mesons have been used for some time now Papazoglou:1998vr ;
Beckmann:2001bu .
The present paper consists of five parts: after the present Introduction, in
Sect. II we define the $SU(3)\times SU(3)$ chiral transformations of three-
quark baryon fields and of the spinless mesons, with special emphasis on the
$SU(3)$ phase conventions. In Sect. III we construct the $SU_{L}(3)\times
SU_{R}(3)$ chirally invariant interactions. In Sect. IV we apply chiral mixing
formalism to the hyperons’ axial currents and then use the chiral interactions
to reproduce the mixing angles. In this way we determine the masses of the
admixed states. Finally, in Sect. V we discuss the results and offer a summary
and an outlook on future developments.
## II Preliminaries: Chiral Transformations of Mesons and Baryons
### II.1 Chiral Transformations of $({\bf 3},\overline{\bf
3})\oplus(\overline{\bf 3},{\bf 3})$ Spinless Mesons
We follow the same definition of chiral transformation in Ref. Chen:2008qv :
$\displaystyle\bf{U(1)_{V}}$ $\displaystyle:$ $\displaystyle
q\to\exp(i{\lambda^{0}\over 2}a_{0})q=q+\delta q\,,$
$\displaystyle\bf{SU(3)_{V}}$ $\displaystyle:$
$\displaystyle{q}\to\exp(i{\vec{\lambda}\over
2}\cdot\vec{a}){q}=q+\delta^{\vec{a}}q\,,$ (1) $\displaystyle\bf{U(1)_{A}}$
$\displaystyle:$ $\displaystyle q\to\exp(i\gamma_{5}{\lambda^{0}\over
2}b_{0})q=q+\delta_{5}q\,,$ $\displaystyle\bf{SU(3)_{A}}$ $\displaystyle:$
$\displaystyle{q}\to\exp(i\gamma_{5}{\vec{\lambda}\over
2}\cdot\vec{b}){q}=q+\delta_{5}^{\vec{b}}q\,.$
We define the scalar and pseudoscalar mesons in the $SU(3)$ space:
$\displaystyle\sigma^{a}$ $\displaystyle=$
$\displaystyle\bar{q}_{A}\lambda_{AB}^{a}q_{B}\,,$ (2) $\displaystyle\pi^{a}$
$\displaystyle=$ $\displaystyle\bar{q}_{A}\lambda_{AB}^{a}i\gamma_{5}q_{B}\,,$
(3)
where the index $a$ goes from 0 to 8, and the zero component of Gell-Mann
matrices is $\lambda^{0}=\sqrt{2\over 3}{\bf 1}$.
The nucleon fields belong to the chiral representation of $({\bf
3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})$, and their combination
transforms as:
$\displaystyle\delta_{5}^{\vec{b}}(\sigma^{b}+i\gamma_{5}\pi^{b})$
$\displaystyle=$
$\displaystyle-i\gamma_{5}b^{a}d_{abc}(\sigma^{c}+i\gamma_{5}\pi^{c})\,,$ (4)
$\displaystyle\delta_{5}^{\vec{b}}(\sigma^{b}-i\gamma_{5}\pi^{b})$
$\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}d_{abc}(\sigma^{c}-i\gamma_{5}\pi^{c})\,,$
where $d_{abc}$ and $f_{abc}$ are defined to contain the 0 index:
$\\{\lambda^{a},\lambda^{b}\\}=2d^{abc}\lambda^{c}\,,[\lambda^{a},\lambda^{b}]=2if^{abc}\lambda^{c}\,.$
(5)
We note here that in these equations we do not have the $\delta^{ab}$ factors
which are necessary in the usual equation
$\\{\lambda^{a},\lambda^{b}\\}=2d^{abc}\lambda^{c}+{4\over
3}\delta^{ab}\,,(a,b=1,\cdots 8)\,.$ (6)
The nonzero $f$ and $d$ coefficients are:
$\displaystyle\begin{array}[]{cc}abc&f^{abc}\\\ \hline\cr 123&1\\\ 147&1/2\\\
156&-1/2\\\ 246&1/2\\\ 257&1/2\\\ 345&1/2\\\ 367&-1/2\\\ 458&\sqrt{3}/2\\\
678&\sqrt{3}/2\\\ \hline\cr\end{array}\hskip
85.35826pt\begin{array}[]{cc|cc|cc}abc&d^{abc}&abc&d^{abc}&abc&d^{abc}\\\
\hline\cr 000&\sqrt{2/3}&118&{1/\sqrt{3}}&355&1/2\\\
011&\sqrt{2/3}&146&1/2&366&-1/2\\\ 022&\sqrt{2/3}&157&1/2&377&-1/2\\\
033&\sqrt{2/3}&228&{1/\sqrt{3}}&448&-1/(2\sqrt{3})\\\
044&\sqrt{2/3}&247&-1/2&558&-1/(2\sqrt{3})\\\
055&\sqrt{2/3}&256&1/2&668&-1/(2\sqrt{3})\\\
066&\sqrt{2/3}&338&{1/\sqrt{3}}&778&-1/(2\sqrt{3})\\\
077&\sqrt{2/3}&344&1/2&888&-1/\sqrt{3}\\\ 088&\sqrt{2/3}&&\\\
\hline\cr\end{array}$ (27)
To simplify our calculations sometimes we use the “physical” basis, whose
definitions are:
$\displaystyle\left(\begin{array}[]{c}M^{1}\\\ M^{2}\\\ M^{3}\\\ M^{4}\\\
M^{5}\\\ M^{6}\\\ M^{7}\\\ M^{8}\\\ M^{9}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccc}1&0&0&0&0&0&0&0&0\\\
0&{1\over\sqrt{2}}&-{i\over\sqrt{2}}&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\
0&{1\over\sqrt{2}}&{i\over\sqrt{2}}&0&0&0&0&0&0\\\
0&0&0&0&{1\over\sqrt{2}}&-{i\over\sqrt{2}}&0&0&0\\\
0&0&0&0&{1\over\sqrt{2}}&{i\over\sqrt{2}}&0&0&0\\\
0&0&0&0&0&0&{1\over\sqrt{2}}&-{i\over\sqrt{2}}&0\\\
0&0&0&0&0&0&{1\over\sqrt{2}}&{i\over\sqrt{2}}&0\\\
0&0&0&0&0&0&0&0&1\end{array}\right)\left(\begin{array}[]{c}\sigma^{0}+i\gamma_{5}\pi^{0}\\\
\sigma^{1}+i\gamma_{5}\pi^{1}\\\ \sigma^{2}+i\gamma_{5}\pi^{2}\\\
\sigma^{3}+i\gamma_{5}\pi^{3}\\\ \sigma^{4}+i\gamma_{5}\pi^{4}\\\
\sigma^{5}+i\gamma_{5}\pi^{5}\\\ \sigma^{6}+i\gamma_{5}\pi^{6}\\\
\sigma^{7}+i\gamma_{5}\pi^{7}\\\
\sigma^{8}+i\gamma_{5}\pi^{8}\end{array}\right)\,.$ (55)
In this basis:
$\displaystyle M^{1}=\sigma_{0}+i\gamma_{5}\eta_{0}\,,$ (56) $\displaystyle
M^{2}=a_{0}^{+}+i\gamma_{5}\pi^{+}\,,M^{3}=a_{0}^{0}+i\gamma_{5}\pi^{0}\,,M^{4}=a_{0}^{-}+i\gamma_{5}\pi^{-}\,,$
$\displaystyle
M^{5}=\kappa^{+}+i\gamma_{5}K^{+}\,,M^{6}=\kappa^{-}+i\gamma_{5}K^{-}\,,M^{7}=\kappa^{0}+i\gamma_{5}K^{0}\,,M^{8}=\bar{\kappa}^{0}+i\gamma_{5}\bar{K}^{0}\,,$
$\displaystyle M^{9}=f_{0}+i\gamma_{5}\eta_{8}\,.$
We have classified the baryon interpolating fields in our previous paper
Chen:2008qv . We found that the baryon interpolating fields
$N_{+}^{a}=N^{a}_{1}+N^{a}_{2}$ belong to the chiral representation
$(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$; $\Lambda$ and
$N_{-}^{a}=N^{a}_{1}-N^{a}_{2}$ belong to the chiral representation
$(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{\overline{3}},\mathbf{3})$;
$N^{a}_{\mu}$ and $\Delta^{P}_{\mu}$ belong to the chiral representation
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$; and
$\Delta^{P}_{\mu\nu}$ belong to the chiral representation
$(\mathbf{10},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})$. Here $N^{a}_{1}$ and
$N^{a}_{2}$ are the two independent kinds of nucleon fields. $N^{a}_{1}$
contains the “scalar diquark” and $N^{a}_{2}$ contains the “pseudoscalar
diquark”. Moreover, we calculated their chiral transformations in Ref.
Chen:2008qv . In the following sections, we will use these baryon fields
together with one meson field to construct the chiral invariant Lagrangians.
### II.2 Chiral Transformations of Baryons
#### II.2.1 Chiral Transformations of $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$ Baryons
The baryon field $N_{(18)}=(N_{\mu},\Delta_{\mu})^{T}$ belongs to the chiral
representation $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$:
$\displaystyle
N^{1}=p\,,N^{2}=n\,,N^{3}=\Sigma^{+}\,,N^{4}=\Sigma^{0}\,,N^{5}=\Sigma^{-}\,,N^{6}=\Xi^{0}\,,N^{7}=\Xi^{-}\,,N^{8}=\Lambda_{8}\,,$
(57) $\displaystyle
N^{9}=\Delta^{++}\,,N^{10}=\Delta^{+}\,,N^{11}=\Delta^{0}\,,N^{12}=\Delta^{-}\,,$
$\displaystyle
N^{13}=\Sigma^{+}\,,N^{14}=\Sigma^{0}\,,N^{15}=\Sigma^{-}\,,N^{16}=\Xi^{0}\,,N^{17}=\Xi^{-}\,,N^{18}=\Omega\,,$
and we can write out their chiral transformation:
$\delta_{5}^{\vec{b}}N_{(18)}=i\gamma_{5}b^{a}{\bf
F}_{(18)}^{a}N_{(18)}=i\gamma_{5}b^{a}\left(\begin{array}[]{cc}{\bf
D}_{(8)}^{a}+{2\over 3}{\bf F}_{(8)}^{a}&{2\over\sqrt{3}}{\bf
T}_{(8/10)}^{a}\\\ {2\over\sqrt{3}}{\bf T}^{\dagger a}_{(8/10)}&{1\over 3}{\bf
F}_{(10)}^{a}\end{array}\right)\left(\begin{array}[]{c}N_{\mu}\\\
\Delta_{\mu}\end{array}\right)\,.$ (58)
where the matrices ${\bf D}^{a}_{(8)}$, ${\bf F}_{(8)}^{a}$, ${\bf
F}_{(10)}^{a}$ and ${\bf T}_{(8/10)}^{a}$ are calculated in our previous paper
Chen:2009sf .
#### II.2.2 Chiral Transformations of $[({\bf 3},\overline{{\bf
3}})\oplus(\overline{{\bf 3}},{\bf 3})]$ Baryons
This chiral representation contains the flavor octet and singlet
representations $\mathbf{\bar{3}}\otimes\mathbf{3}=\mathbf{8}\oplus\mathbf{1}$
$\sim N_{(9)}=(\Lambda,N_{-})^{T}$:
$\displaystyle
N^{1}=\Lambda_{0}\,,N^{2}=p\,,N^{3}=n\,,N^{4}=\Sigma^{+}\,,N^{5}=\Sigma^{0}\,,N^{6}=\Sigma^{-}\,,N^{7}=\Xi^{0}\,,N^{8}=\Xi^{-}\,,N^{9}=\Lambda_{8}\,,$
(59)
and their chiral transformations are
$\delta_{5}^{\vec{b}}N_{(9)}=i\gamma_{5}b^{a}{\bf
F}_{(9)}^{a}N_{(9)}=i\gamma_{5}b^{a}\left(\begin{array}[]{cc}0&\sqrt{2\over
3}{\bf T}^{a}_{1/8}\\\ \sqrt{2\over 3}{\bf T}^{\dagger a}_{1/8}&{\bf
D}_{(8)}^{a}\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1}\\\
N_{-}\end{array}\right)\,.$ (60)
#### II.2.3 Chiral Transformations of $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf
8})]$ Baryons
This chiral representation $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$
contains the flavor octet representation
$\mathbf{8}\otimes\mathbf{1}=\mathbf{8}$ $\sim N_{(8)}=N_{+}$. The chiral
transformation is
$\displaystyle\delta_{5}^{\vec{b}}N_{(8)}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}{\rm\bf F}_{(8)}^{a}N_{(8)}\,.$ (61)
#### II.2.4 Chiral Transformations of $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf
10})]$ Baryons
This chiral representation $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})]$
contains the flavor decuplet representation
$\mathbf{10}\otimes\mathbf{1}=\mathbf{10}$ $\sim N_{(10)}=\Delta_{\mu\nu}$.
The chiral transformation is
$\displaystyle\delta_{5}^{\vec{b}}N_{(10)}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}{\rm\bf F}_{(10)}^{a}N_{(10)}\,.$ (62)
## III Chiral Interactions
In this Section we propose a new method for the construction of $N_{f}$=3
chiral invariants that differs from the one proposed for $N_{f}$=2 in Ref.
Nagata:2008xf and used in Refs. Dmitrasinovic:2009vp ; Dmitrasinovic:2009vy .
### III.1 Diagonal Interactions: Mass Terms
#### III.1.1 Chiral $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ Baryons
Diagonal Interactions
Our aim is to construct a chiral invariant Lagrangian:
$\bar{N}_{(18)}^{a}M^{c}N_{(18)}^{b}{\bf C}^{abc}_{(18)}\,,$ (63)
where the indices $a$ and $b$ run from 1 to 18, and the index $c$ just runs
from 1 to 9. By performing the chiral transformation to this Lagrangian, we
can obtain many equations. For example we have:
$\displaystyle\delta_{5}^{1}\big{(}\bar{p}M^{2}n{\bf C}^{122}_{(18)}\big{)}$
$\displaystyle=$ $\displaystyle{5\over 6}{\bf
C}^{122}_{(18)}\bar{n}M^{2}(i\gamma_{5}b_{1})n+\cdots\,,$ (64)
$\displaystyle\delta_{5}^{1}\big{(}\bar{\Delta}^{+}M^{2}n{\bf
C}^{10,2,2}_{(18)}\big{)}$ $\displaystyle=$ $\displaystyle-{\sqrt{2}\over
3}{\bf C}^{10,2,2}_{(18)}\bar{n}M^{2}(i\gamma_{5}b_{1})n+\cdots\,,$
$\displaystyle\delta_{5}^{1}\big{(}\bar{n}M^{2}\Delta^{-}{\bf
C}^{2,12,2}_{(18)}\big{)}$ $\displaystyle=$ $\displaystyle\sqrt{2\over 3}{\bf
C}^{2,12,2}_{(18)}\bar{n}M^{2}(i\gamma_{5}b_{1})n+\cdots\,,$
$\displaystyle\delta_{5}^{1}\big{(}\bar{n}M^{1}n{\bf C}^{221}_{(18)}\big{)}$
$\displaystyle=$ $\displaystyle{1\over\sqrt{3}}{\bf
C}^{221}_{(18)}\bar{n}M^{2}(i\gamma_{5}b_{1})n+\cdots\,,$
$\displaystyle\delta_{5}^{1}\big{(}\bar{n}M^{9}n{\bf C}^{229}_{(18)}\big{)}$
$\displaystyle=$ $\displaystyle{1\over\sqrt{6}}{\bf
C}^{229}_{(18)}\bar{n}M^{2}(i\gamma_{5}b_{1})n+\cdots\,.$
These are all the fields that are transformed to
$\bar{n}M^{2}(i\gamma_{5}b_{1})n$. If the Lagrangian (63) is chiral invariant,
this sum should be zero:
$\displaystyle{5\over 6}{\bf C}^{122}_{(18)}-{\sqrt{2}\over 3}{\bf
C}^{10,2,2}_{(18)}+\sqrt{2\over 3}{\bf C}^{2,12,2}_{(18)}+{1\over\sqrt{3}}{\bf
C}^{221}_{(18)}+{1\over\sqrt{6}}{\bf C}^{229}_{(18)}=0\,.$ (65)
Solving these equations for ${\bf C}^{abc}_{(18)}$ together with the
hermiticity condition, we find that there is only one solution. The uniqueness
of the solution is guaranteed by the fact that there is only one way to form
the chiral singlet combination out of the baryon field $[({\bf 6},{\bf
3})\oplus({\bf 3},{\bf 6})]$ and the meson field $[({\bf 3},\overline{\bf
3})\oplus(\overline{\bf 3},{\bf 3})]$. This solution can be written out much
more easily using ${\bf D}_{(18)}^{c}$ in the following form:
$g_{(18)}\bar{N}_{(18)}^{a}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
D}_{(18)}^{c})_{ab}N_{(18)}^{b}\,,$ (66)
where $g_{(18)}$ is the coupling constant, and the matrices ${\bf D}_{(18)}$
are solved to be:
$\displaystyle{\bf D}^{0}_{(18)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left(\begin{array}[]{cc}{\bf 1}_{8\times
8}&0\\\ 0&-2\times{\bf 1}_{10\times 10}\end{array}\right)\,,$ (69)
$\displaystyle{\bf D}^{a}_{(18)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}{\rm{\bf D}_{(8)}^{a}+{2\over 3}{\bf
F}_{(8)}^{a}}&-{1\over\sqrt{3}}{\rm{\bf T}_{(8/10)}^{a}}\\\
-{1\over\sqrt{3}}{\rm{\bf T}^{\dagger a}_{(8/10)}}&-{2\over 3}{\bf
F}_{(10)}^{a}\end{array}\right)\,.$ (72)
Besides the Lagrangian (63), its mirror part
$g_{(18)}\bar{N}_{(18m)}^{a}(\sigma^{c}-i\gamma_{5}\pi^{c})({\bf
D}_{(18)}^{c})_{ab}N_{(18m)}^{b}\,,$ (73)
is also chiral invariant. Using these solutions, and performing the chiral
transformation, we can obtain the following relation:
$\displaystyle{\bf F}^{a\dagger}_{(18)}{\bf D}^{b}_{(18)}+{\bf
D}^{b}_{(18)}{\bf F}^{a}_{(18)}-d_{abc}{\bf D}^{c}_{(18)}=0\,,$ (74)
where ${\bf F}^{a}_{(18)}$ and ${\bf D}^{b}_{(18)}$ are defined in the
previous Eqs. (58) and (69).
The solution in the physical basis ($\bar{N}_{(18)}^{a}M^{c}N_{(18)}^{b}{\bf
C}^{abc}_{(18)}$) can be obtained by the following relations:
$\displaystyle{\bf C}^{ab1}_{(18)}=({\bf D}^{0}_{(18)})_{ab}\,,{\bf
C}^{ab3}_{(18)}=({\bf D}^{3}_{(18)})_{ab}\,,{\bf C}^{ab9}_{(18)}=({\bf
D}^{8}_{(18)})_{ab}\,,$ (75) $\displaystyle{1\over\sqrt{2}}({\bf
C}^{ab2}_{(18)}+{\bf C}^{ab4}_{(18)})=({\bf
D}^{1}_{(18)})_{ab}\,,{i\over\sqrt{2}}(-{\bf C}^{ab2}_{(18)}+{\bf
C}^{ab4}_{(18)})=({\bf D}^{2}_{(18)})_{ab}\,,$
$\displaystyle{1\over\sqrt{2}}({\bf C}^{ab5}_{(18)}+{\bf
C}^{ab6}_{(18)})=({\bf D}^{4}_{(18)})_{ab}\,,{i\over\sqrt{2}}(-{\bf
C}^{ab5}_{(18)}+{\bf C}^{ab6}_{(18)})=({\bf D}^{5}_{(18)})_{ab}\,,$
$\displaystyle{1\over\sqrt{2}}({\bf C}^{ab7}_{(18)}+{\bf
C}^{ab8}_{(18)})=({\bf D}^{6}_{(18)})_{ab}\,,{i\over\sqrt{2}}(-{\bf
C}^{ab7}_{(18)}+{\bf C}^{ab8}_{(18)})=({\bf D}^{7}_{(18)})_{ab}\,.$
#### III.1.2 Chiral $[({\bf 3},\overline{{\bf 3}})\oplus(\overline{{\bf
3}},{\bf 3})]$ Baryons Diagonal Interactions
Following the same procedure of the previous section, we find that the
Lagrangian $\bar{N}_{(9)}^{a}M^{c}N_{(9)}^{b}{\bf C}^{abc}_{(9)}$ can not be
chiral invariant, which means that their is no solution for ${\bf
C}^{abc}_{(9)}$. However, we can still get a chiral invariant Lagrangian
through “different” fields. There are two possible ways:
1. 1.
We use the meson field $\sigma^{a}-i\gamma_{5}\pi^{a}$:
$\displaystyle\delta_{5}^{\vec{b}}(\sigma^{b}-i\gamma_{5}\pi^{b})=i\gamma_{5}b^{a}d_{abc}(\sigma^{c}-i\gamma_{5}\pi^{c})\,.$
(76)
2. 2.
We use the mirror field of $N_{(9)}$:
$\delta_{5}^{\vec{b}}N_{(9m)}=-i\gamma_{5}b^{a}{\bf
F}_{(9)}^{a}N_{(9m)}=i\gamma_{5}b^{a}\left(\begin{array}[]{cc}0&-\sqrt{2\over
3}{\bf T}^{a}_{(1/8)}\\\ -\sqrt{2\over 3}{\bf T}^{\dagger a}_{(1/8)}&-{\bf
D}_{(8)}^{a}\end{array}\right)N_{(9m)}\,.$ (77)
Then we can construct the chiral invariant Lagrangians:
$\bar{N}_{(9m)}^{a}M^{c}N_{(9m)}^{b}{\bf C}^{abc}_{(9)}\,.$ (78)
or its mirror part
$\bar{N}_{(9)}^{a}(M^{+})^{c}N_{(9)}^{b}{\bf C}^{abc}_{(9)}\,,$ (79)
Assuming that they are hermitian, we find that there is only one solution for
${\bf C}^{abc}_{(9)}$. The solution for the coefficients ${\bf C}^{abc}_{(9)}$
in these two Lagrangians is the same, and it can be written out in the
following form:
$g_{(9)}\bar{N}_{(9m)}^{a}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
D}_{(9)}^{c})_{ab}N_{(9m)}^{b}\,,$ (80)
where the solution is
$\displaystyle{\bf D}^{0}_{(9)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left(\begin{array}[]{cc}-2&{\bf 0}_{1\times
8}\\\ {\bf 0}_{8\times 1}&{\bf 1}_{8\times 8}\end{array}\right)\,,$ (83)
$\displaystyle{\bf D}^{a}_{(9)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&{1\over\sqrt{6}}{\rm{\bf
T}^{a}}_{(1/8)}\\\ {1\over\sqrt{6}}{\rm{\bf T}_{(1/8)}^{\dagger a}}&-\rm{\bf
D}^{a}_{(8)}\end{array}\right)\,.$ (86)
The uniqueness of the solution is guaranteed by the fact that there is only
one way to form the chiral singlet combination out of the baryon field $[({\bf
3},\overline{{\bf 3}})\oplus(\overline{{\bf 3}},{\bf 3})]$ and the meson field
$[({\bf 3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})]$. The
coefficients ${\bf C}^{abc}_{(9)}$ can be similarly obtained like Eq. (75).
From this Lagrangian, we can obtain another relation:
$\displaystyle{\bf F}^{a\dagger}_{(9)}{\bf D}^{b}_{(9)}+{\bf D}^{b}_{(9)}{\bf
F}^{a}_{(9)}+d_{abc}{\bf D}^{c}_{(9)}=0\,.$ (87)
#### III.1.3 Chiral $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$ Baryons
Diagonal Interactions
Simply adding one $[({\bf 3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf
3})]$ meson field to two $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$ baryon
fields can not produce a chirally invariant Lagrangian. By adding two $[({\bf
3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})]$ meson fields, however,
there are several possible ways to construct chirally invariant Lagrangians
Jido09 . First we can write out the group structures:
$\displaystyle\big{(}(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})\big{)}^{2}\otimes\big{(}(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})\big{)}^{2}$
$\displaystyle\rightarrow$
$\displaystyle\big{(}(\mathbf{1},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})\big{)}\otimes\big{(}(\mathbf{1},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})\big{)}\rightarrow\big{(}(\mathbf{1},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})\big{)}----------(1)$
$\displaystyle\rightarrow$
$\displaystyle\Big{(}2\times\big{(}(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})\big{)}\Big{)}\otimes\big{(}(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})\big{)}\rightarrow
2\times\big{(}(\mathbf{1},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})\big{)}-----(2)$
$\displaystyle\rightarrow$
$\displaystyle\Big{(}4\times\big{(}(\mathbf{8},\mathbf{8})\oplus(\mathbf{8},\mathbf{8})\big{)}\Big{)}\otimes\big{(}(\mathbf{8},\mathbf{8})\oplus(\mathbf{8},\mathbf{8})\big{)}\rightarrow
4\times\big{(}(\mathbf{1},\mathbf{1})\oplus(\mathbf{1},\mathbf{1})\big{)}-----(3)$
Here we just give the Lagrangian for the simplest case (1), which is
$M^{+a}M^{a}\bar{N}^{b}_{(8)}\gamma_{5}N^{b}_{(8m)}+h.c.$. The others can be
obtained by using $M$, $M^{+}$, $N_{(8)}$ and $N_{(8m)}$ as well as related
coefficients $d_{abc}$ and $f_{abc}$.
#### III.1.4 Chiral $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})]$ Baryons
Diagonal Interactions
We find that simply adding one $[({\bf 3},\overline{\bf
3})\oplus(\overline{\bf 3},{\bf 3})]$ meson field to two $[({\bf 10},{\bf
1})\oplus({\bf 1},{\bf 10})]$ baryon fields can not produce a chirally
invariant Lagrangian.
### III.2 Chiral Mixing Interactions
#### III.2.1 Chiral Mixing Interaction $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$ \- $[({\bf 3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})]$
The mixing of $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ with
$[(\overline{\bf 3},{\bf 3})\oplus({\bf 3},\overline{\bf 3})]$ (we note that
this is a mirror baryon) together a meson field can be a chiral singlet. So
from this section we will study the five nontrivial off-diagonal Lagrangians.
The simple form made from the “naive” baryons $N_{(18)}\sim[({\bf 6},{\bf
3})\oplus({\bf 3},{\bf 6})]$ and $N_{(9)}\sim[({\bf 3},\overline{\bf
3})\oplus(\overline{\bf 3},{\bf 3})]$ , $N^{a}_{(9)}M^{c}N_{(18)}^{b}{\bf
C}^{abc}_{(9/18)}+h.c.$ can not be chiral invariant. We need to use the mirror
field $N_{(9m)}\sim[(\overline{\bf 3},{\bf 3})\oplus({\bf 3},\overline{\bf
3})]$(mir), and find the following form of field
$\displaystyle\bar{N}^{a}_{(9m)}M^{c}N^{b}_{(18)}{\bf C}^{abc}_{(9/18)}+h.c.$
(89)
as well as its mirror part can be chiral invariant. Again we turn to the
following form
$\displaystyle
g_{(9/18)}\bar{N}^{a}_{(9m)}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
T}^{c}_{(9/18)})_{ab}N^{b}_{(18)}+h.c.$ (90)
We find that the only solution is
$\displaystyle{\bf T}^{0}_{(9/18)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left(\begin{array}[]{cc}{\bf 0}_{1\times
8}&{\bf 0}_{1\times 10}\\\ {\bf 1}_{8\times 8}&{\bf 0}_{8\times
10}\end{array}\right)\,,$ (93) $\displaystyle{\bf T}^{a}_{(9/18)}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-{1\over\sqrt{6}}{\bf
T}^{a}_{(1/8)}&{\bf 0}_{1\times 10}\\\ {1\over 3}{\bf
F}^{a}_{(8)}&{1\over\sqrt{3}}{\bf T}^{a}_{(8/10)}\end{array}\right)\,.$ (96)
The coefficients ${\bf C}^{abc}_{(9/18)}$ can be similarly obtained as in Eq.
(75), and we have the following relation:
$\displaystyle-{\bf F}^{a\dagger}_{(9)}{\bf T}^{b}_{(9/18)}+{\bf
T}^{b}_{(9/18)}{\bf F}^{a}_{(18)}-d_{abc}{\bf T}^{c}_{(9/18)}=0\,.$ (97)
#### III.2.2 Chiral Mixing Interaction $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$ – $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$
The mixing of a mirror baryon $[({\bf 3},{\bf 6})\oplus({\bf 6},{\bf
3})]$(mir) with $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$ together a meson
field can be a chiral singlet, and we find the following form of field:
$\displaystyle\bar{N}^{a}_{(8)}M^{c}N^{b}_{(18m)}{\bf C}^{abc}_{(9/18)}+h.c.$
(98)
and its mirror part can be chiral invariant. Again we turn to the basis
$\displaystyle g_{(8/18)}\bar{N}^{a}_{(8)}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
T}^{c}_{(8/18)})_{ab}N^{b}_{(18m)}+h.c.$ (99)
and the only solution is
$\displaystyle{\bf T}^{0}_{(8/18)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left({\bf 1}_{8\times 8},{\bf 0}_{8\times
10}\right)\,,$ (100) $\displaystyle{\bf T}^{a}_{(8/18)}$ $\displaystyle=$
$\displaystyle\left(-{1\over 2}{\bf D}_{(8)}^{a}+{1\over 6}{\bf
F}_{(8)}^{a},-{1\over\sqrt{3}}{\bf T}^{a}_{(8/10)}\right)\,.$ (101)
The coefficients ${\bf C}^{abc}_{(8/18)}$ can be similarly obtained as in Eq.
(75). And we have the following relation:
$\displaystyle-{\bf F}^{a\dagger}_{(8)}{\bf T}^{b}_{(8/18)}+{\bf
T}^{b}_{(8/18)}{\bf F}^{a}_{(18)}+d_{abc}{\bf T}^{c}_{(8/18)}=0\,.$ (102)
#### III.2.3 Chiral Mixing Interaction $[({\bf 3},\overline{\bf
3})\oplus(\overline{\bf 3},{\bf 3})]$ \- $[({\bf 8},{\bf 1})\oplus({\bf
1},{\bf 8})]$
The mixing of $[({\bf 3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})]$
with $[({\bf 8},{\bf 1})\oplus({\bf 1},{\bf 8})]$ together a meson field can
be a chiral singlet, and we find that there are two possibilities. One is the
following form of Lagrangian:
$\displaystyle\bar{N}^{a}_{(8)}M^{c}N^{b}_{(9)}{\bf C}^{abc}_{(8/9)}+h.c.$
(103)
and its mirror part can be chiral invariant. Again we turn to the basis
$\displaystyle g_{(8/9)}\bar{N}^{a}_{(8)}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
T}^{c}_{(8/9)})_{ab}N^{b}_{(9)}+h.c.$ (104)
and the only solution is
$\displaystyle{\bf T}^{0}_{(8/9)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left({\bf 0}_{8\times 1},{\bf 1}_{8\times
8}\right)\,,$ (105) $\displaystyle{\bf T}^{a}_{(8/9)}$ $\displaystyle=$
$\displaystyle\left({1\over\sqrt{6}}{\bf T}^{\dagger a}_{(1/8)},{1\over 2}{\bf
D}^{a}_{(8)}+{1\over 2}{\bf F}^{a}_{(8)}\right)\,.$ (106)
The coefficients ${\bf C}^{abc}_{(8/9)}$ can be similarly obtained like Eq.
(75). and we have the following relation:
$\displaystyle-{\bf F}^{a\dagger}_{(8)}{\bf T}^{b}_{(8/9)}-{\bf
T}^{b}_{(8/9)}{\bf F}^{a}_{(9)}+d_{abc}{\bf T}^{c}_{(8/9)}=0\,.$ (107)
The other possibility is the following form of Lagrangian, and the mixing of
$[({\bf 3},\overline{\bf 3})\oplus(\overline{\bf 3},{\bf 3})]$ with $[({\bf
1},{\bf 8})\oplus({\bf 8},{\bf 1})]$(mir)
$\displaystyle\bar{N}^{a}_{(8m)}M^{c}N^{b}_{(9)}{\bf C}^{abc}_{(8/9)}+h.c.$
(108)
This and its mirror image part can both be chiral invariant. Again we turn to
the particle basis
$\displaystyle g_{(B)}\bar{N}^{a}_{(8m)}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
T}^{c}_{(B)})_{ab}N^{b}_{(9)}+h.c.$ (109)
The only solution is
$\displaystyle{\bf T}^{0}_{B}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left({\bf 0}_{8\times 1},{\bf 1}_{8\times
8}\right)\,,$ (110) $\displaystyle{\bf T}^{a}_{B}$ $\displaystyle=$
$\displaystyle\left({1\over\sqrt{6}}{\bf T}^{\dagger a}_{(1/8)},{1\over 2}{\bf
D}^{a}_{(8)}-{1\over 2}{\bf F}^{a}_{(8)}\right)\,.$ (111)
Since we find that this is the only case which violate the $U_{A}(1)$
symmetry, we use the subscript $B$. The coefficients ${\bf C}^{abc}_{(8/9)}$
can be similarly obtained as in Eq. (75), and we have the following relation:
$\displaystyle{\bf F}^{a\dagger}_{(8)}{\bf T}^{b}_{B}-{\bf T}^{b}_{B}{\bf
F}^{a}_{(9)}+d_{abc}{\bf T}^{c}_{B}=0\,.$ (112)
#### III.2.4 Chiral Mixing Interaction $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$ \- $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})]$
For completeness’ sake we also show the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$ \- $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})]$ chiral mixing
interaction. The $[({\bf 10},{\bf 1})\oplus({\bf 1},{\bf 10})]$ decuplet
baryon field can only mix with $[({\bf 3},{\bf 6})\oplus({\bf 6},{\bf
3})]$(mir) to compose a chiral singlet, and we find the following form of
Lagrangian:
$\displaystyle\bar{N}^{a}_{(10)}M^{c}N^{b}_{(18m)}{\bf
C}^{abc}_{(10/18)}+h.c.$ (113)
and its mirror part can be chiral invariant. Again we turn to the basis
$\displaystyle
g_{(10/18)}\bar{N}^{a}_{(10)}(\sigma^{c}+i\gamma_{5}\pi^{c})({\bf
T}^{c}_{(10/18)})_{ab}N^{b}_{(18m)}+h.c.$ (114)
and the only solution is
$\displaystyle{\bf T}^{0}_{(10/18)}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left({\bf 0}_{10\times 8},{\bf 1}_{10\times
10}\right)\,,$ (115) $\displaystyle{\bf T}^{a}_{(10/18)}$ $\displaystyle=$
$\displaystyle\left(-{1\over\sqrt{3}}{\bf T}^{\dagger a}_{(8/10)},{1\over
3}{\bf F}^{a}_{(10)}\right)\,.$ (116)
The coefficients ${\bf C}^{abc}_{(8/9)}$ can be similarly obtained like Eq.
(75). and we have the following relation:
$\displaystyle-{\bf F}^{a\dagger}_{(10)}{\bf T}^{b}_{(10/18)}+{\bf
T}^{b}_{(10/18)}{\bf F}^{a}_{(18)}+d_{abc}{\bf T}^{c}_{(10/18)}=0\,.$ (117)
### III.3 Brief Summary of Interactions
Altogether we have the following form of chiral invariant Lagrangian:
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\overline{N}_{(8m)}&\overline{N}_{(9m)}&\overline{N}_{(18)}&\overline{N}_{(10m)}\end{array}\right)\Bigg{(}(\sigma^{a}+i\gamma_{5}\pi^{a})\left(\begin{array}[]{cccc}{\bf
0}_{8\times 8}&{\bf 0}_{8\times 9}&{\bf 0}_{8\times 18}&{\bf 0}_{8\times
10}\\\ {\bf 0}_{9\times 8}&g_{(9)}{\bf D}^{a}_{(9)}&g_{(9/18)}{\bf
T}^{a}_{(9/18)}&{\bf 0}_{9\times 10}\\\ {\bf 0}_{18\times
8}&g^{*}_{(9/18)}{\bf T}^{\dagger a}_{(9/18)}&g_{(18/18)}{\bf
D}^{a}_{(18)}&{\bf 0}_{18\times 10}\\\ {\bf 0}_{10\times 8}&{\bf 0}_{10\times
9}&{\bf 0}_{10\times 18}&{\bf 0}_{10\times 10}\end{array}\right)$ (132)
$\displaystyle+(\sigma^{a}-i\gamma_{5}\pi^{a})\left(\begin{array}[]{cccc}{\bf
0}_{8\times 8}&g_{(8/9)}{\bf T}^{a}_{(8/9)}&g_{(8/18)}{\bf
T}^{a}_{(8/18)}&{\bf 0}_{8\times 10}\\\ g_{(8/9)}^{*}{\bf T}^{\dagger
a}_{(8/9)}&{\bf 0}_{9\times 9}&{\bf 0}_{9\times 18}&{\bf 0}_{9\times 10}\\\
g_{(8/18)}^{*}{\bf T}^{\dagger a}_{(8/18)}&{\bf 0}_{18\times 9}&{\bf
0}_{18\times 18}&g_{(10/18)}^{*}{\bf T}^{\dagger a}_{(10/18)}\\\ {\bf
0}_{10\times 8}&{\bf 0}_{10\times 9}&g_{(10/18)}{\bf T}^{a}_{(10/18)}&{\bf
0}_{10\times
10}\end{array}\right)\Bigg{)}\left(\begin{array}[]{c}{N}_{(8m)}\\\
{N}_{(9m)}\\\ {N}_{(18)}\\\ {N}_{(10m)}\end{array}\right)\,,$
its mirror part is also chiral invariant:
$\displaystyle\mathcal{L}_{(m)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\overline{N}_{(8)}&\overline{N}_{(9)}&\overline{N}_{(18m)}&\overline{N}_{(10)}\end{array}\right)\Bigg{(}(\sigma^{a}-i\gamma_{5}\pi^{a})\left(\begin{array}[]{cccc}{\bf
0}_{8\times 8}&{\bf 0}_{8\times 9}&{\bf 0}_{8\times 18}&{\bf 0}_{8\times
10}\\\ {\bf 0}_{9\times 8}&g^{\prime}_{(9)}{\bf
D}^{a}_{(9)}&g^{\prime}_{(9/18)}{\bf T}^{a}_{(9/18)}&{\bf 0}_{9\times 10}\\\
{\bf 0}_{18\times 8}&g^{\prime*}_{(9/18)}{\bf T}^{\dagger
a}_{(9/18)}&g^{\prime}_{(18/18)}{\bf D}^{a}_{(18)}&{\bf 0}_{18\times 10}\\\
{\bf 0}_{10\times 8}&{\bf 0}_{10\times 9}&{\bf 0}_{10\times 18}&{\bf
0}_{10\times 10}\end{array}\right)$ (147)
$\displaystyle+(\sigma^{a}+i\gamma_{5}\pi^{a})\left(\begin{array}[]{cccc}{\bf
0}_{8\times 8}&g^{\prime}_{(8/9)}{\bf T}^{a}_{(8/9)}&g^{\prime}_{(8/18)}{\bf
T}^{a}_{(8/18)}&{\bf 0}_{8\times 10}\\\ g_{(8/9)}^{\prime*}{\bf T}^{\dagger
a}_{(8/9)}&{\bf 0}_{9\times 9}&{\bf 0}_{9\times 18}&{\bf 0}_{9\times 10}\\\
g_{(8/18)}^{\prime*}{\bf T}^{\dagger a}_{(8/18)}&{\bf 0}_{18\times 9}&{\bf
0}_{18\times 18}&g_{(10/18)}^{\prime*}{\bf T}^{\dagger a}_{(10/18)}\\\ {\bf
0}_{10\times 8}&{\bf 0}_{10\times 9}&g^{\prime}_{(10/18)}{\bf
T}^{a}_{(10/18)}&{\bf 0}_{10\times
10}\end{array}\right)\Bigg{)}\left(\begin{array}[]{c}{N}_{(8)}\\\ {N}_{(9)}\\\
{N}_{(18m)}\\\ {N}_{(10)}\end{array}\right)\,.$
Besides these, there is another single piece of Lagrangian which is also
chiral invariant:
$\displaystyle\mathcal{L}_{(B)}$ $\displaystyle=$ $\displaystyle
g_{(B)}\overline{N}_{(8)}(\sigma^{a}-i\gamma_{5}\pi^{a}){\bf
T}^{a}_{(B)}N_{(9m)}+h.c.\,,$
together with its mirror part
$\displaystyle\mathcal{L}_{(Bm)}$ $\displaystyle=$ $\displaystyle
g^{\prime}_{(B)}\overline{N}_{(8m)}(\sigma^{a}+i\gamma_{5}\pi^{a}){\bf
T}^{a}_{(B)}N_{(9)}+h.c.\,.$
At the same time, we have also proven that this is the only possible case.
Moveover, we can easily verify that this Lagrangian is also invariant under
$U_{A}(1)$ chiral transformation, except $\mathcal{L}_{(B)}$ and
$\mathcal{L}_{(Bm)}$. All these information is listed in Table 1. Besides
these Lagrangians, we still have the naive combinations:
${m_{(8)}}\overline{N}_{(8m)}\gamma_{5}N_{(8)}$,
${m_{(9)}}\overline{N}_{(9m)}\gamma_{5}N_{(9)}$,
${m_{(18)}}\overline{N}_{(18m)}\gamma_{5}N_{(18)}$ and
${m_{(10)}}\overline{N}_{(10m)}\gamma_{5}N_{(10)}$. There are no meson fields,
but these Lagrangians are still chiral $SU_{L}(3)\times SU_{R}(3)$ invariant
and chiral $U(1)_{A}$ invariant. This information is listed in Table 2.
Table 1: Allowed chiral invariant terms with one meson field. The $\surd$ denotes that the symmetries are conserved, while $\times$ denotes not. ($SU_{A}(3)$, $U_{A}(1)$) | $(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})$[mir] | $(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})$[mir] | $(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$ | $(\mathbf{1},\mathbf{10})\oplus(\mathbf{10},\mathbf{1})$[mir]
---|---|---|---|---
$(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})$[mir] | N/A | ($\surd$, $\surd$) | ($\surd$, $\surd$) | N/A
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$[mir] | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$) | N/A
$(\mathbf{\bar{6}},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{\bar{6}})$ | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$)
$(\mathbf{1},\mathbf{\overline{10}})\oplus(\mathbf{\overline{10}},\mathbf{1})$[mir] | N/A | N/A | ($\surd$, $\surd$) | N/A
($SU_{A}(3)$, $U_{A}(1)$) | $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$ | $(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$ | $(\mathbf{3},\mathbf{6})\oplus(\mathbf{6},\mathbf{3})$[mir] | $(\mathbf{10},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})$
$(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$ | N/A | ($\surd$, $\surd$) | ($\surd$, $\surd$) | N/A
$(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})$ | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$) | N/A
$(\mathbf{\bar{3}},\mathbf{\bar{6}})\oplus(\mathbf{\bar{6}},\mathbf{\bar{3}})$[mir] | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$) | ($\surd$, $\surd$)
$(\mathbf{\overline{10}},\mathbf{1})\oplus(\mathbf{1},\mathbf{\overline{10}})$ | N/A | N/A | ($\surd$, $\surd$) | N/A
($SU_{A}(3)$, $U_{A}(1)$) | $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$ | $(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})$[mir] | |
$(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})$ | N/A | ($\surd$, $\times$) | |
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$[mir] | ($\surd$, $\times$) | N/A | |
These results stand in marked contrast to the two-flavor case
Dmitrasinovic:2009vp ; Dmitrasinovic:2009vy , where the $SU_{L}(2)\times
SU_{R}(2)$ symmetric interactions have both a $U_{A}(1)$ symmetry-conserving
and a $U_{A}(1)$ symmetry-breaking version. Thus, the three-flavor chiral
symmetry is more restrictive than the two-flavor one.
Table 2: Allowed chirally invariant terms without meson field (the so-called mirror-mass terms). The $\surd$ denotes that the symmetries are conserved, while $\times$ denotes not. ($SU_{A}(3)$, $U_{A}(1)$) | $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$ | $(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$ | $(\mathbf{3},\mathbf{6})\oplus(\mathbf{6},\mathbf{3})$[mir] | $(\mathbf{10},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})$
---|---|---|---|---
$(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})$[mir] | ($\surd$, $\surd$) | N/A | N/A | N/A
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$[mir] | N/A | ($\surd$, $\surd$) | N/A | N/A
$(\mathbf{\bar{6}},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{\bar{6}})$ | N/A | N/A | ($\surd$, $\surd$) | N/A
$(\mathbf{1},\mathbf{\overline{10}})\oplus(\mathbf{\overline{10}},\mathbf{1})$[mir] | N/A | N/A | N/A | ($\surd$, $\surd$)
## IV Chiral mixing
In this section we establish the phenomenologically preferable mixing
pattern(s) and then we use the allowed chiral interactions to reproduce some
of them. First we summarize the salient features of chiral mixing and axial
couplings from Ref. Chen:2009sf .
There are three admissible scenarios (i.e. choices of pairs of chiral
multiplets admixed to the
$[(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})]$ one that lead to real
mixing angles) when fitting the $g_{A}^{(0)}$ and $g_{A}^{(3)}$ that yield the
values of $F$ and $D$. Similarly, when we fit $g_{A}^{(3)}$ and $g_{A}^{(8)}$,
or equivalently $F$ and $D$, we predict the values for $g_{A}^{(0)}$ and
$g_{A}^{(3)}$.
This is due to the fact that all three-quark baryon fields satisfy the
relation $g_{A}^{(0)}=3F-D=\sqrt{3}g_{A}^{(8)}$. Manifestly, in this way one
cannot satisfy both $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$ and $g_{A~{}\rm
expt.}^{(8)}=0.34\pm 0.07$. Thus we are left with two possible scenarios:
1. 1.
Fit $g_{A}^{(0)}$ and $g_{A}^{(3)}$ and predict $F$ and $D$. In Ref.
Chen:2009sf we found that there are three possible mixing patterns. Now the
chiral selection rules from Sect. III allow only two of them: the case III-I
mixing (See Table 3): $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$–$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$
and the case IV-I mixing: $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$–$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$.
However, the latter mixing violates $U_{A}(1)$ symmetry.
2. 2.
Fit $g_{A}^{(3)}$ and $g_{A}^{(8)}$ and predict $g_{A}^{(0)}$. Since we have
$g_{A}^{(0)}=\sqrt{3}g_{A}^{(8)}$ for all the three-quark nucleon fields
Chen:2008qv , the results here can be obtained by simple refitting of the
previous case. Fitting $(F,D)$ has not been a problem, so we leave this
exercise out of this paper because it generally overpredicts the $g_{A}^{(0)}$
by a factor of roughly $\sqrt{3}=1.73$.
Thus we determine the mixing angles in Sect. IV.1, which we then translate
into statements about the admixed fields’ masses in Sect. IV.2.
Table 3: The Abelian and the non-Abelian axial charges and the non-Abelian chiral multiplets of $J^{P}=\frac{1}{2}$, Lorentz representation $(\frac{1}{2},0)$ nucleon and $\Delta$ fields, see Refs. Nagata:2007di ; Nagata:2008zzc ; Dmitrasinovic:2009vp ; Dmitrasinovic:2009vy . case | field | $g_{A}^{(0)}$ | $g_{A}^{(3)}$ | $\sqrt{3}g_{A}^{(8)}$ | $F$ | $D$ | $SU_{L}(3)\times SU_{R}(3)$
---|---|---|---|---|---|---|---
I | $N_{-}=N_{1}-N_{2}$ | $-1$ | $+1$ | $-1$ | $~{}~{}0$ | $+1$ | $(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$
II | $N_{+}=N_{1}+N_{2}$ | $+3$ | $+1$ | $+3$ | $+1$ | $~{}~{}0$ | $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$
III | $N_{-}^{\prime}$ ($N_{-}^{(m)}$) | $+1$ | $-1$ | $+1$ | $~{}~{}0$ | $-1$ | $(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})$
IV | $N_{+}^{\prime}$ ($N_{+}^{(m)}$) | $-3$ | $-1$ | $-3$ | $-1$ | $~{}~{}0$ | $(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})$
0 | $\partial_{\mu}N^{\mu}$ | $+1$ | $+\frac{5}{3}$ | $+1$ | $+\frac{2}{3}$ | $+1$ | $({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})$
We note here that the relation $g_{A}^{(8)}=\frac{1}{\sqrt{3}}(3F-D)$ is a
general $SU(3)$ result valid for octet fields, whereas $g_{A}^{(0)}=3F-D$ is a
result that depends on our specific choice of three-quark interpolating fields
being admixed to the $({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})$ one. The
latter relation changes when one considers “exotic” interpolating fields, such
as certain five-quark (“pentaquark”) ones for example, and that allows a
simultaneous fit of $g_{A}^{(0)},g_{A}^{(3)}$ and $g_{A}^{(8)}$, which topic
is beyond the scope of this paper.
### IV.1 Phenomenology of the Axial Coupling Constants
A basic feature of the linear chiral realization is that the axial couplings
are determined by the chiral representations. For the nucleon (proton and
neutron), the three-quark chiral representations of $SU_{L}(3)\times
SU_{R}(3)$, $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$,
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$ and
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$ provide the nucleon
isovector axial coupling $g_{A}^{(3)}=1$, 1 and $5/3$ respectively. Therefore,
the mixing of chiral $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$,
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$ and
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$ nucleons leads to the
axial coupling
$\displaystyle 1.267$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(3)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(3)}~{}\sin^{2}\theta$
(148) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(3)}~{}\cos^{2}\theta+\frac{5}{3}~{}\sin^{2}\theta\,,$
where $g_{A~{}(\frac{1}{2},0)}^{(3)}$ represents the coupling of either
$(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$ or
$(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})$, and
$g_{A~{}(1,\frac{1}{2})}^{(3)}$ represents the coupling of
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$. The coupling
$g_{A~{}(1,\frac{1}{2})}^{(3)}$ is needed because only the coupling of
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$ is larger than the
experimental value $1.267$. We list the results of the mixing angles for all
the four cases in Table 4.
Table 4: The values of the baryon isoscalar axial coupling constant predicted from the naive mixing and $g_{A~{}\rm expt.}^{(3)}=1.267$; compare with $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.03\pm 0.05$, $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.571\pm 0.005$, Ref. Yamanishi:2007zza . case | $g_{A~{}\rm expt.}^{(3)}$ | $\theta_{i}$ | $g_{A~{}\rm mix.}^{(0)}$ | $\sqrt{3}g_{A~{}\rm mix.}^{(8)}$ | $F$ | $D$ | $F$/$D$
---|---|---|---|---|---|---|---
I | 1.267 | $39.3^{o}$ | $-0.20$ | $-0.20$ | 0.267 | 1 | 0.267
II | 1.267 | $39.3^{o}$ | 2.20 | 2.20 | 0.866 | 0.401 | 2.16
III | 1.267 | $67.2^{o}$ | 1.00 | 1.00 | 0.567 | 0.700 | 0.81
IV | 1.267 | $67.2^{o}$ | 0.40 | 0.40 | 0.417 | 0.850 | 0.491
Three-quark nucleon interpolating fields in QCD have well-defined $U_{A}(1)$
chiral transformation properties, see Table 3, that can be used to predict the
flavor singlet axial coupling $g_{A~{}\rm mix.}^{(0)}$ and the $F$ and $D$
values
$\displaystyle g_{A~{}\rm mix.}^{(0)}$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(0)}~{}\sin^{2}\theta$
(149) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+\sin^{2}\theta,$ $\displaystyle
F$ $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+F_{(1,\frac{1}{2})}^{(1)}~{}\sin^{2}\theta,$
(150) $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\frac{2}{3}~{}\sin^{2}\theta\,,$
$\displaystyle D$ $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+D_{(1,\frac{1}{2})}~{}\sin^{2}\theta$
(151) $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\sin^{2}\theta\,.$
The mixing angle $\theta$ is extracted from Eq. (148), where we used the bare
$F$ and $D$ values for different chiral multiplets as listed in Table 3. Due
to the different (bare) non-Abelian $g_{A}^{(3)}$ and Abelian $g_{A}^{(0)}$
axial couplings, see Table 3, the mixing formulae Eq. (149) give substantially
different predictions from one case to another, see Table 4. We can see in
Table 4 that the two best candidates are cases I and IV, with
$g_{A}^{(0)}=-0.2$ and $g_{A}^{(0)}=0.4$, respectively, the latter being
within the error bars of the measured value $g_{A~{}\rm expt.}^{(0)}=0.33\pm
0.08$ Bass:2007zzb ; Ageev:2007du . Selection rules from Sect. III allow the
case III and the case IV. And so the case IV is the best candidate so long as
we consider just the mixing of two nucleon fields Dmitrasinovic:2009vp .
Manifestly, a linear superposition of any three fields (except for the
mixtures of cases II and III, IV above, which yield complex mixing angles)
gives a perfect fit to the central values of the experimental axial couplings
$g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$ and $g_{A~{}\rm expt.}^{(3)}=1.267$ and
predict the $F$ and $D$ values, or vice versa: one may fit $g_{A}^{(3)}$ and
$g_{A}^{(8)}$, (or equivalently $F$ and $D$) and thus predict $g_{A}^{(0)}$.
This has been done in Ref. Chen:2009sf , and where there were three allowed
cases: I-II, I-III and I-IV. The selection rules from Sect. III indicate that
only two of them are possible in the one-meson approximation: (1) the case
I-III and (2) the case I-IV. In the former case the $U_{A}(1)$ symmetry is
conserved, whereas in the latter the $U_{A}(1)$ is violated.
Such a three-field admixture introduces two new free parameters, besides the
already introduced mixing angles, e.g. $\theta_{3}$ and $\theta_{1}$(=0),
(which we may set to vanish in the present approximation). For the case I-III
(we shall call it here case III-I for reasons soon to be clarified) we have
the relative/mutual mixing angle $\theta_{31}=\varphi$, as the two nucleon
fields III and I mix due to the off-diagonal interaction Eq. (104). Thus we
find two equations with two unknowns of the general form:
$\displaystyle\frac{5}{3}\,{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(3)}({\rm
III}){\cos}^{2}\varphi+g_{A}^{(3)}({\rm I}){\sin}^{2}\varphi\right)$
$\displaystyle=1.267\,,$ (152)
$\displaystyle{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(0)}({\rm
III}){\cos}^{2}\varphi+g_{A}^{(0)}({\rm I})\,{\sin}^{2}\varphi\right)$
$\displaystyle=0.33\pm 0.08\,.$ (153)
The solutions to these equations (the values of the mixing angles
$\theta,\varphi$) provide, at the same time, input for the prediction of $F$
and $D$:
$\displaystyle\cos^{2}\theta\,\left(F({\rm III})\,{\cos}^{2}\varphi+F({\rm
I})\,{\sin}^{2}\varphi\right)+\frac{2}{3}~{}\sin^{2}\theta$
$\displaystyle=F\,,$ (154) $\displaystyle\cos^{2}\theta\,\left(D({\rm
III})\,{\cos}^{2}\varphi+D({\rm I})\,{\sin}^{2}\varphi\right)+\sin^{2}\theta$
$\displaystyle=D\,.$ (155)
The values of the mixing angles ($\theta,\varphi$) obtained from this
straightforward fit to the baryon axial coupling constants are shown in Table
5. We also show the result of the case I-IV as well as IV-I in this Table.
Besides these cases, the cases I-II and II-I can also be used to produce the
experimental $g_{A}^{(0)}$ and $g_{A}^{(3)}$, which are however not allowed
from Sect III.
Table 5: The values of the mixing angles obtained from the simple fit to the baryon axial coupling constants and the predicted values of axial $F$ and $D$ couplings. The experimental values are $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.575\pm 0.005$ and $g_{A}^{(8)}=0.33\pm 0.01$, Ref. Ratcliffe:1990dh . The most recent analysis of experimental values leads to $F=0.477\pm 0.001$ and $D=0.835\pm 0.001$ and $g_{A}^{(8)}=0.344\pm 0.001$ in Ref. Yamanishi:2007zza . Note that these values are more than 2-$\sigma$ away from the old ones, and that the new $F$,$D$ add up to $F+D$ = 1.312 $\neq 1.269\pm 0.002$. Also $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$. case | $g_{A~{}\rm expt.}^{(3)}$ | $g_{A}^{(0)}$ | $g_{A}^{(8)}$ | $\theta$ | $\varphi$ | $F$ | $D$ | $F$/$D$
---|---|---|---|---|---|---|---|---
I-III | 1.267 | $0.33\pm 0.08$ | $0.19\pm 0.05$ | $50.7^{o}\pm 1.8^{o}$ | $23.9^{o}\pm 2.9^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
III-I | 1.267 | $0.33\pm 0.08$ | $0.19\pm 0.05$ | $50.7^{o}\pm 1.8^{o}$ | $66.1^{o}\pm 2.9^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
I-IV | 1.267 | $0.33\pm 0.08$ | $0.19\pm 0.05$ | $63.2^{o}\pm 4.0^{o}$ | $54^{o}\pm 23^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
IV-I | 1.267 | $0.33\pm 0.08$ | $0.19\pm 0.05$ | $63.2^{o}\pm 4.0^{o}$ | $36^{o}\pm 23^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
### IV.2 Baryon Masses
The next step is to try and reproduce this phenomenological mixing starting
from a model interaction, rather than per fiat. As the first step in that
direction we must look for a dynamical source of mixing. One such mechanism is
the simplest chirally symmetric non-derivative one-$(\sigma,\pi)$-meson
interaction Lagrangian, which induces baryon masses via its $\sigma$-meson
coupling. Chiral symmetry is spontaneously broken through the “condensation”
of the sigma field
$\sigma\rightarrow\sigma_{0}=\langle\sigma\rangle_{0}=f_{\pi}$, which leads to
the dynamical generation of baryon masses, as can be seen from the linearized
chiral invariant interaction Lagrangians Eqs. (66) and (80).
In this section, we study the masses of the octet baryons. There are
altogether six types of octet baryon fields: $N_{+}$ ($N_{(8)}$), $N_{-}$
(contained in $N_{(9)}$) and $N_{\mu}$ (contained in $N_{(18)}$), as well as
their mirror fields $N_{+}^{\prime}$ ($N_{(8m)}$), $N_{-}^{\prime}$ (contained
in $N_{(9m)}$), $N^{\prime}_{\mu}$ (contained in $N_{(18m)}$). The nucleon
mass matrix is already in a simple block-diagonal form when the nucleon fields
form the following mass matrix:
$\displaystyle M$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\bar{N}\left(\begin{array}[]{ccc|ccc}0&g_{(8/9)}&g_{(8/18)}&m_{(8)}\gamma_{5}&g_{B}&0\\\
g^{*}_{(8/9)}&g_{(9/9)}&g_{(9/18)}&g^{*}_{B}&m_{(9)}\gamma_{5}&0\\\
g^{*}_{(8/18)}&g^{*}_{(9/18)}&g_{(18/18)}&0&0&m_{(18)}\gamma_{5}\\\ \hline\cr
m_{(8)}\gamma_{5}&g^{\prime}_{B}&0&0&g^{\prime}_{(8/9)}&g^{\prime}_{(8/18)}\\\
g^{\prime*}_{B}&m_{(9)}\gamma_{5}&0&g^{\prime*}_{(8/9)}&g^{\prime}_{(9/9)}&g^{\prime}_{(9/18)}\\\
0&0&m_{(18)}\gamma_{5}&g^{\prime*}_{(8/18)}&g^{\prime*}_{(9/18)}&g^{\prime}_{(18/18)}\end{array}\right)N\,,$
(162)
where
$\displaystyle
N=(N_{+}^{\prime},N_{-}^{\prime},N_{\mu},N_{+},N_{-},N_{\mu}^{\prime})^{T}\,.$
(163)
Since there are three nucleon fields as well as their mirror fields, there can
be a nonzero phase angle. However, for simplicity, we assume all the axial
couplings are real.
### IV.3 Masses due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$
mixing
We use the results of Sect. III: the chirally invariant diagonal, Eqs. (66)
and (80) and off-diagonal, Eq. (90) meson-baryon-baryon interactions involving
$\displaystyle(B_{1},\Lambda)$ $\displaystyle\in$
$\displaystyle(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})[{\rm
mir}]\,,$ $\displaystyle(B_{2},\Delta)$ $\displaystyle\in$ $\displaystyle({\bf
6},{\bf 3})\oplus({\bf 3},{\bf 6})\,,$ (164) $\displaystyle(\sigma,\pi)$
$\displaystyle\in$
$\displaystyle(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})\,.$
Here all baryons have spin 1/2, while the isospin of $B_{1}$ and $B_{2}$ is
1/2 and that of $\Delta$ is 3/2. The $\Delta$ field is then represented by an
isovector-Diracspinor field $\Delta^{i}$, ($i=1,2,3$).
In writing down the Lagrangians Eqs. (66), (80) and (90), we have implicitly
assumed that the parities of $B_{1}$, $B_{2}$, $\Lambda$ and $\Delta$ are the
same. In principle, they are arbitrary, except for the ground state nucleon,
which must be even. For instance, if $B_{2}$ has odd parity, the first term in
the interaction Lagrangian Eq. (90) must include another $\gamma_{5}$ matrix
Jido:2001nt . Here we assume the ground state nucleon is contained in either
$[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ or
$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$, and so
at least one of $B_{1}$ and $B_{2}$ has even parity. Next we consider all
possible cases for the parities of $B_{2}$, $\Lambda$ and $\Delta$. The
results are similar to the two-flavor ones shown in Ref. Dmitrasinovic:2009vp
; Dmitrasinovic:2009vy (because we assumed good $SU(3)$ symmetry here).
Having established the mixing interaction Eq. (90), as well as the diagonal
terms Eqs. (66) and (80), we calculate the masses of the baryon states, as
functions of the pion decay constant/chiral order parameter and the coupling
constants $g_{1}\sim g_{(9)}$, $g_{2}\sim g_{(18)}$ and $g_{3}\sim
g_{(9/18)}$:
$\displaystyle\mathcal{L}_{(9)}$ $\displaystyle=$ $\displaystyle-
g_{1}\Big{(}\bar{B}_{1}\sigma
B_{1}-2\bar{\Lambda}\sigma\Lambda\Big{)}+\cdots\,,$
$\displaystyle\mathcal{L}_{(18)}$ $\displaystyle=$ $\displaystyle-
g_{2}\Big{(}\bar{B}_{2}\sigma
B_{2}-2\bar{\Delta}^{i}\sigma\Delta^{i}\Big{)}+\cdots\,,$ (165)
$\displaystyle\mathcal{L}_{(9/18)}$ $\displaystyle=$ $\displaystyle-
g_{3}\Big{(}\bar{B}_{1}\sigma B_{2}\Big{)}+\cdots\,,$
Altogether we have
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle-
f_{\pi}(\bar{B}_{1},\bar{B}_{2})\left(\begin{array}[]{cc}g_{1}&g_{3}\\\
g_{3}&g_{2}\end{array}\right)\left(\begin{array}[]{c}B_{1}\\\
B_{2}\end{array}\right)+2g_{1}f_{\pi}\bar{\Lambda}\Lambda+2g_{2}f_{\pi}\bar{\Delta}^{i}\Delta^{i}$
(170)
We diagonalize the mass matrix and express the mixing angle in terms of
diagonalized masses
$\displaystyle N(N^{*})$ $\displaystyle=$ $\displaystyle\cos\theta
B_{1}+\sin\theta B_{2}\,,$ (171) $\displaystyle N^{*}(N)$ $\displaystyle=$
$\displaystyle-\sin\theta B_{1}+\cos\theta B_{2}\,.$
We find the following double-angle formulas for the mixing angles
$\theta_{1,\cdots,8}$ between $B_{1}$ and $B_{2}$ in the eight different
parities scenarios
$\displaystyle\tan 2\theta_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N+\Delta)(2N^{*}+\Delta})}{(N+N^{*}+\Delta)}=-\frac{\sqrt{-(2N+\Lambda)(2N^{*}+\Lambda})}{(N+N^{*}+\Lambda)},$
(172) $\displaystyle\tan 2\theta_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N+\Delta)(2N^{*}-\Delta})}{(N-N^{*}+\Delta)}=-\frac{\sqrt{(2N+\Lambda)(2N^{*}-\Lambda})}{(N-N^{*}+\Lambda)},$
(173) $\displaystyle\tan 2\theta_{3}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N+\Delta)(2N^{*}+\Delta})}{(N+N^{*}+\Delta)}=-\frac{\sqrt{-(2N-\Lambda)(2N^{*}-\Lambda})}{(N+N^{*}-\Lambda)},$
(174) $\displaystyle\tan 2\theta_{4}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N+\Delta)(2N^{*}-\Delta})}{(N-N^{*}+\Delta)}=\frac{\sqrt{(2N-\Lambda)(2N^{*}+\Lambda})}{(-N+N^{*}+\Lambda)},$
(175) $\displaystyle\tan 2\theta_{5}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N-\Delta)(2N^{*}-\Delta})}{(N+N^{*}-\Delta)}=-\frac{\sqrt{-(2N+\Lambda)(2N^{*}+\Lambda})}{(N+N^{*}+\Lambda)},$
(176) $\displaystyle\tan 2\theta_{6}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N-\Delta)(2N^{*}+\Delta})}{(N-N^{*}-\Delta)}=-\frac{\sqrt{(2N+\Lambda)(2N^{*}-\Lambda})}{(N-N^{*}+\Lambda)},$
(177) $\displaystyle\tan 2\theta_{7}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N-\Delta)(2N^{*}-\Delta})}{(N+N^{*}-\Delta)}=-\frac{\sqrt{-(2N-\Lambda)(2N^{*}-\Lambda})}{(N+N^{*}-\Lambda)},$
(178) $\displaystyle\tan 2\theta_{8}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N-\Delta)(2N^{*}+\Delta})}{(N-N^{*}-\Delta)}=\frac{\sqrt{(2N-\Lambda)(2N^{*}+\Lambda})}{(N-N^{*}-\Lambda)},$
(179)
where $N$, $N^{*}$, $\Lambda$ and $\Delta$ represent the masses of the
corresponding particles. The four angles correspond to the eight possible
parities; $\theta_{1}:(N^{*+},\Lambda^{+},\Delta^{+})$,
$\theta_{2}:(N^{*-},\Lambda^{+},\Delta^{+})$,
$\theta_{3}:(N^{*+},\Lambda^{-},\Delta^{+})$,
$\theta_{4}:(N^{*-},\Lambda^{-},\Delta^{+})$,
$\theta_{5}:(N^{*+},\Lambda^{+},\Delta^{-})$,
$\theta_{6}:(N^{*-},\Lambda^{+},\Delta^{-})$,
$\theta_{7}:(N^{*+},\Lambda^{-},\Delta^{-})$,
$\theta_{8}:(N^{*-},\Lambda^{-},\Delta^{-})$, where $\pm$ indicate the parity
of the state. Note that the angle $\theta_{1}$, $\theta_{3}$ and $\theta_{5}$
is necessarily imaginary so long as the $\Delta$, $\Lambda$ and $N^{*}$ masses
are physical (positive), and that the reality of the mixing angle(s) imposes
stringent limits on the $\Delta,N^{*}$ resonance masses in other cases, as
well.
In the present study we have three model parameters $g_{1},g_{2}$ and $g_{3}$,
which can be determined by different set of inputs. We can use two baryon
masses and the mixing angle as inputs and predicts the third baryon mass
(Inverse prediction). We use the formulas Eqs. (172)-(179) for the (double)
mixing angles $\theta_{1,...,8}$ together with the two observed nucleon masses
and the mixing angle $\theta=67.2^{o}$ as shown in Table 4 to predict the
$\Delta$ masses shown in the Table 6.
Table 6: The values of the $\Delta$ baryon masses predicted from the isovector axial coupling $g_{A~{}\rm mix.}^{(1)}=g_{A~{}\rm expt.}^{(1)}=1.267$ and $g_{A~{}\rm mix.}^{(0)}=0.4$ vs. $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$. $(N^{*P},\Lambda^{P^{\prime}},\Delta^{P^{{}^{\prime\prime}}})$ | ($N,~{}~{}N^{*}$) | $\Lambda$ (MeV) | $\Lambda_{\rm expt.}$ (MeV) | $\Delta$ (MeV) | $\Delta_{\rm expt.}$ (MeV)
---|---|---|---|---|---
$(-,+,+)$ | N(940), R(1535) | 2330 | - | 2330 | 1910
$(-,-,+)$ | N(940), R(1535) | 1140 | 1405 | 2330 | 1910
$(-,+,-)$ | N(940), R(1535) | 2330 | - | 1140 | -
$(+,-,-)$ | N(940), R(1440) | 2030,2730 | - | 2030,2730 | -
$(-,-,-)$ | N(940), R(1535) | 1140 | 1405 | 1140 | -
We see that only the $(N^{*-},\Delta^{+})$ parity combination leads to a
realistic prediction of the baryon masses. Otherwise, at least one of the
predicted baryon masses is off by a factor of of order two. Indeed, the case
$(N^{*P},\Lambda^{P^{\prime}},\Delta^{P^{{}^{\prime\prime}}})=(-,-,+)$
predicts the (odd-parity) $SU(3)$ flavor-singlet $\Lambda$ at 1140 MeV,
somewhat below the measured value (1405 MeV) and $\Delta(2330)$, the nearest
known candidate state being the (four star PDG, Ref. Amsler:2008zzb )
$P_{31}(1910)$ resonance. It is curious that the flavor-singlet
$\Lambda(1140)$ state lies (considerably) below the flavor-octet state
$N^{*}(1535)$ even in the good flavor $SU(3)$ symmetry limit; the predicted
mass difference might/ought to be improved by introducing explicit $SU(3)$
symmetry breaking strange-up/down quark mass difference.
### IV.4 Masses due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$–$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$
mixing
To improve our analysis, we may add a third chiral multiplet nucleon field. As
in the previous section III, we consider baryon fields
$\displaystyle(B_{1},\Lambda_{1})$ $\displaystyle\in$
$\displaystyle(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})[{\rm
mir}]\,,$ $\displaystyle(B_{2},\Delta)$ $\displaystyle\in$ $\displaystyle({\bf
6},{\bf 3})\oplus({\bf 3},{\bf 6})\,,$ (180)
$\displaystyle(B_{3},\Lambda_{2})$ $\displaystyle\in$
$\displaystyle(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})\,.$
As discussed above, the case III-I allows one to reproduce the experimental
couplings $g_{A}^{(0)}$ and $g_{A}^{(3)}$. To study this mixing, we need to
use the previous Lagrangian Eq. (IV.3) as well as the new ones:
$\displaystyle\mathcal{L}^{\prime}_{(9)}$ $\displaystyle=$ $\displaystyle-
g_{4}\Big{(}\bar{B}_{3}\sigma
B_{3}-2\bar{\Lambda}_{1}\sigma\Lambda_{1}\Big{)}+\cdots\,,$
$\displaystyle\mathcal{L}_{(9/9)}$ $\displaystyle=$ $\displaystyle-
g_{5}f_{\pi}\bar{B}_{1}B_{3}-g_{5}f_{\pi}\bar{\Lambda}_{1}\Lambda_{2}+\cdots\,,$
(181)
that follow from Eq. (80), where the the third nucleon field $B_{3}$ is a
mirror image of $B_{1}$. We note that $B_{1}$ and $B_{3}$ couple with each
other through the naive combinations:
${m_{(9)}}\overline{N}_{(9m)}\gamma_{5}N_{(9)}$. Chiral symmetry is
spontaneously broken through the “condensation” of the sigma field
$\sigma\to\sigma_{0}=\langle\sigma\rangle_{0}=f_{\pi}$, which leads to the
dynamical generation of baryon masses:
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle-
f_{\pi}(\bar{B}_{1},\bar{B}_{3},\bar{B}_{2})\left(\begin{array}[]{ccc}g_{1}&g_{5}&g_{3}\\\
g_{5}&g_{4}&0\\\
g_{3}&0&g_{2}\end{array}\right)\left(\begin{array}[]{c}B_{1}\\\ B_{3}\\\
B_{2}\end{array}\right)-f_{\pi}(\bar{\Lambda}_{1},\bar{\Lambda}_{2})\left(\begin{array}[]{cc}-2g_{1}&g_{5}\\\
g_{5}&-2g_{4}\end{array}\right)\left(\begin{array}[]{c}\Lambda_{1}\\\
\Lambda_{2}\end{array}\right)+2g_{2}f_{\pi}\bar{\Delta}^{i}\Delta^{i}$ (192)
To solve this system in its full generality seems both too complicated and not
very useful. However, since $g_{6}$ of $g_{6}\bar{B}_{3}B_{2}$ vanishes, we
only need five conditions to solve this system. Therefore, we just use the
three nucleon candidates $N(940)$, $N(1440)$ and $N^{*}(1535)$ as well as the
two mixing angles $\theta^{o}=63.2^{o}$ and $\phi=36^{o}$. Finally we find
that there are two possibilities as shown in Table 7.
Table 7: The values of the $\Delta$ and $\Lambda$ baryon masses predicted from the isovector axial coupling $g_{A~{}\rm mix.}^{(1)}=g_{A~{}\rm expt.}^{(1)}=1.267$ and $g_{A~{}\rm mix.}^{(0)}=0.33\pm 0.08$ due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ – $[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$ – $[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$ mixing . No. | $g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{5}$ | $\Lambda_{1}^{P}$ (MeV) | $\Lambda_{2}^{P}$ (MeV) | $\Delta^{P}$ (MeV)
---|---|---|---|---|---|---|---|---
1 | $-4.7$ | 8.4 | $-3.4$ | 2.9 | 9.8 | $1370^{-}$ | $1850^{+}$ | $2170^{-}$
2 | $-7.2$ | 4.6 | 7.9 | 9.1 | $-4.2$ | $1940^{+}$ | $2430^{-}$ | $1200^{-}$
Once again, the odd-parity $\Delta$ option appears as the better one. Now, the
first flavor-singlet $\Lambda$ lies at 1370 MeV, substantially closer to 1405
MeV than before. A second flavor-singlet $\Lambda$ lies at 1850 MeV, very
close to the (three star PDG, Ref. Amsler:2008zzb ) $P_{01}(1810)$ resonance.
This is our best candidate in the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$–$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$
mixing scenario.
### IV.5 Masses due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$ mixing
We can also study the baryon masses due to $[({\bf 6},{\bf 3})\oplus({\bf
3},{\bf 6})]$ – $[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$
mixing
$\displaystyle B_{1}$ $\displaystyle\in$
$\displaystyle(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})[{\rm
mir}]\,,$ $\displaystyle(B_{2},\Delta)$ $\displaystyle\in$ $\displaystyle({\bf
6},{\bf 3})\oplus({\bf 3},{\bf 6})\,.$ (194)
Having established the mixing interaction Eq. (99), as well as the diagonal
terms Eq. (66), we calculate the masses of the baryon states, as functions of
the pion decay constant/chiral order parameter and the coupling constants
$g_{2}\sim g_{(18)}$ and $g_{3}\sim g_{(8/18)}$:
$\displaystyle\mathcal{L}_{(18)}$ $\displaystyle=$ $\displaystyle-
g_{2}\Big{(}\bar{B}_{2}\sigma
B_{2}-2\bar{\Delta}^{i}\sigma\Delta^{i}\Big{)}+\cdots\,,$ (195)
$\displaystyle\mathcal{L}_{(8/18)}$ $\displaystyle=$ $\displaystyle-
g_{3}\Big{(}\bar{B}_{1}\sigma B_{2}\Big{)}+\cdots\,,$
Note that $g_{1}\sim g_{(8)}$ is zero now. We diagonalize the mass matrix and
express the mixing angle in terms of diagonalized masses. We find the
following double-angle formulas for the mixing angles $\theta_{1,\cdots,4}$
between $B_{1}$ and $B_{2}$ in the four different parities scenarios
$\displaystyle\tan 2\theta_{1}$ $\displaystyle=$
$\displaystyle-2i\frac{\sqrt{NN^{*}}}{N^{*}+N}\,,\Delta=-2(N^{*}+N)\,,$ (196)
$\displaystyle\tan 2\theta_{2}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{NN^{*}}}{N^{*}-N}\,,\Delta=2(N^{*}-N)\,,$ (197)
$\displaystyle\tan 2\theta_{3}$ $\displaystyle=$
$\displaystyle-2i\frac{\sqrt{NN^{*}}}{N^{*}+N}\,,\Delta=2(N^{*}+N)\,,$ (198)
$\displaystyle\tan 2\theta_{4}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{NN^{*}}}{N^{*}-N}\,,\Delta=-2(N^{*}-N)\,,$ (199)
where $N$, $N^{*}$ and $\Delta$ represent the masses of the corresponding
particles. The four angles correspond to the four possible parities;
$\theta_{1}:(N^{*+},\Delta^{+})$, $\theta_{2}:(N^{*-},\Delta^{+})$,
$\theta_{3}:(N^{*+},\Delta^{-})$, $\theta_{4}:(N^{*-},\Delta^{-})$, where
$\pm$ indicate the parity of the state. Note that only $\theta_{2}$ leads to a
physical result. We can use the mixing angle $\theta=67.2^{o}$ and the nucleon
mass 940 MeV to predict the excited nucleon mass and $\Delta$ mass, see Table
8.
Table 8: The values of the $\Delta$ baryon masses predicted from the isovector axial coupling $g_{A~{}\rm mix.}^{(1)}=g_{A~{}\rm expt.}^{(1)}=1.267$ and $g_{A~{}\rm mix.}^{(0)}=0.4$ vs. $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$ due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ – $[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$ mixing without additional two-meson interactions. $(N^{*P},\Delta^{P^{{}^{\prime}}})$ | $N$ | $N^{*}$ | $N^{*}_{\rm expt.}$ (MeV) | $\Delta$ (MeV) | $\Delta_{\rm expt.}$ (MeV)
---|---|---|---|---|---
$(-,+)$ | N(940) | 5320 | - | 8760 | -
This gives predictions of no practical value. To get a practically useful
result, we need to add one of the two-meson interaction Lagrangians from Sect.
III.1.3, and thus a non-zero $g_{1}$ term:
$\displaystyle\mathcal{L}_{(8)}$ $\displaystyle=$ $\displaystyle-{g_{1}\over
f_{\pi}}\bar{B}_{1}\sigma^{2}B_{1}+\cdots\,,$
and we have four new different parities scenarios:
$\displaystyle\tan 2\theta_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N+\Delta)(2N^{*}+\Delta})}{(N+N^{*}+\Delta)}\,,$
(200) $\displaystyle\tan 2\theta_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N+\Delta)(2N^{*}-\Delta})}{(N-N^{*}+\Delta)}\,,$
(201) $\displaystyle\tan 2\theta_{3}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-(2N-\Delta)(2N^{*}-\Delta})}{(N+N^{*}-\Delta)}\,,$
(202) $\displaystyle\tan 2\theta_{4}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2N-\Delta)(2N^{*}+\Delta})}{(N-N^{*}-\Delta)}\,.$
(203)
Note that only $\theta_{1}$ is imaginary for positive baryon masses, i.e.
unphysical. We can use the mixing angle $\theta=67.2^{o}$ and the two nucleon
masses to predict the $\Delta$ mass, see Table 9. The nearest known candidate
for the $\Delta(2330)$ state is the (four star PDG, Ref. Amsler:2008zzb )
$P_{31}(1910)$ resonance.
Table 9: The values of the $\Delta$ baryon masses predicted from the isovector axial coupling $g_{A~{}\rm mix.}^{(1)}=g_{A~{}\rm expt.}^{(1)}=1.267$ and $g_{A~{}\rm mix.}^{(0)}=0.4$ vs. $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.08$ due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ – $[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$ mixing with additional two-meson interactions. $(N^{*P},\Delta^{P^{{}^{\prime}}})$ | ($N,~{}~{}N^{*}$) | $\Delta$ (MeV) | $\Delta_{\rm expt.}$ (MeV)
---|---|---|---
$(-,+)$ | N(940), R(1535) | 2330 | 1910
$(+,-)$ | N(940), R(1440) | 2030,2730 | -
$(-,-)$ | N(940), R(1535) | 1140 | -
### IV.6 Masses due to $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$–$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$
mixing
To improve our analysis, we can add a third field, and altogether we consider
$\displaystyle B_{1}$ $\displaystyle\in$
$\displaystyle(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})[{\rm
mir}]\,,$ $\displaystyle(B_{2},\Delta)$ $\displaystyle\in$ $\displaystyle({\bf
6},{\bf 3})\oplus({\bf 3},{\bf 6})\,,$ (204) $\displaystyle(B_{3},\Lambda)$
$\displaystyle\in$ $\displaystyle({\bf
3},{\bf\bar{3}})\oplus({\bf\bar{3}},{\bf 3})\,.$ (205)
As discussed above, the case IV-I is possible to produce the experimental
couplings $g_{A}^{(0)}$ and $g_{A}^{(3)}$, although this is $U_{A}(1)$
violated. To study this mixing, we need to use the previous Lagrangian Eq.
(195) as well as the new ones:
$\displaystyle\mathcal{L}^{\prime}_{(9)}$ $\displaystyle=$ $\displaystyle-
g_{4}\Big{(}\bar{B}_{3}\sigma
B_{3}-2\bar{\Lambda}\sigma\Lambda\Big{)}+\cdots\,,$
$\displaystyle\mathcal{L}_{(B)}$ $\displaystyle=$ $\displaystyle-
g_{5}\bar{B}_{1}\sigma B_{3}+\cdots\,,$ (206)
that follow from Eqs. (80) and (109). Chiral symmetry is spontaneously broken
through the “condensation” of the sigma field
$\sigma\to\sigma_{0}=\langle\sigma\rangle_{0}=f_{\pi}$, which leads to the
dynamical generation of baryon masses:
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle-
f_{\pi}(\bar{B}_{1},\bar{B}_{3},\bar{B}_{2})\left(\begin{array}[]{ccc}g_{1}&g_{5}&g_{3}\\\
g_{5}&g_{4}&0\\\
g_{3}&0&g_{2}\end{array}\right)\left(\begin{array}[]{c}B_{1}\\\ B_{3}\\\
B_{2}\end{array}\right)+2g_{4}f_{\pi}\bar{\Lambda}\Lambda+2g_{2}f_{\pi}\bar{\Delta}^{i}\Delta^{i}$
(213)
Since $g_{6}$ of $g_{6}\bar{B}_{3}B_{2}$ vanishes, we only need five
conditions to solve this system. Therefore, we may use the three lowest-lying
nucleon states $N(940)$, $N(1440)$ and $N^{*}(1535)$ as well as the two mixing
angles $\theta^{o}=50.7^{o}$ and $\phi=66.1^{o}$. Finally we find that there
are two real possibilities as shown in Table 10.
Table 10: The values of the $\Delta$ and $\Lambda$ baryon masses predicted from the isovector axial coupling $g_{A~{}\rm mix.}^{(1)}=g_{A~{}\rm expt.}^{(1)}=1.267$ and $g_{A~{}\rm mix.}^{(0)}=0.33\pm 0.08$ and the mass fit to $N(940)$, $N(1440)$ and $N^{*}(1535)$. No. | $g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$ | $g_{5}$ | $\Lambda^{P}$ (MeV) | $\Delta^{P}$ (MeV)
---|---|---|---|---|---|---|---
1 | $4.6$ | 8.0 | $-1.8$ | $-6.1$ | 9.7 | $1580^{+}$ | $2070^{-}$
2 | $-8.4$ | 4.3 | 7.1 | 10.6 | $-2.4$ | $2750^{-}$ | $1124^{-}$
3 | $-1.3$ | 10.2 | 2.1 | $-2.5$ | 9.8 | $640^{+}$ | $2660^{-}$
4 | $-8.7$ | 8.1 | 7.3 | 7.1 | 2.9 | $1850^{-}$ | $2110^{-}$
Once again, the two odd-parity $\Delta$ options appear as the best ones.
First, even-parity flavor-singlet $\Lambda(1580)$, lies very close to the
(three star PDG, Ref. Amsler:2008zzb ) $P_{01}(1600)$ resonance. Second, the
odd-parity flavor-singlet $\Lambda$ lies at 1850 MeV, also very close to the
(three star PDG, Ref. Amsler:2008zzb ) $S_{01}(1800)$ resonance. These are our
best candidates in the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf
6})]$–$[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$ mixing
scenario, that shows that this option is open.
### IV.7 Baryon masses and chiral restoration
Note that, starting from the above mass formulas one may study the behavior of
baryon masses in the chiral restoration limit, i.e., as $f_{\pi}\to 0$. We do
not wish to go into this subject in any depth here, except to point out
several more-or-less immediate consequences of our results.
First we note that in the two-flavor case one often finds nucleon parity
doublets in the chiral restoration limit $f_{\pi}\to 0$ Dmitrasinovic:2009vp .
That, however, is generally a consequence of the assumptions made about the
number and kind of chiral multiplets that are being mixed: If one assumes, as
in our studies above, that more than two multiplets are mixed, then, of
course, there will be no parity doublets, but triplets, or generally as many
states as there are admixed multiplets. Moreover, if there are more than two
degenerate states, such as in our studies above, then at least two will have
the same parity, i.e. the concept of “parity doublets” ceases to be meaningful
and “parity multiplets” ought to be introduced. Finally, if two different
flavor $SU(3)$ multiplets form one chiral multiplet, such as the ${\bf 8}$ and
${\bf 10}$ in the $[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$, then the two
flavor $SU(3)$ multiplets may form a mass-degenerate “parity doublet” in the
chiral restoration limit, even though most of the states in such doublets do
not have the same flavor quantum numbers.
Various conjectures have been made about the potential relation between the
observed parity doublets high in the baryon spectrum and chiral symmetry
restoration, especially the restoration of the (otherwise explicitly broken)
$U_{A}(1)$ symmetry (see Ref. Jaffe:2006jy and references therein). Our
results above viz. that there are two basic allowed scenarios that differ in
the $U_{A}(1)$ (non)symmetry of their interactions, show immediately that the
$U_{A}(1)$ symmetry need not play a role in the baryon spectra. In this regard
we agree with the conclusions of Ref. Jaffe:2005sq ; Jaffe:2006jy , who used
only a two-flavor model, however. Such conclusions were also previously
reached in the two-flavor case in Ref. Dmitrasinovic:2009vy and in Ref.
Christos , only in the more restricted case of just one $SU(2)$ parity doublet
and without mirror fields. The first, limited, attempts at the three-flavor
case were made in Refs. Christos2 ; Zheng:1992mn .
## V Summary and Outlook
We have used the results of our previous paper Chen:2009sf to construct the
$SU_{L}(3)\times SU_{R}(3)$ chiral invariant interactions based on the
phenomenological facts regarding the baryon axial currents, of the chiral
$[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ multiplet mixing with other non-
exotic baryon field multiplets, such as the
$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$ and
$[(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})]$.
The existence of these multiplets is not limited to three-quark interpolators:
they are present in the the $SU(3)_{L}\times SU(3)_{R}$ Clebsch-Gordan series
for the 5-quark interpolating fields, as well as the 7-quark ones, etc..
Indeed, these are the only non-exotic chiral multiplets, as they consist of
only non-exotic flavor $SU(3)$ multiplets. The “ordinary” (vector) $SU(3)$
multiplet content of a chiral multiplet is determined by the Clebsch-Gordan
series for the tensor product of the right- and left- $SU(3)$ multiplets: thus
$\mathbf{1}\oplus\mathbf{8}\in(\mathbf{3},\mathbf{\bar{3}});~{}\mathbf{8}\in(\mathbf{8},\mathbf{1});~{}\mathbf{8}\oplus\mathbf{10}\in({\bf
6},{\bf 3})$. Introducing multiple fields with identical chiral contents would
lead to double counting, however. That is to say that the effects of multi-
quark fields are implicitly accounted for, unless these fields differ from the
ones we assumed in some respect other than the non-Abelian chiral multiplet.
Introduction of exotic chiral multiplets, on the other hand, would lead to
exotic flavor $SU(3)$ multiplets in the spectrum, which are absent
experimentally, however. Thus, we may conclude that these three chiral
multiplets, together with their mirror images, are the only ones consistent
with the present experimental knowledge, and that no additional chiral mixing
is phenomenologically allowed, without further explanation.
The results of the three-field (“two-angle”) mixing are curious insofar as all
phenomenologically permissible combinations of interpolating fields lead to
the same $F$,$D$ values, that are in reasonable agreement with experiment.
This (unexpected) equivalence of results is a consequence of the relation
$g_{A}^{(0)}=3F-D$ between the flavor singlet axial coupling $g_{A}^{(0)}$ and
the (previously unrelated) flavor octet $F$ and $D$ values. That relation is a
benchmark feature of the three-quark interpolating fields and any (potential)
departures from it may be attributed to interpolating fields with a number of
quarks that is higher than three.
We constructed all $SU_{L}(3)\times SU_{R}(3)$ chirally symmetric baryon-one-
meson interactions that mix the three basic baryon chiral multiplets (and
their mirror images). All of these interactions, with only one exception, obey
the $U_{A}(1)$ symmetry as well. We used these interactions to relate the
mixing angles to the masses of physical (“mixed”) baryons. Then we tried to
reproduce the phenomenological mixing angles based on observed baryon spectra.
Once the number of admixed fields exceeds three there is too much freedom,
i.e. too many mixing angles, in the most general form of such a mixing
procedure to be constrained by only three measured numbers. That assumption
can be relaxed, if/when more detailed studies become necessary if/when new
observables are measured in the future.
For the purpose of simplification we used the two lowest-lying nucleon states
and then “fit” the phenomenological values of the mixing angles and thus
predicted (at least) one high-lying resonance, which we then searched for in
the PDG tables, Ref. Amsler:2008zzb . This has led us to (at least) two
allowed scenarios. In this way we have made the first tentative assignments of
observed baryon states to chiral multiplets. As explained above, this
procedure does not necessarily lead to unique results, however. The two basic
allowed scenarios differ primarily in the number of predicted flavor-singlet
$\Lambda$ hyperons and in the $U_{A}(1)$ (non)symmetry of their interactions.
At this moment in time we have no reason to prefer one solution to another,
other than aesthetical ones, such as the $U_{A}(1)$ symmetry breaking.
Manifestly, the good $U_{A}(1)$ symmetry limit is sufficient to reproduce the
nucleon axial couplings and the low-lying spectrum, as shown in the first
scenario ($[({\bf 6},{\bf 3})\oplus({\bf 3},{\bf 6})]$ –
$[(\mathbf{\bar{3}},\mathbf{3})\oplus(\mathbf{3},\mathbf{\bar{3}})]$ –
$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$ mixing),
but it is not necessary, as shown in the second scenario ($[({\bf 6},{\bf
3})\oplus({\bf 3},{\bf 6})]$ –
$[(\mathbf{1},\mathbf{8})\oplus(\mathbf{8},\mathbf{1})]$ –
$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$ mixing).
This result stands in contrast to the two-flavor case Dmitrasinovic:2009vp ;
Dmitrasinovic:2009vy , where all $SU_{L}(2)\times SU_{R}(2)$ symmetric
interactions have both a $U_{A}(1)$ symmetry-conserving and a $U_{A}(1)$
symmetry-breaking version. Thus, the three-flavor chiral symmetry is more
restrictive and consequently more instructive than the two-flavor one.
As a simple corollary of this result follows one of our conclusions: the mass
degeneracy of opposite-parity baryon resonances is not necessarily a
consequence of the explicit $U_{A}(1)$ symmetry restoration in agreement with
the conclusions drawn from the two-flavor model calculations, Ref.
Jaffe:2005sq ; Jaffe:2006jy . Moreover, the parity doubling need be neither
one of, nor the only consequence of the spontaneous $SU_{L}(3)\times
SU_{R}(3)$ symmetry restoration.
This result also shows that the “$U_{A}(1)$ anomaly” in QCD may still, but
need not be the underlying source of the “spin problem” Bass:2007zzb , as was
once widely thought Zheng:1991pk . In all likelihood it provides only a
relatively small part of the solution, the largest part coming from the chiral
structure of the nucleon.
The main line of applications of these results lies in the non-zero
density/temperature physics: all previous attempts, see Refs.
Papazoglou:1997uw ; Beckmann:2001bu included only the
$[(\mathbf{3},\mathbf{\bar{3}})\oplus(\mathbf{\bar{3}},\mathbf{3})]$ baryon
chiral multiplet, which naturally led to axial couplings that differ from the
measured ones. Another step, left for the future, is to include the explicit
chiral symmetry breaking.
## Acknowledgments
We wish to thank Profs. Daisuke Jido, Akira Ohnishi and Makoto Oka for
valuable conversations regarding the present work. One of us (V.D.) wishes to
thank the RCNP, Osaka University, under whose auspices this work was begun,
and the Yukawa Institute for Theoretical Physics, Kyoto, where it was
finished, for kind hospitality and financial support under the YITP Molecule
workshop “Algebraic aspect of chiral symmetry for the study of excited
baryons” (11/2-20 (2009)) program. The work of one of us (V.D.) was supported
by the Serbian Ministry of Science and Technological Development under grant
number 141025.
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|
arxiv-papers
| 2010-09-13T15:31:38 |
2024-09-04T02:49:12.872144
|
{
"license": "Public Domain",
"authors": "Hua-Xing Chen, V. Dmitrasinovic and Atsushi Hosaka",
"submitter": "Hua-Xing Chen",
"url": "https://arxiv.org/abs/1009.2422"
}
|
1009.2507
|
# $\mathrm{{}^{56}Ni}$ Production in Double Degenerate White Dwarf Collisions
Cody Raskin11affiliation: School of Earth and Space Exploration, Arizona State
University, P.O. Box 871404, Tempe, AZ, 85287-1404 , Evan
Scannapieco11affiliation: School of Earth and Space Exploration, Arizona State
University, P.O. Box 871404, Tempe, AZ, 85287-1404 , Gabriel
Rockefeller22affiliation: Los Alamos National Laboratories, Los Alamos, NM
87545 , Chris Fryer22affiliation: Los Alamos National Laboratories, Los
Alamos, NM 87545 , Steven Diehl22affiliation: Los Alamos National
Laboratories, Los Alamos, NM 87545 , & F.X. Timmes11affiliation: School of
Earth and Space Exploration, Arizona State University, P.O. Box 871404, Tempe,
AZ, 85287-1404 33affiliation: The Joint Institute for Nuclear Astrophysics
###### Abstract
We present a comprehensive study of white dwarf collisions as an avenue for
creating type Ia supernovae. Using a smooth particle hydrodynamics code with a
13-isotope, $\alpha$-chain nuclear network, we examine the resulting
$\mathrm{{}^{56}Ni}$ yield as a function of total mass, mass ratio, and impact
parameter. We show that several combinations of white dwarf masses and impact
parameters are able to produce sufficient quantities of $\mathrm{{}^{56}Ni}$
to be observable at cosmological distances. We find the $\mathrm{{}^{56}Ni}$
production in double-degenerate white dwarf collisions ranges from sub-
luminous to the super-luminous, depending on the parameters of the collision.
For all mass pairs, collisions with small impact parameters have the highest
likelihood of detonating, but $\mathrm{{}^{56}Ni}$ production is insensitive
to this parameter in high-mass combinations, which significantly increases
their likelihood of detection. We also find that the $\mathrm{{}^{56}Ni}$
dependence on total mass and mass ratio is not linear, with larger mass
primaries producing disproportionately more $\mathrm{{}^{56}Ni}$ than their
lower mass secondary counterparts, and symmetric pairs of masses producing
more $\mathrm{{}^{56}Ni}$ than asymmetric pairs.
## 1 Introduction
While the preferred mechanism for type Ia supernovae (SNeIa) involves a single
white dwarf star accreting material from a non-degenerate companion (Whelan &
Iben 1973; Nomoto 1982; Hillebrandt & Niemeyer 2000), recent observational
evidence suggests a non-negligible fraction of observed SNeIa may derive from
double-degenerate progenitor scenarios. Scalzo et al. (2010) observed the
supernova SN 2007if photometrically, and assuming no host galaxy extinction,
they found 1.6$\pm$0.1$\rm\thinspace M_{\odot}\thinspace$of
$\mathrm{{}^{56}Ni}$ with 0.3-0.5$\rm\thinspace M_{\odot}\thinspace$of
unburned carbon and oxygen forming an envelope. This $\mathrm{{}^{56}Ni}$
yield implies a progenitor mass of 2.4$\pm$0.2$\rm\thinspace
M_{\odot}\thinspace$, which is well above the Chandrasekhar limit - the
maximum mass for a non-rotating white dwarf (Chandrasekhar 1931, Pfannes et
al. 2010, Yoon & Langer 2004, Yoon & Langer 2005). It follows that two white
dwarfs must have been involved in the event that produced SN 2007if, since a
single white dwarf cannot accrete enough material to reach this mass without
either exploding as a SNIa or collapsing to form a neutron star (Yoon et al.
2007). Furthermore, spectroscopic observations by Tanaka et al. (2010) suggest
SN 2009dc produced $\apprge 1.2$$\rm\thinspace M_{\odot}\thinspace$of
$\mathrm{{}^{56}Ni}$, depending on the assumed dust absorption. This also
implies a progenitor mass $>1.4$$\rm\thinspace M_{\odot}\thinspace$as
0.92$\rm\thinspace M_{\odot}\thinspace$of $\mathrm{{}^{56}Ni}$ is the greatest
yield a Chandrasekhar mass can produce (Khokhlov et al. 1993).
Howell et al. (2006) inferred from their observations of SN 2003fg that $\sim
1.3$$\rm\thinspace M_{\odot}\thinspace$of $\mathrm{{}^{56}Ni}$ was produced,
as did Hicken et al. (2007) in their observations of SN 2006gz. There is a
growing body of evidence supporting double-degenerate SNeIa progenitor
systems. Since any supernova arising from a double-degenerate progenitor
scenario may not fit the standard templates for SNeIa, these transients must
be filtered out if SNeIa are to remain as premier cosmological tools. To that
end, we must develop models that give clear and detectable signatures of
double-degenerate SNeIa to distinguish them from standard SNeIa.
Currently, models of double-degenerate progenitors are split into two,
dynamically different scenarios. In the first, white dwarfs in close binaries
lose angular momenta through gravitational radiation, ultimately merging into
a thermally supported super-Chandrasekhar object or detonating outright (Iben
& Tutukov 1984; Webbink 1984; Benz et al. 1989a; Yoon et al. 2007; Pakmor et
al. 2010). In the second, two white dwarfs collide in dense stellar systems
such as globular cluster cores (Raskin et al. 2009; Rosswog et al. 2009; Lor
n-Aguilar et al. 2009; Lor n-Aguilar et al. 2010), where white dwarf number
densities can be as high as $\approx 10^{4}$ pc-3. This follows from
conservative estimates for the average globular cluster mass,
$10^{6}$$\rm\thinspace M_{\odot}\thinspace$(Brodie & Strader 2006), and for
the average globular cluster core radius, 1.5 pc (Peterson & King 1975), taken
together with the Salpeter IMF (Salpeter 1955). Assuming cluster velocity
dispersions on the order of 10 km s-1, this allows for $10-100$, $z\lesssim 1$
collisions per year. Observations by Chomiuk et al. (2008) of globular
clusters in the nearby S0 galaxy NGC 7457 have detected what is likely to be a
SNIa remnant. Given the difficulty in distinguishing SNeIa as residing in
galaxy field stars or in globular clusters in front of or behind their host
galaxies (Pfahl et al. 2009), the frequency with which these can occur
warrants investigation.
Numerical simulations of white dwarf collisions were pioneered in Benz et al.
(1989b) using a smooth particle hydrodynamics (SPH) code. They concluded from
their results that white dwarf collisions were of little interest as the
$\mathrm{{}^{56}Ni}$ yields were small. However, their simulations employed an
approximate equation of state for white dwarfs and resolutions were low,
relative to what is possible with current computing resources. Moreover, as
will be discussed, the infall velocities and velocity gradients play a crucial
role in the final $\mathrm{{}^{56}Ni}$ yields.
More recently, Raskin et al. (2009), Rosswog et al. (2009), and Lor n-Aguilar
et al. (2010) revisited collisions using up-to-date SPH codes and vastly more
particles ($8\times 10^{5}$, $2\times 10^{6}$, and $4\times 10^{5}$,
respectively). In Raskin et al. (2009), a single mass pair (0.6$\rm\thinspace
M_{\odot}\thinspace$$\times 2$) was explored with three impact parameters,
whereas in Rosswog et al. (2009), several mass pairs were examined in direct,
head-on collisions. Both of these papers aimed at establishing double-
degenerate collisions as SNeIa progenitors, finding that $\mathrm{{}^{56}Ni}$
is indeed produced prodigiously in such collisions, lending credence to their
candidacy. Lor n-Aguilar et al. (2010) examined one mass pair
(0.6$\rm\thinspace M_{\odot}\thinspace$\+ 0.8$\rm\thinspace
M_{\odot}\thinspace$), but at a number of different impact parameters, ranging
from those that resulted in direct collisions to those that resulted in
eccentric binaries, aimed at establishing the parameters of white dwarf
coalescence arising from collisional dynamics.
In this paper, we revisit the three impact parameters studied in Raskin et al.
(2009) using a variety of mass pairings. Using 22 combinations of masses and
impact parameters, we aim to answer five key questions; how does
$\mathrm{{}^{56}Ni}$ production depend on
* •
the total mass of the system?
* •
the mass ratio of the two stars?
* •
the impact parameter?
* •
the infall velocities of the constituent stars?
* •
tidal effects?
While the last two of these questions can be eliminated with robust initial
conditions, they are nevertheless important details that are sometimes
overlooked. Armed with this information, we will be able to make some
conclusions about the observability of different combinations of collision
parameters, and to determine whether the resulting transient of any particular
collision is as luminous as a SNIa.
The structure of this paper is as follows. In §2, we discuss the details of
our initial conditions and our new hybrid burning nuclear network. In §3, we
give the details of the results of each simulation that resulted in a
detonation along with a study of the effect of numerical parameters on the
$\mathrm{{}^{56}Ni}$ yield in §3.1.2, and in §4, we discuss those that
resulted in remnants. Finally, in §5, we summarize our results and
conclusions.
## 2 Method
### 2.1 Particle Setups & Initial Conditions
As in Raskin et al. (2009), we employ a version of a 3D SPH code called SNSPH
(Fryer et al. 2006). SPH codes are particularly well suited to these kinds of
simulations as the white dwarf stars involved are very dense and moving very
rapidly. Advecting rapidly moving, isothermal, cold white dwarfs in Eulerian,
grid-based codes introduces perturbations that can be challenging to overcome.
Moreover, because many of our simulations are grazing impacts, conservation of
angular momentum is crucial to the final outcomes, for which SNSPH excels
(Fryer et al. 2006).
In our previous work, we used a Weighted Voronoi Tessellations method (WVT,
Diehl & Statler 2006) for our particle setups. This method arranges particles
in a pseudo-random spatial distribution with thermodynamic quantities that are
consistent with the chosen equation of state (EOS). The default operation for
this method is to allow the masses of particles to vary in order to keep their
sizes, or smoothing lengths ($h$), constant throughout the initial setup. This
approach has its advantages when it comes to spatial arrangement, but one
disadvantage is that it produces a uniform level of refinement in the initial
conditions regardless of where most of the mass resides. The result is that
much higher particle counts are required to reach convergence.
To remedy this, we modified the WVT method to keep mass fixed, varying $h$
consistent with the density profiles of white dwarf stars. This has the effect
of concentrating resolution where most of the mass resides, vastly reducing
the required particle counts for convergence. In fact, whereas in Raskin et
al. (2009), we showed that convergence of the $\mathrm{{}^{56}Ni}$ yield was
reached at approximately $10^{6}$ particles, using constant mass particles, we
reach convergence with only 200,000. A convergence test on particle count of
our fiducial case, 0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$ with zero
impact parameter, is discussed in §3.1.2.
A further modification we have added to our previous approach is an
isothermalization step in our initial conditions. When mapping 1D profiles for
cold white dwarfs onto a resolution limited, 3D particle setup, there is often
a relaxation time, during which the stars oscillate before finding
equilibrium. For white dwarf masses of $\approx 0.6$$\rm\thinspace
M_{\odot}\thinspace$, this settling time is short, but for larger masses, the
oscillations can continue for several minutes or hours. These repeated
gravitational contractions heat the interiors of the stars until they can no
longer be considered “cold” white dwarfs. Therefore, we relaxed each
individual star in a modified version of SNSPH that artificially cools the
stars by keeping each particle at a constant temperature during the stars’
oscillations until they reach a cold equilibrium. Figure 1 shows a temperature
profile for a 0.64$\rm\thinspace M_{\odot}\thinspace$white dwarf that has been
passed through this isothermalization routine, indicating an isothermal
temperature of $\approx 10^{7}$K throughout.
Figure 1: Temperatures and densities of particles lying on the x-axis in a
0.64$\rm\thinspace M_{\odot}\thinspace$white dwarf. This star was created
using WVT and isothermalized to $10^{7}$K after $\sim 5$ minutes.
As in our previous work, the initial conditions for the positions and
velocities of the white dwarf stars in our simulations were generated using a
fourth-order Runge-Kutta solver with an adaptive time-step that integrates
simple kinematic equations. The impactor star was initially given a small
velocity comparable to the velocity dispersion of globular cluster cores,
$\sigma=10$km/s. The solver places the stars at
0.1R$\rm\thinspace{}_{\odot}\thinspace$apart with the proper velocity vectors
expected for free-fall from large initial separations with a given velocity
dispersion.
Figure 2 compares the initial conditions of the 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$, head-on collision to the velocity gradient
that is introduced by tidal forces. The relative velocity of the centers of
mass can be predicted analytically for a zero impact parameter collision with
$v_{\rm c}=\left[{2G(M_{1}+M_{2})}/{\Delta r}\right]^{1/2}$, where $M_{i}$ are
the masses of the constituent white dwarfs and $\Delta r$ is the separation of
their centers of mass.
Figure 2: The velocity evolution from our initial conditions to the moment of
first contact, indicating strong velocity gradients induced by tidal forces.
The shaded area denotes the spread in relative $x$-velocities, and $v_{\rm c}$
is the relative velocity of the centers of mass.
As Figure 2 demonstrates, when the stars are allowed to free-fall in SNSPH
from larger separations, such as the separation of
0.1R$\rm\thinspace{}_{\odot}\thinspace$used throughout this paper, the
velocity gradients that arise from tidal distortions are non-negligible. As
will be shown in §3.1.2, the magnitudes of these velocities and their spreads
play an important role in the final outcomes and the $\mathrm{{}^{56}Ni}$
yields of each simulation as they determine how much kinetic energy is
converted to thermal energy, and thusly, when carbon-ignition occurs. A
shooting-method was used to determine the necessary, initial vertical
separation that resulted in the final impact parameter that we desired at the
moment of impact. Initial velocities for all of our collision scenarios with
zero impact parameter are given in Table 1.
Table 1: Initial velocities of each component star in the head-on cases of each mass pair for initial separations of 0.1R$\rm\thinspace{}_{\odot}\thinspace$. All velocities are relative to the center of mass. # | $m_{1}$ [$\rm\thinspace M_{\odot}\thinspace$] | $m_{2}$ [$\rm\thinspace M_{\odot}\thinspace$] | $-v_{1}$ [$\times 10^{3}$km/s] | $v_{2}$ [$\times 10^{3}$km/s]
---|---|---|---|---
1 | 0.64 | 0.64 | 1.10 | 1.10
2 | 0.64 | 0.81 | 1.31 | 1.03
3 | 0.64 | 1.06 | 1.58 | 0.95
4 | 0.81 | 0.81 | 1.24 | 1.24
5 | 0.81 | 1.06 | 1.51 | 1.15
6 | 0.96 | 0.96 | 1.35 | 1.35
7 | 1.06 | 1.06 | 1.41 | 1.41
8 | 0.50 | 0.50 | 0.97 | 0.97
Likewise, all of our stars are initialized with 50% $\mathrm{{}^{12}C}$ and
50% $\mathrm{{}^{16}O}$ throughout. This approximates typical carbon-oxygen
white dwarf compositions, and we use the Helmholtz free-energy EOS (Timmes &
Arnett 1999; Timmes & Swesty 2000).
### 2.2 Hybrid Burner
Most large, hydrodynamic codes use some form of a hydrostatic nuclear network
(e.g. Eggleton 1971; Weaver et al. 1978; Arnett 1994; Fryxell et al. 2000;
Starrfield et al. 2000; Herwig 2004; Young & Arnett 2005; Nonaka et al. 2008).
That is, the thermodynamic conditions present at the start of a burn
calculation are not altered until the next hydrodynamic time step, which often
times is controlled by abundance or energy changes from the burn calculation
rather than a pure Courant condition. The effect of this is to fix the
temperature-dependent reaction rates throughout the hydrodynamic time step to
what they were at its start.
There are several reasons for running a simulation this way, the most
important of which is to avoid a decoupling of the nuclear network from the
hydrodynamic calculation. However, for limited spatial, mass and time
resolutions, this approximation - that the thermodynamic conditions do not
change rapidly enough during a burn to warrant a sub-cycle recalculation of
the nuclear reaction rates - fails in regimes where the nuclear reactions are
strongly temperature dependent, such as at temperatures where photo-
disintegration is the dominant nuclear process. As Figure 3 shows below, the
nuclear statistical equilibrium (NSE) state for material with $\rho=1\times
10^{6}$ g cm-3 at $T_{9}\approx 7$ has most of the heavy isotopes photo-
disintegrating back to 4He.
Figure 3: NSE distributions for $\rho=$1e7 g cm-3 and Ye=0.5 in an
$\alpha$-centric nuclear network. Proton and neutron mass fractions are
plotted for reference. At $T_{9}\approx 6$, 4He begins to dominate the isotope
distribution.
In such a regime, the material undergoing photo-disintegration experiences
what amounts to an abrupt phase change through a strongly endothermic
reaction. In nature, this reaction should rapidly cool the material before
complete photo-disintegration, allowing these liberated $\alpha$-particles to
react with other isotopes. However, a hydrostatic burn will overestimate the
time-scale for this cooling as it assumes a full hydrodynamic time step is
necessary for relevant pressure or temperature changes. With the temperature
remaining fixed over an artificially long time, this approach results in the
nuclear network removing far too much internal energy, $u$, to be physical.
Typically, one attempts to limit the impact of such a phase change by relying
on a global time-step minimization scheme of the form
$\Delta t_{n+1}=\min\left[\Delta t_{c},\Delta t_{n}\times
f_{u}\times\left(\frac{u^{i}_{n-1}}{u^{i}_{n}-u^{i}_{n-1}}\right)\right],$ (1)
where the subscript $n$ refers to the iteration number, $\Delta t_{c}$ is the
Courant time, $u^{i}$ is the specific internal energy of the $i^{\rm th}$
particle, and $f_{u}$ is a dimensionless parameter which constrains the
maximum allowable change in energy. Our global time-step is also controlled in
this manner. In practice, the conditions immediately prior to photo-
disintegration will fix the next time-step, $\Delta t_{n+1}$, to of order
$10^{-5}$s for $f_{u}=0.30$. However, even this time-step is too large to
capture the relevant temperature changes effecting the reaction rates on time-
scales of order $10^{-12}$s.
The alternative approach to a hydrostatic burn is to use a “self-
heating/cooling” nuclear network that simultaneously integrates an energy
equation and the abundance equation self-consistently (see e.g. Müller 1986).
When applied to a particle code like SPH, this type of calculation keeps
$\rho$ fixed, but updates temperature in a fashion that is consistent with the
equation of state and the new internal energy at each sub-cycle.
The ordinary differential equation that governs a hydrostatic burn calculation
of the abundance of an isotope $Y_{i}$, assuming the mass diffusion gradients
are negligible, is of the form
$\displaystyle\dot{Y_{i}}$ $\displaystyle=$
$\displaystyle\sum_{j}C_{i}R_{j}Y_{j}$ (2) $\displaystyle+$
$\displaystyle\sum_{j,k}\frac{C_{i}}{C_{j}!C_{k}!}\rho N_{A}R_{j,k}Y_{j}Y_{k}$
$\displaystyle+$
$\displaystyle\sum_{j,k,l}\frac{C_{i}}{C_{j}!C_{k}!C_{l}!}\rho^{2}N_{A}^{2}R_{j,k,l}Y_{j}Y_{k}Y_{l},$
where the coefficients $C_{i..l}$ specify how many particles of the $i^{\rm
th}$ species are created or destroyed, and $R_{i..l}$ are the temperature-
dependent reaction rates for each of the different reaction types. The first
term describes weak reactions ($\beta$-decays and electron captures) and
photo-disintegrations, the second describes two-body reactions of the type
$\mathrm{{}^{12}C}$($\alpha$,$\gamma$)$\mathrm{{}^{16}O}$, and the third term
describes three-body reactions, such as
$\mathrm{{}^{4}He}$(2$\alpha$,$\gamma$)$\mathrm{{}^{12}C}$. The energy
generation ODE takes the form
$\dot{\epsilon}=-N_{A}\sum_{i}\dot{Y_{i}}m_{i}c^{2},$ (3)
where $m_{i}$ is the rest-mass of the $i^{\rm th}$ isotope (see e.g. Benz et
al. 1989b). The density and temperature equations are simply $\dot{\rho}=0$
and $\dot{T}=0$, respectively, in a hydrostatic burn.
A self-heating at constant density calculation modifies only the temperature
equation, starting from the first law of thermodynamics in specific mass
units,
$\frac{du}{dt}-\frac{P}{\rho^{2}}\frac{d\rho}{dt}=T\frac{ds}{dt},$ (4)
where $du/dt$ is the change in specific energy and $ds/dt$ is the change in
specific entropy. In accordance with $\dot{\rho}=0$ and employing the identity
$T\dot{s}=\dot{\epsilon}$, this reduces to
$\displaystyle\frac{\partial u}{\partial T}\frac{dT}{dt}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}$ $\displaystyle\dot{T}$ $\displaystyle=$
$\displaystyle\frac{\dot{\epsilon}}{c_{\rm v}},$ (5)
where $c_{\rm v}$ is the specific heat capacity at a constant volume.
Equations (2), (3), and (5) are evolved simultaneously and self-consistently
(Müller 1986).
At very high spatial resolutions and small time-steps, the self-heating
approach would be identical to the hydrostatic approach. As Figure 4 shows for
the energy, temperature and composition of a single particle over a finite and
relatively large time-step as determined by Equation (1) with $f_{u}=0.30$,
these two burning calculations reach very different conclusions about the
final energy and composition of the particle after photo-disintegration.
Figure 4: Calculations of the energy, temperature, and composition of a
particle after a representative time-step as determined by Equation (1) with
$f_{u}=0.30$. Solid lines show the implicit integrations from a hydrostatic
calculation, while dotted lines indicate those for a self-heating calculation.
In both cases, $\rho$ is kept constant, while in the self-heat calculation,
the temperature, and thus the nuclear reaction rates, are recalculated at each
implicit integration step, consistent with the first law of thermodynamics.
To capture the relevant temperature changes using a hydrostatic approach would
require a global time-step $\sim 10^{-12}$s, where the temperatures of the two
calculations have diverged by $\approx 5\%$. This is problematic for two
reasons: 1) such a time-step cannot be predicted from the conditions
immediately prior to photo-disintegration, and 2) having such a small global
time-step exceeds the limit of machine precision for many hydrodynamic codes,
SNSPH included. When SNSPH attempts to calculate velocities for the next time-
step using $\Delta t\sim 10^{-12}$s, it often fails or returns zero.
Unfortunately, a self-heating nuclear network can also expose the weakness of
a mass resolution limit. For a typical simulation of $10^{6}$ particles, each
particle has a mass of $\approx 10^{27}$g. A self-heating nuclear calculation
for carbon-burning of such large masses becomes rapidly explosive on time-
scales approaching the Courant limit. Without any mechanism for energy
transport on such short time-scales, the assumption of a homogeneous burn of
all $10^{27}$g begins to break down. The vigorous burning of so much material
rapidly liberates more energy than the binding energy of the star.
Our “middle-path” solution to these two extremes is a hybrid-burning scheme
wherein a combination of these two approaches is used under different
circumstances. Since the hydrostatic approach is a better approximation for
exothermic reactions at our resolution limit, the self-heating/cooling
approach is only employed for particles that undergo strong, net endothermic
reactions such as photo-disintegration. This allows these particles to
smoothly “step over” the photo-disintegration phase change without
artificially losing too much energy. We apply this approach, along with the
time-step minimization of Equation (1), to the $\alpha$-chain aprox13 nuclear
network (Timmes 1999; Timmes et al. 2000) by imposing the condition that if
$\dot{\epsilon}<0$ after a hydrostatic burn, the burn is recalculated
employing Equation (5).
While stepping through a photo-disintegration process is an interesting
wrinkle for numerical simulations of double-degenerate white dwarf collisions,
it is not a significant factor for $\mathrm{{}^{56}Ni}$ production. In all of
our simulated cases, we found that on average $<2$% of particles experienced
this phase change. For the most part, the local conditions for a number of
particles wherein a single particle might undergo photo-disintegration are
sufficiently high-energy that the neighboring particles will have already
initiated a detonation. Based on the results of our simulations, we do not
expect that collisions with yet higher kinetic energies than those attempted
here would paradoxically yield less $\mathrm{{}^{56}Ni}$ due to photo-
disintegration effects.
## 3 Results & Analysis I - Detonations
Our previous work narrowed the range of pertinent impact parameters to three
scenarios, which we revisited for each of our mass combinations. We simulated
head-on impacts, partially grazing collisional impacts, and fully
grazing/glancing impacts. Table 2 summarizes the $\mathrm{{}^{56}Ni}$ yields
of each of our simulations. In this table, the impact parameter, $b$ (the
vertical separation between the cores of both white dwarfs at the moment of
impact) is given as the fraction of the radius of the primary white dwarf.
Thus the $b=0$ column shows the yields for head-on impacts, the $b=1$ column
indicates a full white dwarf radius and $b=2$ indicates 2 white dwarf radii,
or a fully grazing impact.
Table 2: Simulation $\mathrm{{}^{56}Ni}$ yields for various mass combinations and parameters. Values in bold are super-Chandrasekhar masses, and values indicated with a (¤) are those simulations that resulted in remnants. Dashes (–) indicate combinations of parameters we did not simulate. All simulations listed here used $f_{u}=0.30$ and 200k particles. # | $m_{1}$ [$\rm\thinspace M_{\odot}\thinspace$] | $m_{2}$ [$\rm\thinspace M_{\odot}\thinspace$] | $m_{tot}[\hbox{$\rm\thinspace M_{\odot}\thinspace$}]$ | $b=0$ | $b=1$ | $b=2$
---|---|---|---|---|---|---
1 | 0.64 | 0.64 | 1.28 | 0.51 | 0.47 | ¤
2 | 0.64 | 0.81 | 1.45 | 0.14 | 0.53 | ¤
3 | 0.64 | 1.06 | 1.70 | 0.26 | ¤ | ¤
4 | 0.81 | 0.81 | 1.62 | 0.84 | 0.84 | 0.65
5 | 0.81 | 1.06 | 1.87 | 0.90 | 1.13 | ¤
6 | 0.96 | 0.96 | 1.92 | 1.27 | 1.32 | 1.33
7 | 1.06 | 1.06 | 2.12 | 1.71 | 1.72 | 1.61
8 | 0.50 | 0.50 | 1.00 | 0.00 | – | –
Dursi & Timmes (2006) examined the shock-ignited detonation criteria for
carbon in a white dwarf using numerical models. They derived a relationship
between the density of the carbon fuel and the minimum radius of a burning
region that will launch a detonation. For a carbon abundance of
$X_{{}^{12}{\rm C}}=0.5$ and densities typically found in the white dwarfs
used in our simulations, $\rho\sim 10^{7}$g cm-3, their results suggest a
minimum burning region, or match head size of $r_{b}\sim 10^{4}$cm. This is
three orders of magnitude smaller than our smallest particle size, and
properly resolving this criterion would require $\sim 10^{12}$ particles. Such
a high-resolution study is too expensive with our current computing resources,
and therefore, we acknowledge that we cannot resolve the precise detonation
mechanism in our simulations.
The criteria established in Dursi & Timmes (2006) would suggest that a single
particle in any of our simulations can initiate a detonation. However, in
order for a detonation to be sustained, the energy that the initiating
particle deposits in its neighbors must be sufficient to cause those neighbors
to liberate an equal amount of nuclear energy. This somewhat softens the
ability of a single particle to initiate a sustainted detonation. Indeed, in
all of our simulations, we found that several particles ignited nearly
simultaneously, or at least outside of causal contact with one another in
order to initiate a sustained detonation. Moreover, the pressure gradient
established by particles neighboring those that reached ignition needed to be
favorable for a significant and rapid energy deposition.
In SPH, energy is shared between particles via $PdV$ work with
$\dot{u}_{ij}=\frac{P_{i}}{\rho_{i}^{2}}m_{j}\Delta
v_{ij}\cdot\nabla_{i}W_{ij},$ (6)
where $P_{i}$ and $\rho_{i}$ are the pressure and density of particle $i$,
$m_{j}$ is the mass of particle $j$, $\Delta v_{ij}$ is the difference in
velocities of particles $i$ and $j$, and $W_{ij}$ is the SPH smoothing kernel.
For each particle $i$, there is an implied sum over all particles $j$. In SPH
formalism, the condition for a sustained detonation would require that this
quantity is large enough to ignite explosive burning in particle $i$. Put
another way, if particle $i$ generates energy $\epsilon_{i}$ at time $t_{0}$,
$\dot{u}_{ji}$ must be sufficient such that at time $t_{0}+\Delta t$,
$\dot{\epsilon}_{j}\approx\dot{\epsilon}_{i}$, where $\Delta t$ is the Courant
time. This requires a proportionality between the energy generation rate in
particle $i$ and the pressure gradient with its nearest neighbors, and to
first order, this criterion reduces to
$\dot{\epsilon_{i}}\geq\frac{P_{i}}{c_{s}\rho_{i}^{2}}\nabla P_{ij}.$ (7)
For a given energy generation rate, large and positive pressure gradients can
inhibit a detonation breakout. In situations where particles ignited carbon-
burning, but were nevertheless unable to deposit enough energy into their
neighbors to cause them to also ignite, the material settled into a slow-burn
regime rather than detonating. While we cannot resolve the detonation
mechanism to the desired precision, we compared one-dimensional ZND detonation
profiles (see e.g. Fickett & Davis 1979) with detonation profiles from one-
dimensional slices through the SNSPH models and concluded that our collision
calculations are resolving the detonation widths and structures to within 20%.
### 3.1 Mass Pair 1 - 0.64$\hbox{$\rm\thinspace M_{\odot}\thinspace$}\times
2$
#### 3.1.1 Fiducial Case
We recalculated the $\approx 0.6$$\rm\thinspace M_{\odot}\thinspace$equal mass
case as in Raskin et al. (2009) to establish a baseline comparison with our
equal-mass particle configuration and hybrid-burner technique. Empirical white
dwarf mass functions (e.g. , Williams, Bolte, & Koester 2004) suggest
collisions with this mass pair are expected to be among the most common. With
our equal-mass particle constraint, the final mass of the star used in the
simulations came to 0.64$\rm\thinspace M_{\odot}\thinspace$.
The right-top panel of Figure 5 shows that when the stars first collide the
infall speeds, $v_{x}$ of material entering the shocked region are greater
than the sounds speeds, $c_{s}$, resulting in a stalled shock in that region.
The conditions in the center plane of this shocked region (the $y$-$z$ plane,
$\rho\approx 10^{6.5-7}$ g cm-3 and $T_{9}\approx 1$) are sufficient to ignite
carbon with an energy-generation rate scaling roughly as
$\dot{\epsilon}\sim\rho T^{22}$, burning up to silicon at $T_{9}\approx 3$.
The separation of material into three distinct phases is clearest in the left
two panels of Figure 5, which plots particle number density in the $\rho$-$T$
plane. The lower, more populated region is unshocked, carbon-oxygen material
and is indicated in green. The less populated middle region, also represented
with green at $T_{9}\approx 1$, is shocked material that has yet to reach the
critical conditions for carbon ignition, and the upper, sparse region is
material that has begun burning carbon to silicon-group elements, represented
in red.
Figure 5: Left Panels: Density vs. temperature for all particles from each
constituent star in mass pair 1, 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$ and $b=0$. Each point is colored indicating the
isotope(s) with the greatest abundance and by the particle number density at
each $\rho-T$ coordinate. Green indicates high concentrations of carbon and
oxygen, red indicates silicon group elements, and blue indicates iron-peak
elements, most predominantly $\mathrm{{}^{56}Ni}$. The darker the color in
each group, the higher the particle number density. Right-top Panel: Sound
speed, infall velocity, and density for particles lying on the $x$-axis.
Right-bottom Panel: A slice in the $x$-$y$ plane of particle densities.
The pressure gradient slopes positively in all directions away from the
geometric center where this early burning begins, which is in fact at lower
densities than the surrounding shocked medium. While the whole of the shocked
region continues to heat up, causing more material to ignite near the center,
the steep pressure gradient prevents the energy liberated by burning to
initiate a detonation. For these burning particles, with silicon ash behind
them and higher pressure carbon in front, the energy they deposit in their
neighbors is insufficient to greatly alter their energy-generation rate.
Instead, the burning region grows only as fast as material is heated to the
critical temperatures needed for carbon ignition, $T_{9}\approx 1$, by the
conversion of kinetic energy to thermal energy.
Approximately two seconds after the stars first collide, sufficiently high
temperatures and densities are reached at the edges of the shocked region to
initiate carbon-burning. In these locations, the pressure gradient slopes
negatively in all directions. The liberated energy is free to break out,
initiating detonations at the ignition points. The sound speeds in these zones
are raised higher than the infall speeds due to the rapid rise in temperature,
and $\mathrm{{}^{56}Ni}$ begins to appear in large quantities, indicated in
blue in the left panels of Figure 6.
Figure 6: Same format as Figure 5, at a later time in the simulation.
Sustained detonation fronts then propagate through the unburned material, as
well as the silicon “ash” that lies in the shocked region. As Figure 7 shows,
significantly more $\mathrm{{}^{56}Ni}$ is produced during this phase. In
Figure 7, it is also evident that the shocks overtake one another inside the
contact zone, shocking the material a second time and producing yet more
$\mathrm{{}^{56}Ni}$. Less than one second after the detonations began, the
entire system has become unbound, freezing out the nuclear reactions, as can
be seen in Figure 8. The final $\mathrm{{}^{56}Ni}$ yield for this simulation
was 0.51$\rm\thinspace M_{\odot}\thinspace$.
Figure 7: Same format as Figure 5, at a later time in the simulation. Figure
8: Same format as Figure 5, at a later time in the simulation.
In the $b=1$ simulation, the added angular momentum distorted the shocked
region between the two stars, resulting in detonations lighting off-center and
off-axis as compared to the $b=0$ case. As Figure 9 shows, the detonation
waves traveling through the densest portions of the shocked regions where the
sound speed is highest, twist the material into a uniquely anisotropic
configuration. Moreover, because much of the material is traveling nearly
perpendicular to the shock, the density in the pre-detonation, shocked region
is lower than in the $b=0$ case. This reduces $\mathrm{{}^{56}Ni}$ production
by about 7% to 0.47$\rm\thinspace M_{\odot}\thinspace$.
Figure 9: A 2D slice of interpolated densities through the $x$-$y$ plane of
the $b=1$ case of two 0.64$\rm\thinspace M_{\odot}\thinspace$white dwarfs
colliding. Four snapshots at different times are shown. Arrows in the top-left
panel indicate the directions of motion of each star.
Since the post-explosion, expansion phase is homologous, the pattern of
isotopes present at the moment the system becomes unbound is not altered by
the expansion. Therefore, the velocities plotted in Figure 10 for several
isotopes in the $b=0$ and $b=1$ cases are directly related to the radial
distribution of the isotopes.
Figure 10: Masses of several isotopes at logarithmically spaced velocity bins
for the $b=0$ and $b=1$ cases of mass pairing 1, 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$.
The velocity structure preserves the isotopic segregation expected behind the
burning front, with a progression from complete burning of carbon & oxygen to
iron-peak elements, though silicon-group elements, and finally ending with an
unburned or only partially burned carbon & oxygen envelope. This layered
structure is in agreement with the observations of Scalzo et al. (2010) and
others of type Ia SNe suspected of having been produced from double-degenerate
progenitor scenarios.
The $b=2$ scenario for this mass pair did not feature a detonation, and
instead, resulted in a hot remnant embedded in a disk. Details of this
simulation and its outcome will be discussed in §4.
#### 3.1.2 Variations on Parameters
In order to assess the impact of the time-step on the nuclear yields, we
compared three simulations of the 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$, $b=0$ case varying the value of $f_{u}$ in
Equation (1); one with a value of $f_{u}=0.50$, another with $f_{u}=0.30$, and
finally, one with $f_{u}=0.25$. As the results in table 3 show for the
$\mathrm{{}^{56}Ni}$ yields, changes in the value of $f_{u}$ below 0.5 have
little discernible impact on the final outcomes.
Table 3: $\mathrm{{}^{56}Ni}$ yields for 0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$, $b=0$ simulations with variations on the parameter $f_{u}$ and particle count. $f_{u}$ | Particle Count | $\mathrm{{}^{56}Ni}$
---|---|---
0.50 | $2\times 10^{5}$ | 0.51
0.30 | $2\times 10^{5}$ | 0.51
0.25 | $2\times 10^{5}$ | 0.51
0.30 | $1\times 10^{4}$ | 0.21
0.30 | $4\times 10^{4}$ | 0.31
0.30 | $4\times 10^{5}$ | 0.49
0.30 | $2\times 10^{6}$ | 0.53
The detonations on either side of the shocked region are unique to the
0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$ mass pairing and the
0.50$\rm\thinspace M_{\odot}\thinspace$$\times 2$ mass pairing described in
§3.7. This is due, in large part, to the kinetic energy at impact, which is
related directly to the infall speed. With greater infall speeds, the shocked
region heats sufficiently to initiate a detonation earlier, and the
detonations begin nearer to the central region (the $y$-$z$ plane).
We tested this mechanism with a 0.64$\rm\thinspace M_{\odot}\thinspace$$\times
2$ collision scenario by giving the constituent stars an artificially high
infall velocity to reproduce the kinematic energies associated with collisions
of larger masses. In that test, the critical temperatures for carbon ignition
were reached in locations nearer to the $y$-$z$ plane, but still displaced
enough that the pressure gradient was favorable to a detonation. In this case,
the $\mathrm{{}^{56}Ni}$ production was actually depressed, resulting in only
0.39$\rm\thinspace M_{\odot}\thinspace$, due to an early detonation coupled
with altered shock conditions.
We also tested the effect of velocity gradients (tidal distortions) on the
final $\mathrm{{}^{56}Ni}$ yield by running a 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$, $b=0$ collision with an initial separation of
only 0.048R$\rm\thinspace{}_{\odot}\thinspace$with the commensurate relative
velocities. In that simulation, the $\mathrm{{}^{56}Ni}$ yield was also
depressed, slightly, to 0.48$\rm\thinspace M_{\odot}\thinspace$. The
combination of infall velocity and tidal distortions are evidently critical
for $\mathrm{{}^{56}Ni}$ production.
However, by far the most important parameter effecting the convergence of the
$\mathrm{{}^{56}Ni}$ yield is resolution. We carried out a convergence test of
the $\mathrm{{}^{56}Ni}$ yield in mass pair 1, $b=0$, using equal-mass
particle setups. We varied particle counts from $10^{4}$ particles total, to
$2\times 10^{6}$. As Figure 11 demonstrates, convergence was reached at
$2\times 10^{5}$ particles. This compares favorably to previous convergence
estimates in Raskin et al. (2009) that concluded $\sim 10^{6}$ particles were
needed for convergence using equal-$h$ particle setups.
Figure 11: Convergence of the $\mathrm{{}^{56}Ni}$ yield with particle count
for simulations employing equal mass particles (blue,
$0.64\hbox{$\rm\thinspace M_{\odot}\thinspace$}\times 2$) and equal h
particles (red, $0.6\hbox{$\rm\thinspace M_{\odot}\thinspace$}\times 2$ from
Raskin et al. 2009). The dashed, vertical line indicates the number of
particles used in simulations throughout this paper.
Early work on numerical simulations of white dwarf collisions carried out by
Benz et al. (1989b) did not have the benefit of modern computational resources
to reach these kinds of resolutions. Consequently, the $\mathrm{{}^{56}Ni}$
yields in those simulations were comparatively quite low.
### 3.2 Mass Pair 2 - 0.64$\rm\thinspace M_{\odot}\thinspace$\+
0.81$\rm\thinspace M_{\odot}\thinspace$
For asymmetrical collisions involving 0.64$\rm\thinspace
M_{\odot}\thinspace$and 0.81$\rm\thinspace M_{\odot}\thinspace$white dwarfs,
the higher kinetic energy with which they collide results in several, almost
immediate detonations near the center in the $b=0$ scenario. As Figure 12
shows, these detonation shocks superimpose to form a single, nearly spherical
shock front that raises the sound speed above the infall speed for material in
the 0.64$\rm\thinspace M_{\odot}\thinspace$star, but the pressure gradient
leftward of the detonation (region 1) stalls the detonation shock, which can
be seen as a higher-density, laminar shock at the rightmost edge of region 1
in Figure 12.
Figure 12: Same format as Figure 5 for mass pair 2, 0.64$\rm\thinspace
M_{\odot}\thinspace$\+ 0.81$\rm\thinspace M_{\odot}\thinspace$, and $b=0$.
Material shocked by the collision is labeled as region 1. Material behind the
first detonation shock is labeled as region 2.
However, region 1 does not maintain its lenticular shape as the two stars are
moving at different speeds relative to this shocked region. This allows the
detonation shock to travel through this region at $\approx$ Mach 1, eventually
reaching fresh carbon outside of region 1. This fresh carbon ignites
explosively, creating a second detonation (region 3 in Figure 13), which sends
leading shocks back through region 1 and into region 2, shocking it a second
time and eventually catching up with the first detonation shock.
Figure 13: Same format as Figure 5 for mass pair 2, 0.64$\rm\thinspace
M_{\odot}\thinspace$\+ 0.81$\rm\thinspace M_{\odot}\thinspace$, and $b=0$, at
a later time in the simulation. Material shocked by the collision is labeled
as region 1. Material behind the first detonation shock is labeled as region
2, and material behind the second detonation shock is labeled region 3.
Most of the $\mathrm{{}^{56}Ni}$ in this scenario is produced in the more
massive star, as Table 4 demonstrates. Since only low-density portions of the
0.64$\rm\thinspace M_{\odot}\thinspace$star had entered the shocked region
before the detonation, most of its contribution to the total output is in Si-
group elements.
Table 4: Isotope yields for the $b=0$ and $b=1$ cases of mass pairing 2, 0.64$\rm\thinspace M_{\odot}\thinspace$\+ 0.81$\rm\thinspace M_{\odot}\thinspace$. $b$ | Isotope | 0.81 [$\rm\thinspace M_{\odot}\thinspace$] | 0.64 [$\rm\thinspace M_{\odot}\thinspace$] | Total [$\rm\thinspace M_{\odot}\thinspace$]
---|---|---|---|---
0 | $\mathrm{{}^{12}C}$ | 0.21 | 0.03 | 0.24
$\mathrm{{}^{16}O}$ | 0.25 | 0.14 | 0.39
$\mathrm{{}^{28}Si}$ | 0.12 | 0.27 | 0.39
$\mathrm{{}^{56}Ni}$ | 0.13 | 0.02 | 0.14
1 | $\mathrm{{}^{12}C}$ | 0.02 | 0.03 | 0.05
$\mathrm{{}^{16}O}$ | 0.07 | 0.14 | 0.21
$\mathrm{{}^{28}Si}$ | 0.12 | 0.25 | 0.37
$\mathrm{{}^{56}Ni}$ | 0.49 | 0.04 | 0.53
In the $b=1$ case, the pre-detonation, shocked region reaches much higher
densities, and the oblique angle at which the white dwarf stars enter the
shocked region allows more material to become strongly shocked by the
detonation. The detonation shock also twists around the peculiar density
contours inside the shocked region, shocking much of the material several
times, as is seen in the bottom-right panel of Figure 14. The
0.81$\rm\thinspace M_{\odot}\thinspace$star experiences a near complete burn
of all of its carbon and oxygen. However, as in the $b=0$ case, most of the
0.64$\rm\thinspace M_{\odot}\thinspace$star remains unshocked at the time of
the detonation breakout. As before, the 0.64$\rm\thinspace
M_{\odot}\thinspace$star contributes mostly Si-group elements to the total
output, as shown in Table 4.
Figure 14: A 2D slice of interpolated densities through the $x$-$y$ plane of
the $b=1$ case of a 0.64$\rm\thinspace M_{\odot}\thinspace$white dwarf
colliding with a 0.81$\rm\thinspace M_{\odot}\thinspace$white dwarf. Four
snapshots at different times are shown. Arrows in the top-left panel indicate
the directions of motion of each star.
The velocity profiles for the $b=0$ and $b=1$ cases of mass pair 2, shown in
Figure 15, reinforce the observation that $\mathrm{{}^{56}Ni}$ is created in a
confined region in the $b=0$ case, mainly in the densest portions of the
shocked material from the 0.81$\rm\thinspace M_{\odot}\thinspace$star. Carbon
and oxygen, together, dominate the total output by mass, while in the $b=1$
case, $\mathrm{{}^{56}Ni}$ is the dominant isotope, followed by
$\mathrm{{}^{28}Si}$.
Figure 15: Masses of several isotopes at logarithmically spaced velocity bins
for the $b=0$ and $b=1$ cases of mass pairing 2, 0.64$\rm\thinspace
M_{\odot}\thinspace$\+ 0.81$\rm\thinspace M_{\odot}\thinspace$.
### 3.3 Mass Pair 3 - 0.64$\rm\thinspace M_{\odot}\thinspace$\+
1.06$\rm\thinspace M_{\odot}\thinspace$
As with mass pair 2, the $b=0$ case of mass pair 3 experiences a detonation of
material in the 0.64$\rm\thinspace M_{\odot}\thinspace$star very quickly after
the stars first collide. However, owing to the greater potential well into
which the 0.64$\rm\thinspace M_{\odot}\thinspace$star is falling, the sound
speed of the material shocked by the detonation is less than the infall
velocity as shown in the top-left panel of Figure 16.
Figure 16: Same format as Figure 5 for mass pair 3, 0.64$\rm\thinspace
M_{\odot}\thinspace$\+ 1.06$\rm\thinspace M_{\odot}\thinspace$, and $b=0$.
After $\approx 0.7$s, as the core of the 1.06$\rm\thinspace
M_{\odot}\thinspace$star enters the shocked region, a second detonation lights
on the left edge of the shocked region. This powers a shock that travels
through both stars, catching up with the shock from the first detonation in
the 0.64$\rm\thinspace M_{\odot}\thinspace$star. The material in the
0.64$\rm\thinspace M_{\odot}\thinspace$star burns mostly to
$\mathrm{{}^{28}Si}$, while what burns in the 1.06$\rm\thinspace
M_{\odot}\thinspace$star burns almost entirely to $\mathrm{{}^{56}Ni}$, due to
its higher density. The contributions from each star to the total elemental
abundances are given in Table 5.
Table 5: Isotope yields for the $b=0$ case of mass pairing 3, 0.64$\rm\thinspace M_{\odot}\thinspace$\+ 1.06$\rm\thinspace M_{\odot}\thinspace$. Isotope | 1.06 [$\rm\thinspace M_{\odot}\thinspace$] | 0.64 [$\rm\thinspace M_{\odot}\thinspace$] | Total [$\rm\thinspace M_{\odot}\thinspace$]
---|---|---|---
12C | 0.39 | 0.02 | 0.41
16O | 0.40 | 0.15 | 0.55
28Si | 0.05 | 0.24 | 0.29
$\mathrm{{}^{56}Ni}$ | 0.19 | 0.07 | 0.26
In simulations of mass pair 3 that introduced a non-zero impact parameter, the
1.06$\rm\thinspace M_{\odot}\thinspace$star was simply too compact to be
significantly disrupted by a collision with a 0.64$\rm\thinspace
M_{\odot}\thinspace$star. In both the $b=1$ and $b=2$ cases, most of the
1.06$\rm\thinspace M_{\odot}\thinspace$star survived the collision, while
completely disrupting the 0.64$\rm\thinspace M_{\odot}\thinspace$star. Details
of those simulations are given in §4.
### 3.4 Mass Pair 4 - 0.81$\hbox{$\rm\thinspace M_{\odot}\thinspace$}\times
2$
The symmetrical mass pair, 0.81$\rm\thinspace M_{\odot}\thinspace$$\times 2$
is quite unlike the 0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$ mass
pairing discussed above. For the 0.81$\rm\thinspace
M_{\odot}\thinspace$$\times 2$ mass pair with $b=0$, several detonations occur
in the $y$-$z$ plane simultaneously and almost immediately after impact, owing
to the higher temperatures reached in the shocked region from the higher
infall speeds. As Figure 17 shows, these detonations superimpose and produce
copious amounts of $\mathrm{{}^{56}Ni}$ as they travel through the much denser
material present inside the 0.81$\rm\thinspace M_{\odot}\thinspace$stars. This
denser material allows for a significantly greater conversion of carbon and
oxygen to $\mathrm{{}^{56}Ni}$. Therefore, with only a 26% increase in total
mass of the system over the 0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$
scenario, there is a 64% increase in $\mathrm{{}^{56}Ni}$ production to
0.84$\rm\thinspace M_{\odot}\thinspace$.
Figure 17: Same format as Figure 5 for mass pair 4, 0.81$\rm\thinspace
M_{\odot}\thinspace$$\times 2$, and $b=0$.
Having denser and more compact stars also reduces the sensitivity of the
$\mathrm{{}^{56}Ni}$ yield to impact parameter. Indeed, with two
0.81$\rm\thinspace M_{\odot}\thinspace$stars, both the $b=1$ and $b=2$
simulations resulted in detonations and significant $\mathrm{{}^{56}Ni}$
production, 0.84$\rm\thinspace M_{\odot}\thinspace$and 0.65$\rm\thinspace
M_{\odot}\thinspace$, respectively. Differences in the $\mathrm{{}^{56}Ni}$
yield for the two non-zero impact parameter simulations stem mainly from the
amount of material that burns to 28Si before the detonations occur, with the
$b=2$ scenario featuring much more material burning at lower temperatures to
silicon before the detonation. The high activation energy of 28Si prevents
much of that material from being converted to $\mathrm{{}^{56}Ni}$.
### 3.5 Mass Pair 5 - 0.81$\rm\thinspace M_{\odot}\thinspace$\+
1.06$\rm\thinspace M_{\odot}\thinspace$
The 0.81$\rm\thinspace M_{\odot}\thinspace$\+ 1.06$\rm\thinspace
M_{\odot}\thinspace$mass pair follows a very similar pattern to that of mass
pair 2 (0.64$\rm\thinspace M_{\odot}\thinspace$\+ 0.81$\rm\thinspace
M_{\odot}\thinspace$). Intermediate impact parameters allow more material to
enter the shocked region before detonation, and so there is a rise in
$\mathrm{{}^{56}Ni}$ production in the $b=1$ case over $b=0$. However, because
both stars involved in the collision are denser than their counterparts in
mass pair 2, much more $\mathrm{{}^{56}Ni}$ is produced overall. Contributions
to the total yield in the $b=0$ and $b=1$ simulations are given in Table 6
below.
Table 6: Isotope yields for the $b=0$ and $b=1$ cases of mass pairing 5, 0.81$\rm\thinspace M_{\odot}\thinspace$\+ 1.06$\rm\thinspace M_{\odot}\thinspace$. $b$ | Isotope | 1.06 [$\rm\thinspace M_{\odot}\thinspace$] | 0.81 [$\rm\thinspace M_{\odot}\thinspace$] | Total [$\rm\thinspace M_{\odot}\thinspace$]
---|---|---|---|---
0 | 12C | 0.17 | 0.01 | 0.18
16O | 0.19 | 0.09 | 0.28
28Si | 0.06 | 0.22 | 0.28
$\mathrm{{}^{56}Ni}$ | 0.58 | 0.32 | 0.90
1 | 12C | 0.05 | 0.02 | 0.07
16O | 0.07 | 0.09 | 0.16
28Si | 0.06 | 0.22 | 0.28
$\mathrm{{}^{56}Ni}$ | 0.82 | 0.31 | 1.13
### 3.6 Mass Pairs 6 & 7 - 0.96$\hbox{$\rm\thinspace
M_{\odot}\thinspace$}\times 2$ & 1.06$\rm\thinspace
M_{\odot}\thinspace$$\times 2$
The 0.96$\rm\thinspace M_{\odot}\thinspace$$\times 2$ and 1.06$\rm\thinspace
M_{\odot}\thinspace$$\times 2$ simulations were essentially similar to the
0.81$\rm\thinspace M_{\odot}\thinspace$$\times 2$ simulations with the
exception that the greater the mass of the constituent stars, the less
sensitive the $\mathrm{{}^{56}Ni}$ yield was to impact parameter. Indeed, both
mass pairs 6 and 7 produced almost the same yield in all three tested
collision scenarios.
What distinguishes the 1.06$\rm\thinspace M_{\odot}\thinspace$$\times 2$ mass
pair from all the others attempted is that the $\mathrm{{}^{56}Ni}$ yield is
super-Chandrasekhar in all cases. Were such explosions observed, there would
be no doubt that a double-degenerate progenitor scenario of some kind was
responsible. The resulting 1.71$\rm\thinspace M_{\odot}\thinspace$of
$\mathrm{{}^{56}Ni}$ from the 1.06$\rm\thinspace M_{\odot}\thinspace$$\times
2$ simulations appear strikingly similar to the 1.7$\rm\thinspace
M_{\odot}\thinspace$of $\mathrm{{}^{56}Ni}$ derived from the observations of
Scalzo et al. (2010).
### 3.7 Mass Pair 8 - 0.50$\hbox{$\rm\thinspace M_{\odot}\thinspace$}\times
2$
Finally, we studied symmetric collisions of low-mass, 0.50$\rm\thinspace
M_{\odot}\thinspace$white dwarfs. Table 2 demonstrates that the $b=0$
collision scenario for this mass pairing produced less than
$0.01\hbox{$\rm\thinspace M_{\odot}\thinspace$}$ of $\mathrm{{}^{56}Ni}$
despite having resulted in a detonation. In this case, the energy generated
from even mild carbon-burning was sufficient to unbind the stars. As in the
0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$, $b=0$ scenario, the lower
velocities with which the stars collide results in a late detonation. The
shocked region slowly heats up until carbon-burning at its edges ignites a
detonation.
It is clear from the velocity profile of the most abundant isotopes from this
collision, given in Figure 18, that carbon and oxygen remain mostly unburned
in this scenario. This seems to suggest that collisions of low-mass white
dwarfs ($M\apprle 0.6\hbox{$\rm\thinspace M_{\odot}\thinspace$}$) of the CO
variety would not produce observable transients. Other simulations introducing
impact parameters with this mass pair were not attempted with carbon-oxygen
white dwarfs as the $b=0$ simulation yielded essentially a non-result.
However, further investigation involving Helium white dwarfs is warranted.
5.25
Figure 18: Masses of several isotopes at logarithmically spaced velocity bins
for the $b=0$ case of mass pairing 8, 0.50$\rm\thinspace
M_{\odot}\thinspace$$\times 2$.
## 4 Results & Analysis II - Remnants
As seen in Raskin et al. (2009), the $b=2$ case of mass pair 1
(0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$) did not feature a
detonation and instead formed a hot remnant of thermally-supported carbon and
oxygen with some carbon-burning products. Figure 19 illustrates the dynamics
of this collision, starting with a glancing case that leads to the constituent
stars spinning off from each other before coalescing into a single hot object.
Figure 19: Snapshots of density isosurfaces at six different times for the
$b=2$ simulation of mass pair 1, 0.64$\rm\thinspace
M_{\odot}\thinspace$$\times 2$. After first colliding, the stars separate
before coalescing into a single object.
The compact remnant core after 100s featured a nearly constant density of
$\rho\sim 10^{6}$ g cm-3. It was surrounded by a thick, Keplerian disk
$\approx 2.0\times 10^{10}$ cm in radius with a scale radius $r_{0}\approx
2.3\times 10^{9}$ cm. The compact object at the center of the disk is not
strictly a white dwarf since much of its pressure support is thermal
($T\approx 5\times 10^{8}$ K). Indeed, since degeneracy pressure support
necessitates that more massive white dwarfs are smaller than less massive
ones, this object, at $\approx 0.8$$\rm\thinspace M_{\odot}\thinspace$is far
too large to be wholly degenerate ($r_{rem}\approx 2.5\times 10^{9}$ cm);
larger than the 0.64$\rm\thinspace M_{\odot}\thinspace$white dwarfs that
entered into the collision ($r_{0.64}=6.98\times 10^{8}$ cm).
Carbon ignition nominally takes place at approximately 7-8$\times$108 K (e.g.
Gasques et al. 2007), but recent phenomenological models (e.g. Jiang et al.
2007) have suggested a strongly reduced, low-energy astrophysical S-factors
for carbon fusion reactions that potentially reduce carbon ignition
temperatures to $\approx$ 3$\times$108 K, especially at densities of 109 g
cm-3. A lower carbon burning threshold would be of interest to future studies
of collision remnants.
Since the system started in a bound state, ($T\lesssim-V$, where $T$ in this
case is total kinetic energy and $V$ is total gravitational potential energy)
and since any energy gained from nuclear processes is negligible, most of the
material cannot escape the system and the disk remains bound to the compact
core. It will eventually cool and collapse onto the surface of the compact
object. However, the hot core may accelerate parts of the disk to escape
velocity via radiative processes, and so the calculation of the final mass of
the resultant white dwarf is beyond the scope of this paper. Suffice it to
say, the final mass will not exceed the Chandrasekhar limit as only
1.28$\rm\thinspace M_{\odot}\thinspace$of material is available.
For the $b=2$ case of mass pair 2, 0.64$\rm\thinspace M_{\odot}\thinspace$\+
0.81$\rm\thinspace M_{\odot}\thinspace$, the compact remnant was slightly less
massive at $\approx 0.75$$\rm\thinspace M_{\odot}\thinspace$. However, since
the total mass of the system is super-Chandrasekhar, the final remnant mass
may result in a super-Chandrasekhar white dwarf. Again, this final mass will
depend greatly on radiative processes, and the likelihood of producing a SNIa
will hinge on the accretion rate of the disk onto the core.
The simulations of mass pair 3, 0.64$\rm\thinspace M_{\odot}\thinspace$\+
1.06$\rm\thinspace M_{\odot}\thinspace$, resulted in remnants in both the
$b=1$ and $b=2$ cases as the 1.06$\rm\thinspace M_{\odot}\thinspace$white
dwarf was too compact for the star to be much affected by a grazing collision
with a 0.64$\rm\thinspace M_{\odot}\thinspace$white dwarf. In the $b=1$ case,
some of the atmosphere of the 1.06$\rm\thinspace M_{\odot}\thinspace$star was
stripped away to join the material from the disrupted 0.64$\rm\thinspace
M_{\odot}\thinspace$star in the disk, while in the $b=2$ case, the
1.06$\rm\thinspace M_{\odot}\thinspace$star was nearly unaffected by the
collision. Figure 20 illustrates that the remnant core of the $b=2$ simulation
masses $\approx 1.0\hbox{$\rm\thinspace M_{\odot}\thinspace$}$, and the
central densities are essentially unchanged from the 1.06$\rm\thinspace
M_{\odot}\thinspace$progenitor.
Figure 20: Temperature and density profiles of the $b=1$ and $b=2$ simulation
remnants of mass pair 3, $0.64\hbox{$\rm\thinspace
M_{\odot}\thinspace$}+1.06\hbox{$\rm\thinspace M_{\odot}\thinspace$}$.
Some features worth noting in the $b=2$ profiles in Figure 20 are the
indications of a cold core ($T_{\rm core}\approx 2\times 10^{8}$ K) surrounded
by a hot envelope ($T_{\rm env.}\approx 9\times 10^{8}$ K), and the presence
of a strong overdensity in a part of the disk, which causes large spreads in
density and temperature for the disk material. This suggests that while the
0.64$\rm\thinspace M_{\odot}\thinspace$star is no longer a gravitationally
bound object, it has nevertheless not been completely disrupted.
The radial profiles of the $b=2$ simulation of the 0.81$\rm\thinspace
M_{\odot}\thinspace$\+ 1.06$\rm\thinspace M_{\odot}\thinspace$mass pair, shown
in Figure 21, indicate a very similar structure, with a high-temperature
envelope surrounding a cold core and an overdensity in a part of the disk.
These inhomogeneities in the disk would most likely vanish after many
crossing-times, however, it is highly unlikely that radiative processes or
collisional excitations within the disk could remove as much as
0.4$\rm\thinspace M_{\odot}\thinspace$. This suggests that after a Kelvin-
Helmholtz timescale, the remnant from the $b=2$ collision of mass pair 5 will
almost certainly be super-Chandrasekhar.
Figure 21: Temperature and density profiles of the $b=2$ simulation remnant of
mass pair 5, $0.81\hbox{$\rm\thinspace
M_{\odot}\thinspace$}+1.06\hbox{$\rm\thinspace M_{\odot}\thinspace$}$.
Some of the properties of these remnants are very similar to the simulated
remnant explored in Yoon et al. (2007), which was produced by the merger of
0.9$\rm\thinspace M_{\odot}\thinspace$and 0.6$\rm\thinspace
M_{\odot}\thinspace$white dwarfs. In that simulation, the remnant also
featured a cold core ($T_{\rm core}\approx 1\times 10^{8}$ K) surrounded by a
hot envelope ($T_{\rm env.}\approx 6\times 10^{8}$ K), embedded inside a
thick, Keplerian disk. Using a 1D stellar evolution code, they found that such
systems can indeed evolve on timescales $\sim 10^{5}$yr toward becoming SNeIa.
## 5 Discussion
White dwarf collisions are not typically regarded as SNeIa progenitors, and
therefore, they have been relatively unexplored theoretically. Here we have
conducted a comprehensive suite of simulations of such collisions examining
the dependence of their $\mathrm{{}^{56}Ni}$ yield on total mass, mass ratio,
and impact parameter. Our results suggest that white dwarf collisions are a
viable avenue for producing SNeIa with brightnesses that range from sub-
luminous to super-luminous.
In fact, in more than 75% of our simulations, collisions resulted in
detonations, and in all but the least massive combination of stars,
significant amounts of $\mathrm{{}^{56}Ni}$ were produced. We found that even
mass pairs that are below the Chandraskehar limit featured explosive nuclear
burning, with the 0.64$\rm\thinspace M_{\odot}\thinspace$$\times 2$ mass pair
producing $\mathrm{{}^{56}Ni}$ in quantities comparable to standard SNeIa.
Moreover, the most massive combinations of stars produced super-luminous
quantities of $\mathrm{{}^{56}Ni}$, regardless of the impact parameter,
greatly increasing their likelihood of detection. The $\mathrm{{}^{56}Ni}$
yields from these collisions are consistent with those of observed SNeIa with
super-Chandrasekhar mass progenitors.
Asymmetric mass pairs generally produced less $\mathrm{{}^{56}Ni}$ in head-on
collisions than symmetric pairs. At middling impact parameters, much more
$\mathrm{{}^{56}Ni}$ was produced, however, at high impact parameters, there
was little or none. This is due primarily to the delicate balance which must
be struck between the dynamics of the impact and the binding energy of the
less massive star in order to establish a stalled shock region that can lead
to a detonation. At high impact parameters, the less massive star is typically
unbound by the collision before much or any $\mathrm{{}^{56}Ni}$ is produced.
For combinations of masses and impact parameters that did not detonate, the
end result always featured a compact, semi-degenerate object surrounded by a
bound, thick disk of carbon and oxygen. Many of these systems were super-
Chandrasekhar, and over Kelvin-Helmholtz time scales, these, too, are
candidate progenitors for producing SNeIa.
Our results have shown that $\mathrm{{}^{56}Ni}$ production in white dwarf
collisions is a non-linear process that depends on several factors, including
infall velocities and tidal distortion effects. Foremost among parameters to
be explored in future studies is the composition of the constituent white
dwarfs. Helium has a much lower activation energy than carbon or oxygen, and
combinations of stars that include helium white dwarfs would almost certainly
produce interesting and different results. Other avenues to be explored
include the impact of more detailed modeling of the isotopic profiles in the
progenitor stars, and the possibility of sparse hydrogen atmospheres. The
results of these studies will shed further light on the contribution of
double-degenerate collisions to the observed population of SNeIa.
## Acknowledgments
This work was supported by the National Science Foundation under grant AST
08-06720, by the National Aeronautics and Space Administration under NESSF
grant PVS0401, and by a grant from the Arizona State University chapter of the
GPSA. All simulations were conducted at the Ira A. Fulton High Performance
Computing Center at Arizona State University. We thank James Rhoads and Sumner
Starrfield for insightful discussions and our anonymous referee for useful
suggestions and feedback.
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|
arxiv-papers
| 2010-09-13T20:01:59 |
2024-09-04T02:49:12.882767
|
{
"license": "Public Domain",
"authors": "Cody Raskin, Evan Scannapieco, Gabriel Rockefeller, Chris Fryer,\n Steven Diehl, F.X. Timmes",
"submitter": "Cody Raskin",
"url": "https://arxiv.org/abs/1009.2507"
}
|
1009.2536
|
# Maximally efficient quantum thermal machines: The basic principles
Sandu Popescu H. H. Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, United Kingdom
###### Abstract
Following the result by Skrzypczyk et al., arXiv:1009.0865, that certain self-
contained quantum thermal machines can reach Carnot efficiency, we discuss the
functioning of self-contained quantum thermal machines and show, in a very
general case, that they can reach the Carnot efficiency limit. Most
importantly, the full analytical solution for the functioning of the machines
is not required; the efficiency can be deduced from a very small number of
fundamental and highly intuitive equations which capture the core of the
problem.
In a very recent work lps two fundamental questions were raised about thermal
machines. The first question was whether or not there exists a fundamental
limitation to the size of (quantum) thermal machines (where size is measured
in the number of quantum states the machine). The second question refers to
the efficiency of the machines: is there any complementarity between size and
efficiency? That is, can the Carnot efficiency be reached by machines with
only very few quantum states? The first question was answered in lps where
the smallest refrigerator was designed: there is effectively no lower limit on
the number of states. The second question was answered in carnot where it was
shown that there is no trade-off between size and efficiency and that the
smallest possible refrigerator can reach the Carnot limit.
The results in carnot however are based on rather complicated computations,
involving solving for the exact analytical solution. All these computation
mysteriously simplify in the end. Here we revisit the problem and show that
finding the entire analytical solution (which depends on all the parameters of
the problem and on the details of the interaction with the environment) is not
necessary. Instead we formulate main principles that govern the functioning of
our quantum thermal machines; these principles capture the core of the problem
and lead to the Carnot efficiency in a clear, straightforward and very
intuitive manner.
Obviously, the above results do not come into a vacuum. During the last couple
of years there has been a lot of interest in the functioning of quantum
thermal machines I1 -Bard01 , with major results being obtained. Here however
we are specifically interested in accounting for all the degrees of freedom of
the machine, for all its states. Hence we are considering fully self-contained
machines and we do not allow, explicitly or implicitly, for any external
source of work. In particular, we do not allow for time dependent Hamiltonians
or prescribed unitary transformations. All that our machines are allowed is
access to heat baths.
## I A quantum self-contained refrigerator
The Model The refrigerator we consider consists of three qubits, 1,2 and 3,
each in contact with a different heat bath $T_{1}>T_{2}>T_{3}$. Qubit 1 is in
contact with the ”hot” bath $T_{1}$, qubit 2 is in contact with a ”room
temperature” bath, $T_{2}$, and qubit 3 is in contact with the ”cold” bath
$T_{3}$. The refrigerator works by taking heat from the cold bath and dumping
it at the room temperature. Free energy to run the fridge is provided by the
hot bath in conjunction with the room temperature one: energy flows from the
hot bath into the room temperature one.
The defining characteristics of the refrigerator are:
* •
The free hamiltonians of each qubit are chosen such that the energy
differences $E_{i}$ between the excited state $|1\rangle_{i}$ and ground state
$|0\rangle_{i}$ of each qubit are chosen such that
$E_{1}+E_{3}=E_{2}.$ (1)
Hence the states $|010\rangle$ and $|101\rangle$ are degenerate. (Here we used
the simplified notation $|0\rangle_{1}|1\rangle_{2}|0\rangle_{3}=|010\rangle$
and so on.)
* •
The interaction Hamiltonian is weak relative to the free Hamiltonian. Apart
from this, the interaction hamiltonian is completely arbitrary.
Given that the interaction Hamiltonian is weak relative to the free one, we
can take with a very good approximation the free Hamiltonian to define the
total energy of each qubit. Furthermore,since the interaction Hamiltonian is
weak, all it can do is to produce transitions between the two degenerate
energy levels $|010\rangle$ and $|101\rangle$. Even if the interaction
Hamiltonian has terms that couple different states, in the weak coupling
regime all other transitions are suppressed.
We note however, that even though the interaction Hamiltonian is weak, this
doesn’t mean that the state of the system is necessarily approximately equal
to the state in the absence of interaction. In fact the state can be
significantly different; it all depends on the exact parameters describing the
interaction of the qubits with the heat baths. A particular example is
explicitly calculated in lps .
Working regime The whole idea of the refrigerator is to try and cool spin 3 to
a temperature lower than that of the cold bath with which it is in contact.
When this is done, it will draw heat from the cold bath.
Cooling spin 3 is tantamount to increasing the probability to find it in the
ground state. Hence all we have to do is to try and enhance the probability of
the transition $|101\rangle\rightarrow|010\rangle$ (in which qubit 3 goes from
the excited state to the ground) over the probability of the reverse
transition $|010\rangle\rightarrow|101\rangle$. This is done by arranging that
the probability to find the system in the state $|101\rangle$ in the absence
of interaction is larger than the probability to find the system in state
$|010\rangle$, i.e. when
$e^{-{{E_{1}}\over{kT_{1}}}}e^{-{{E_{3}}\over{kT_{3}}}}>e^{-{{E_{2}}\over{kT_{2}}}}$
(2)
which leads to
${{E_{1}}\over{T_{1}}}+{{E_{3}}\over{T_{3}}}<{{E_{2}}\over{T_{2}}}.$ (3)
Equation (3) defines thus the working regime.
We note that equations (2) and (3) refer to the populations of the different
levels in the absence of interaction. When the interaction is turned on, the
populations change. In particular, the actual temperatures of the three qubits
will be different from the ones of their surroundings. One may therefore
wander why it is the population in the absence of interaction that defines the
working regime and not what happens when the interaction is on. The reason is
that if the initial populations are as stated, when the interaction is turned
on, the refrigerator starts cooling. Qubit 3 will become colder than the cold
bath, so draws heat from it, qubit 2 becomes warmer than room temperature
(since it is pushed into the excited state) so it will dump heat into the room
and qubit 1 will become cooler than the hot bath, so it extracts heat from it,
to keep the refrigerator working. Due to the interaction qubits 3 and 1 will
continue to cool while qubit 2 will continue to warm-up until, after a
transient period, they will each stabilize at a working temperature which
depend on the parameters of the problem. Whatever the details of these final
temperatures are, they are however in the right order relative to their
environments so that the system works as a refrigerator.
Heat flow The qubits exchange energy with their environment as well as with
each other. What interests us here is the energy that is exchanged between
them - at equilibrium the total energy of each qubit is constant, so the
energy a qubit extracts from the environment is equal to the energy passed to
the other qubits. Due to the fact that the only thing the interaction does is
to make transitions between the states $|101\rangle$ and $|010\rangle$, the
energy gains and loses of the qubits due to interaction are constraint:
Whenever qubit 1 losses energy $E_{1}$, qubit 3 must lose energy $E_{3}$ and
qubit 2 must gain energy $E_{2}$ and vice-versa, when qubit 1 gains $E_{1}$,
qubit 3 gains $E_{3}$ and qubit 2 loses $E_{2}$. How often a forward or a
backward transition between $|101\rangle$ and $|010\rangle$ occurs again
depends on the parameters of the problem: the exact interaction Hamiltonian,
the strength of coupling of each qubit with its thermal bath, the exact model
of the interaction with the bath. Hence, without these details we cannot tell
what the heat exchange rates are. That is, we cannot tell how much heat is
extracted or dumped into the baths per unit time.
However, and this is the key element of the entire argument, due to the
constraint on the energy exchanges between the qubits, the ratio of the heat
exchanges with the baths is independent from all the details and it is simply
the same as the ratio of the energy exchanges between the qubits, which is the
ratio between the energy levels:
$Q_{1}:Q_{2}:Q_{3}=E_{1}:E_{2}:E_{3}$ (4)
Reversibility The main question raised in this paper is that of the maximum
efficiency of the refrigerator. As in all thermal machines, maximal efficiency
is obtained at reversibility. For this we must ensure that the forward
transition, that results in cooling qubit 3, is infinitesimally close to the
reverse transition, that warms qubit 3. Hence the reversibility condition
means that the $>$ sign in equations (2) and (3) has to be replaced by
equality. That is, at reversibility
${{E_{1}}\over{T_{1}}}+{{E_{3}}\over{T_{3}}}={{E_{2}}\over{T_{2}}}.$ (5)
Incidentally we note that just running the refrigerator more slowly by simply
making the interaction Hamiltonian weaker is not enough. This will not ensure
reversibility. The actual design of the fridge (i.e. the energy levels) has to
be matched to the working temperatures, as in (5). This is similar to the case
of macroscopic devices. Indeed, consider a refrigerator consisting of a
cylinder with a piston and containing a gas. The refrigerator extracts heat
from a cold bath at temperature $T_{1}$ when we expand the gas by pulling out
the piston. The cylinder is then disconnected from the cold bath and isolated
thermally. The gas is then compressed until achieves temperature $T_{2}$ equal
to that of the room and it is then put in thermal contact with the room and it
is further compressed slowly, releasing heat into the room. The movement of
the piston during the entire process has to be slow, as not to produce
pressure and heat waves inside the gas, but this is not enough. We must also
ensure that the point at which the gas is compressed while the cylinder is
thermally isolated is precisely engineered to correspond to the gas reaching
room temperature.
## II Carnot efficiency for the refrigerator
The upshot of the above arguments is that as far as the question of efficiency
in the reversible regime is concerned, most of the details of the refrigerator
are irrelevant and it all reduces to three simple equations: Equation (1) that
describes the basic built of the fridge, equation (4) that describes the
connection between the heat flows and the basic construction of the
refrigerator and equation (5) that describes the reversible regime.
From (1) and (5) we obtain
${{E_{1}}\over{T_{1}}}+{{E_{3}}\over{T_{3}}}={{E_{1}+E_{3}}\over{T_{2}}}$ (6)
which then, using (4) leads to
${{Q_{1}}\over{T_{1}}}+{{Q_{3}}\over{T_{3}}}={{Q_{1}+Q_{3}}\over{T_{2}}}$ (7)
which is the relation that connects $Q_{3}$, the heat extracted by the fridge
from the cold bath and $Q_{1}$ the heat extracted from the hot bath that
drives the refrigerator. This expression is identical to that of a classical
reversible refrigerator that works between these temperatures (see lps ).
## III A quantum self-contained heat engine
In heat_engine a model for a quantum self-contained heat engine was proposed.
The engine consists by two qubits, 1 and 2, in contact with a hot bath,
$T_{1}$ and a cold bath, $T_{2}<T_{1}$. The energy separation between the
ground state $|0\rangle_{i}$ and the excited state $|1\rangle_{i}$ of qubit
$i$ is $E_{i}$. The engine lifts a weight in equal steps; the energy
difference between two subsequent positions, $|n\rangle_{w}$ and
$|n+1\rangle_{w}$ is $E_{3}$. The weight is not connected to any heat bath.
The engine is constructed such that
$E_{1}=E_{2}+E_{3}$ (8)
.
Due to the above constraint, the states $|10n\rangle$ and $|01n+1\rangle$ are
degenerate where $|10n\rangle$ stands for
$|1\rangle_{1}|0\rangle_{2}|n\rangle_{w}$ and so on. Again, an interaction
Hamiltonian is added, of magnitude smaller than that of the free hamiltonians.
Due to this, it can only induce transitions between the degenerate
eigenstates.
Again, the basic idea of the engine is to make the transition
$|10n\rangle\rightarrow|01n+1\rangle$ in which the weight is lifted more
favorable than the reverse transition. The condition for this is
$e^{-{{E_{1}}\over{kT_{1}}}}>e^{-{{E_{2}}\over{kT_{2}}}}$ (9)
which leads to
${{E_{1}}\over{T_{1}}}<{{E_{2}}\over{T_{2}}}.$ (10)
As in the case of the refrigerator, the reversibility condition is when the
froward and backward transitions are equally probable, i.e.
${{E_{1}}\over{T_{1}}}={{E_{2}}\over{T_{2}}}.$ (11)
Finally, the interaction imposes the constraint that whenever the wight is
lifted and it gains energy $E_{3}$, qubit 2 gains energy $E_{2}$ and qubit 1
loses energy $E_{1}$ and vice versa. While we cannot tell anything about the
time scales involved without more information about the parameters of the
device, it is clear that the ratio between the heat $Q_{1}$ extracted by qubit
1 from the hot bath, the heat $Q_{2}$ dumped by qubit 2 into the cold bath and
the work gained by the weight are in the same ratio as the energies gained and
lost in one transition:
$Q_{1}:Q_{2}:W=E_{1}:E_{2}:E_{3}$ (12)
Putting all this together we obtain
${{Q_{1}}\over{T_{1}}}={{Q_{1}-W}\over{T_{2}}}$ (13)
which can be arranged to read
${W\over{Q_{1}}}=1-{{T_{2}}\over{T_{1}}}$ (14)
the well known expression for the Carnot efficiency of a simple classical heat
engine.
## IV Conclusion
To conclude, we presented the basic principles for the functioning of quantum
self-contained heat machines. While to find out the exact working parameters
outside of the reversible regime is complicated and depends on all the
parameters of the problem, all these details become irrelevant in the
reversible regime. The three simple and very intuitive equations are enough to
tell that the quantum engines reach ideal Carnot efficiency
## V Further considerations
Our general considerations above also allow us to immediately tell a couple of
things about the working point far from reversibility.
Suppose that the thermalisation model is the one in lps in which with
probability $p_{i}$ per unit time the state of the qubit gets replaced by the
thermal state $\tau_{i}$. Furthermore suppose that the steady state is also
diagonal in the same basis, $|0\rangle_{i}$, $|1\rangle_{i}$. The heat
extracted per unit time is $p_{i}$ times the energy gained/lost when a
replacement is made. This energy is equal to the difference of the mean
energies in the steady state $\rho^{S}_{i}$ and $\tau_{i}$; which is equal to
$E_{i}$ times $\delta q_{i}$ the change in probability for the excited state.
Hence
$Q_{i}=p_{i}E_{i}\delta q_{i}$ (15)
But taking into account that the ratio of heat flows is the same as the ratio
of energies (4) we obtain that
$p_{1}\delta q_{1}=p_{2}\delta q_{2}=p_{3}\delta q_{3}$ (16)
which is confirmed by the explicit form deduced in lps .
## References
* (1) N. Linden, S. Popescu and P. Skrzypczyk
* (2) P. Skrzypczyk, N. Brunner, N. Linden and S. Popescu, arXiv:1009.0865
* (3) S. Popescu et al. in preparation.
* (4) R. Landauer, IBM J. Res. Develop. 5, 183 (1961).
* (5) C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).
* (6) G. Gemma, M. Michel and G. Mahler, Quantum Thermodynamics, Springer (2004).
* (7) E. P. Gyftopoulos and G. P. Beretta, Thermodynamics: Foundations and Applications, Dover (2005).
* (8) F. Tonner and G. Mahler, Phys. Rev. E 72, 066118 (2005).
* (9) M. J. Henrich, M. Michel and G. Mahler, Europhys. Lett. 76, 1057 (2006).
* (10) F. Rempp, M. Michel and G. Mahler, Phys. Rev. A 76, 032325 (2007).
* (11) M. J. Henrich, G. Mahler and M. Michel, Phys. Rev. E 75, 051118 (2007).
* (12) D. Janzing et. al., Int J. Th. Phys. 39, 2717 (2000).
* (13) A. E. Allehverdyan and Th. M. Nieuwenhuizen, Phys. Rev. Lett. 85, 1779 (2000).
* (14) E. Geva and R. Kosloff, J. Chem. Phys. 96, 3054 (1992).
* (15) E. Geva and R. Kosloff, J. Chem. Phys. 104, 7681 (1996).
* (16) T. Feldmann and R. Kosloff, Phys. Rev. E 61, 4774 (2000).
* (17) J. P. Palao, R. Kosloff and J. M. Gordon, Phys. Rev. E 64, 056130 (2001).
* (18) D. Segal and A. Nitzan, Phys. Rev. E 73, 026109 (2006).
* (19) T. Feldmann, E. Geva and P. Salamon, Am. J. Phys. 64, 485 (1996).
* (20) T. Feldmann and R. Kosloff, Phys. Rev. E 68, 016101 (2003).
* (21) H. T. Quan et. al., Phys. Rev. E 76, 031105 (2007).
* (22) T. Feldmann and R. Kosloff, arXiv:quant-ph/0906.0986.
* (23) G. P. Beretta, arXiv:quant-ph/0703.3261.
* (24) For a review see: F. Bardou et. al., Lévy statistics and laser cooling, Cambridge University Press (2001).
|
arxiv-papers
| 2010-09-13T22:56:48 |
2024-09-04T02:49:12.892655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sandu Popescu",
"submitter": "Sandu Popescu",
"url": "https://arxiv.org/abs/1009.2536"
}
|
1009.2548
|
# New 3-mode squeezing operator and squeezed vacuum state in 3-wave mixing
††thanks: Work was supported by the National Natural Science Foundation of
China under grants 10775097 and the Key Programs Foundation of Ministry of
Education of China (No. 210115).
Xue-xiang Xu1,2, Hong-yi Fan1, Li-yun Hu2,and Hong-chun Yuan1
1Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China
2College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China E-mail:hlyun2008@126.com.
###### Abstract
In a 3-wave mixing process occurring in some nonlinear optical medium when
$a_{1}^{\dagger}$mode interacts with both $a_{2}^{\dagger}$mode and
$a_{3}^{\dagger}$mode, we theoretically study the squeezing effect generated
by the operator
$S_{3}\equiv\exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})]$.
The new 3-mode squeezed vacuum state in Fock space is derived, and the
uncertainty relation for it is demonstrated, It turns out that $S_{3}$may
exhibit enhanced squeezing. By virtue of the technique of integration within
an ordered product (IWOP) of operators, we also derive $S_{3}$’s normally
ordered expansion. The Wigner function of new 3-mode squeezed vacuum state is
calculated by using the Weyl ordering invariance under similar
transformations.
PACS 42.50.-p – Quantum optics
PACS 03.65.-w – Quantum mechanics
## 1 Introduction
Nowadays quantum entanglement is the focus of quantum information research and
attracts many interests due to its wide applications in quantum communication
[1, 2]. Entangled states have brought much attention and interests of
physicists [3, 4]. The usual two-mode squeezed state, generated from a
parametric amplifier [5], not only exhibits squeezing, but also quantum
entanglement between the idle-mode and the signal-mode in frequency domain.
Therefore, it is simultaneously a typical entangled state of continuous
variable. Theoretically, the two-mode squeezed state is constructed by acting
a two-mode squeezing operator
$S_{2}=\exp[\lambda(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})]$ [6, 7] on the
two-mode vacuum state $\left|00\right\rangle$, i.e.
$S_{2}\left|00\right\rangle=$sech$\lambda\exp[-a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda]\left|00\right\rangle$,
where $\lambda\ $is a squeezing parameter, and $a_{i}$($a_{j}^{\dagger}$) Bose
annihilation (creation) operator satisfying
$[a_{i},a_{j}^{\dagger}]=\delta_{ij}$. Using the relation between Bose
operators ($a_{i},a_{i}^{\dagger}$) and the coordinate and momentum operators
$Q_{i}=\frac{a_{i}+a_{i}^{\dagger}}{\sqrt{2}},\
P_{i}=\frac{a_{i}-a_{i}^{\dagger}}{\sqrt{2}\mathtt{i}},$ (1)
one can recast $S_{2}$ into the form
$S_{2}=\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\right],$
(2)
noting
$\begin{array}[c]{c}\left[Q_{1}P_{2},Q_{2}P_{1}\right]=\mathtt{i}\left(Q_{2}P_{2}-Q_{1}P_{1}\right),\\\
\left[Q_{1}P_{2},\mathtt{i}\left(Q_{2}P_{2}-Q_{1}P_{1}\right)\right]=2Q_{1}P_{2},\\\
\left[Q_{2}P_{1},\mathtt{i}\left(Q_{2}P_{2}-Q_{1}P_{1}\right)\right]=-2Q_{2}P_{1},\end{array}$
(3)
thus there involves a $SU(1,1)$ algebraic stricture. In the state
$S_{2}\left|00\right\rangle$, the variances of the two-mode quadrature
operators of light field,
$\mathfrak{X}=\frac{Q_{1}+Q_{2}}{2},\text{
}\mathfrak{P}=\frac{P_{1}+P_{2}}{2},$ (4)
satisfying the commutation relation
$[\mathfrak{X},\mathfrak{P}]=\frac{\mathtt{i}}{2}$, exhibiting the standard
squeezing, i.e.,
$\left\langle
00\right|S_{2}^{\dagger}\mathfrak{X}^{2}S_{2}\left|00\right\rangle=\frac{1}{4}e^{-2\lambda},\left\langle
00\right|S_{2}^{\dagger}\mathfrak{P}^{2}S_{2}\left|00\right\rangle=\frac{1}{4}e^{2\lambda},$
(5)
which satisfy $(\Delta\mathfrak{X})(\Delta\mathfrak{P})=\frac{1}{4}$.
An interesting question naturally arises: if $a_{1}^{\dagger}$ mode in a
nonlinear optical medium, interacting with both $a_{2}^{\dagger}$ mode and
$a_{3}^{\dagger}$ mode (e.g., a three-wave mixing), and the corresponding
three-mode exponential operator is introduced as
$S_{3}\equiv\exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})].$
(6)
Using Eq.(1) we can recast $S_{3}$ into the form
$S_{3}\equiv\exp\left[\mathtt{i}\mu\left(Q_{2}P_{1}+Q_{1}P_{2}\right)+\mathtt{i}\nu\left(Q_{3}P_{1}+Q_{1}P_{3}\right)\right],$
(7)
where $\mu,$ $\nu$ are two different interaction parameters, then what is its
squeezing effect for the 3-mode quadratures of light field?
To answer this question we must know what is the state
$S_{3}\left|000\right\rangle$ ($\left|000\right\rangle$ is the 3-mode vacuum
state) in Fock space, for this aim, we should know what is the normally
ordered expansion of $S_{3}$. But how to disentangle the exponential operator
$S_{3}?$ Because there is no simple $SU(1,1)$ algebraic structure among
$Q_{2}P_{1},Q_{1}P_{2},Q_{3}P_{1}$ and $Q_{1}P_{3},$ the disentangling seems
hard. Thus we turn to appeal to Dirac’s coordinate representation and the
technique of integration within an ordered product (IWOP) of operators [8, 9,
10, 11] to solve this problem. Our work is arranged as follows: firstly we
derive the explicit form of$\ S_{3}\left|000\right\rangle,$ then we
demonstrate that it really satisfies the Heisenberg uncertainty relation and
may exhibit squeezing enhancement. We also employ the technique of integration
within an ordered product (IWOP) of operators to derive the normally ordered
expansion of $S_{3}$. The Wigner function of $S_{3}\left|000\right\rangle$ is
calculated by using the Weyl ordering invariance under similar transformations
[12, 13, 14].
## 2 New 3-mode squeezed vacuum state
For the sake of convenience, we rewrite $S_{3}$ in Eq.(7) as the following
compact form,
$S_{3}=\exp[\mathtt{i}Q_{i}\Lambda_{ij}P_{j}],i,j=1,2,3,$ (8)
where the repeated indices imply the Einstein summation notation, and
$\Lambda=\left(\begin{array}[c]{ccc}0&\mu&\nu\\\ \mu&0&0\\\
\nu&0&0\end{array}\right),$ (9)
thus
$e^{\Lambda}=\allowbreak\left(\begin{array}[c]{ccc}\cosh r&\cos\theta\sinh
r&\sin\theta\sinh r\\\ \cos\theta\sinh r&\sin^{2}\theta+\cos^{2}\theta\cosh
r&\frac{\sin 2\theta}{2}\left(\cosh r-1\right)\\\ \sin\theta\sinh r&\frac{\sin
2\theta}{2}\left(\cosh r-1\right)&\sin^{2}\theta\cosh
r+\cos^{2}\theta\end{array}\right),$ (10)
its inverse is
$e^{-\Lambda}\allowbreak=\allowbreak\left(\begin{array}[c]{ccc}\cosh
r&-\cos\theta\sinh r&-\sin\theta\sinh r\\\ -\cos\theta\sinh
r&\sin^{2}\theta+\cos^{2}\theta\cosh r&\frac{\sin 2\theta\left(\cosh
r-1\right)}{2}\\\ -\sin\theta\sinh r&\frac{\sin 2\theta\left(\cosh
r-1\right)}{2}&\sin^{2}\theta\cosh r+\cos^{2}\theta\end{array}\right),$ (11)
where we have set
$r=\sqrt{\mu^{2}+\nu^{2}},\cos\theta=\frac{\mu}{r},\sin\theta=\frac{\nu}{r},$
(12)
noting that $\Lambda$ is a symmetric matrix. Using the Baker-Hausdorff
formula,
$\displaystyle e^{A}Be^{-A}$
$\displaystyle=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]$
$\displaystyle+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots,$
(13)
we see that $S_{3}$ causes the following transformation
$S_{3}^{-1}Q_{k}S_{3}=(e^{-\Lambda})_{ki}Q_{i},\text{\
}S_{3}^{-1}P_{k}S_{3}=(e^{\Lambda})_{ki}P_{i}.$ (14)
It then follows that $S_{3}^{-1}a_{k}S_{3}=(e^{-\lambda\Lambda})_{ki}a_{i}$,
i.e.
$\displaystyle S_{3}^{-1}a_{1}S_{3}$ $\displaystyle=a_{1}\cosh
r-a_{2}^{{\dagger}}\cos\theta\sinh r-a_{3}^{{\dagger}}\sin\theta\sinh r,$
$\displaystyle S_{3}^{-1}a_{2}S_{3}$
$\displaystyle=-a_{1}^{{\dagger}}\cos\theta\sinh
r+a_{2}\left(\sin^{2}\theta+\cos^{2}\theta\cosh r\right)$
$\displaystyle+\frac{1}{2}a_{3}\left(\cosh r-1\right)\sin 2\theta,$ (15)
$\displaystyle S_{3}^{-1}a_{3}S_{3}$
$\displaystyle=-a_{1}^{{\dagger}}\sin\theta\sinh r+\frac{1}{2}a_{2}\left(\cosh
r-1\right)\sin 2\theta$ $\displaystyle+a_{3}\left(\sin^{2}\theta\cosh
r+\cos^{2}\theta\right).$
Noticing that $S_{3}^{\dagger}=S_{3}^{-1}$ and
$S_{3}^{\dagger}\left(\mu,\nu\right)=S_{3}\left(-\mu,-\nu\right)$, from
Eq.(15) we also have
$\displaystyle S_{3}a_{1}S_{3}^{-1}$ $\displaystyle=a_{1}\cosh
r+a_{2}^{{\dagger}}\cos\theta\sinh r+a_{3}^{{\dagger}}\sin\theta\sinh r,$
$\displaystyle S_{3}a_{2}S_{3}^{-1}$
$\displaystyle=a_{1}^{{\dagger}}\cos\theta\sinh
r+a_{2}\left(\sin^{2}\theta+\cos^{2}\theta\cosh r\right)$
$\displaystyle+\frac{1}{2}a_{3}\left(\cosh r-1\right)\sin 2\theta$ (16)
$\displaystyle S_{3}a_{3}S_{3}^{-1}$
$\displaystyle=a_{1}^{{\dagger}}\sin\theta\sinh r+\frac{1}{2}a_{2}\left(\cosh
r-1\right)\sin 2\theta$ $\displaystyle+a_{3}\left(\sin^{2}\theta\cosh
r+\cos^{2}\theta\right).$
For convenience to write, we set
$S_{3}\left|000\right\rangle=\left\|000\right\rangle$. In order to obtain the
explicit form of $\left\|000\right\rangle$, using Eq.(15) and
$a_{1}\left|000\right\rangle=0$, we operate $a_{1}$ on
$\left\|000\right\rangle$ and obtain
$\displaystyle a_{1}\left\|000\right\rangle$
$\displaystyle=S_{3}S_{3}^{-1}a_{1}S_{3}\left|000\right\rangle$
$\displaystyle=S_{3}(a_{1}\cosh r-a_{2}^{{\dagger}}\cos\theta\sinh
r-a_{3}^{{\dagger}}\sin\theta\sinh r)\left|000\right\rangle$
$\displaystyle=S_{3}(-a_{2}^{{\dagger}}\cos\theta\sinh
r-a_{3}^{{\dagger}}\sin\theta\sinh r)S_{3}^{-1}S_{3}\left|000\right\rangle$
$\displaystyle=-S_{3}(a_{2}^{{\dagger}}\cos\theta\sinh
r+a_{3}^{{\dagger}}\sin\theta\sinh r)S_{3}^{-1}\left\|000\right\rangle,$ (17)
then we continue to use Eq.(16) to derive
$\displaystyle a_{1}\left\|000\right\rangle$
$\displaystyle=-\\{[a_{1}\cos\theta\sinh
r+a_{2}^{\dagger}\left(\sin^{2}\theta+\cos^{2}\theta\cosh r\right)$
$\displaystyle+\frac{1}{2}a_{3}^{\dagger}\left(\cosh r-1\right)\sin
2\theta]\cos\theta\sinh r$ $\displaystyle+[a_{1}\sin\theta\sinh
r+\frac{1}{2}a_{2}^{\dagger}\left(\cosh r-1\right)\sin 2\theta$
$\displaystyle+a_{3}^{\dagger}\left(\sin^{2}\theta\cosh
r+\cos^{2}\theta\right)]\sin\theta\sinh r\\}\left\|000\right\rangle$
$\displaystyle=-(a_{1}\sinh^{2}r+a_{2}^{\dagger}\allowbreak\cos\theta\cosh
r\sinh r$ $\displaystyle+\frac{1}{2}a_{3}^{\dagger}\sin\theta\sinh
2r)\left\|000\right\rangle,$ (18)
so we reach the equation
$a_{1}\left\|000\right\rangle=-\tanh
r(a_{2}^{\dagger}\allowbreak\cos\theta+a_{3}^{\dagger}\sin\theta)\left\|000\right\rangle.$
(19)
Similarly, operating $a_{2}$ on $\left\|000\right\rangle$ and using Eqs.(15)
and (16) yields
$\displaystyle a_{2}\left\|000\right\rangle$
$\displaystyle=S_{3}S_{3}^{-1}a_{2}S_{3}\left|000\right\rangle=S_{3}(-a_{1}^{{\dagger}}\cos\theta\sinh
r)\left|000\right\rangle$
$\displaystyle=S_{3}(-a_{1}^{{\dagger}}\cos\theta\sinh
r)S_{3}^{-1}\left\|000\right\rangle$ $\displaystyle=-(a_{1}^{{\dagger}}\cosh
r+a_{2}\cos\theta\sinh r$ $\displaystyle+a_{3}\sin\theta\sinh
r)\cos\theta\sinh r\left\|000\right\rangle,$ (20)
which leads to
$\displaystyle[a_{2}\left(1+\cos^{2}\theta\sinh^{2}r\right)+\frac{1}{2}a_{3}\sin
2\theta\sinh^{2}r]\left\|000\right\rangle$
$\displaystyle=-\frac{1}{2}a_{1}^{{\dagger}}\cos\theta\sinh
2r\left\|000\right\rangle.$ (21)
On the other hand, operating $a_{3}$ on $\left\|000\right\rangle$ and using
Eqs.(15) and (16) yields
$\displaystyle a_{3}\left\|000\right\rangle$
$\displaystyle=S_{3}S_{3}^{-1}a_{3}S_{3}\left|000\right\rangle=S_{3}(-a_{1}^{{\dagger}}\sin\theta\sinh
r)\left|000\right\rangle$
$\displaystyle=S_{3}(-a_{1}^{{\dagger}}\sin\theta\sinh
r)S_{3}^{-1}\left\|000\right\rangle$ $\displaystyle=-(a_{1}^{{\dagger}}\cosh
r+a_{2}\cos\theta\sinh r$ $\displaystyle+a_{3}\sin\theta\sinh
r)\sin\theta\sinh r\left\|000\right\rangle,$ (22)
i.e.,
$\displaystyle[a_{3}\left(1+\sin^{2}\theta\sinh^{2}r\right)+\frac{1}{2}a_{2}\sin
2\theta\sinh^{2}r]\left\|000\right\rangle$
$\displaystyle=-\frac{1}{2}a_{1}^{{\dagger}}\sin\theta\sinh
2r\left\|000\right\rangle.$ (23)
Combining Eqs.(21) and (23) we have
$a_{2}\left\|000\right\rangle=-a_{1}^{{\dagger}}\tanh
r\cos\theta\left\|000\right\rangle,$ (24)
and
$a_{3}\left\|000\right\rangle=-a_{1}^{{\dagger}}\tanh
r\sin\theta\left\|000\right\rangle.$ (25)
From Eqs.(19),(24) and (25), we may predict that $\left\|000\right\rangle$ has
the following explicit form:
$\left\|000\right\rangle=N\exp[-(a_{2}^{{\dagger}}\cos\theta+a_{3}^{{\dagger}}\sin\theta)a_{1}^{{\dagger}}\tanh
r]\left|000\right\rangle,$ (26)
where $N$ is the normalization constant, which can be determined by
$\left\langle 000\right.\left\|000\right\rangle=1$, and we calculate
$N=\sec$h$r.$
## 3 Squeezing property and quantum fluctuation in $\left\|000\right\rangle$
Squeezing is an important phenomenon in quantum theory and has many
applications in various areas in quantum optics and quantum information [15].
In this section, we examine the quadrature squeezing effects of
$\left\|000\right\rangle$. The quadratures in the 3-mode case are defined as
$X_{1}=\frac{1}{\sqrt{6}}\sum_{i=1}^{3}Q_{i},\text{
}X_{2}=\frac{1}{\sqrt{6}}\sum_{i=1}^{3}P_{i},$ (27)
which satisfy the relation $[X_{1},X_{2}]=\frac{\mathtt{i}}{2}.$ Their
variances are $\left(\Delta X_{i}\right)^{2}=\left\langle
X_{i}^{2}\right\rangle-\left\langle X_{i}\right\rangle^{2}$, $i=1,2.$ Noting
the expectation values of $X_{1}$ and $X_{2}$ in the state
$\left\|000\right\rangle$ is $\left\langle X_{1}\right\rangle=\left\langle
X_{2}\right\rangle=0$. With the help of Eq.(15), we can calculate that the
corresponding variances in the state $\left\|000\right\rangle$: (noting
$\Lambda$ is symmetric)
$\displaystyle\left(\triangle X_{1}\right)^{2}$ $\displaystyle=\left\langle
000\right|S_{3}^{-1}X_{1}^{2}S_{3}\left|000\right\rangle$
$\displaystyle=\frac{1}{6}\sum_{i=1}^{3}\sum_{j=1}^{3}(e^{-\Lambda})_{ki}(e^{-\Lambda})_{jl}\left\langle
000\right|Q_{k}Q_{l}\left|000\right\rangle$
$\displaystyle=\frac{1}{12}\sum_{i=1}^{3}\sum_{j=1}^{3}(e^{-\Lambda})_{ki}(e^{-\Lambda})_{jl}\left\langle
000\right|a_{k}a_{l}^{\dagger}\left|000\right\rangle$
$\displaystyle=\frac{1}{12}\sum_{i=1}^{3}\sum_{j=1}^{3}(e^{-\Lambda})_{ki}(e^{-\Lambda})_{jl}\delta_{kl}$
$\displaystyle=\frac{1}{12}\underset{i,j}{\sum^{3}}(e^{-2\Lambda})_{ij},$ (28)
and
$\left(\triangle X_{2}\right)^{2}=\left\langle
000\right|S_{3}^{-1}X_{2}^{2}S_{3}\left|000\right\rangle=\frac{1}{12}\underset{i,j}{\sum^{3}}(e^{2\Lambda})_{ij}.$
(29)
The explicit form of the matrices $e^{2\Lambda}$ and $e^{-2\Lambda}$ can be
derived from Eq.(10) and (11), so we can obtain
$\displaystyle\left(\triangle X_{1}\right)^{2}$
$\displaystyle=\frac{1}{12}[(2\cosh 2r+1)+\sin 2\theta(\allowbreak\cosh 2r-1)$
$\displaystyle+2\left(\cos\theta+\sin\theta\right)\sinh 2r],$ (30)
and
$\displaystyle\left(\triangle X_{2}\right)^{2}$
$\displaystyle=\frac{1}{12}[(2\cosh 2r+1)+\sin 2\theta(\allowbreak\cosh 2r-1)$
$\displaystyle-2\left(\cos\theta+\sin\theta\right)\sinh 2r].$ (31)
We can successfully verify
$\displaystyle(\triangle X_{1})(\triangle X_{2})$
$\displaystyle=\frac{1}{12}\sqrt{\left(4\cosh
2r+4\right)+\left(1-2\sinh^{2}r\sin 2\theta\right)^{2}}$
$\displaystyle\geqslant\frac{1}{12}\sqrt{\left(4\cosh
2r+4\right)+\left(1-2\sinh^{2}r\right)^{2}}$
$\displaystyle=\frac{1}{12}\sqrt{\frac{1}{2}\cosh
4r+\frac{17}{2}}\geqslant\frac{1}{4},$ (32)
which confirms the uncertainty relation of quantum mechanics.
To see the trend of squeezing effects in the $X_{1}-$ or $X_{2}-$direction, we
plot $\left(\Delta X_{1}\right)^{2}$ and $\left(\Delta X_{2}\right)^{2}$ as
the function of parameter $\mu$ for different $\nu$ in Fig.1. When $\nu=0,$ it
exhibits the usual two mode squeezing effect depending on the varying $\mu$,
$\left(\Delta X_{1}\right)^{2}$ increases accompanying $\left(\Delta
X_{2}\right)^{2}$ decreases; when $\nu=0.5$, $\left(\Delta X_{1}\right)^{2}\
$increases more than the case of $\nu=0$, which exhibits enhanced squeezing in
certain domain of $\mu$. In Fig.2, we plot the uncertainty value $(\triangle
X_{1})(\triangle X_{2})$ as the function of $r$ for different $\theta$.
## 4 Normally ordered form of $S_{3}$
We calculate the normally ordered form (denoted by $\colon\colon$) of $S_{3}$
by inserting the completeness relation of coherent state
$S_{3}=\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{3}}{\pi^{3}}S_{3}\left|z_{1}z_{2}z_{3}\right\rangle\left\langle
z_{1}z_{2}z_{3}\right|.$ (33)
where $\left|z_{1}z_{2}z_{3}\right\rangle$ is the three-mode coherent state
and
$\left|z_{i}\right\rangle=\exp[-\frac{\left|z_{i}\right|^{2}}{2}+z_{i}a_{i}^{{\dagger}}]\left|0_{i}\right\rangle$,
$i=1,2,3$.
Using the relations in Eqs.(15) and (16), we have the explicit relation of
$S_{3}\left|z_{1}z_{2}z_{3}\right\rangle\allowbreak$
$\displaystyle S_{3}\left|z_{1}z_{2}z_{3}\right\rangle$
$\displaystyle=\exp(-\sum_{i=1}^{3}\frac{\left|z_{i}\right|^{2}}{2})S_{3}\exp\left(z_{1}a_{1}^{\dagger}+z_{2}a_{2}^{\dagger}+z_{3}a_{3}^{\dagger}\right)S_{3}^{-1}S_{3}\left|000\right\rangle$
$\displaystyle=\frac{1}{\cosh
r}\exp(-\sum_{i=1}\frac{\left|z_{i}\right|^{2}}{2}+z_{1}z_{2}\cos\theta\tanh
r+z_{1}z_{3}\sin\theta\tanh r)$
$\displaystyle\times\exp\allowbreak\\{\frac{1}{\cosh
r}[a_{1}^{{\dagger}}z_{1}+\allowbreak(a_{2}^{{\dagger}}\left(\sin^{2}\theta\cosh
r+\cos^{2}\theta\right)-\frac{1}{2}a_{3}^{{\dagger}}\left(\cosh r-1\right)\sin
2\theta)z_{2}$
$\displaystyle+(a_{3}^{{\dagger}}\left(\sin^{2}\theta+\cos^{2}\theta\cosh
r\right)-\frac{1}{2}a_{2}^{{\dagger}}\left(\cosh r-1\right)\sin
2\theta)z_{3}]\\}$ $\displaystyle\times\exp[-a_{1}^{{\dagger}}\tanh
r(a_{2}^{{\dagger}}\cos\theta+a_{3}^{{\dagger}}\sin\theta)]\left|000\right\rangle.$
(34)
Substituting Eq.(34) into Eq.(33), noticing that
$\left|000\right\rangle\left\langle
000\right|=\colon\exp(-a_{1}^{{\dagger}}a_{1}-a_{2}^{{\dagger}}a_{2}-a_{3}^{{\dagger}}a_{3})\colon$,
and using the following formula
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{ \ \
}\mathtt{Re}\left(\zeta\right)<0,$ (35)
as well as the IWOP technique, we can obtain the explicit normally ordered
expansion of $S_{3}$:
$\displaystyle S_{3}$ $\displaystyle=\frac{1}{\cosh
r}\exp[-a_{1}^{{\dagger}}(a_{2}^{{\dagger}}\cos\theta+a_{3}^{{\dagger}}\sin\theta)\tanh
r]$ $\displaystyle\times\colon\exp[\frac{1-\cosh r}{\cosh
r}(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\cos^{2}\theta$
$\displaystyle+a_{3}^{{\dagger}}a_{3}\sin^{2}\theta+\frac{1}{2}a_{2}a_{3}^{{\dagger}}\sin
2\theta+\frac{1}{2}a_{2}^{{\dagger}}a_{3}\sin 2\theta)]\colon$
$\displaystyle\times\exp[a_{1}(a_{2}\cos\theta+\allowbreak
a_{1}\sin\theta)\tanh r].$ (36)
## 5 Wigner function of $\left\|000\right\rangle$
Wigner distribution function of quantum states [16, 17, 18] is widely studied
in quantum statistics and quantum optics and is very important tool for a
global description of nonclassical effect in the quantum system, which can be
measured by various means such as photon counting experiment and homodyne
tomography. Now we derive the Wigner function of $\left\|000\right\rangle$ by
using a new method.
Recalling that in Ref. [14] we have introduced the Weyl ordering form of
single-mode Wigner operator $\Delta_{1}\left(q_{1},p_{1}\right)$,
$\Delta_{1}\left(q_{1},p_{1}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(37)
where the symbols$\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$
denote the Weyl ordering, while its normal ordering form is
$\Delta_{1}\left(q_{1},p_{1}\right)=\frac{1}{\pi}\colon\exp\left[-\left(q_{1}-Q_{1}\right)^{2}-\left(p_{1}-P_{1}\right)^{2}\right]\colon.$
(38)
Thus the Wigner function for $\left|0\right\rangle$ can be easily expressed as
$\left\langle
0\right|\Delta_{1}\left(q_{1},p_{1}\right)\left|0\right\rangle=\frac{1}{\pi}\exp(-q_{1}^{2}-p_{1}^{2})$.
Note that the order of Bose operators $a_{1}$ and $a_{1}^{\dagger}$ within a
normally ordered product (or a Weyl ordered product) can be permuted. That is
to say, even though $[a_{1},a_{1}^{\dagger}]=1$, we can have $\colon
a_{1}a_{1}^{\dagger}\colon=\colon a_{1}^{\dagger}a_{1}\colon$
and$\genfrac{}{}{0.0pt}{}{:}{:}a_{1}a_{1}^{\dagger}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}a_{1}^{\dagger}a_{1}\genfrac{}{}{0.0pt}{}{:}{:}.$
The Weyl ordering of operators has a remarkable property, i.e., the Weyl-
ordering invariance of operators under similar transformations, which means
$U\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}U^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}U\left(\circ\circ\circ\right)U^{-1}\genfrac{}{}{0.0pt}{}{:}{:},$
(39)
as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did
not exist when $U$ operates.
For 3-mode case, the Weyl ordering form of the Wigner operator is
$\Delta_{3}\left(\mathbf{q},\mathbf{p}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\mathbf{q}-\mathbf{Q}\right)\delta\left(\mathbf{p}-\mathbf{P}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(40)
where $\mathbf{Q}=(Q_{1},Q_{2},Q_{3})^{T}$,
$\mathbf{P}=(P_{1},P_{2},P_{3})^{T}$, $\mathbf{q}=(q_{1},q_{2},q_{3})^{T}$ and
$\mathbf{p}=(p_{1},p_{2},p_{3})^{T}$. Then according to the Weyl ordering
invariance under similar transformations and using Eq.(14), we have
$\displaystyle S_{3}^{-1}\Delta_{3}\left(\mathbf{q},\mathbf{p}\right)S_{3}$
$\displaystyle=S_{3}^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\mathbf{q}-\mathbf{Q}\right)\delta\left(\mathbf{p}-\mathbf{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}S_{3}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{k}-(e^{-\Lambda})_{ki}Q_{i}\right)\delta\left(p_{k}-(e^{\Lambda})_{ki}P_{i}\right)\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left((e^{\Lambda})_{ki}q_{i}-Q_{k}\right)\delta\left((e^{-\Lambda})_{ki}p_{i}-P_{k}\right)\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\mathbf{q}^{\prime}-\mathbf{Q}\right)\delta\left(\mathbf{p}^{\prime}-\mathbf{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=\Delta_{3}\left(\mathbf{q}^{\prime},\mathbf{p}^{\prime}\right),$
(41)
where $q_{k}^{\prime}=(e^{\Lambda})_{ki}q_{i},$
$p_{k}^{\prime}=(e^{-\Lambda})_{ki}p_{i}$. Thus the Wigner function of
$\left\|000\right\rangle$ is
$\displaystyle\left\langle
000\right|S_{3}^{-1}\Delta_{3}\left(\mathbf{q},\mathbf{p}\right)S_{3}\left|000\right\rangle$
$\displaystyle=\left\langle
000\right|\Delta_{3}\left(\mathbf{q}^{\prime},\mathbf{p}^{\prime}\right)\left|000\right\rangle$
$\displaystyle=\frac{1}{\pi^{3}}\exp\left(-\mathbf{q}^{T}e^{2\Lambda}\mathbf{q}-\mathbf{p}^{T}e^{-2\Lambda}\mathbf{p}\right),$
(42)
where $e^{2\Lambda}$ and $e^{-2\Lambda}$ are given by $e^{\Lambda}$ in Eq.(10)
and $e^{-\Lambda}$ in Eq.(11), respectively.
In summary, we have shown that the operator
$S_{3}\equiv\exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})]$
is a new 3-mode squeezed operator by calculating the quantum fluctuation for
3-mode quadratures. We have obtained the new 3-mode squeezed vacuum state and
derived the normally ordered expansion of $S_{3}$. The IWOP technique brings
convenience in our derivation.
## References
* [1] Nielsen M. A. and Chuang I. L., Quantum Computation and Quantum Information (Cambridge University Press) 2000.
* [2] Bouwmeester D. et al., The Physics of Quantum Information (Springer, Berlin) 2000.
* [3] Fan H.-y., and Yu G.-c. Phys. Rev. A 65 (2002) 033829.
* [4] Fan H.-y. and Klauder J. R., Phys. Rev. A, 49 (1994) 704.
* [5] Mandel L. and Wolf E., Optical Coherence and Quantum Optics (Cambridge:Cambridge University Press) 1995.
* [6] Loudon R. and Knight P. L., J. Mod. Opt., 34 (1987) 709.
* [7] Dodonov V. V., J. Opt. B: Quantum Semiclass. Opt., 4 (2002) R1.
* [8] Fan H.-y., Europhys. Lett., 23 (1993) 1.
* [9] Fan H.-y., Europhys. Lett., 17 (1992) 285; 19 (1992) 443.
* [10] Hu L.-y. and Fan H.-y.,Europhys. Lett., 85 (2009) 60001.
* [11] Fan H.-y., J. Opt. B: Quantum Semiclass. Opt., 5 (2003) R147.
* [12] Fan H.-y., J. Phys. A, 25 (1992) 3443; Fan H.-y. and Fan Y., Int. J. Mod. Phys. A, 17 (2002) 701.
* [13] Fan H.-y., Mod. Phys. Lett. A, 15 (2000) 2297.
* [14] Fan H.-y., Ann. Phys. (N.Y.), 323 (2008) 500; 1502.
* [15] El-Orany F. A. A., et al., Opt. Commu. 283 (2010) 3158
* [16] Wigner E. P., Phys. Rev., 40 (1932) 749.
* [17] O’Connell R. F. and Wigner E. P., Phys. Lett. A, 83 (1981) 145.
* [18] Schleich W., Quantum Optics (Wiley, New York) 2001.
Figure 1: (Colour online) The quantity $(\triangle X_{1})^{2}$ and $(\triangle
X_{2})^{2}$ as the function of squeezing parameter $\mu$ for different case
$\nu=0$ and $\nu=0.5$. Figure 2: (Colour online) The uncertainty value
$(\triangle X_{1})(\triangle X_{2})$ as the function of $r$ for different
$\theta$.
|
arxiv-papers
| 2010-09-14T00:56:29 |
2024-09-04T02:49:12.898058
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-xiang Xu, Hong-yi Fan, Li-yun Hu, and Hong-chun Yuan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1009.2548"
}
|
1009.2593
|
# Remarks on the Sequential Products††thanks: This project is supported by
Zhejiang Innovation Program for Graduates (YK2009002) and Natural Science
Foundation of China (10771191 and 10471124) and Natural Science Foundation of
Zhejiang Province of China (Y6090105).
Liu Weihua, Wu Zhaoqi, Wu Junde
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
Corresponding author E-mail: wjd@zju.edu.cn
Abstract. In this paper, we show that those sequential products which were
proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. 42, 185206
(2009), J. Phys. A: Math. Theor. 42, 345203 (2009)] are just unitary
equivalent to the sequential product $A\circ
B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$.
Key words. Hilbert space, Lüders operation, Sequential product.
Pacs. 03.65.Aa, 03.65.Db
Quantum measurement theory is one of the key problems in quantum theory, it
contains a great many of mathematical problems and philosophical problems.
Also it has applications in quantum information theory and quantum correction
theory. The essential difference between quantum measurement and classical
measurement is that the quantum measurement would make the system collapsed.
It has the follows four characteristics:
(1). Randomness. It is unpredictable and uncontrollable.
(2). Irreversibility. In general, measurement is entropy-increasing procedure.
(3). Decoherence. Eliminate all the coherence of the original state.
(4). Nonlocality. The collapse of the wave function is nonlocal.
In history, Heinsenberg, von Neumann, Birkhoff published some important far-
reaching fundamental works. In order to state our main results, now, we need
to recall some elementary notations.
Let $\cal L$ be a quantum-mechanical system and it be represented by a complex
Hilbert space $H$. Each self-adjoint operator $A$ on $H$ satisfies that $0\leq
A\leq I$ is said to be a quantum effect ([1-2]). Quantum effects represent
yes-no measurements that may be unsharp. The set of quantum effects on $H$ is
denoted by ${\cal E}(H)$. The subset ${\cal P}(H)$ of ${\cal E}(H)$ consisting
of orthogonal projection operators represents sharp yes-no measurements. Let
${\cal T}(H)$ be the set of trace class operators on $H$ and ${\cal S}(H)$ the
set of density operators on $H$, i.e., the state set of quantum system $\cal
L$.
As we knew, a quantum measurement can be described as a quantum operation
which is a completely positive linear mapping $\Phi:{\cal
T}(H)\rightarrow{\cal T}(H)$ such that for each $T\in{\cal S}(H)$, $0\leq
tr[\Phi(T)]\leq 1$ ([3-5]). For each $P\in{\cal P}(H)$, the so-called Lüders
operation $\Phi_{L}^{P}$ is defined by $T\rightarrow PTP$, in physics, it
implied that if the quantum-mechanical system $\cal L$ is in state $W\in{\cal
S}(H)$, then the probability that the measurement $P$ is observed is given by
$p_{W}(P)=tr(PWP)$, moreover, the resulting state after the measurement $P$ is
observed is $\frac{PWP}{tr(PWP)}$ whenever $tr(PWP)\neq 0$ ([4]). Each quantum
effect $B\in{\cal E}(H)$ gives to a general Lüders operation
$\Phi_{L}^{B}:T\rightarrow B^{\frac{1}{2}}TB^{\frac{1}{2}}$. If $A,B\in{\cal
E}(H)$, then the composition operation $\Phi_{L}^{B}\circ\Phi_{L}^{A}$ defines
a new operation and is called a sequential operation as it is obtained by
performing first $\Phi_{L}^{A}$ and then $\Phi_{L}^{B}$. It is easily to prove
that
$\Phi_{L}^{B}\circ\Phi_{L}^{A}=\Phi_{L}^{A^{\frac{1}{2}}BA^{\frac{1}{2}}}$
$([5,P_{26-27}])$. Let us denote $A^{\frac{1}{2}}BA^{\frac{1}{2}}$ by $A\circ
B$, then $A\circ B\in{\cal E}(H)$ and $\circ$ has the following important
properties ([6-7]):
(S1). The map $B\rightarrow A\circ B$ is additive for each $A\in{\cal E}(H)$,
that is, if $B+C\leq I$,
then $(A\circ B)+(A\circ C)\leq I$ and $(A\circ B)+(A\circ C)=A\circ(B+C)$.
(S2). $I\circ A=A$ for all $A\in{\cal E}(H)$.
(S3). If $A\circ B=0$, then $A\circ B=B\circ A$.
(S4). If $A\circ B=B\circ A$, then $A\circ(I-B)=(I-B)\circ A$ and
$A\circ(B\circ C)=(A\circ B)\circ C$
for all $C\in{\cal E}(H)$.
(S5). If $C\circ A=A\circ C$, $C\circ B=B\circ C$, then $C\circ(A\circ
B)=(A\circ B)\circ C$
and $C\circ(A+B)=(A+B)\circ C$ whenever $A+B\leq I$.
Professor Gudder called $A\circ B$ the sequential product of $A$ and $B$, it
represents the quantum effect produced by first measuring $A$ then measuring
$B$ ([6-7]). In [8], Gudder asked: is $A\circ
B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$ the only operation on ${\cal E}(H)$ which
satisfies the properties (S1)-(S5) ? In [9], Liu and Wu showed that if $H$ is
a finite dimensional complex Hilbert space, $f_{z}(u)$ is the complex-valued
function defined on $[0,1]$, where $f_{z}(u)=\exp z(\ln u)$ if $u\in(0,1]$ and
$f_{z}(0)=0$, and denote $A^{i}=f_{i}(A),\,\,A^{-i}=f_{-i}(A)$, then
$A\circ_{1}B=A^{1/2}A^{i}BA^{-i}A^{1/2}$ defined a new sequential product
which satisfies the properties (S1)-(S5), thus, Gudder’s problem was answered
negatively.
Note that the sequential product $A\circ
B=A^{\frac{1}{2}}BA^{\frac{1}{2}}=A^{\frac{1}{2}}B(A^{\frac{1}{2}})^{*}$ of
$A$ and $B$ can only describe the instantaneous measurement, that is, the
measurement $B$ is completed at once after the measurement $A$ is performed.
In order to describe a more complicated process where we allow a duration
between the measurement $A$ with the measurement $B$, then we need to replace
$A^{\frac{1}{2}}$ with $f(A)$, $(A^{\frac{1}{2}})^{*}$ with $(f(A))^{*}$,
where $f(A)$ is a function of $A$ which describe the change of $A$ was made by
the duration between $B$ with $A$. Thus, we need to consider the following
general sequential product $f(A)B(f(A))^{*}$.
By the above motivation, in [10], Shen and Wu proved the following result:
Theorem 1. Let $H$ be a finite dimensional complex Hilbert space, for each
$A\in{\cal E}(H)$, $sp(A)$ the spectra of $A$ and ${\cal B}(sp(A))$ the set of
all bounded complex Borel functions on $sp(A)$. Take a $f_{A}\in{\cal
B}(sp(A))$. Define $A\diamond B=f_{A}(A)B(f_{A}(A))^{*}$ for $B\in{\cal
E}(H)$. Then $\diamond$ has the properties (S1)-(S5) iff the set
$\\{f_{A}\\}_{A\in{\cal E}(H)}$ satisfies the following conditions:
(i) For every $A\in{\cal E}(H)$ and $t\in sp(A)$, $|f_{A}(t)|=\sqrt{t}$;
(ii) For any $A,B\in{\cal E}(H)$, if $AB=BA$, then there exists a complex
constant $\xi$ such that $|\xi|=1$ and $f_{A}(A)f_{B}(B)=\xi f_{AB}(AB)$.
Note that for each $A\in{\cal E}(H)$, we can take many $f_{A}\in{\cal
B}(sp(A))$ satisfies the conditions (i) and (ii), so, Theorem 1 told us that
for each given finite dimensional complex Hilbert space $H$, there are many
sequential products on $({\cal E}(H),0,I,\oplus)$.
In this note, we show that these sequential products are unitary equivalent to
the sequential product $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$.
Firstly, we need the following:
Lemma 1.1 ([10]). If $\\{f_{A}\\}_{A\in{\cal E}(H)}$ satisfies the conditions
(i) and (ii) of Theorem 1, then we have
(1) $f_{A}(A)\overline{f_{A}}(A)=\overline{f_{A}}(A)f_{A}(A)=A$,
$(f_{A}(A))^{*}=\overline{f_{A}}(A)$.
(2) If $0\in sp(A)$, then $f_{A}(0)=0$.
(3) If $A=\sum\limits^{n}_{k=1}\lambda_{k}E_{k}$, where
$\\{E_{k}\\}^{n}_{k=1}$ are pairwise orthogonal projections and
$\lambda_{k}\neq 0$, then
$f_{A}(A)=\sum\limits^{n}_{k=1}f_{A}(\lambda_{k})E_{k}$.
Our main result is:
Theorem 2. Let $H$ be a finite dimensional complex Hilbert space. Then the
sequential product $f_{A}(A)B(f_{A}(A))^{*}$ on $({\cal E}(H),0,I,\oplus)$ is
unitary equivalent to the sequential product $A\circ
B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$.
Proof. Let $A=\sum\limits^{n}_{k=1}\lambda_{k}E_{k}$ be the spectra
decomposition of $A$, where $\\{E_{k}\\}^{n}_{k=1}$ be pairwise orthogonal
projection operators and $\lambda_{k}>0,$ $k=1,2,\cdots,n$. By condition (i)
of Theorem 1, we have $|f_{A}(\lambda_{k})|=\sqrt{\lambda_{k}}$, so
$f_{A}(\lambda_{k})=\sqrt{\lambda_{k}}e^{i\theta_{k}}$ for some real number
$\theta$. Let $E_{0}=I-\sum_{k=1}^{n}E_{k}$ and
$U=\sum_{k=1}^{n}e^{i\theta_{k}}E_{k}+E_{0}$. Then $U$ is an unitary operator
and it is easy to see that $AU=UA$, so by Lemma 1.1, we have
$f_{A}(A)=A^{1/2}U$. Thus, $A\diamond
B=f_{A}(A)B\overline{f_{A}}(A)=A^{1/2}UB(A^{1/2}U)^{*}=U(A^{1/2}BA^{1/2})U^{*}=U(A\circ
B)U^{*}$ and the conclusion is proved.
References
[1] G. Ludwig, Foundations of Quantum Mechanics (I-II), Springer, New York,
1983
[2] G. Ludwig, An Axiomatic Basis for Quantum Mechanics (II), Springer, New
York, 1086
[3] K. Kraus, Effects and Operations, Springer-Verlag, Beilin, 1983
[4] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976
[5] P. Busch, M. Grabowski M and P. J. Lahti, Operational Quantum Physics,
Springer-Verlag, Beijing Word Publishing Corporation, 1999
[6] S. Gudder, G. Nagy, J. Math. Phys. 42, 5212 (2001)
[7] S. Gudder, R. Greechie. Rep. Math. Phys. 49, 87 (2002)
[8] S. Gudder, Inter. J. Theory. Phys. 44, 2219 (2005)
[9] Liu W. H., Wu J. D. J. Phys. A: Math. Theor. 42, 185206 (2009)
[10] Shen J., Wu J. D., J. Phys. A: Math. Theor. 42, 345203 (2009)
|
arxiv-papers
| 2010-09-14T08:55:25 |
2024-09-04T02:49:12.903700
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liu Weihua, Wu Zhaoqi, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/1009.2593"
}
|
1009.2641
|
# Hidden possibilities in controlling optical soliton in fiber guided doped
resonant medium
Anjan Kundu
Theory Group & CAMCS, Saha Institute of Nuclear Physics, Calcutta, INDIA
Fiber guided optical signal propagating in a Erbium doped nonlinear resonant
medium is known to produce cleaner solitonic pulse, described by the self
induced transparency (SIT) coupled to nonlinear Schrödinger equation. We
discover two new possibilities hidden in its integrable structure, for
amplification and control of the optical pulse. Using the variable soliton
width permitted by the integrability of this model, the broadening pulse can
be regulated by adjusting the initial population inversion of the dopant
atoms. The effect can be enhanced by another innovative application of its
constrained integrable hierarchy, proposing a system of multiple SIT media.
These theoretical predictions are workable analytically in details, correcting
a well known result.
I. INTRODUCTION
Optical communication through fiber has achieved phenomenal development over
the last two decades [1]. Dissipation and dispersion in the media, which are
the main hindrances in signal transmission, are usually attempted to be solved
by the dispersion management techniques and devices [1, 2]. On the other hand,
in soliton based optical communication, mediated by the nonlinear Schrödinger
(NLS) equation proposed much earlier [3], the group velocity dispersion can be
countered by the self phase modulation in the nonlinear fiber medium [4].
However, the experiments revealed insufficiency of the model for its efficient
practical application [5]. Another proposal with improved solitonic
transmission was due to the self-induced transparency (SIT), produced by the
coherent response of the medium to an ultra short optical pulse [6, 7].
Finally, the benefits of both the NLS and the SIT systems, were combined in a
coupled NLS-SIT model [8], by transmitting the optical soliton through an
Erbium doped nonlinear resonant medium [9, 10].
However the solitonic communication, in spite of its favorable features and
theoretical advantages due to its underlying integrability, did not receive
the needed response. Our aim here therefore, is to revisit the NLS-SIT model
for exploring new possibilities hidden in its integrable structures and use
them for the control and amplification of the solitonic pulse. Though the
soliton usually moves with a constant velocity or speed, the integrable
property of this coupled NLS-SIT system, as we find here, allows the soliton
speed to be a tunable function. And since the soliton width in this case is
related to its speed, which in turn is linked here to the initial population
inversion, the pulse width can be regulated by manipulating the population
inversion profile of the dopant atoms. This controlling effect can be further
enhanced by exploring another specialty of this integrable system, namely its
constrained integrable hierarchy, through a novel use of coupled multiple SIT
media, in place of the conventional single doped medium. The details can be
worked out exactly due to the integrability of the model, detecting the
limitation of a well known result on NLS-SIT soliton [9, 10].
Propagation of a stable optical pulse through a fiber medium, serving as a
dispersive and nonlinear wave guide with Kerr nonlinearity [3], can be
described by the optical electric field $E(z,t)$ satisfying the well known NLS
equation
$\ iE_{z}-E_{tt}-2|E|^{2}E=0,\ $ (1)
with space and time variables being interchanged as customary in nonlinear
optics [4]. On the other hand, ultra short optical pulse producing SIT in the
medium can be described by the Maxwell-Bloch equation [6, 7]
$iE_{z}=2p,\ \ ip_{t}=2NE,\ iN_{t}=-(Ep^{*}-E^{*}p),$ (2)
with the induced polarization $p$ and the population inversion $N$ of the
medium, contributed by the Bloch equation. Fascinatingly, it is possible to
combine these two effects by transmitting the stable nonlinear pulse produced
in the fiber wave guide through a doped medium with coherent response,
governed by a coupled NLS-SIT system given by a deformed NLS equation
$iE_{z}-E_{tt}-2|E|^{2}E=2p_{1},\ $ (3)
together with the SIT equations
$ip_{1t}=2(N_{1}E-w_{0}p_{1}),\ iN_{1t}=-(Ep_{1}^{*}-E^{*}p_{1}),$ (4)
representing a nonholonomic constraint [12]. In (4) , $\
p_{1}=\nu\tilde{\nu}^{*}$ is the induced polarization and
$N_{1}=|\tilde{\nu}|^{2}-|\nu|^{2},\ -1\leq N_{1}\leq 1,\ $ is the population
inversion of the two level dopant atoms with normalized wave functions $\nu,\
\tilde{\nu}$ for the ground and the excited states, respectively and $w_{0}$
is the natural frequency of these resonant ions. Assuming a homogeneous
broadening of the frequency spread with a sharp resonance at $\Delta
w=w-w_{0}$, we have taken the symmetric distribution $g(\Delta
w)=\delta(\Delta w)$ and replaced the average value $<p_{1}>=\int dwg(\Delta
w)p_{1}(z,t,w)$ appearing in (3) by $p_{1}=p_{1}(z,t,w_{0})$ and normalized
the coupling constants to ensure the integrability of the model.
II. INTEGRABILITY AND SOLITON SOLUTION
Recall that, an integrable nonlinear equation may be associated with a linear
system $\Phi_{t}=U\Phi,\ \Phi_{z}=V\Phi,$ defined through a Lax pair
$U(\lambda),V(\lambda)$, which are matrices with their elements depending on
the basic fields and a parameter $\lambda$, known as the spectral parameter.
Through compatibility of the linear Lax equations, inducing flatness
condition: $U_{z}-V_{t}+[U,V]=0,$ the Lax pair yield the given nonlinear
equation and at the same time can be used to extract its exact solutions
through the inverse scattering method (ISM) [11]. We remind again that, the
space $z$ and time $t$ are interchanged here in the context of nonlinear
optics. It is noteworthy that, while the set of coupled NLS-SIT equations
(3-4) generalize both the NLS (1) and the SIT (2) equations, the associated
Lax pair $U_{nls:sit},V_{nls:sit}$ contain these subsystems as the constituent
parts:
$U_{nls:sit}(\lambda)=U_{nls}(\lambda)=U_{sit}(\lambda),\ \ \
V_{nls:sit}(\lambda)=V_{nls}(\lambda)+V_{sit}(\lambda)$ (5)
where $U_{nls},V_{nls}$ and $U_{sit},V_{sit}$ are the Lax pairs related to the
NLS and the SIT equations, respectively. The NLS Lax pair is well known as
[11]
$\displaystyle U_{nls}(\lambda)$ $\displaystyle=$ $\displaystyle
i(\sigma^{3}\lambda+U^{(0)}),\ U^{(0)}=E\sigma^{+}+E^{*}\sigma^{-}$ (6)
$\displaystyle V_{nls}(\lambda)$ $\displaystyle=$ $\displaystyle
V^{(0)}+V^{(1)}\lambda+V^{(2)}\lambda^{2},\ $ $\displaystyle V^{(0)}$
$\displaystyle=$ $\displaystyle(U_{x}^{(0)}-i(U^{(0)})^{2})\sigma^{3},\
V^{(1)}=2iU^{(0)},\ V^{(2)}=2i\sigma^{3},$ (7)
where $\sigma^{3},\ \sigma^{\pm}=\frac{1}{2}(\sigma^{1}\pm i\sigma^{2})$ are
the $2\times 2$ Pauli matrices. The SIT Lax pair can be given by the same
time-Lax operator as that of the NLS: $U_{sit}(\lambda)=U_{nls}(\lambda),$
while the space-Lax operator
$V_{sit}(\lambda)=i(\lambda-w_{0})^{-1}G_{1},\
G_{1}=N_{1}\sigma^{3}+p_{1}\sigma^{+}+p^{*}_{1}\sigma^{-},$ (8)
can be linked to the nonholonomic deformation [12]. We check easily that, the
flatness condition of (6,7) yields the NLS equation (1), while (6,8) the SIT
equation (2) and similarly, the Lax pair (5) would yield the coupled NLS-SIT
equations (3,4)
The time-Lax operator $U(\lambda)$ plays the central role in the ISM for
finding the exact solutions of the nonlinear equation [11] and therefore,
since $U(\lambda)$ is the same for NLS (1), SIT (2) and NLS-SIT (3,4)
equations as seen from (5), the form of soliton solutions and the ISM
procedure are remarkably similar for all the three equations. We therefore
present soliton solutions for all of them in an unified way following the ISM
[11], which though an involved method, gives the 1-soliton solution in an
amazingly simple form:
$E=c\ \frac{g}{1+|g|^{2}},\ \ g={\rm exp}[2i({\rm u}^{\infty}t+\tilde{\rm
v}^{\infty}z+\phi)],$ (9)
with $c,\phi=$ constants. Note that, the crucial elements ${\rm
u}^{\infty},\tilde{\rm v}^{\infty}=\frac{1}{z}\int^{z}dz{\rm v}^{\infty}$ in
(9), though linked to the Lax pair of the given system, need information only
about their asymptotic properties: ${\rm
u}^{\infty}=\sigma^{3}U(\lambda_{1})|_{t=-\infty},\ {\rm
v}^{\infty}=\sigma^{3}V(\lambda_{1})|_{t=-\infty},$ at discrete spectral
parameter $\lambda_{1}$. Therefore, fixing the initial condition of the basic
fields involved in the NLS-SIT equations as
$E(z,t=-\infty)\to 0,\ p_{1}(z,t=-\infty)\to 0,N_{1}(z,t=-\infty)=N_{0}(z)$
(10)
with an arbitrary function $N_{0}(z),$ we can easily derive from the Lax pair
(5-8):
$\displaystyle\sigma^{3}{\rm u}_{nls:sit}^{\infty}$ $\displaystyle=$
$\displaystyle
U_{nls:sit}(\lambda_{1})|_{t=-\infty}=U_{nls}(\lambda_{1})|_{t=-\infty}=U_{sit}(\lambda_{1})|_{t=-\infty}=2i\sigma^{3}\lambda_{1},$
$\displaystyle\sigma^{3}{\rm v}_{nls:sit}^{\infty}$ $\displaystyle=$
$\displaystyle
V_{nls:sit}(\lambda)|_{t=-\infty}=(V_{nls}(\lambda)+V_{nls}(\lambda))|_{t=-\infty}=i\sigma^{3}(2\lambda_{1}^{2}+(\lambda_{1}-w_{0})^{-1}N_{0}(z)),$
(11)
which at the discrete spectral parameter with complex value:
$\lambda_{1}=k+i\eta,$ take the explicit form
${\rm u}_{nls:sit}^{\infty}=i\lambda_{1},\ {\rm v}_{nls:sit}^{\infty}={\rm
v}_{nls}^{\infty}+{\rm v}_{sit}^{\infty},\ {\rm
v}_{nls}^{\infty}=2i\lambda_{1}^{2},\ {\rm
v}_{sit}^{\infty}=i(\lambda_{1}-w_{0})^{-1}N_{0}(z).$ (12)
Inserting the needed complex valued expressions (12) in (9) and grouping its
real (Re) and imaginary (Im) parts we get the 1-soliton solution for the
optical field in NLS-SIT equations (3,4) in the familiar $sech$\- form
$E=-2i\eta\ {\rm sech}2\zeta e^{2i\theta},\ \zeta=\eta(t-t_{0}-vz),\
\theta=\omega z+kt+\phi_{0},$ (13)
where $t_{0}$ and $\phi_{0}$ are constant time and phase shift. Inverse speed
$v$ and phase rotation $\omega$ for the NLS-SIT soliton (13) are given by the
superposition
$v=v_{nls}+v_{sit},\ \omega=\omega_{nls}+\omega_{sit},$ (14)
of the corresponding parameters from the NLS and the SIT subsystems, derived
from (12) using $\lambda_{1}=k+i\eta$ as
$\displaystyle v_{nls}=-\frac{1}{\eta}{Im[{\rm v}_{nls}^{\infty}]}=-4k,\
\omega_{nls}=Re[\tilde{\rm v}_{nls}^{\infty}]=2(k^{2}-\eta^{2}),$ (15)
$\displaystyle v_{sit}=-\frac{1}{\eta}Im[\tilde{\rm
v}_{sit}^{\infty}]=\frac{1}{\rho}{f(z)},\ \omega_{sit}=Re[\tilde{\rm
v}_{sit}^{\infty}]=\frac{\tilde{k}}{\rho}{f(z)},$ (16)
with $\tilde{k}=k-w_{0},\ \rho={\tilde{k}}^{2}+\eta^{2},$ and
$f(z)=\frac{1}{z}\int^{z}N_{0}(z^{\prime})dz^{\prime}$.
It is intriguing to note that, since the z-evolution of the optical field $E$
in the NLS-SIT model follows the superposition rule (14) contributed
separately by the NLS and the SIT parts, the term $iE_{z}$ in equation (3),
evolving according to solution (13), breaks up into two parts: one follows the
NLS contribution with parameters (15) and satisfies the pure NLS part of the
equation in the left hand side, while the other part equates to the SIT
deformation $2p_{1}$ in the right hand side of (3) involving the related
parameters (16). Using this dynamics we derive the soliton solution for the
dipole $p_{1}$ from (13), in the form
$p_{1}=\ \frac{\eta}{\rho}\ N_{0}{\rm sech}2\zeta(i\tilde{k}-\eta{\rm
tanh}2\zeta)e^{2i\theta},\\\ $
with $\zeta,\theta$ as expressed in (13). Inserting solutions (13, Hidden
possibilities in controlling optical soliton in fiber guided doped resonant
medium ) for $E$ and $p_{1}$ in (4) and integrating by $t$ we derive further
the solution for population inversion
$N_{1}=N_{0}(1-\frac{\eta^{2}}{\rho}\ {\rm sech}^{2}2\zeta),$ (17)
again in the solitonic form with arbitrary function
$N_{0}(z)=N_{1}(t\to-\infty)$, adjusted by the integration constant. We obtain
thus the complete set of exact soliton solutions to the NLS-SIT equations
(3-4) as (13) for the optical field $E$, ( Hidden possibilities in controlling
optical soliton in fiber guided doped resonant medium ) for the dipole $p_{1}$
and (17) for the population inversion $N_{1}$. A beautiful interaction pattern
can be noticed in these solutions, manifested in the superposition relations:
$v=v_{nls}+v_{sit},\ \omega=\omega_{nls}+\omega_{sit}$, for the solitonic
parametersappearing in (13,14). Intriguingly, in the absence of the SIT system
with $p_{1}=N_{1}=0$, when the coupled NLS-SIT equations reduce to the NLS
equation (1) for the field $E$, one recovers from (13) the well known NLS
soliton by simply putting $v_{sit}=\omega_{sit}=0$ due to the vanishing of
(8). Therefore, the NLS soliton takes exactly the same form as (13), though
the parameters are reduced to pure NLS case: $v=v_{nls},\
\omega=\omega_{nls}$. Similarly, we can directly get the soliton solution for
the pure SIT equations (2) in the same form (13, Hidden possibilities in
controlling optical soliton in fiber guided doped resonant medium ,17), but
with soliton parameters reducing to $v=v_{sit},\ \omega=\omega_{sit}$, due to
switching off the NLS influence: $v_{nls}=\omega_{nls}=0$. Thus our exact NLS-
SIT soliton can reproduce the solutions for both the NLS and the SIT equations
in a unified way, consistent with the ISM. However this rich interaction
picture seems to have been missed in a well known earlier work [9, 10],
leading to wrong conclusions in the general case. In particular, the soliton
solution for the NLS-SIT equation presented in [9, 10] gives the expression
for the pulse delay as $\delta=\frac{n}{c}(1+\gamma)$ ((4.9) in [10]), which
is equivalent to the inverse soliton speed for the SIT ((2.22) in [10]), i.e.
$v\equiv\delta=v_{sit}$, in our notation [13]. Similarly, the phase rotation
in [9, 10] is given as $\alpha=2\eta^{2}$, meaning
$\omega\equiv\alpha=-\omega_{nls}$, (at $k=0$, see (15)) in our notation [13].
Both these results for the coupled NLS-SIT equations appear to be incomplete,
when compared with our exact result (14). It is clear that, the solution of
[9, 10] can be justified only in a very limited sense, when
$v_{nls}=\omega_{sit}=0$ and therefore unlike our soliton solution can not
interpolate between the solutions of the NLS and the SIT equations.
This partial result unfortunately led to wrong conclusions, for the NLS-SIT
system in general, stating that (sect. IV [10]), the normalized speed (i.e.
$\delta^{-1}$) of the NLS-SIT soliton is determined only by the SIT effect
and similarly, the z dependence of the phase of the dipole (i.e. $\alpha$) is
determined solely by the nonlinear phase change due to the NLS soliton. Our
exact solutions for the optical field $E$ (13) and the dipole $p_{1}$ ( Hidden
possibilities in controlling optical soliton in fiber guided doped resonant
medium ) with correct expressions (14), conclude on the other hand that, only
a part (i.e. $v_{sit}$) in the normalized speed
$v^{-1}=(v_{nls}+v_{sit})^{-1}$ of the NLS-SIT soliton is determined by the
SIT effect, while there is an additional contribution coming from the NLS part
$v_{nls}$. Similarly, the z dependence of the phase of the dipole and the
input optical field gets contribution from both the NLS and the SIT parts as
$\omega=\omega_{nls}+\omega_{sit}$, consistent with the interaction picture in
the coupled NLS-SIT system.
III. CONTROLLING OPTICAL SOLITON EXPLOITING INTEGRABLE STRUCTURES
Based on the integrable structures underlying the NLS-SIT system describing
the propagation of optical soliton in fiber guided doped medium, we propose
two possible ways for controlling the amplitude and width of the optical
pulses.
A. Soliton control by regulating initial population inversion
It is commonly believed that, the exact soliton solution of a homogeneous
equation always moves with a constant speed, width and frequency, as in the
case of the NLS soliton (15) with constant values for $v_{nls},\omega_{nls}$.
However, it is crucial to note that, for the NLS-SIT soliton the parameters
$(v,\omega)$, as evident from (14,16) can become variable functions, depending
on the initial population inversion $N_{0}(z)$ (10). Due to this peculiarity
of integrable structure of the NLS-SIT system, hidden in the expressions like
(5,8,12,16), the soliton speed: $v^{-1}$ and width: $(\eta v)^{-1}$, as
defined from the soliton argument $\zeta$ (13), can be variable and linked to
a controllable arbitrary function $N_{0}(z)$.
We show that, this important observation embedded in the integrability of the
NLS-SIT system can open up a new avenue for controlling the optical soliton
propagating through the doped medium, by regulating its initial population
inversion profile $N_{0}(z)$. This fact however remained unexplored in earlier
investigations [9, 10, 14, 15], due to the restriction to a fixed initial
profile $N_{0}(z)=-1$. Note that, at this particular value giving $f(z)=-1$,
our more general result (16) reduces to the simplified expressions obtained
earlier:
$v_{sit}=-\frac{1}{\rho},\ \omega_{sit}=-\frac{\tilde{k}}{\rho},$ (18)
The choice for the initial population inversion in the NLS-SIT model as an
arbitrary function $N_{0}(z)>-1$, that we propose here, gives us the needed
freedom for obtaining the excited and the ground state occupancies at the
initial moment as $|\tilde{\nu}|^{2}=\frac{1}{2}(1+{N_{0}(z)})$ and
$|\nu|^{2}=\frac{1}{2}(1-{N_{0}(z)})$, respectively. Therefore, for
$N_{0}>-1$, giving $|\tilde{\nu}|^{2}>0$, we can prepare the dopant atoms
initially in an excited state by optical prepumping, resulting to the creation
of a laser-active amplifying medium with its intensity determined by $N_{0}$.
Note that, only in such a case when more active dopant atoms are in the
excited state, the optical soliton can gain net energy [1].
In addition to the soliton pulse amplification, variable initial profile
$N_{0}(z)>-1$, permitted by the integrability of the NLS-SIT system, can play
a crucial role in controlling the shape and dynamics of the optical soliton.
It is possible, as we see below, to address the important problem of pulse
broadening by regulating the initial profile of the dopant atoms. For example,
a solitonic pulse governed by the NLS equation under small perturbation by a
term $-i\frac{\Gamma}{2}E$ with $\ \Gamma<<1$, would suffer broadening by a
factor $(4k\eta(z))^{-1}$, which can be worked out through the variational
perturbation method as $\eta(z)=\eta\ e^{-\Gamma z}$[4], which is valid
however upto the range $\Gamma z\approx 1$ along $z$. Beyond this range with
$z>>1$, as shown by some other method, the broadening of the pulse width
follows a different rule, by increasing linearly with $z$ at a rate slower
than the linear medium [4]. Though an attenuation with intensity loss would
also occur simultaneously, the broadening leads to more serious problem of
information loss and bandwidth limitation. Therefore we concentrate here only
on the broadening problem of the perturbed NLS soliton, due to the increasing
solitonic width $\frac{1}{4k\eta}\ e^{\Gamma z}$ along $z$, as shown in Fig 1.
As stated above for $z>>1$ it would follow a different rule. We show that, by
transmitting this solitonic pulse through a doped resonant medium, described
by an interacting NLS-SIT model (3-4), it is possible to control the pulse
broadening, by suitably preparing the initial population inversion profile
$N_{0}(z)$. Fig 2a shows this controlling effect, where the broadening of the
solitonic pulse suffered in Fig 1, is countered by the narrowing of the pulse
due to variable width $V(z)=(v_{nls}+N_{0}(z)/(\rho\Gamma))^{-1}$, by taking
$N_{0}(z)\sim\eta(z)^{-1}$. Note that the profile $N_{0}(z)$ has to be
adjusted differently at different ranges, as mentioned above, to control the
broadening in the respective regions for a wide range of $z$. The soliton
dynamics would also change to a variable speed $V(z)$, possible due to the
energy supplied by optical prepumping.
This potential opportunity for controlling the pulse width, hidden in the
integrable property of the NLS-SIT system, as explained above, was missed in
earlier investigations [9, 10, 14, 15], since the initial atoms are usually
taken in their ground state: $\ |\nu|^{2}=1,\ |\tilde{\nu}|^{2}=0,$ by
restricting to $N_{0}(z)=-1$.
Figure 1: Broadening of the perturbed NLS soliton $|E(z,t)|$ along the fiber,
moving with a constant speed with parameter choice $k=0.25,\ \eta=0.50,\
\Gamma=0.28$ Figure 2: a) Broadening NLS soliton pulse is controlled by a
coupled NLS-SIT system with $\ N_{0}(z)=-0.11e^{\Gamma z}$ and $w_{0}=0.3$.
Variable speed of the soliton is evident from its bending in the ($z,t$)-
plane. b) Additional control is achieved by coupling to a second SIT system
with $N^{(2)}_{0}(z)=0.11e^{\Gamma z},\ $ showing an efficient restoration of
the soliton width.
B. Enhanced soliton control through multiple doping
Another promising opportunity in managing optical soliton in fiber
communication, emerging also from the integrability of the NLS-SIT model, is
overlooked completely in earlier investigations. This is the proposal of
enhancing the effect of amplification and control of the optical soliton by
replacing the conventional single SIT system, the only case considered in the
literature, by a coupled multiple SIT system, using recursively the
constrained integrable hierarchy in the NLS-SIT model (see Fig. 3). The
physical meaning of coupling the NLS equation to such multi SIT system can be
given through a novel proposal of using coupled multiple doped resonant media,
in place of a single doped medium.
For generating the governing hierarchal equations and showing their
integrability, we extend Lax operator $V(\lambda)$ (5,8) by adding more
deforming terms $V_{sitM}=i\sum_{j}^{M}(\lambda-w_{0})^{-j}G_{j}$, linked to
the $M$-th constrained hierarchy [12] in the NLS-SIT system, fixed at level
$M$ from the possible infinite sequence : $j=1,2,\ldots$. In analogy with
$G_{1}$ (8) we can express the deforming matrices $G_{j}$, through dipole
moment $p_{j}$ and population inversion $N_{j}$ of the $j$-th doped resonant
medium. For explicit demonstration we restrict to the next higher level $M=2$
in the constrained hierarchy, by considering only an additional SIT system to
the original NLS-SIT set. Compatibility of the Lax pair thus defined would
generate an extended set of equations given by the same deformed NLS (3)
coupled however to a double SIT system
$\displaystyle ip_{1t}$ $\displaystyle=$ $\displaystyle
2(N_{1}E-w_{0}p_{1}-p_{2}),\ iN_{1t}=-(Ep_{1}^{*}-E^{*}p_{1}),$ $\displaystyle
ip_{2t}$ $\displaystyle=$ $\displaystyle 2(N_{2}E-w_{0}p_{2}),\
iN_{2t}=-(Ep_{2}^{*}-E^{*}p_{2}),$ (19)
with induced polarization $p_{2}$ and population inversion $N_{2}$, linked to
the additional doped medium described by the second SIT system. We find
intriguingly that, the exact soliton solution for the optical pulse $E$ in
this extended NLS-SIT model (3,19), can be expressed again in the same form
(13), where the soliton parameters are to be modified with contributions from
all its interacting parts, i.e. from the NLS as well as from the multiple SIT
system as $\ v=v_{nls}+v_{sit1}+v_{sit2},\ \
\omega=\omega_{nls}+\omega_{sit1}+\omega_{sit2}.\ $ Parameters
$v_{nls},\omega_{nls}$ and $v_{sit1},\omega_{sit1}$ have the same expressions
as found already in (15,16), while the additional SIT contribution
$v_{sit2},\omega_{sit2}$, can be derived following a similar argument as (16)
in the form
$\displaystyle v_{sit2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\eta}Im[(\lambda_{1}-w_{0})^{-2}]{f_{2}(z)}=2\frac{\tilde{k}}{\rho^{2}}{f_{2}(z)},$
$\displaystyle\omega_{sit2}$ $\displaystyle=$ $\displaystyle
Re[(\lambda_{1}-w_{0})^{-2}]{f_{2}(z)}=2\frac{\tilde{k}^{2}-\eta^{2}}{\rho^{2}}{f_{2}(z)},$
(20)
with $f_{2}(z)=\frac{1}{z}\int^{z}N_{0}^{(2)}(z^{\prime})dz^{\prime},$
involving an additional arbitrary function
$N_{0}^{(2)}(z)=N_{2}(z,t=-\infty).$ It opens up therefore another novel way,
hidden again in the integrable structure of the NLS-SIT system, for an
enhanced control of the soliton width and dynamics, by adding a coupled second
SIT system, as shown in Fig. 2b.
This process of coupling the NLS equation to the set of multiple SIT equations
can be continued within the framework of the integrable system, as mentioned
above, creating a form of directional connected network with feedback, as
shown in Fig. 3. In particular, as evident from the coupled equations (3,19),
the input optical pulse $E$ would influence the dipole field $p_{j}$ and the
population inversion $N_{j}$ in all the resonant SIT media with
$j=1,2,\ldots,M$, while only $p_{1}$ from the first medium gives feed back to
the field $E$. On the other hand, $p_{j+1}$ are coupled sequentially to
$p_{j}$, across the media, while $N_{j}$ are mutually coupled only with
$p_{j}$ from the same medium, in the multiple SIT system with $j\in[1,M]$.
This network, would exhibit more and more manipulative power for control over
width and amplification of the optical pulse, enhanced sequentially by
choosing a set of initial condition
$N_{0}^{(j)}(z)=N_{j}(z,t=-\infty),j=1,2,\ldots M$ and is based on the notion
of constrained hierarchy of the integrable NLS-SIT system (see Fig. 3).
Figure 3: Connected network of the NLS and the multiple SIT system with $E$
as the input optical field, $p_{j}$ as the induced polarization and $N_{j}$ as
the population inversion of the $j$-th doped resonant medium with
$j=1,2,3,\ldots,M$. The arrows show the directions of coupling with equations
(3,19) describing this network in the particular case of $M=2$. Sequential
enhancement of the control of width and amplification are predicted by this
network, which is consistent with the constrained integrable hierarchy of the
NLS-SIT system.
This theoretical prediction, as presented schematically in Fig 3, is an
experimental challenge to incorporate the contribution of coupled multiple SIT
systems. Repeating the idea of available experimental realization of single
doped fiber medium, either to a series of doped media coupled through induced
polarization, or to multiple doping with parallel coupling in a single medium,
such experimental set up is likely to be organized.
IV. CONCLUDING REMARKS
Exploring the integrability of the coupled NLS-SIT system we have given novel
proposals for controlling its solitonic pulse. The broadening problem of the
optical pulse can be addressed by adjusting initial population inversion of
the dopant atoms, linked to the soliton width, by choosing more general
function $N_{0}(z)>-1$ for the initial profile, in place of the traditional
restriction to $N_{0}(z)=-1$. This also allows amplification of the signal
through initial excitation by prepumping energy. The controlling effect can be
refined further by using another integrable property of the coupled NLS-SIT
model given by its constrained hierarchy. The idea is to replace the
conventional single SIT system by a network of sequentially coupled multiple
SIT media with doping. Each additional SIT medium can bring in a new tunable
function $N^{(j)}_{0}(z)>-1,j=2,3,\ldots$ in the form of initial population
inversion of additional dopant atoms, providing more manipulative power for
controlling the shape and dynamics of the optical soliton. One set of dopant
atoms in the resonant medium is coupled to another set by induced
polarization, with all SIT media interacting in turn with the input optical
field. This network of interacting systems described by the constrained
hierarchy of the integrable NLS-multiSIT equations is predicted to have
enhanced control over solitonic width and amplitude, which can increase
sequentially with the number of coupled SIT media. In such a multi-doped media
requiring higher threshold intensity for the formation of solitonic pulse, one
could possibly use a multi-level dopant like neodymium (Nd3+), where with more
than two available levels the energy can be pumped throughout the process,
unlike in two levels, resulting to a higher gain [1].
Both of our theoretical proposals with applicable potentials can be worked out
analytically in minute details through ISM, due to the underlying
integrability of the system.
## References
* [1] G. P. Agarwal, Fiber Optic Communication Systems, (John Wiley, NY, 2002) ; V. Alwyn Fiber Optic Technologies (Cisco Press, 2004); Encyclopedia of Laser Physics & Technology, (Virtual Web-Library, RP Photonics Consulting).
* [2] B. J. Eggleton et al, J. Lightwave Tech. 18, 1418 (2000); X. F. Chen et al, Photonics. Tech. Lett. IEEE 12 , 1013 (2000); F. Poletti et al, Photonics. Tech. Lett. IEEE 20 , 1449 (2008).
* [3] L. F. Mollenauer et al, , R.H. Stolen and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980); A. Hasegawa and F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973); A. Hasegawa, Optical Fiber Solitons (Springer,Berlin, 1989)
* [4] G. P. Agarwal, Nonlinear Fiber Optics (Acad. Press, N.Y., 2007).
* [5] F. M. Mitshke and L. F. Mollenauer, Opt. Lett. 11, 657 (1986).
* [6] S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967); Phys. Rev. 183, 457 (1969).
* [7] G. L. Lamb Jr., Rev. Mod. Phys. 43, 99 (1971).
* [8] A. I. Maimistov and E. A. Manyakin, Sov. Phys. JETP 58, 685 (1983).
* [9] M. Nakazawa, E. Yamada and H. Kubota, Phys. Rev. Lett. 66, 2625 (1991).
* [10] M. Nakazawa, E. Yamada and H. Kubota, Phys. Rev. A 44, 5973 (1991).
* [11] M. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Stud. Appl. Math. 53, 294 (1974); M. Ablowitz and H. Segur, Solitons and Inverse Scattering Transforms (SIAM, Philadelphia, 1981); S. Novikov et al, Theory of Solitons (Consultants Bureau, NY, 1984).
* [12] A. Kundu, J. Math Phys. 50, 102702 (2009).
* [13] Comparing NLS-SIT soliton (4.4) in [10] with our (13) we identify pulse delay $\delta$ with our $v$, phase rotation $\alpha$ with our $\omega$, inverse soliton speed $\frac{n}{c}(1+\gamma)$ with our $v_{sit}$ and $2\eta^{2}$ with our $\omega_{sit}$ at $k=0$.
* [14] S. Kakei and J Satsuma, J. Phys. Soc. Jpn. 63, 885 (1994).
* [15] K. Porsezian and K. Nakkeeran, Phys. Rev. Lett. 74, 2941 (1995).
|
arxiv-papers
| 2010-09-14T12:42:16 |
2024-09-04T02:49:12.908638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anjan Kundu",
"submitter": "Anjan Kundu",
"url": "https://arxiv.org/abs/1009.2641"
}
|
1009.2672
|
# Quantum Maxwell’s Demon in Thermodynamic Cycles
H. Dong, D.Z. Xu and C.P. Sun suncp@itp.ac.cn http://power.itp.ac.cn/
suncp/index.html Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing, 100190, China
###### Abstract
We study the physical mechanism of Maxwell’s Demon (MD) helping to do extra
work in thermodynamic cycles, by describing measurement of position, insertion
of wall and information erasing of MD in a quantum mechanical fashion. The
heat engine is exemplified with one molecule confined in an infinitely deep
square potential inserted with a movable solid wall, while the MD is modeled
as a two-level system (TLS) for measuring and controlling the motion of the
molecule. It is discovered that the the MD with quantum coherence or on a
lower temperature than that of the heat bath of the particle would enhance the
ability of the whole work substance formed by the system plus the MD to do
work outside. This observation reveals that the role of the MD essentially is
to drive the whole work substance being off equilibrium, or equivalently
working with an effective temperature difference. The elaborate studies with
this model explicitly reveal the effect of finite size off the classical limit
or thermodynamic limit, which contradicts the common sense on Szilard heat
engine (SHE). The quantum SHE’s efficiency is evaluated in detail to prove the
validity of second law of thermodynamics.
###### pacs:
03.67.-a,05.70.Ln,05.30.-d,03.65.Ta
## I Introduction
Maxwell’s demon (MD) has been a notorious being since its existence could
violate the second law of thermodynamics (SLoT) MD_book ; Nori2009 : the MD
distinguishes the velocities of the gas molecules, and then controls the
motions of molecules to create a difference of temperatures between the two
domains. In 1929, Leo Szilard proposed the “one molecular heat engine”(we call
Szilard heat engine(SHE)) Szilard1929 as an alternative version of heat
engine assisted by MD. The MD firstly measures which domain, the single
molecule stays in and then gives a command to the system for extracting work
according to the measurement results. In a thermodynamic cycle, the molecule
seems to extract heat from a single heat bath at temperature $T$, and thus do
work $k_{\mathrm{B}}T\ln 2$ without evoking other changes. This consequence
obviously violates the SLoT.
The first revival of the studies of MD is due to the recognition about the
trade-off between information and entropy in the MD-controlled thermodynamic
cycles. The milestone discovery is the “Landauer principle” Landauer1961 ,
which reveals that erasing one bit information from the memory in computing
process would inevitably accompany an increasing entropy of the environment.
In the SHE, the erasing needs work $k_{\mathrm{B}}T\ln 2$ done by the external
agent. It gives a conceptual solution for the MD paradox Bennett1982 by
considering the MD as a part of the whole work substance, thus the erasing
information stored in the demon’s memory is necessary to restart a new
thermodynamic cycle. This observation about erasing the information of the MD
finally saves the SLoT.
The recent revival of the studies of MD is due to the development of quantum
information science. The corresponding quantum thermodynamics concerns the
quantum heat engines (QHEs) Kieu2004 ; htquan utilizing quantum coherent
system serving as the work substance. A quantum work substance is a quantum
system off the thermodynamic limit, which perseveres its quantum coherence
Scully2003 ; htquan2006 to some extent, and obviously has tremendous
influence on the properties of QHEs. Especially, when quantum MD is included
in the thermodynamical cycle Zurek1984 ; lloyd1997 ; Quan2006 , the whole work
substance formed by the work substance plus MD would be off the thermodynamic
limit and possesses some quantum coherence. There are many attempts to
generalize the SHE by quantum mechanically approaching the measurement process
Zurek1984 , the motion control Quan2006 , the insertion and the expansion
process Ueda2010 . However, to our best knowledge, a fully quantum approach
for all actions in the SHE integrated with a quantum MD intrinsically is still
lack. The quantum-classical hybrid description of the SHE may result in some
notorious observations about MD assisted thermodynamic process, which seems to
challenge the common senses in physics. Therefore, we need a fully quantum
theory for the MD-assisted thermodynamics.
In this paper, we propose a quantum SHE assisted by MD with a finite
temperature different from that of the system. In this model, we give a
consistent quantum approach to the measurement process without using the von
Neumann projection Pzhang . Then we calculate the works done by the insertion
of the movable wall in the framework of quantum mechanics. The controlled gas
expansion is treated with the quantum conditional dynamics. Furthermore, we
also consider the process of removing wall to complete a thermodynamic cycles.
With these necessary subtle considerations, the quantum approach for the MD-
assisted thermodynamic cycle will go much beyond the conventional theories
about the SHE. We show that the system off the thermodynamic limit exhibits
uncommon observable quantum effects due to the finite size of system , which
results in the discrete energy levels that could be washed out by the heat
fluctuation. Quantum coherence can assist the MD to extract more work by
reducing effective temperature, while thermal excitation of the MD at a finite
temperature would reduce its abilities for quantum measurement and conditional
control of the expansion. It means that, only cooled to the absolute zero
temperature, could the MD help the molecule to do maximum work outside.
Our paper is organized as follows: In Sec. II we firstly give a brief review
of classical version of SHE, and then present our model in quantum version
with MD included intrinsically. The role of quantum coherence of MD is
emphasized with the definition of the effective temperature for arbitrary two
level system. In the Sec. III of main part, we consider the quantum SHE with a
quantum MD at finite temperature, doing measurement for the position of the
particle confined in a one dimensional infinite deep well. The whole cycle
consists four steps: insertion, measurement, expansion and removing. Detailed
descriptions are performed subsequently in the whole cycle of SHE. We
calculate the work done and heat exchange in every sub-step. In Sec. IV, we
discuss quantum SHE’s operation efficiency in comparison with the Carnot heat
engine. We restore the well-known results in the classical case by tuning the
parameters in the quantum version, such as the width of the potential well.
Conclusions and remarks are given in Sec. VI.
## II Quantum Maxwell’s Demon in Szilard Heat Engine
In this section, we firstly revisit Szilard’s single molecular heat engine
(SHE) in brief. As illustrated in Fig. 1(a), the whole thermodynamic cycle
consists three steps: insertion(i-ii), measurement(ii-iii) and controlled
expansion(iii-iv) by the MD. The demon inserts a piston isothermally in the
center of the chamber. Then, it finds which domain, the single molecule stays
in and changes its own state to register the information of the system.
Without losing generality, we assume the demon initially is in the state $0$.
Finding the molecule is on the right, namely $L/2<x<L$, the demon changes its
own memory to state $1$, while it does not change if the molecule is on the
left $\left(0<x<L/2\right)$. According to the information acquired in the
measurement process, the demon controls the expansion of the domain with the
single molecule: allowing the isothermal expansion with the piston moving from
$L/2$ to $L$ if its memory registers $0$, and moving from $L/2$ to $0$, if the
register is on the state $1$. In each thermodynamic cycle, the system does
work $W=k_{\mathrm{B}}T\ln 2$ to the outside agent in the isothermal
expansion. In an overall looking, the system extracts heat from a single heat
bath to do work, thus it would violate the SLoT if the MD were not treated as
a part of the work substance in the SHE. However, after the cycle, MD stores
one bit information as its final state and is in the mixture of $0$ and $1$
states with equal probability. Thus it does not return to its initial state.
Landauer’s principle states that to erase such a bit of information at
temperature $T$ requires the dissipation of energy at least
$k_{\mathrm{B}}T\ln 2$. The work extracted by the system just compensates the
energy for erasing the information. Therefore, the SLoT is saved. In this
sense, the classical version of MD paradox is only a misunderstanding, due to
the ignorance of some roles of the MD Bennett1982 .
Figure 1: Classical and quantum Szilard’s single molecular heat engine. (a)
Classical version: (i-ii) A piston is inserted in the center of a chamber.
(ii-iii) The demon finds which domain, the single molecule stays in. (iii-iv)
The demon controls the system to do work according to its memory; (b) Quantum
version: The demon is modeled as a two level system with two energy levels
$\left|g\right\rangle$ and $\left|e\right\rangle$ and energy spacing $\Delta$.
The chamber is quantum mechanically described as an infinite potential with
width $L$. (I-II) An impenetrable wall is inserted at arbitrary position in
the potential. (II-III) The demon measures the state of the system and then
record the results in its memory by flipping its own state or no action taken.
The measurement may result in the wrong results illustrated in the green
dashed rectangle. (III-IV) The demon controls the expansion for the single
molecule according to the measurement. (IV-V) The wall is removed from the
potential.
In the most of previous investigations about the MD paradox, it is usually
assumed the system and the MD possess the same heat bath. Thus the whole work
substance formed by the system plus the MD is in equilibrium, and no quantum
coherence exists. If the demon is in contact with a lower temperature heat
bath while the system’s environment possesses higher temperature $T$, the work
needed in the erasing process is smaller than $k_{B}T\ln 2$ Quan2006 . Under
this circumstance, we actually construct a quantum heat engine with non-
equilibrium or an equilibrium working substance work between two different
heat baths. Furthermore, when the MD is initially prepared with quantum
coherence, the quantum nature of the whole work substance results in many
exotic functions for QHE.
To tackle this problem, we study here a quantum version of Szilard’s model
with an MD accompanying it. In this model, the chamber is modeled as an
infinite square potential well with the width $L$, as illustrated in Fig.
1(b). And the demon is realized by a single two-level atom with energy levels
$\left|g\right\rangle$, $\left|e\right\rangle$ and level spacing $\Delta$.
Initially, the system is in thermal state with inverse temperature $\beta$.
And the demon has been in contact with the low temperature bath at the inverse
temperature $\beta_{D}$. Namely, the demon is initial prepared in the
equilibrium state
$\rho_{D}=p_{g}\left|g\right\rangle\left\langle
g\right|+p_{e}\left|e\right\rangle\left\langle e\right|,$ (1)
with the probability $p_{e}=1-p_{g}$ in the excited state and one in ground
state
$p_{g}=1/\left[1+\exp\left(-\beta_{D}\Delta\right)\right].$
Actually, the inverse temperature $\beta_{D}$ could represent an effective
inverse temperature of the MD with quantum coherence. For an environment being
a mesoscopic system, the number of its degrees of freedom is not so large.
Under this circumstance, the strong coupling to the MD leaves finite off-
diagonal elements in the reduced density matrixDong2007 . This remnant of
coherence can be utilized to improve the apparent efficiency of the heat
engine Scully2003 ; htquan2006 . For the demon with coherence, the density
matrix usually reads as
$\rho_{D}=\left[\begin{array}[]{cc}p_{g}&F\\\
F^{\ast}&p_{e}\end{array}\right],$ (2)
where the off-diagonal element $F$ measures the quantum coherence. The eigen-
values of the above reduced density matrix represent two effective population
probabilities as
$\displaystyle p_{+}\left(F\right)$ $\displaystyle\simeq$ $\displaystyle
p_{e}-\coth\left(\frac{\Delta}{2}\beta_{D}\right)\left|F\right|^{2},$
$\displaystyle p_{-}\left(F\right)$ $\displaystyle\simeq$ $\displaystyle
p_{g}+\coth\left(\frac{\Delta}{2}\beta_{D}\right)\left|F\right|^{2}.$ (3)
We can define an effective inverse temperature $\beta_{\mathrm{eff}}=\ln
p_{+}\left(F\right)/p_{-}\left(F\right)$ for the two-level MD, namely,
$\beta_{\mathrm{eff}}=\beta_{D}+\frac{4\left|F\right|^{2}}{\Delta}\cosh^{2}\left(\frac{\Delta}{2}\beta_{D}\right)\coth\left(\frac{\Delta}{2}\beta_{D}\right).$
(4)
The effective temperature $T_{\mathrm{eff}}=1/\beta_{\mathrm{eff}}$ here is
lower than the bath temperature $T_{D}$. As shown as follows, it is the lowing
of the effective temperature of the MD that results in an increasing of the
heat engine efficiency.
As for the modeling of the chamber as an infinite square potential well, the
eigenfunctions of the confined single molecule are
$\left\langle
x\right.\left|\psi_{n}\left(L\right)\right\rangle=\sqrt{\frac{2}{L}}\sin\left[n\pi
x/L\right],$ (5)
with the corresponding eigen-energies $E_{n}\left(L\right)=\left(\hbar
n\pi\right)^{2}/\left(2mL^{2}\right)$, where the quantum number $n$ ranges
from $1$ to $\infty$.
On this bases, the initial state of the total system is expressed as a product
state
$\rho_{0}=\frac{1}{Z\left(L\right)}\sum_{n}e^{-\beta
E_{n}\left(L\right)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes\rho_{0}^{D},$
(6)
where
$Z\left(L\right)=\sum_{n}\exp\left[-\beta E_{n}\left(L\right)\right]$ (7)
is the partition function of the system.
Here, we remark that the discrete spectrum of the system results from the
finite size of the width $L.$ As $L\rightarrow\infty$, the spectrum becomes
continuous as the energy level spacings is proportional to $1/L^{2}$. Then
heat excitation characterized by $k_{B}T$ can wash out the quantum effect so
that the system approaches a classical limit. Some of finite size effect based
quantum phenomenon could also disappear as $T\rightarrow\infty.$
With the above modelings, the MD-assisted thermodynamic cycle for the quantum
SHE is divided as four steps illustrated in Fig. 1(b): (I-II) the insertion of
a mobile solid wall into the potential well at a position $x=l$ (the origin is
$x=0$ ); (II-III) the measurement done by the MD to create the quantum
entanglement of its two internal states to the spatial wave functions of the
confined molecule; (III-IV) quantum control for the mobile wall to move the
according to the record in the demon’s memory; (IV-V) removing the wall so
that the next thermodynamic cycle can be restarted. Their descriptions will be
discussed subsequently in the next section and detailed calculations will be
found in the Appendix.
## III Quantum Thermodynamic Cycles with Measurement
In this section we analyze in details the thermodynamic cycle of the molecule
confined in an infinite square potential well. The molecule’s position is
monitored and then controlled by the MD. The MD may have quantum coherence as
in Eq.2, or equivalently, possesses a lower temperature
$T_{\mathrm{D}}=1/\beta_{\mathrm{D}}$ than $T=1/\beta$ of the confined
molecule’s heat bath. In each step, we will evaluate the work done by outside
agent and heat exchange in detail. In order to concentrate on the physical
properties, we put the calculations in the Appendix.
Figure 2: (Color Online) Probability $P_{L}$ and the corresponding classical
one $P_{L}^{C}$ vs temperature $1/\beta$ for different piston position $l=1/3$
and $l=1/4$. Without losing generality, we set the parameters as $L=1$,
$m=\pi^{2}/2$ and $\hbar=1$.
### Step1: Quantum Insertion (I-II)
In the first process, the system is in contact with the heat bath $\beta$,
then a piston is inserted isothermally into the potential at position $l$. The
potential is then divided into two domains, denoted simply as $L$ and $R$,
with the length $l$ and $L-l$ respectively. The eigenstates change into the
following two sets as
$\displaystyle\left\langle x\right.\left|\psi_{n}^{R}(L-l)\right\rangle$
$\displaystyle=$
$\displaystyle\begin{cases}\sqrt{\frac{2}{L-l}}\sin\left[\frac{n\pi\left(x-l\right)}{L-l}\right]&l\leq
x\leq L\\\ 0&0\leq x\leq l\end{cases},$ (8)
and
$\left\langle x\right.\left|\psi_{n}^{L}(l)\right\rangle=\begin{cases}0&l\leq
x\leq L\\\ \sqrt{\frac{2}{l}}\sin(n\pi x/l)&0\leq x\leq l\end{cases},$
with the corresponding eigen-values $E_{n}\left(L-l\right)$ and
$E_{n}\left(l\right)$. In the following discussions we use the free
Hamiltonian $H_{T}=H+H_{D}$ for
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}[E_{n}\left(l\right)\left|\psi_{n}\left(l\right)\right\rangle\left\langle\psi_{n}\left(l\right)\right|$
$\displaystyle+E_{n}(L-l)\left|\psi_{n}\left(L-l\right)\right\rangle\left\langle\psi_{n}\left(L-l\right)\right|]$
for $0\leq l\leq L$ and $H_{D}=\Delta\left|e\right\rangle\left\langle
e\right|.$ Here, we take its ground state energy as the zero point of energy
of atom.
At the end of the insertion process, the system is still in the thermal state
with the temperature $\beta$ and the MD is on its own state without any
changes. With respect to the above splitted bases, the state of the whole
system is rewritten in terms of the new bases as
$\rho_{\mathrm{ins}}=[P_{L}\left(l\right)\rho^{L}\left(l\right)+P_{R}\left(l\right)\rho^{R}\left(L-l\right)]\otimes\rho_{0}^{D},$
(9)
where
$\rho^{L}\left(l\right)=\sum_{n}\frac{e^{-\beta
E_{n}\left(l\right)}}{Z\left(l\right)}\left|\psi_{n}^{L}(l)\right\rangle\left\langle\psi_{n}^{L}(l)\right|,$
(10)
and
$\rho^{R}\left(L-l\right)=\sum_{n}\frac{e^{-\beta
E_{n}\left(L-l\right)}}{Z\left(L-l\right)}\left|\psi_{n}^{R}(L-l)\right\rangle\left\langle\psi_{n}^{R}(L-l)\right|,$
(11)
refer to the system localized in the left and right domain respectively. With
respect to the their sum
$\mathcal{Z}\left(l\right)=Z\left(l\right)+Z\left(L-l\right),$ the temperature
dependent ratios
$P_{L}\left(l\right)=Z\left(l\right)/\mathcal{Z}\left(l\right)$
and
$P_{R}\left(l\right)=Z\left(L-l\right)/\mathcal{Z}\left(l\right).$
are the probabilities to find the single molecule on the left and the right
side respectively. For simplicity, we denote $P_{L}\left(l\right)$ and
$P_{R}\left(l\right)$ by $P_{L}$ and $P_{R}$ respectively in the following
discussions. We emphasize that the probabilities are different from the
classical probabilities, $P_{L}^{c}=l/L$ and $P_{L}^{c}=\left(L-l\right)/L$,
finding single molecule on the left and right side that is proportional to the
volume. We numerically illustrate this discrepancy between this classical
result and ours in Fig. 2 for different insertion position $l=1/3$ and
$l=1/4$. It is clearly in Fig. 2 that the probabilities $P_{L}$ approaches to
the corresponding classical ones $P_{L}^{c}$, as the temperature increases to
the high temperature limit. However, a large discrepancy is observed at low
temperature. This deviate from the classical one is due to the discreteness of
the energy levels of the potential well with finite width, which disappears as
level spacing becomes small with $L\rightarrow\infty$. In this case, the heat
excitation will erase all the quantum feature of the system and the classical
limit is approached.
In this step, work should be done to the system. In the isothermal process,
the work done by the outside agent can be expressed as
$W_{\mathrm{ins}}=\Delta U_{\mathrm{ins}}-T\Delta S_{\mathrm{ins}}$, with the
internal energy change
$\Delta
U_{\mathrm{ins}}=\mathrm{Tr}\left[\left(\rho_{\mathrm{ins}}-\rho_{0}\right)H_{T}\right]$
and the total entropy change
$\Delta
S_{\mathrm{ins}}=\mathrm{Tr}\left(-\rho_{\mathrm{ins}}\ln\rho_{\mathrm{ins}}+\rho_{0}\ln\rho_{0}\right).$
During this isothermal process, the work done by outside just compensates the
change of the free energy as
$W_{\mathrm{ins}}=T\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (12)
The same result has been obtained in Ref. Ueda2010 . By taken inverse
temperature $\beta=1$ and $L=1$, we illustrate the work needed for the
insertion of the piston into the potential in Fig. 3. It is shown that to
insert the piston at the center of the potential needs the maximum work to be
done. Another reasonable fact is that no work is needed to insert the piston
at position $l=0$ and $l=L$. Classically, it is well known that no work should
be paid for inserting the piston at any position, while for a fixed $L$, we
notice that $W_{\mathrm{ins}}\rightarrow-\infty$ as $T\rightarrow\infty$. The
discrete property of the system due to the finite width of the potential well
results in the typical quantum effect, even at a high temperature, namely,
$\lim_{T\rightarrow\infty}W_{\mathrm{ins}}\neq 0$ and
$\lim_{T\rightarrow\infty}Q_{\mathrm{ins}}\neq 0$. This finite size induced
quantum effect is typical for mesoscopic system. To restore the classical
results, we simply take the limit $L\rightarrow\infty$ to make the spectrum
continuous, rather than $T\rightarrow\infty$. Under this limit
$L\rightarrow\infty$, we have
$\mathcal{Z}\left(l\right)/Z\left(L\right)\rightarrow 1$, which just recovers
the classical result that
$\lim_{L\rightarrow\infty}W_{\mathrm{ins}}=0,$ (13)
as illustrated in Fig. 3(b) for different insertion positions $l=0.1L$, $0.3L$
and $0.5L$.
Figure 3: (Color Online) Work done by the outside agent. (a)$W_{\mathrm{ins}}$
vs $l$ for different system inverse temperature $\beta=1$, $0.5$ and $0.1$.
Here, we choose the same parameter as that in Fig. 2. (b) $W_{\mathrm{ins}}$
vs $L$ for different insertion position $l=0.1L$, $0.3L$ and $0.5L$.
After the insertion of piston, the entropy of the system changes. The system
exchanges heat with the heat bath during this isothermal reversible process.
The heat is obtained by $Q_{\mathrm{ins}}=-T\Delta S_{\mathrm{ins}}$ as
$Q_{\mathrm{ins}}=\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (14)
Similar to the asymptotic properties of the work in Eq. (13),
$Q_{\mathrm{ins}}$ approaches to zero when $L\rightarrow\infty$.
### Step2: Quantum Measurement (II-III)
In the second step, the system is isolated from the heat bath. The MD finds
which domain, the single molecule stays in and registers the result into its
own memory. In the classical way, the memory can also be imaged as a chamber
with single molecule. The classical state of single molecule on the right and
left side are denoted as the state $0$ and $1$. And the memory is architecture
always by two bistable states with no energy difference $\Delta=0$ and no
energy is needed in the measurement process. This setup based on “chamber ”
argument seems to exclude the possibility for quantum coherence in a
straightforward way. Therefore, we adopt the TLS as the memory to allow the
quantum coherence to take the role, as discussed in Sec. II. In the scheme
here, the demon performs the controlled-NOT operation Quan2006 . If the
molecule is on the left side, no operation is done. And the demon flips its
state, when finding the molecule on the right. This operation is realized by
the following unitary operator,
$\displaystyle U$ $\displaystyle=$
$\displaystyle\sum_{n}\left|\psi_{n}^{L}\left(l\right)\right\rangle\left\langle\psi_{n}^{L}\left(l\right)\right|\otimes\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)$ (15)
$\displaystyle+\left|\psi_{n}^{R}\left(L-l\right)\right\rangle\left\langle\psi_{n}^{R}\left(L-l\right)\right|\otimes\left(\left|e\right\rangle\left\langle
g\right|+\mathrm{h.c}\right).$
After the measurement, the MD and the system are correlated. This correlation
enables the MD to control the system to perform work to the outside agent. The
density matrix of the whole system after measurement is
$\displaystyle\rho_{\mathrm{mea}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l\right)+P_{R}p_{e}\rho^{R}\left(L-l\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$ (16)
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l\right)+P_{R}p_{g}\rho^{R}\left(L-l\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$
If the temperature of the demon is zero, namely $T_{D}=0$, the measurement
actually results in a perfect correlation between the system and the MD,
$\rho_{\mathrm{mea}}=P_{L}\rho^{L}\left(l\right)\otimes\left|g\right\rangle\left\langle
g\right|+P_{R}\rho^{R}\left(L-l\right)\otimes\left|e\right\rangle\left\langle
e\right|.$ (17)
Then the demon can distinguish exactly the domain where the single molecule
stays, e.g. state $\left|g\right\rangle$ representing the molecule on left
side and vice visa. At a finite temperature, this correlation gets ambiguous.
As illustrated in the dashed green box in Fig. 1(b), the demon actually gets
the wrong information about the domain, where the single molecule stays. For
example, the demon thinks the molecule is on the left with memory registering
$\left|g\right\rangle$, while the molecule is actually on the right. The MD
loses a certain amount of information about the system and lowers its ability
to extract work. For case $\Delta\neq 0$ at finite temperature, the above
imperfect correlation leads to a condition for the MD’s temperature, under
which the total system could extract positive work.
The worst case is that, when we first let the MD to become degenerate, i.e.,
$\Delta=0,$ then the temperature to approach zero. In this sense the demon is
prepared in s mixing state
$\rho_{0}^{D}\left(\Delta=0\right)=\frac{1}{2}\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)$
and the state of the whole system after the measurement reads
$\rho_{\mathrm{mea}}=\left[\rho^{L}\left(l\right)+\rho^{R}\left(L-l\right)\right]\otimes\rho_{0}^{D}\left(\Delta=0\right).$
(18)
Thus, no information is obtained by MD. There exists another limit process
that the non-degenerate MD is firstly prepared in the zero-temperature
environment, and then let $\Delta$ approach zero. Thus, the state of the MD is
broken into $\left|g\right\rangle\left\langle g\right|$ of
$\rho_{0}^{D}\left(\Delta=0\right)$. In this case, we get a more cleaver MD as
mentioned above. The physical essence of the difference between the two limit
processes lies on the symmetry breakingjqliao2009 (we will discuss this again
later). With such symmetry breaking, the degenerate MD could also make an
ideal measurement. A intuitive understanding for the zero-temperature MD
helping to do work is that a more calm MD can see the states of the molecule
more clear, thus control it more effectively.
Next we calculate the work done in the measurement process by assuming the
total system being isolated from the heat bath of the molecule. The heat
exchange here is exactly zero, namely $Q_{\mathrm{mea}}=0$, since the
operation is unitary and the total entropy is not changed during this process.
However, the total internal energy changes, which merely results from the work
done by the outside agent
$W_{\mathrm{mea}}=P_{R}\left(p_{g}-p_{e}\right)\Delta.$
to register the MD’s memory. The work needed is actually a monotonous function
of the demon’s bath temperature $T_{D}$. If the temperature of the demon is
zero ( the MD is prepared in a pure state ), namely $T_{D}=0$, the work
reaches its maximum $W_{\mathrm{mea}}^{\mathrm{max}}=P_{R}\Delta$. The demon
can distinguish exactly the domain, where the single molecule stays, state
$\left|g\right\rangle$ representing molecule on the left, and vice visa. As
discussed as follows, the work done by the outside agent here is the main
factor to low down the efficiency of the heat engine. However, the low
temperature results in a more perfect quantum correlation between the MD and
the system, thus enables the MD to extract more work. Requirement of the work
done in the measurement and the ability of controlling free expansion are two
competing factors of the QHE. Finally, we prove that a low temperature of the
demon results in the high efficiency of quantum heat engine in the following
section. It is clear that less work is needed, if the insertion position is
closer to the right boundary of the potential. And the work needed in the
measurement process approaches to zero, namely $W_{mea}\rightarrow 0$, when
$l\rightarrow L$. Thus, the efficiency is promoted to reach the corresponding
Carnot efficiency when $l=L$ for this measurement.
### Step3: Controlled Expansion (III-IV)
In the third step, the system is brought into contact with the heat bath with
temperature $\beta$. Then the expansion is performed slowly enough to enable
the process to be reversible and isothermal. The expansion is controlled by
the demon according to its memory. Finding its state on
$\left|g\right\rangle$, the outside agent allows the piston to move right,
thus the single molecule performs work to the outside. However, the agent pays
some work to move piston to the right if the MD’s memory is inaccurate, e.g.
the situation in the green dashed box in Fig. 1(b). If in state
$\left|e\right\rangle$, the piston is allowed to move to the left side. Under
this description, we avoid the conventional heuristic discussion with adding
an object in the classical version of SHE. Here, we choose two arbitrary final
positions of the controlled expansion as $l_{g}$ and $l_{e}$ for the
corresponding MD’s state $\left|g\right\rangle$ and $\left|e\right\rangle$. We
will prove later that the total work extracted is independent on the expansion
position chosen here. After the expansion process, the density matrix of the
whole system is expressed as
$\displaystyle\rho_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l_{g}\right)+P_{R}p_{e}\rho^{R}\left(L-l_{g}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$ (19)
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l_{e}\right)+P_{R}p_{g}\rho^{R}\left(L-l_{e}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$
During the expansion, the system performs work $-W_{\mathrm{exp}}\geq 0$ to
the outside agent,
$\displaystyle W_{\mathrm{exp}}$ $\displaystyle=$ $\displaystyle
T\left[\ln\mathcal{Z}\left(l\right)+P_{L}\ln P_{L}+P_{R}\ln P_{R}\right.$ (20)
$\displaystyle-P_{L}p_{g}\ln Z\left(l_{g}\right)-P_{R}p_{e}\ln
Z\left(L-l_{g}\right)$ $\displaystyle\left.-P_{L}p_{e}\ln
Z\left(l_{e}\right)-P_{R}p_{g}\ln Z\left(L-l_{e}\right)\right].$
For a perfect correlation ($p_{g}=1$), the piston is moved to the side of the
potential, namely $l_{g}=L$ and $l_{e}=0$, and the work is simply
$W_{\mathrm{exp}}=T\left(P_{L}\ln P_{L}+P_{R}\ln
P_{R}\right)-W_{\mathrm{ins}},$
which is the maximum work one can be extracted in this process. In the
classical limit $L\rightarrow\infty$, and the work is
$W_{\mathrm{exp}}=T\left(P_{L}\ln P_{L}+P_{R}\ln P_{R}\right).$
We restore the well known result $W_{\mathrm{exp}}=-k_{\mathrm{B}}T\ln 2$,
when the piston is inserted in the center of the potential. If the demon is
not perfectly correlated to the position of the single molecule ($p_{g}<1$),
the work extracted $-W_{\mathrm{exp}}$ would be less. Therefore, it is clear
that the ability of MD to extract work closely depends on the accuracy of the
measurement.
In this step, the heat exchange is related to the change of entropy as
$\displaystyle\\!\\!\\!\\!Q_{\mathrm{exp}}=$ $\displaystyle
P_{L}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L-l_{e}\right)\right]$
$\displaystyle\\!\\!\\!\\!+P_{R}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L-l\right)-p_{e}\ln Z\left(L-l_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$ (21)
### Step4: Removing(IV-V)
To complete the thermodynamic cycle, the system and the MD should be reset to
their own initial states respectively. As for the system, the piston inserted
in the first step should be removed. In the previous studies, this process is
neglected, since the measurements are always ideal and the piston is moved to
an end boundary of the chamber. Thus no work is required to remove piston.
However, in an arbitrary process, we can show the importance of removing
piston in the whole cycle. During this process, the system keeps contact with
the heat bath with inverse temperature $\beta$ and the removing is performed
isothermally. The density matrix of the total system after removing the piston
reads
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\rho_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{e^{-\beta
E_{n}\left(L\right)}}{Z(L)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes$
(22)
$\displaystyle\\!\\!\\!\\!\\!\left[\left(P_{L}p_{g}\\!+\\!P_{R}p_{e}\right)\left|g\right\rangle\\!\\!\left\langle
g\right|\\!+\\!\left(P_{L}p_{e}\\!+\\!P_{R}p_{g}\right)\left|e\right\rangle\\!\\!\left\langle
e\right|\right].$
In this process, the work done and the heat absorbed by the outside are
$\displaystyle\\!\\!\\!\\!\\!\\!\\!W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\left(\rho_{\mathrm{rev}}-\rho_{\mathrm{exp}}\right)\left(H+H_{D}\right)\right]$
(23)
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right],$
and
$Q_{\mathrm{rev}}=-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right],$
(24)
respectively. We refer the Appendix for the exact expression of those two
formula. The MD now is no longer entangled with the system. And the density
matrix of the demon is factorized out as
$\rho_{\mathrm{rev}}^{D}=\left(P_{L}p_{g}\\!+\\!P_{R}p_{e}\right)\left|g\right\rangle\\!\\!\left\langle
g\right|\\!+\\!\left(P_{L}p_{e}\\!+\\!P_{R}p_{g}\right)\left|e\right\rangle\\!\\!\left\langle
e\right|.$ (25)
In the ideal case $T_{D}=0$, the demon is on the state
$\rho_{\mathrm{rev}}^{D}=P_{L}\left|g\right\rangle\left\langle
g\right|+P_{e}\left|e\right\rangle\left\langle e\right|$
with entropy
$S_{\mathrm{rev}}^{D}=-P_{L}\ln P_{L}-P_{R}\ln P_{R}.$
According to Landauer’s Principal, erasing the memory of the MD dissipates at
least $T_{D}S_{\mathrm{rev}}^{D}=0$ work into the environment. In this sense,
we can extracted $k_{B}T\ln 2$ work with MD’s help. However, we does not
violate the SLoT, since the whole system functionalizes as a heat engine
working between high temperature bath and zero temperature bath. Actually, the
increase of entropy in the zero temperature bath is exactly
$S_{\mathrm{rev}}^{D}$. Therefore, the energy dissipated actually depends on
the temperature of environment, where the information is erased. In the
previous studies, people always set the same temperature for the system and
MD. Thus the exactly mechanism of MD was not clear to certain extent,
especially for SHE. Let’s consider another special case $\Delta=0$, which
directly results in $p_{e}=p_{g}=1/2$. MD is prepared on its maximum entropy
state $\rho_{0}^{D}\left(\Delta=0\right)$. At the end of the cycle, MD
actually is on the same state, namely
$\rho_{\mathrm{rev}}^{D}=\rho_{0}^{D}\left(\Delta=0\right)$. Thus, no work is
paid to erase the memory.
After this procedure, the MD is decoupled from the system and brought into
contact with its own thermal bath with inverse temperature $\beta_{D}$. Since
$P_{L}p_{e}+P_{R}p_{g}\geq p_{e},$ (26)
the MD releases energy into its heat bath. We will not discuss this
thermalization process here in details. The MD and the system are reset to
their own initial states $\rho_{0}$, which allows a new cycle to start.
Figure 4: (Color Online) Work vs insertion position $l$ and MD’s inverse
temperature $\beta_{D}$. (a) Total work as a function of $\beta_{D}$ for
different $l=0.2$, $0.5$ and $0.8$. (b)Total work as a function of insertion
position $l$ for different $\beta_{D}=2.0$, $3.0$ and $4.0$. (c) Contour plot
for total work as function of $l$ and $\beta_{D}$. The position for maximum
work extracted is denoted as white dashed line. Figure 5: (Color Online)
Efficiency vs insertion position $l$ and O inverse temperature $\beta_{D}$.
(a) Efficiency as a function of $\beta_{D}$ for different $l=0.4$, $0.5$ and
$0.6$. (b) Efficiency as a function $l$ for different $\beta_{D}=2$, $3$ and
$4$. (c) Contour plot of efficiency vs $l$ and $\beta_{D}$.
## IV Efficiency of Szilard Heat Engine
For quantum version of the SHE, the quantum coherent based on the finite size
of the chamber results in various different properties from the classical one.
Work is required during the insertion and removing processes, while the same
process can be done freely in the classical version. The microscopic model
here relates the efficiency of the measurement by MD to the temperature of the
heat bath. In the whole thermodynamic cycle, the work done by the system to
outside is the sum of all the work done in each process,
$\displaystyle W_{\mathrm{tot}}$ $\displaystyle=$
$\displaystyle-\left(W_{\mathrm{ins}}+W_{\mathrm{mea}}+W_{\mathrm{exp}}+W_{\mathrm{rev}}\right)$
(27) $\displaystyle=$ $\displaystyle T\left[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\right.$
$\displaystyle-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\left.-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right]$
$\displaystyle-P_{R}\left(p_{g}-p_{e}\right)\Delta.$
To enable the system to do work outside, the temperature of the MD should be
low enough to make sure $W_{\mathrm{tot}}\geq 0$, which is known as the
positive-work condition(PWC) htquan . To evaluate the efficiency of QHE, we
need to obtain the heat absorbed from the high temperature heat bath.
Different from the classical one, the exchange of heat with high temperature
source persists in each step. The total heat absorbed from the high
temperature source is the sum over that of all the four steps,
$\displaystyle Q_{\mathrm{tot}}$ $\displaystyle=$
$\displaystyle-\left(Q_{\mathrm{ins}}+Q_{\mathrm{mea}}+Q_{\mathrm{exp}}+Q_{\mathrm{rev}}\right)$
(28) $\displaystyle=$ $\displaystyle T\left[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\right.$
$\displaystyle-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\left.-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right].$
Here, the absorbed energy is used to perform work to the outside, while only
the measurement process wastes $W_{\mathrm{mea}}$, which is released to the
low temperature heat bath. It is very interesting to notice that
$W_{\mathrm{mea}}\rightarrow 0$ as $\Delta\rightarrow 0$, while the total heat
$Q_{\mathrm{tot}}\rightarrow 0$ and $W_{\mathrm{tot}}\rightarrow 0$. To check
the validity of SLoT, one should concern the the efficiency of this heat
engine in a cycle,
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
1-\frac{P_{R}\left(p_{g}-p_{e}\right)\Delta}{Q_{\mathrm{tot}}}.$ (29)
As an example, we consider the special case $l=L/2$, which is similar to the
case of the ordinary SHE with the piston inserted in the center of the
chamber. In this special case, the probabilities for the single molecule
staying at the two sides are the same as that of the classical one, namely
$P_{L}=P_{R}=1/2$. The total work extracted here can be written in a simple
form
$W_{\mathrm{tot}}=T\left(\ln 2+p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)-\left(p_{g}-p_{e}\right)\Delta/2.$ (30)
In this special case, to make the system capable to do work on the outside,
there is a requirement to the temperature of the demon (low temperature bath).
For example, when we choose $\beta=1$ and $\Delta=0.5$, the PWC is
$\beta_{D}\geq 2.09$. This requirement is more strict than that of Carnot heat
engine, $\beta_{D}>1$. And the efficiency of this heat engine reads
$\eta=1-\frac{\left(p_{g}-p_{e}\right)\Delta}{2T\left(\ln 2+p_{e}\ln
p_{e}+p_{g}\ln p_{g}\right)},$ (31)
which is lower than the corresponding Carnot efficiency
$\eta_{\mathrm{Carnot}}=1-\frac{T_{D}}{T}.$
Here, the efficiency is a monotonic function of the energy spacing $\Delta$
and reaches its maximum
$\eta_{\mathrm{max}}=1-\frac{2T_{D}}{T}\leq\eta_{\mathrm{Carnot}}$
with $\Delta=0$.
In the general case, we show the work done by the system and efficiency of the
heat engine vs the position of the wall $l$ and the temperature of demon
$\beta_{D}$ in Fig. 4 and Fig. 5. As illustrated in Fig. 4(a), for small
insertion position, e.g. $l=0.16$ and $0.36$, the system can not extract
positive work. There exists a critical insertion position $l_{\mathrm{cri}}$
to extract positive work, namely
$T\left(P_{L}^{\mathrm{cri}}\ln P_{L}^{\mathrm{cri}}+P_{R}^{\mathrm{cri}}\ln
P_{R}^{\mathrm{cri}}\right)+P_{R}^{\mathrm{cri}}\Delta=0,$ (32)
where $P_{R}^{\mathrm{cri}}=P_{R}\left(l_{\mathrm{cri}}\right)$ and
$P_{L}^{\mathrm{cri}}=P_{L}\left(l_{\mathrm{cri}}\right)$. This critical value
of insertion position here is $l_{\mathrm{cri}}=0.447$ for the typical
parameter chosen here. Due to the requirement of work in the measurement
process, the work extracted is not a symmetric function of the insertion
piston $l$, namely $W_{\mathrm{tot}}\left(0.5-l\right)\neq
W_{\mathrm{tot}}\left(0.5+l\right)$, as illustrated in Fig. 4(b,c). Since the
high energy state $\left|e\right\rangle$ of the demon is utilized to register
the right side for single molecule, more work is need when $l<L/2$. Due to the
requirement of work done by outside agent in the measurement process, the
optimal position to extract maximum work is not at the center of the
potential. The maximum work can be extracted for a given MD’s inverse
temperature is reached, when
$\frac{P_{L}^{\mathrm{wmax}}p_{e}+P_{R}^{\mathrm{wmax}}p_{g}}{P_{L}^{\mathrm{wmax}}p_{g}+P_{R}^{\mathrm{wmax}}p_{e}}=e^{-\beta\Delta},$
(33)
where $P_{L}^{\mathrm{wmax}}=P_{L}\left(l_{\mathrm{wmax}}\right)$ and
$P_{R}^{\mathrm{wmax}}=P_{R}\left(l_{\mathrm{wmax}}\right)$. It is clear that
the position for the maximum work depends on the temperature of the demon
$\beta_{D}$.
In Fig. 5, we show the efficiency of this single molecular heat engine. We
consider only the positive work situation, and set efficiency as $0$ for all
the negative work area. Fig. 5(a) shows the monotonous behavior of efficiency
as the MD’s inverse temperature. Efficiency is also a monotonous function of
the insertion position $l$, illustrated in Fig. 5(b,c), which is not similar
to the total work extracted. It worth noticing that the efficiency reaches its
maximum at $l=1$, while no work can be extracted. Since the measurement is the
only way of wasting energy, it is the only way to improve the efficiency by
reducing $W_{\mathrm{mea}}$ with decreasing $P_{R}$. The efficiency of QHE
reaches the well-known Carnot efficiency $\eta_{\mathrm{Carnot}}$, when
$P_{R}=0$. At the same time, the total work extracted approaches to zero,
namely $W_{\mathrm{tot}}=0$. We meet this dilemma, since the measurement
results in an imperfect correlation between MD and the system.
Before concluding this paper, we draw our attention to two limit
processesjqliao2009 again
$\displaystyle\lim_{\beta_{D}\rightarrow+\infty}\lim_{\Delta\rightarrow
0}\rho_{D}$ $\displaystyle=\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)/2,$ (34)
$\displaystyle\lim_{\Delta\rightarrow
0}\lim_{\beta_{D}\rightarrow+\infty}\rho_{D}$
$\displaystyle=\left|g\right\rangle\left\langle g\right|.$ (35)
Note that taking the two limits in different orders leads to completely
different results, the latter being a reflection of the spontaneous symmetry
breaking phenomenon. This difference for the MD’s initial state results in the
different work extracted, namely,
$\displaystyle\lim_{\beta_{D}\rightarrow+\infty}\lim_{\Delta\rightarrow
0}W_{\mathrm{tot}}$ $\displaystyle=0,$ (36)
$\displaystyle\lim_{\Delta\rightarrow
0}\lim_{\beta_{D}\rightarrow+\infty}W_{\mathrm{tot}}$ $\displaystyle=k_{B}T\ln
2.$ (37)
The former one means that MD actually gets no information about the position
of molecule and extracts no work, while the latter one show that MD obtains
the exact information on the position of the molecule and enables the system
to perform maximum work to the outside agent. The same phenomenon has also
been revealed in the process of dynamic thermalizationjqliao2009 .
## V Conclusions
In summary, we have studied a quantum version of SHE with a quantum MD with
lower finite temperature than that of the system. We overall simplified the MD
as a two-level system, which carries out measurement in quantum fashion and
controlling the system to do work to the out-side agent. In this sense, the MD
assisted thermodynamic cycle are clarified as the four steps, insertion,
measurement, expansion and removing, which are all described in the framework
of quantum mechanics. In each step, we also consider the special case to
restore the well-known results in classical version of SHE. We explicitly
analyzed the total work extracted and the corresponding efficiency. To resolve
the MD paradox, we compared the obtained efficiency of the heat engine with
that of Carnot heat engine. It is found the efficiency is always below that of
Carnot since the quantum MD is included as the a part of the the whole work
substance and its functions are also correctly “quantized”. Thus nothing
violates the SLoT.
In comparison with the classical version of SHE, the following quantum natures
were discovered in the quantum thermodynamic cycles: (1) The finite size
effect of the potential well was found as reason for the non-vanishing work
required in the insertion and removing of the middle walls, while the
corresponding manipulations could be achieved freely in the classical case;
(2) The quantum coherence is allowed to exist in the MD’s density matrix. It
is the decrease of effective temperature caused by this coherence that
actually improves the efficiency of SHE; (3) In the measurement process, the
finite temperature of MD actually results in the incorrect decision to control
the single molecule’s motion. This incorrectness decreased the MD’s ability to
extract work. To our best knowledge, even for in the classical case, the
similar investigation has never been carried out; (4) In the whole
thermodynamic cycle, the removing process is necessary in returning to the
initial state for the whole work substance. This fact is neglected in the
previous studies even for the classical SHE.
Finally, we should stress that the model studied here could help to resolve
many paradoxical observations due to heuristic arguments with hybridization of
classical-quantum points of views about thermodynamics. For instance, it could
be recognized that the conventional argument about the MD paradox only
concerns a classical version of MD at the same temperature as that of the
system. Our results can enlighten the comprehensive understandings about some
fundamental problems in thermodynamics, such as the relationship between
quantum unitarity and SLoTDong2010 .
###### Acknowledgements.
HD would like to thank J.N Zhang for helpful discussion. This work was
supported by NSFC through grants 10974209 and 10935010 and by the National 973
program (Grant No. 2006CB921205).
## Appendix
In this appendix, we present a detailed calculation for the work done and
efficiency of SHE. Following the calculations for the four steps listed in the
context step by step, the reader can deeply understand the physical essences
of the MD in some subtle fashion.
Step 1: Insertion. In this process, the changes of internal energy $\Delta
U_{\mathrm{int}}=\mathrm{Tr}\left[\left(H+H_{D}\right)\left(\rho_{\mathrm{ins}}-\rho_{0}\right)\right]$
and total entropy $\Delta
S_{\mathrm{ins}}=\mathrm{Tr}\left[-\rho_{\mathrm{ins}}\ln\rho_{\mathrm{ins}}\right]-\mathrm{Tr}\left[-\rho_{\mathrm{0}}\ln\left(\rho_{0}\right)\right]$
is explicitly given by
$\displaystyle\Delta U_{\mathrm{int}}$
$\displaystyle=\sum_{n}p_{n}\left(l\right)E_{n}\left(l\right)+$
$\displaystyle\sum_{n}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)-\sum_{n}p_{n}\left(L\right)E_{n}\left(L\right)$
(38) $\displaystyle=\frac{\partial}{\partial\beta}\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right],$
where $L^{\prime}=L-l$ and
$\displaystyle\Delta S_{\mathrm{ins}}$
$\displaystyle=\left(\ln\mathcal{Z}\left(l\right)-\ln Z\left(L\right)\right)+$
$\displaystyle\beta\sum_{n}\left[\begin{array}[]{c}p_{n}\left(l\right)E_{n}\left(l\right)\\\
+p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)-p_{n}\left(L\right)E_{n}\left(L\right)\end{array}\right]$
(41)
$\displaystyle=\left(1-\beta\frac{\partial}{\partial\beta}\right)\left(\ln\mathcal{Z}\left(l\right)-\ln
Z\left(L\right)\right),$ (42)
where
$p_{n}\left(y\right)=\frac{\exp\left(-\beta
E_{n}\left(y\right)\right)}{Z\left(y\right)}.$
For the isothermal process, the work done by outside agent and the heat
exchange are simply $W_{\mathrm{ins}}=\Delta U_{\mathrm{int}}-T\Delta
S_{\mathrm{ins}}$ and $Q_{\mathrm{ins}}=-T\Delta S_{\mathrm{ins}}$, namely,
$\displaystyle W_{\mathrm{ins}}$ $\displaystyle=T\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right],$ (43) $\displaystyle
Q_{\mathrm{ins}}$
$\displaystyle=\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (44)
Step2: Measurement. The measurement is realized by a controlled-NOT unitary
operation, which has been illustrated clearly in the Sec. II. After the
measurement process, the density matrix for the total system is
$\displaystyle\rho_{\mathrm{mea}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l\right)+P_{R}p_{e}\rho^{R}\left(L^{\prime}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l\right)+P_{R}p_{g}\rho^{R}\left(L^{\prime}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$
The entropy is not changed in this step. And the work done by outside is
$W_{\mathrm{mea}}=\Delta
U_{\mathrm{mea}}=P_{R}\left(p_{g}-p_{e}\right)\Delta.$ (45)
Step3: Controlled expansion. At the ending of expansion, the state for the
total system reads
$\displaystyle\rho_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l_{g}\right)+P_{R}p_{e}\rho^{R}\left(L_{g}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(L_{e}\right)+P_{R}p_{g}\rho^{R}\left(l_{e}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$
where $L_{g}=L-l_{g}$ and $L_{e}=L-l_{e}$
We move the wall isothermally. And the work done by out-side agent can be
obtain by the same methods used in insertion process as
$\displaystyle W_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\rho_{\mathrm{exp}}\left(H+H_{D}\right)\right]-\mathrm{Tr}\left[\rho_{\mathrm{mea}}\left(H+H_{D}\right)\right]$
(46)
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{mea}}\ln\rho_{\mathrm{mea}}\right]$
$\displaystyle=$
$\displaystyle\sum_{n}[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)$
$\displaystyle+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)]+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-\sum_{n}\left(P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-T\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\ln
P_{L}p_{g}p_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L^{\prime}\right)\ln
P_{R}p_{e}p_{n}\left(L_{g}\right)\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln
P_{L}p_{e}p_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln
P_{R}p_{g}p_{n}\left(l_{e}\right)\right]$
$\displaystyle-T\sum_{n}[p_{g}p_{n}\left(l\right)\ln
p_{g}p_{n}\left(l\right)+p_{e}p_{n}\left(L^{\prime}\right)\ln
p_{e}p_{n}\left(L^{\prime}\right)$
$\displaystyle\qquad\qquad+p_{e}p_{n}\left(l\right)\ln
p_{e}p_{n}\left(l\right)+p_{g}p_{n}\left(L^{\prime}\right)\ln
p_{g}p_{n}\left(L^{\prime}\right)]$ $\displaystyle=$ $\displaystyle
P_{L}T\left[\ln Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{e}\right)\right]+P_{R}T\left[\ln Z\left(L^{\prime}\right)-p_{e}\ln
Z\left(L_{g}\right)-p_{g}\ln Z\left(l_{e}\right)\right].$ (47)
The internal energy changes can be also evaluated as
$\displaystyle\Delta U_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
(48) $\displaystyle+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-\sum_{n}\left(P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle=$
$\displaystyle\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle-\sum_{n}\left[P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right]$
$\displaystyle=$ $\displaystyle P_{L}\frac{\partial}{\partial\beta}\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{e}\right)\right]+P_{R}\frac{\partial}{\partial\beta}\left[\ln
Z\left(L^{\prime}\right)-p_{e}\ln Z\left(L_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$
Then, we obtain the heat exchanges in this process as
$Q_{\mathrm{exp}}=-T\Delta S_{\mathrm{exp}}=W_{\mathrm{exp}}-\Delta
U_{\mathrm{exp}}$ or
$\displaystyle Q_{\mathrm{exp}}$ $\displaystyle=$ $\displaystyle
P_{L}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{g}\right)\right]$
$\displaystyle+P_{R}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L^{\prime}\right)-p_{e}\ln Z\left(L_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$
Step4: Removing. The piston is removed in this process. After that, the system
returns to its initial state and is not entangled with MD as The last step
would be remove the wall in the trap. The system is on the state as
$\displaystyle\rho_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{\exp\left[-\beta
E_{n}\left(L\right)\right]}{Z(L)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes$
$\displaystyle\left[\left(P_{L}p_{g}+P_{R}p_{e}\right)\left|g\right\rangle\left\langle
g\right|+\left(P_{L}p_{e}+P_{R}p_{g}\right)\left|e\right\rangle\left\langle
e\right|\right].$
Then, the work done and the heat absorbed is respectively
$\displaystyle W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\rho_{\mathrm{rev}}\left(H+H_{D}\right)\right]-\mathrm{Tr}\left[\rho_{\mathrm{exp}}\left(H+H_{D}\right)\right]$
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]$
or
$\displaystyle W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}p_{n}\left(L\right)E_{n}\left(L\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
(51)
$\displaystyle-\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle-\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle+T[\sum_{n}p_{n}\left(L\right)\ln
p_{n}\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)]$
$\displaystyle-T\sum_{n}\left\\{P_{L}p_{g}p_{n}\left(l_{g}\right)\ln\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\right]+P_{R}p_{e}p_{n}\left(L_{g}\right)\ln\left[P_{R}p_{e}p_{n}\left(L_{g}\right)\right]\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln\left[P_{L}p_{e}p_{n}\left(L_{e}\right)\right]+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln\left[P_{R}p_{g}p_{n}\left(l_{e}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle T\left[-\ln
Z\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right.$
$\displaystyle\qquad-P_{L}\ln P_{L}-P_{R}\ln P_{R}-p_{e}\ln p_{e}-p_{g}\ln
p_{g}$ $\displaystyle\left.\qquad+P_{L}p_{g}\ln
Z\left(l_{g}\right)+P_{R}p_{e}\ln Z\left(L_{g}\right)+P_{L}p_{e}\ln
Z\left(L_{e}\right)+P_{R}p_{g}\ln Z\left(l_{e}\right)\right].$
and
$\displaystyle Q_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]$
(52) $\displaystyle=$ $\displaystyle T\left[\sum_{n}p_{n}\left(L\right)\ln
p_{n}\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right]$
$\displaystyle-T\sum_{n}\left\\{P_{L}p_{g}p_{n}\left(l_{g}\right)\ln\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\right]+P_{R}p_{e}p_{n}\left(L_{g}\right)\ln\left[P_{R}p_{e}p_{n}\left(L_{g}\right)\right]\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln\left[P_{L}p_{e}p_{n}\left(L_{e}\right)\right]+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln\left[P_{R}p_{g}p_{n}\left(l_{e}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle T\left\\{-\ln
Z\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right.$
$\displaystyle\left.\qquad-P_{L}\ln P_{L}-P_{R}\ln P_{R}-p_{e}\ln
p_{e}-p_{g}\ln p_{g}+P_{L}p_{g}\ln Z\left(l_{g}\right)+P_{R}p_{e}\ln
Z\left(L_{g}\right)+P_{L}p_{e}\ln Z\left(L_{e}\right)+P_{R}p_{g}\ln
Z\left(l_{e}\right)\right\\}$
$\displaystyle-\sum_{n}\left[p_{n}\left(L\right)E_{n}\left(L\right)-P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)-P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)\right.$
$\displaystyle\left.\qquad\qquad-
P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)-P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle=$ $\displaystyle
T\left[\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)-P_{L}\ln
P_{L}-P_{R}\ln P_{R}-p_{e}\ln p_{e}-p_{g}\ln p_{g}\right]$
$\displaystyle-\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L\right)+P_{L}p_{g}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(l_{g}\right)+P_{R}p_{e}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L_{g}\right)$
$\displaystyle+P_{L}p_{e}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L_{e}\right)+P_{R}p_{g}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(l_{e}\right).$
The total work extracted by outside agent is the sum of work extracted in each
step as
$\displaystyle W_{\mathrm{tot}}$
$\displaystyle=-\left(W_{\mathrm{ins}}+W_{\mathrm{mea}}+W_{\mathrm{exp}}+W_{\mathrm{rev}}\right)$
$\displaystyle=T[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\qquad\qquad-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)]-P_{R}\left(p_{g}-p_{e}\right)\Delta.$
(53)
The total heat absorbed can also be obtained as
$\displaystyle Q_{\mathrm{tot}}$
$\displaystyle=-\left(Q_{\mathrm{ins}}+Q_{\mathrm{exp}}+Q_{\mathrm{rev}}\right)$
$\displaystyle=T\left[\begin{array}[]{c}\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\\\
-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)\\\
-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\end{array}\right].$
(57)
## References
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|
arxiv-papers
| 2010-09-14T14:06:46 |
2024-09-04T02:49:12.915918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Dong, D.Z. Xu and C.P. Sun",
"submitter": "H. Dong",
"url": "https://arxiv.org/abs/1009.2672"
}
|
1009.2938
|
# Exploring Mount Neverest
Michiel de Bondt
In one of the columns in the series ‘Perplexities’ in 1922, Henry Ernest
Dudeney formulated the following problem:
> Professor Walkingholme, one of the exploring party, was allotted the special
> task of making a complete circuit of the base of the mountain at a certain
> level. The circuit was exactly a hundred miles in length and he had to do it
> all alone on foot. He could walk twenty miles a day, but he could only carry
> rations for two days at a time, the rations for each day being packed in
> sealed boxes for convenience in dumping. He walked his full twenty miles
> every day and consumed one day’s ration as he walked. What is the shortest
> time in which he could complete the circuit?
This problem can be found in the book ‘536 Puzzles & Curious Problems’ from
Henry Ernest Dudeney, edited by Martin Gardner. This far, I did not find an
optimal solution to the problem. Albeit Martin Gardner is making fun on it,
the right interpretation of the problem is not clear to me at all. Let us
first formulate some ways to tackle the problem.
## Solution of the problem
One way to make the circuit is doing the same as with a straight distance of
one hundred miles. No matter how you interpret this problem, this takes
between $82$ and $87$ days (Dudeney found a solution of $86$ days, but it can
be done in $82\frac{6097}{6144}$ days). A better approach is to walk two round
trips to the $50$ miles distance point on the other side of the mountain,
since it requires between $41$ and $43$ days (Dudeney would have found a
solution of $42\frac{1}{2}$ days if he had considered this problem). But
Dudeney found the following solution:
###### Algorithm 1.
($23\frac{1}{2}$ days)
1. 1.
Dump 5 rations at 90-mile point and return to base (5 days).
2. 2.
Dump 1 at 85 and return to 90 (1 day).
3. 3.
Dump 1 at 80 and return to 90 (1 day).
4. 4.
Dump 1 at 80, return to 85, pick up 1 and dump at 80 (1 day).
5. 5.
Dump 1 at 70 and return to 80 (1 day).
6. 6.
Return to base (1 day). We have thus left one ration at 70 and one at 90.
7. 7.
Dump 1 at 5 and return to base (1 day). If he must walk 20 miles he can do so
by going to 10 and returning to base.
8. 8.
Dump 4 at 10 and return to base (4 days).
9. 9.
Dump 1 at 10 and return to 5; pick up 1 and dump at 10 (1 day).
10. 10.
Dump 2 at 20 and return to 10 (2 days).
11. 11.
Dump 1 at 25 and return to 20 (1 day).
12. 12.
Dump 1 at 30, return to 25, pick up 1 and dump at 30 (1 day).
13. 13.
March to 70 (2 days).
14. 14.
March to base ($1\frac{1}{2}$ days).
Now let us look at step 7. In my opinion, it only takes half a day. You just
throw away half of the content of one of the boxes. Apparently, Dudeney and I
have a disagreement on the interpretation of the problem. The problem might be
that unsealed boxes can not pass through the night, since the forest ants
creep into it. But then, you just can start walking around the mountain in the
middle of the day and do step 7 prior to the others. If Dudeney would have
allowed not to start at dawn, it is likely that he would have formulated his
solution accordingly, so we can add the following items to the analysis of
Gardner:
1. 13.
If he does not finish his unsealed box(es) before the night, the forest ants
will.
2. 14.
Having a long lie-in is a waste of time.
We can divide Dudeney’s solution in three parts:
Part A:
a round trip to the 70 miles point from the base, in which boxes are dumped on
positions 70 and 90 for later use (steps 1 to 6, $10$ days).
Part B:
a one-way trip to the 70 miles point from the base around the mountain (steps
7 to 13, $11\frac{1}{2}$ or $12$ days).
Part C:
walking from 70 to 100, using the boxes dumped in part A (step 14,
$1\frac{1}{2}$ days).
Since it is almost proved that Dudeneys solution can not be optimal if we
count $12$ days for part B, we allow starting on mid-day. Only for part C, it
is immediately clear that is optimal. Later, we will see that part A is
optimal as well, but part B can be improved, to $11\frac{1}{7}$ days exactly.
Before trying to find a better solution, it is always a good idea what others
did. When I searched the internet with google, I found the homepage of a
youngster called Nightvid Cole, who presents better solutions than that of
Dudeney. In the first one, he allows throwing away partially used boxes (and
this solution can easily be adapted to overcome the ants, starting on mid-
day). Albeit only part C of this solution is optimal, it requires
$22\frac{10}{11}$ days instead of $23$. The improvement is that he dumps boxes
on positions $70\frac{10}{11}$ and $90\frac{10}{11}$ rather than $70$ and
$90$. Now part B must reach further, whence it takes $12$ days now.
But probably, he reasoned the other way around: he reserved 12 days for part B
and then thought out an according solution. That was a very good idea, but
since he failed to optimize parts A and B, his solution is not optimal. So I
optimized parts A and B, which resulted in the following solution, a solution
that turned out to be optimal later on:
###### Algorithm 2.
($22\frac{9}{16}$ days)
1. 1.
Dump one ration at $98\frac{3}{4}$ point and return to base ($\frac{1}{8}$
day).
2. 2.
Dump one ration at $97\frac{1}{2}$, return to $98\frac{3}{4}$, pick up one,
dump at $91\frac{1}{4}$ and return to base (1 day).
3. 3.
Dump one ration at $93\frac{3}{4}$, return to $97\frac{1}{2}$, pick up one,
dump at $93\frac{3}{4}$ also and return to base (1 day).
4. 4.
Dump two rations at $90$ and return to base (2 days).
5. 5.
Dump one ration at $86\frac{7}{8}$ and return to $93\frac{3}{4}$ (1 day).
6. 6.
Dump one ration at $82\frac{1}{2}$, return to $86\frac{7}{8}$, pick up one,
dump at $86\frac{1}{4}$ and return to $90$ (1 day).
7. 7.
Dump one ration at $80$, return to $86\frac{1}{4}$, pick up one and get to
$82\frac{1}{2}$ (1 day).
8. 8.
Dump one ration at $71\frac{1}{4}$ and return to $80$ (1 day).
9. 9.
Return to base (1 day). We have thus left one ration at $71\frac{1}{4}$ and
one at $91\frac{1}{4}$.
10. 10.
Dump five rations at $10$ and return to base (5 days).
11. 11.
Dump one ration at $12\frac{1}{2}$, return to $10$, pick up one, dump at
$12\frac{1}{2}$ also and return to $10$ (1 day).
12. 12.
Dump one ration at $20$ and return to $10$ (1 day).
13. 13.
Dump one ration at $20\frac{5}{8}$, return to $20$, pick up one, dump at
$20\frac{5}{8}$ also and return to $12\frac{1}{2}$ (1 day).
14. 14.
Dump one ration at $26\frac{9}{16}$ and return to $20\frac{5}{8}$ (1 day).
15. 15.
Dump one ration at $31\frac{1}{4}$, return to $26\frac{9}{16}$, pick up one
and get to $31\frac{1}{4}$ (1 day).
16. 16.
March to $71\frac{1}{4}$ (2 days).
17. 17.
March to base ($1\frac{7}{16}$ day).
If you look at the above algorithm, then one thing immediately strikes: it
would have been nicer if the circuit would have been 160 kilometers, with a
unit distance of 32 kilometers a day. Dudeney only considered solutions from
which the eating and turning points were a multiple of 5 miles. This is
however impossible for a solution of $22\frac{9}{16}$ days (for
$\frac{20}{5}\cdot 22\frac{9}{16}$ is not integral).
If professor Walkingholme must start at dawn, then the extra $\frac{7}{8}$
days must be used to increase the points where boxes are dumped in part A for
use in part C. Nightvid Cole found a solution of $23\frac{1}{3}$ days in this
context, but again, parts A and B are not optimal. The following optimal
solution not only has ugly positions, but also both part A and part B are
partially done on the first day.
###### Algorithm 3.
($23\frac{25}{116}$ days)
1. 1.
Dump one ration at $8\frac{18}{29}$ and return to base ($\frac{25}{29}$ days).
2. 2.
Dump two rations at $99\frac{9}{29}$ and return to base ($\frac{4}{29}$ days).
3. 3.
Dump one ration at $96\frac{26}{29}$, return to $99\frac{9}{29}$, pick up two
in turn, dump both at $95\frac{25}{29}$ and return to base (1 day).
4. 4.
Dump one ration at $90$ and return to base (1 day).
5. 5.
Dump one ration at $88\frac{28}{29}$, return to $90$, pick up one, dump at
$88\frac{28}{29}$ also and return to $95\frac{25}{29}$ (1 day).
6. 6.
Dump one ration at $82\frac{12}{29}$ and return to $88\frac{28}{29}$ (1 day).
7. 7.
Dump one ration at $75\frac{20}{29}$ and return to $82\frac{12}{29}$ (1 day).
8. 8.
Return to $96\frac{26}{29}$, pick up one, dump at $95\frac{20}{29}$ and return
to base (1 day). We have thus left one ration at $8\frac{18}{29}$, one at
$75\frac{20}{29}$ and another one at $95\frac{20}{29}$.
9. 9.
Dump one ration at $9\frac{9}{29}$, return to $8\frac{18}{29}$, pick up one,
dump at $9\frac{9}{29}$ also and return to base (1 day).
10. 10.
Dump five rations at $10$ and return to base (5 days).
11. 11.
Dump one ration at $12\frac{19}{58}$, return to $10$, pick up one, dump at
$12\frac{19}{58}$ also and return to $9\frac{9}{29}$ (1 day).
12. 12.
Dump one ration at $19\frac{19}{29}$ and return to $10$ (1 day).
13. 13.
Dump one ration at $19\frac{24}{29}$, return to $19\frac{19}{29}$, pick up
one, dump at $19\frac{24}{29}$ also and return to $10$ (1 day).
14. 14.
Dump one ration at $21\frac{19}{116}$ and return to $12\frac{19}{58}$ (1 day).
15. 15.
Dump one ration at $23\frac{18}{29}$, return to $21\frac{19}{116}$, pick up
one, dump at $23\frac{18}{29}$ also and return to $19\frac{24}{29}$ (1 day).
16. 16.
Dump one ration at $31\frac{21}{29}$ and return to $23\frac{18}{29}$ (1 day).
17. 17.
Dump one ration at $35\frac{20}{29}$, return to $31\frac{21}{29}$, pick up one
and get to $35\frac{20}{29}$ (1 day).
18. 18.
March to $75\frac{20}{29}$ (2 days).
19. 19.
March to base ($1\frac{25}{116}$ days).
## Estimates for part A
If professor Walkingholme replaces his unsealed box by a new full box each
time he passes the base, then no rations carried in part A are used in part B
and vice versa. So we can see part A and part B as separate problems.
The optimality of part B of both solutions is almost proved in [1] and [4]. In
both articles, the problem of how far you can get in $N$ days is solved for
integers $N$. But to cross a certain distance, it is very unlikely that you
need an integral number of days. The non-integral case is an easy variation,
however. We will use the techniques of these articles here.
Since part B is almost done in the above references, we only prove the
optimality of part A in full detail. For convenience, we measure the distance
in units of 20 miles from now and indicate the positions the other way around.
But before starting, it is always a good idea to determine what must be done.
In part A, professor Walkingholme must put boxes on positions $\gamma$ and
$\gamma-1$, to be used in part C. If $\gamma>2$, then another box on
$\gamma-2$ and maybe more boxes are needed, but taking $\gamma>2$ is so bad
that remission of the additional costs of getting to the base does not affect
the estimate. If on the other hand $\gamma<1$, then there does not need to be
carried a box to $\gamma-1$, but taking $\gamma<1$ will turn out to be a bad
idea as well.
Suppose for now that $\gamma>1$ and let $m$ be the moment of the first
unsealing after dumping a box on position $\gamma$ to be used in part C, say
at position $r>0$. Assuming that boxes are not carried back and forth
unnecessarily, there is a box on position $\gamma$ and another box on a
position between $0$ and $\gamma-1$ inclusive, to be carried to $\gamma-1$ at
the end.
Let
$0<e_{l}\leq e_{l-1}\leq\cdots\leq e_{2}\leq e_{1}$
be the positions $>0$ where a box is unsealed in part A before dumping a box
at $\gamma$ (prior to $m$), and define
$e_{l+1}=e_{l+2}=\cdots=0$
Suppose that before moment $m$, professor Walkingholme unseals the last box on
position $e_{j}$. Then $(\gamma-e_{j})+(\gamma-r)\leq 1$. Together with
$e_{1}\geq e_{j}$, we get
$\gamma\leq\frac{1}{2}e_{1}+\frac{1}{2}r+\frac{1}{2}$ (1)
This estimates $r$ to be $2\gamma-e_{1}-1$ at least. In case $\gamma\leq 1$,
$r$ might be negative. Therefore we define $r=0$ in case moment $m$ takes
place on a negative position. $e_{1}$ is always nonnegative by definition,
thus (1) is also valid when $\gamma\leq 1$.
Assume that $\gamma>1$ and let $t$ be the time professor Walkingholme uses for
part A. Then professor Walkingholme walks $\frac{1}{2}t$ units in forward
direction in part A, whence by elementary logistics
$\gamma+(\gamma-1)+r+\sum_{i=1}^{k}e_{i}\leq\frac{1}{2}t$ (2)
for all $k$. Since $\gamma-1\leq 0$ when $\gamma\leq 1$, the above estimate is
also valid when $\gamma\leq 1$. This estimate can only be effective when we
show the optimality of an algorithm for part A with $\gamma\geq 1$ and $r\leq
1$ . If $r>1$, then there must be an additional box on position $r-1$ at
least, whence
$\gamma+(\gamma-1)+r+(r-1)+\sum_{i=1}^{k}e_{i}\leq\frac{1}{2}t$
for all $k$. But this estimate is satisfied for $r\leq 1$ as well, since then
you just subtract $1-r$ from the left hand side of (2). It is only not
effective for $r<1$.
If we take the average of the above estimate and (2), we get
$\gamma+(\gamma-1)+r+\frac{r-1}{2}+\sum_{i=1}^{k}e_{i}\leq\frac{1}{2}t$ (3)
for all $k$, which can only be effective for $r=1$. Notice that both (2) and
(3) can only be effective if $k\geq l$.
Put $e_{0}:=r$ and define
$d_{i}:=\frac{1}{2}e_{i}+\frac{1}{2}e_{i+1}+\frac{1}{2}$
for all $i\geq 0$. In order to get more information, we ask the following
question: how many units does professor Walkingholme walk within the interval
$[\beta,\infty)$, before unsealing the box on position $r$ on moment $m$,
where $\beta\geq\frac{1}{2}$? To get the right idea on this question, we
assume that part A must satisfy the Dudeney rule that boxes are only unsealed
at dawn, albeit we need a more general result (which is left to the reader,
the days are too short to write it down).
To answer the question, we slice the total walk up to moment $m$ into parts
$w_{i}$, such that $w_{i}$ ends on the middle of the day that the box at
$e_{i}$ is unsealed (at dawn) and starts on the middle of the day before.
$w_{0}$ is just the last half-day walk and also the first slice does not need
to have length one. Now we can estimate how many units professor Walkingholme
walks within the interval $[\beta,\infty)$ in $w_{i}$ for all $i>0$:
* •
Case $\beta\leq e_{i}$: At most the length of $w_{i}$, i.e. one unit.
* •
Case $\beta-\frac{1}{2}\leq e_{i}\leq\beta$: At most $1-2(\beta-e_{i})$ units,
since a round trip from $\beta$ to $e_{i}$ is $2(\beta-e_{i})$ units.
* •
Case $e_{i}\leq\beta-\frac{1}{2}$: No units.
If $i=0$, then we get half of the above estimates, since $w_{0}$ is only half
a mile. Now take $\beta=e_{2k+2}+\frac{1}{2}$. Then $i\geq 2k+2$ implies the
last case and $i<2k+2$ implies one of the first two cases of the above, i.e.
$2(e_{i}+\frac{1}{2}-\max\\{e_{i},\beta\\})$ units. This makes a total of
$\displaystyle\alpha$ $\displaystyle:=$
$\displaystyle\left(e_{0}+\frac{1}{2}-\max\\{e_{0},\beta\\}\right)+2\sum_{i=1}^{2k+1}\left(e_{i}+\frac{1}{2}-\max\\{e_{i},\beta\\}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{2k+1}\left(e_{i}+\frac{1}{2}-\max\\{e_{i},\beta\\}\right)+\sum_{i=1}^{2k+2}\left(e_{i}+\frac{1}{2}-\max\\{e_{i},\beta\\}\right)$
$\displaystyle=$ $\displaystyle
2\sum_{i=0}^{2k+1}d_{i}-\left(\max\\{e_{0},\beta\\}+2\sum_{i=1}^{2k+2}\max\\{e_{i},\beta\\}-\max\\{e_{2k+2},\beta\\}\right)$
$\displaystyle=$ $\displaystyle
2\sum_{i=0}^{2k+1}d_{i}-\max\\{e_{0},\beta\\}-2\sum_{i=1}^{2k+2}\max\\{e_{i},\beta\\}+\beta$
units.
Scratching all the $[\beta,\infty)$-parts of the $w_{i}$’s together gives a
walk that starts at $\beta$ and ends at $\max\\{e_{0},\beta\\}$, of which
$\frac{\alpha-(\max\\{e_{0},\beta\\}-\beta)}{2}$
units are in backward direction and
$\displaystyle\frac{\alpha+(\max\\{e_{0},\beta\\}-\beta)}{2}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2k+1}d_{i}-\sum_{i=1}^{2k+2}\max\\{e_{i},\beta\\}$
(4) $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{2k+1}d_{i}-\sum_{i=1}^{k}e_{i}-(k+2)\beta$
units are in forward direction. In order to get boxes on
$\gamma,e_{0},e_{1},\ldots$, professor Walkingholme needs to march from
$\beta$ to $\gamma$ if $\gamma>\beta$ and from $\beta$ to $e_{i}$ for all $i$
with $e_{i}>\beta$, whence at least
$\displaystyle(\gamma-\min\\{\gamma,\beta\\})+\sum_{i=0}^{\infty}(e_{i}-\min\\{e_{i},\beta\\})$
(5) $\displaystyle\geq$
$\displaystyle(\gamma-\beta)+\sum_{i=0}^{k}(e_{i}-\beta)+\sum_{i=k+1}^{\infty}(e_{i}-e_{i})$
$\displaystyle=$ $\displaystyle\gamma+r+\sum_{i=1}^{k}e_{i}-(k+2)\beta$
units in forward direction within $[\beta,\infty)$ are required. Combining
(4), (5) and $e_{0}=r$ gives
$\gamma+r+2\sum_{i=1}^{k}e_{i}\leq\sum_{i=0}^{2k+1}d_{i}$ (6)
## Optimality of algorithms 2 and 3
Using (2) for $k=2,3,4$ and (6) for $k=0,1$ gives
$t\geq 12\frac{4}{7}\gamma-9\frac{1}{7}$ (7)
(add variables $C_{k}\geq 0$ on the smaller sides of the inequalities to get
equations), which proves the optimality of part A of the second solution.
Using (3) instead of (2) gives
$t\geq 14\gamma-11$ (8)
which proves the optimality of part A of both Dudeney’s solution and the first
solution.
Next we sketch the optimality of part B. Let $e_{1}\geq e_{2}\geq
e_{3}\geq\cdots$ be the positions where boxes are unsealed in part B before
reaching $\gamma$ from the other side, and define $d_{i}$ as above. In [1],
the last inequality but one reads
$e_{1}+2\sum_{i=2}^{k}e_{i}\leq\sum_{i=1}^{2k-1}d_{i}$ (9)
and the first inequality of lemma B, together with the above definition, looks
like
$e_{1}+2\sum_{i=2}^{k}e_{i}\leq t-1$ (10)
except that the right hand side is $N-1$ instead of $t-1$. But an algorithm
with $N$ boxes takes $N$ days, thus (10) seems correct if $t$ is the time part
B takes in days.
These inequalities can be proved with the techniques of the previous section.
Together with $e_{1}+1=5-\gamma$, we get
$\displaystyle t$ $\displaystyle\geq$ $\displaystyle
13\frac{5}{7}(5-\gamma)-36\frac{6}{7}$ (11) $\displaystyle t$
$\displaystyle\geq$ $\displaystyle 16(5-\gamma)-45$ (12) $\displaystyle t$
$\displaystyle\geq$ $\displaystyle 19\frac{1}{5}(5-\gamma)-56\frac{4}{5}$ (13)
using (9) for $k=1,2,\ldots,n$, and (10) for $k=n+1,\ldots,2n$ to prove
inequality $(n+6)$.
Notice that part C takes $\gamma$ days. Both (11) and (12) prove the
optimality of part B of algorithm 2 individually, but they need to cooperate
to get the optimality of algorithm 2 as a whole. (11) gives the bound
$11\frac{1}{7}$ day on getting to $3\frac{1}{2}$ as well.
If we charge part C for two days, i.e. the number of boxes unsealed in it,
which Nightvid Cole preferred, then it is also clear that algorithm 2 is
optimal among the solutions with $\gamma>1$. But if $\gamma\leq 1$, then it
follows from (13) and (8) that more than
$\left(12\frac{4}{7}\gamma-9\frac{1}{7}\right)+\left(19\frac{1}{7}(5-\gamma)-57\right)+\gamma=29\frac{4}{7}-5\frac{4}{7}\gamma\geq
24$
days are necessary, so algorithm 2 is optimal in Coles’s way of measuring as
well. Furthermore, the straight line solution is also proved to be non-optimal
now.
Since the time used for part A and B together is optimal in the first
solution, the only way to improve it to an optimal solution in Dudeney’s way
of measuring is to decrease $\gamma$. It follows that algorithm 3 is optimal
in Dudeney’s way of measuring.
At last, the round trips to $2\frac{1}{2}$, or actually to some position
$\gamma$. The round trip from the other side is in fact a round trip to
$5-\gamma$. Let $e_{1}$ be the largest position where a box is unsealed. This
can be before or after reaching $\gamma$, but no other box need to be unsealed
in between. Notice that the box for $e_{1}$ can be transported on the road to
$\gamma$, thus $e_{1}$ does not need to be counted for dropping.
Let $r$ be the position where the first box after that on $e_{1}$ and
$e_{2}\geq e_{3}\geq\cdots$ be the positions where the boxes before that on
$e_{1}$ are unsealed. The last box before that on $e_{1}$ is unsealed on a
position $\leq e_{2}$, after which a walk of two units to $r$ which meets
$\gamma$ follows. Hence
$\gamma\leq\frac{e_{2}+r+2}{2}$ (14)
Set $d_{i}:=\frac{1}{2}e_{i}+\frac{1}{2}e_{i+1}+\frac{1}{2}$ for all $i\geq
2$. By way of the techniques for estimating part $A$, one can get the
following inequalities for the round trip to $\gamma$.
$\gamma+r+2\sum_{i=2}^{k}e_{i}\leq\frac{e_{2}+r+2}{2}+\sum_{i=2}^{2k}d_{i}$
(15)
$\gamma+r+(r-1)+2\sum_{i=2}^{k}e_{i}\leq\frac{e_{2}+r+2}{2}+\sum_{i=2}^{2k+1}d_{i}$
(16)
and
$\gamma+r+(r-1)+2\sum_{i=2}^{k}e_{i}\leq\frac{t}{2}$ (17)
Using (14), (15) for $k=2,3,4$, (16) for $k=4,5,6,7,8$, and (17) for
$k=9,10,\allowbreak\ldots,18$, we get
$t\leq 27\gamma-46\frac{7}{8}$
whence at least
$(27\gamma-46\frac{7}{8})+(27(5-\gamma)-46\frac{7}{8})=41\frac{1}{4}$ days are
necessary. This bound can be attained with my way of measuring, with two round
trips to $2\frac{1}{2}$, but in order to take into account the ants as well,
different round trips are necessary, e.g. to $2\frac{1}{2}\pm\frac{1}{72}$,
starting $\frac{1}{3}$ way during the day and ending $\frac{7}{12}$ way during
another day. The reader may verify this.
If the box on $e_{1}$ is not unsealed before reaching $\gamma$, then we have
$\gamma\geq r+1$, and by adding (14), (15) for $k=2,3,\ldots,8$, and (17) for
$k=9,10,\allowbreak\ldots,17$, to the inequality $\gamma\geq r+1$, we can
derive
$t\leq 25\frac{6}{7}\gamma-44$
If we use (16) instead of (15) for $k=8$ (also for $k=7$ and/or $k=6$ when
desired), we get
$t\leq 26\frac{1}{7}\gamma-\frac{313}{7}$
Both bounds on $t$ can be attained simultaneously, if and only if
$e_{1}=\gamma=2\frac{1}{2}$.
## References
* [1] M. de Bondt, The camel-banana problem, Nieuw Archief voor Wiskunde, 14 (1996) 415-426.
* [2] N. Cole, Mount Neverest Problem,
http://users.aristotle.net/${}_{\mbox{\~{}}}$nightvid/neverest.htm
* [3] H. E. Dudeney, 536 Puzzles & Curious Problems, ed. Martin Gardner Charles Scribner’s Sons, New York, 1967.
* [4] G. Rote, G. Zhang, Optimal logistics for expeditions: the jeep problem with complete refilling, Optimierung und Kontrolle, 71 (1996), Spezialforschungsbereich 003, Karl-Franzens-Universität Graz & Technische Universität Graz.
|
arxiv-papers
| 2010-09-13T00:52:37 |
2024-09-04T02:49:12.929833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Michiel de Bondt",
"submitter": "Michiel de Bondt",
"url": "https://arxiv.org/abs/1009.2938"
}
|
1009.2978
|
# The optimal constant in the $L^{2}$ Folland-Stein inequality on the
quaternionic Heisenberg group
Stefan Ivanov University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria ivanovsp@fmi.uni-sofia.bg ,
Ivan Minchev University of Sofia
Sofia, Bulgaria
and Mathematik und Informatik
Philipps-Universität Marburg
Hans-Meerwein-Str. / Campus Lahnberge 35032 Marburg, Germany
minchevim@yahoo.com minchev@fmi.uni-sofia.bg and Dimiter Vassilev Department
of Mathematics and Statistics
University of New Mexico
Albuquerque, New Mexico, 87131-0001 vassilev@math.unm.edu
###### Abstract.
We determine the best (optimal) constant in the $L^{2}$ Folland-Stein
inequality on the quaternionic Heisenberg group and the non-negative functions
for which equality holds.
###### Key words and phrases:
Yamabe equation, quaternionic contact structures
###### 1991 Mathematics Subject Classification:
58G30, 53C17
###### Contents
1. 1 Introduction
2. 2 The model quaternionic contact structures
3. 3 The best constant in the Folland-Stein inequality
## 1\. Introduction
The goal of this note is to determine the best (optimal) constant in the
$L^{2}$ Folland-Stein inequality on the quaternionic Heisenberg group and the
non-negative extremal functions, i.e., the functions for which equality holds.
Alternatively, this is equivalent to finding the Yamabe constant of the
standard quaternionic contact structure of the sphere. The proof relies on the
realization of Branson, Fontana and Morpurgo [2], recently remarkably used
also by Frank and Lieb [6], that the old idea of Szegö [18], see also Hersch
[14], can be used to find the sharp form of (logarithmic) Hardy-Littlewood-
Sobolev type inequalities on the Heisenberg group.
The conformal nature of the problem we consider is key to its solution. The
analysis is purely analytical. In this respect, even though the quaternionic
contact (qc) Yamabe functional is involved, the qc scalar curvature is used in
the proof without much geometric meaning. Rather, it is the conformal sub-
laplacian that plays a central role and the qc scalar curvature appears as a
constant determined by the Cayley transform and the left-invariant sub-
laplacian on the quaternionic Heisenberg group. In this respect, this method
does not give all solutions of the qc Yamabe equation on the quaternionic
contact sphere. The complete solution of the latter problem requires some
additional very non-trivial argument and it is at this place where the
geometric nature of the problem becomes even more important. In the CR
setting, the solution of the CR Yamabe problem was achieved with the help of
an ingenious divergence formula [13]. The other known sub-Riemannian case is
that of the qc Yamabe equation on the seven dimensional standard quaternionic
contact sphere [11]. Another relevant result appeared earlier [7], where the
sub-Riemannian Yamabe equation was solved in the unifying setting of groups of
Iwasawa type under an additional assumption of partial symmetry of the
solution. This result can be used at the final stage of all known proofs after
such symmetry has been shown to exist. We recall that the groups of Iwasawa
type comprise of the complex (=”usual”), quaternion and octonian Heisenberg
groups, which are defined by (1.4) replacing, correspondingly, the quaternions
$\mathbb{H}$ with the complex numbers $\mathbb{C}$, the quaternions
$\mathbb{H}$, and the octonians $\mathbb{O}$.
Given a compact quaternionic contact manifold $M$ of real dimension $4n+3$
with an $\mathbb{R}^{3}$-valued contact form
$\eta=\\{\eta_{1},\eta_{2},\eta_{3}\\}$, i.e. a codimension three horizontal
distribution $H$ determined as the kernel of $\eta$ such that $d\eta_{|_{H}}$
are the fundamental two forms of a quaternionic hermitian structure $({\bf
g},I_{1},I_{2},I_{3})$ on $H$, $(d\eta_{s})_{|_{H}}=2{\bf
g}(I_{s}.,.)=2\omega_{s},s=1,2,3$, a natural question is to determine the qc
Yamabe constant of the conformal class $[\eta]$ of $\eta$ defined as the
infimum
(1.1) $\lambda(M,[\eta])\ =\ \inf\\{\Upsilon(u):\
\int_{M}u^{2^{*}}\,Vol_{\eta}\ =\ 1,\ u>0\\},$
where $Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$
denotes the volume form determined by $\eta$. The qc Yamabe functional of the
conformal class of $\eta$ is defined by
$\Upsilon(u)\ =\ \int_{M}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla u\rvert^{2}\
+\ S\,u^{2}\Bigr{)}Vol_{\eta},\qquad\int_{M}u^{2^{*}}\,Vol_{\eta}\ =\ 1,\
u>0,$
denoting by $\nabla$ the Biquard connection [1] of $\eta$, and $S$ standing
for the qc scalar curvature of $(M,\,\eta)$. This is the so called _qc Yamabe
constant problem_. In this paper we shall find
$\lambda(S^{4n+3},[\tilde{\eta}])$, where $\tilde{\eta}$ is the standard qc
form on the unit sphere $S^{4n+3}$, see (2.1). The question is of course
related to the solvability of the qc Yamabe equation
(1.2) $\mathcal{L}u\ \equiv\ 4\frac{Q+2}{Q-2}\ \triangle u-\ S\,u\ =\ -\
\overline{S}\,u^{2^{*}-1},$
where $\triangle$ is the horizontal sub-Laplacian, $\triangle u\ =\
tr^{g}(\nabla du)$, $S$ and $\overline{S}$ are the qc scalar curvatures
correspondingly of $(M,\,\eta)$ and $(M,\,\bar{\eta})$,
$\bar{\eta}=u^{4/(Q-2)}\eta$, and $2^{*}=\frac{2Q}{Q-2}.$ Here, and throughout
the paper $Q=4n+6$ is the homogeneous dimension. The natural question is to
find all solutions of the qc Yamabe equation. This is the so called _qc Yamabe
problem_ , which is equivalent to finding all qc structures conformal to a
given structure $\eta$ (of constant qc scalar curvature) which also have
constant qc scalar curvature. As usual the two problems are related by noting
that on a compact quaternionic contact manifold $M$ with a fixed conformal
class $[\eta]$ the qc Yamabe equation characterizes the non-negative extremals
of the qc Yamabe functional.
The 4n+3 dimensional sphere is an important example of a locally quternionic
contact conformally flat qc structure characterized locally in [12] with the
vanishing of a curvature-type tensor invariant. From the point of view of the
qc Yamabe problem the sphere plays a role similar to its Riemannian and CR
counterparts. A solution of the qc Yamabe problem on the seven dimensional
sphere equipped with its natural quaternionic contact structure was given in
[11] where more details on the qc Yamabe problem can be found. The main result
of [11] is the following
###### Theorem ([11]).
Let $\tilde{\eta}=\frac{1}{2h}\eta$ be a conformal deformation of the standard
qc-structure $\tilde{\eta}$ on the quaternionic unit sphere $S^{7}$. If $\eta$
has constant qc scalar curvature, then up to a multiplicative constant $\eta$
is obtained from $\tilde{\eta}$ by a conformal quaternionic contact
automorphism. In particular, $\lambda(S^{7})=48\,(4\pi)^{1/5}$ and this
minimum value is achieved only by $\tilde{\eta}$ and its images under
conformal quaternionic contact automorphisms.
Another motivation for studying the qc Yamabe equation and the qc Yamabe
constant of the qc sphere comes from its connection with the determination of
the norm and extremals in a relevant Sobolev-type embedding on the
quaternionic Heisenberg group [7] and [19] and [20]. As well known, the sub-
Riemannian Yamabe equation is also the Euler-Lagrange equation of the
extremals for the $L^{2}$ case of such embedding results. Recall the following
Theorem due to Folland and Stein [5].
###### Theorem (Folland and Stein).
Let $\Omega\subset{G}$ be an open set in a Carnot group ${G}$ of homogeneous
dimension $Q$ and Haar measure $dH$. For any $1<p<Q$ there exists
$S_{p}=S_{p}({G})>0$ such that for $u\in C^{\infty}_{o}(\Omega)$
(1.3) $\left(\int_{\Omega}\ |u|^{p^{*}}\ dH(g)\right)^{1/p^{*}}\leq\ S_{p}\
\left(\int_{\Omega}|Xu|^{p}\ dH(g)\right)^{1/p},$
where $|Xu|=\sum_{j=1}^{m}|X_{j}u|^{2}$ with $X_{1},\dots,X_{m}$ denoting a
basis of the first layer of ${G}$ and $p^{*}=\frac{pQ}{Q-p}$.
Let $S_{p}$ be the best constant in the Folland-Stein inequality, i.e., the
smallest constant for which (1.3) holds.
In [11] we determined all extremals, i.e., solutions of the qc Yamabe
equation, and the best constant in Folland and Stein’s theorem when $p=2$ in
the case of the seven dimensional quaternionic Heisenberg group. In the case
of the complex (i.e. ”usual”) Heisenberg group this was done earlier by
Jerison and Lee [13] who determined _all_ solutions to the CR Yamabe equation
on the CR sphere. In that setting, Frank and Lieb [6] determined the best
constant and found all functions for which the minimum is achieved, thus
simplifying parts of [13] while answering a less general question. However, in
[6] the authors also gave sharp forms of many Hardy-Littlewood-Sobolev type
inequalities on the Heisenberg group.
Following the idea of [6], the main result of this paper determines the best
constant in the Folland and Stein’s theorem when $p=2$ and the functions for
which it is achieved in the case of the quaternionic Heisenberg group
$\boldsymbol{G}$ of any dimension.
As a manifold $\boldsymbol{G}=\mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ with
the group law given by
(1.4) $(q^{\prime},\omega^{\prime})\ =\ (q_{o},\omega_{o})\circ(q,\omega)\ =\
(q_{o}\ +\ q,\omega\ +\ \omega_{o}\ +\ 2\ \text{Im}\ q_{o}\,\bar{q}),$
where $q,\ q_{o}\in\mathbb{H}^{n}$ and
$\omega,\omega_{o}\in\text{Im}\,\mathbb{H}$. The standard quaternionic
contact(qc) structure is defined by the left-invariant quaternionic contact
form
$\tilde{\Theta}\ =\ (\tilde{\Theta}_{1},\ \tilde{\Theta}_{2},\
\tilde{\Theta}_{3})\ =\ \frac{1}{2}\ (d\omega\ -\ q^{\prime}\cdot
d\bar{q}^{\prime}\ +\ dq^{\prime}\,\cdot\bar{q}^{\prime}),$
where $\cdot$ denotes the quaternion multiplication. The purpose of the
present note is to prove the next
###### Theorem 1.1.
a) Let $\boldsymbol{G}\ =\ \mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ be the
quaternionic Heisenberg group. The best constant in the $L^{2}$ Folland-Stein
embedding inequality (1.3) is
$S_{2}\ =\ \frac{\left[2^{-2n}\
\omega_{4n+3}\right]^{-1/(4n+6)}}{2\sqrt{n(n+1)}},$
where $\omega_{4n+3}=2\pi^{2n+2}/(2n+1)!$ is the volume of the unit sphere
$S^{4n+3}\subset\mathbb{R}^{4n+4}$. The non-negative functions for which (1.3)
becomes an equality are given by the functions of the form
(1.5) $\displaystyle F$ $\displaystyle=\ \gamma\left[(1+\lvert
q\rvert^{2})^{2}\ +\
\lvert\omega\rvert^{2}\right]^{-(n+1)},\qquad\gamma=const,$
and all functions obtained from $F$ by translations (3.2) and dilations (3.3).
b) The qc Yamabe constant of the standard qc structure of the sphere is
(1.6) $\lambda(S^{4n+3},[\tilde{\eta}])\ =\
16\,n(n+2)\,\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}.$
These constants are in complete agreement with the ones obtained in [11] and
[7] taking into account the next Remark and the well known formulas involving
the gamma function
$\displaystyle\Gamma(n+1)=n!,\qquad\Gamma(z+n)=z(z+1)\dots(z+n-1)\Gamma(z),\qquad
n\in\mathbb{N},$ $\displaystyle\omega_{n}=2\pi^{n/2}/\Gamma(n/2)\qquad\text{--
volume of unit $(n-1)$--dimensional sphere in }\mathbb{R}^{n},$
$\displaystyle\Gamma(2z)=2^{2z-1}\ \pi^{-1/2}\
\Gamma(z)\,\Gamma\left(z+\frac{1}{2}\right)\quad\text{ -- the Legendre
formula. }$
Our result partially confirms the Conjecture made after [7, Theorem 1.1]. In
addition, the fact that any function of the described form is a solution of
Yamabe problem was first noted in [8] in the setting of groups of Heisenberg
type. Of course, this class of groups is much wider than the class of groups
of Iwasawa type.
###### Remark 1.2.
With the left invariant basis of Theorem 1.1 the quaternionic Heisenberg group
is not a group of Heisenberg type. If we consider it as a group of Heisenberg
type then the best constant in the $L^{2}$ Folland-Stein embedding theorem is,
cf. [7, Theorem 1.6],
$S_{2}\ =\ \frac{1}{\sqrt{4n(4n+4)}}\ 4^{3/(4n+6)}\ \pi^{-(4n+3)/2(4n+6)}\
\left(\frac{\Gamma(4n+3)}{\Gamma((4n+3)/2)}\right)^{1/(4n+6)}.$
and extremals are given by dilations and translations of the function
$F(q,\omega)\ =\ \gamma\ \left[(1+|q|^{2})^{2}\ +\
16|\omega|^{2})\right]^{-(n+1)},\ (q,\omega)\in\boldsymbol{G}.$
It is worth pointing that studying the Yamabe extremals in the sub-Riemannian
setting has applications to sharp inequalities in the Euclidean setting. For
example, in the paper [21] are determined the extremals of some Euclidean
Hardy-Sobolev inequalities involving the distance to a $n-k$ dimensional
coordinate subspace of $\mathbb{R}^{n}$. This is achieved by relating
extremals on the Heisenberg groups to extremals in the Euclidean setting. In
the particular case when $k=n$ one obtains the Caffarelli-Kohn-Nirenberg
inequality, see [3], for which the optimal constant was found in [9].
###### Convention 1.3.
We use the following conventions:
* •
the abbreviation qc will stand for quaternionic contact;
* •
$\boldsymbol{G}$ will denote the qc Heisenberg group;
* •
$\tilde{\eta}$ will denote the standard qc form on the unit sphere $S^{4n+3}$,
see (2.1). Note that this form is actually twice the 3-Sasakain qc form on
$S^{4n+3}$;
* •
$Vol_{\eta}$ will denote the volume form determined by the qc form $\eta$,
thus $Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$,
see [10, Chapter 8].
Acknowledgments. Research was partially supported by Contract “Idei”, DO
02-257/18.12.2008 and DID 02-39/21.12.2009. S.I. and I. M. are partially
supported by the Contract 082/2009 with the University of Sofia
‘St.Kl.Ohridski’.
## 2\. The model quaternionic contact structures
In this section we review the standard quaternionic contact structure on the
quaternionic Heisenberg group and the $4n+3$-dimensional unit sphere. We will
rely heavily on [10], but prefer to repeat some key points in order to make
the current paper somewhat self-contained. Besides serving as a background,
this section will supply some key numerical constants - the qc scalar
curvature and the first eigenvalue of the sub-laplacian of the standard qc
form of the sphere. This will be achieved using the conformal sub-laplacian
and the properties of the Cayley transform.
First let us recall the quaternionic Heisenberg group [10, Section 5.2]. We
remind the following model of the quaternionic Heisenberg group
$\boldsymbol{G}$. Define $\boldsymbol{G}\ =\
\mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ with the group law given by
$(q^{\prime},\omega^{\prime})\ =\ (q_{o},\omega_{o})\circ(q,\omega)\ =\
(q_{o}\ +\ q,\omega\ +\ \omega_{o}\ +\ 2\ \text{Im}\ q_{o}\,\bar{q}),$ where
$q,\ q_{o}\in\mathbb{H}^{n}$ and $\omega,\omega_{o}\in\text{Im}\,\mathbb{H}$.
In coordinates, with $\omega=ix+jy+kz$ and
$q_{\alpha}=t_{\alpha}+ix_{\alpha}+jy_{\alpha}+kz_{\alpha}$, $\alpha=1,\dots
n$, a basis of left invariant horizontal vector fields
$T_{\alpha},X_{\alpha}=I_{1}T_{\alpha},Y_{\alpha}=I_{2}T_{\alpha},Z_{\alpha}=I_{3}T_{\alpha},\alpha=1\dots,n$
is given by
$\displaystyle T_{\alpha}\ =\ {\partial_{t_{\alpha}}}{}\
+2x_{\alpha}{\partial_{x}}{}+2y_{\alpha}{\partial_{y}}{}+2z_{\alpha}{\partial_{z}}{}\,\qquad
X_{\alpha}\ =\ {\partial_{x_{\alpha}}}{}\
-2t_{\alpha}{\partial_{x}}{}-2z_{\alpha}{\partial_{y}}{}+2y_{\alpha}{\partial_{z}}{}$
$\displaystyle Y_{\alpha}\ =\ {\partial_{y_{\alpha}}}{}\
+2z_{\alpha}{\partial_{x}}{}-2t_{\alpha}{\partial_{y}}{}-2x_{\alpha}{\partial_{z}}{}\,\qquad
Z_{\alpha}\ =\ {\partial_{z_{\alpha}}}{}\
-2y_{\alpha}{\partial_{x}}{}+2x_{\alpha}{\partial_{y}}{}-2t_{\alpha}{\partial_{z}}{}\,.$
The above vectors generate the _horizontal space_ , denoted as usual by $H$.
In addition, by declaring them to be an orthonormal basis we obtain a metric
on the horizontal space, which is the so called _horizontal metric_. The
central (vertical) vector fields $\xi_{1},\xi_{2},\xi_{3}$ are described as
follows
$\xi_{1}=2{\partial_{x}}{}\,\quad\xi_{2}=2{\partial_{y}}{}\,\quad\xi_{3}=2{\partial_{z}}{}\,.$
The standard quaterninic contact form, written as a purely imaginary
quaternion valued form
$\tilde{\Theta}=i\tilde{\Theta}_{1}+j\tilde{\Theta}_{2}+k\tilde{\Theta}_{3})$,
is
$2\tilde{\Theta}\ =\ \ d\omega\ -\ q^{\prime}\cdot d\bar{q}^{\prime}\ +\
dq^{\prime}\,\cdot\bar{q}^{\prime},$
where $\cdot$ denotes the quaternion multiplication. The Biquard connection
coincides with the flat left invariant connection on $\boldsymbol{G}$, in
particular the qc scalar curvature vanishes.
Following [10], we give another model of the Heiseneberggroup, which is the
one we will use in this paper. Let us identify $\boldsymbol{G}$ with the
boundary $\Sigma$ of a Siegel domain in $\mathbb{H}^{n}\times\mathbb{H}$,
$\Sigma\ =\ \\{(q^{\prime},p^{\prime})\in\mathbb{H}^{n}\times\mathbb{H}\ :\
\Re{\ p^{\prime}}\ =\ \lvert q^{\prime}\rvert^{2}\\},$
by using the map $(q^{\prime},\omega^{\prime})\mapsto(q^{\prime},\lvert
q^{\prime}\rvert^{2}-\omega^{\prime})$. Since $dp^{\prime}\ =\ q^{\prime}\cdot
d\bar{q}^{\prime}\ +\ dq^{\prime}\,\cdot\bar{q}^{\prime}\ -\
d\omega^{\prime},$ under the identification of $\boldsymbol{G}$ with $\Sigma$
we have also $2\tilde{\Theta}\ =\ -dp^{\prime}\ +\
2dq^{\prime}\cdot\bar{q}^{\prime}.$ Taking into account that $\tilde{\Theta}$
is purely imaginary, the last equation can be written also in the following
form
$4\,\tilde{\Theta}\ =\ (d\bar{p}^{\prime}\ -\ dp^{\prime})\ +\
2dq^{\prime}\cdot\bar{q^{\prime}}\ -\ 2q^{\prime}\cdot d\bar{q}^{\prime}.$
Now, consider the Cayley transform, see [15] and [4], as the map $\
\mathcal{C}:S\mapsto\Sigma\ $ from the sphere $S\ =\ \\{\lvert
q\rvert^{2}+\lvert p\rvert^{2}=1\\}\subset\mathbb{H}^{n}\times\mathbb{H}$
minus a point to the Heisenberg group $\Sigma$, with $\mathcal{C}$ defined by
$(q^{\prime},p^{\prime})\ =\ \mathcal{C}\ \Big{(}(q,p)\Big{)},\qquad
q^{\prime}\ =\ (1+p)^{-1}\ q,\qquad p^{\prime}\ =\ (1+p)^{-1}\ (1-p)$
and with an inverse map $(q,p)\ =\
\mathcal{C}^{-1}\Big{(}(q^{\prime},p^{\prime})\Big{)}$ given by
$q\ =\ \ 2(1+p^{\prime})^{-1}\ q^{\prime},\qquad p\ =\ (1+p^{\prime})^{-1}\
(1-p^{\prime}).$
The Cayley transform maps $S^{4n+3}\setminus\\{(-1,0)\\}$,
$(-1,0)\in\mathbb{H}^{n}\times\mathbb{H}$, to $\Sigma$ since
$\Re{\ p^{\prime}}\ =\ \Re{\frac{(1+\bar{p})(1-p)}{\lvert 1+p\,\rvert^{2}}}\
=\ \Re{\frac{1-\lvert p\rvert}{\lvert 1+p\,\rvert^{2}}}\ =\ \frac{\lvert
q\rvert^{2}}{\lvert 1+p\,\rvert^{2}}\ =\ \lvert q^{\prime}\rvert^{2}.$
Writing the Cayley transform in the form $(1+p)q^{\prime}\ =\ \
q,\quad(1+p)p^{\prime}\ =\ 1-p,$ gives
$dp\cdot q^{\prime}\ +\ (1+p)\cdot dq^{\prime}\ =\ dq,\hskip 36.135ptdp\cdot
p^{\prime}\ +\ (1+p)\cdot dp^{\prime}\ =\ -dp,$
from where we find
$\displaystyle dp^{\prime}$ $\displaystyle=\ -2(1+p)^{-1}\cdot
dp\cdot(1+p)^{-1}$ $\displaystyle dq^{\prime}$ $\displaystyle=\
(1+p)^{-1}\cdot[dq\ -\ dp\cdot(1+p)^{-1}\cdot q].$
The Cayley transform is a conformal quaternionic contact diffeomorphism
between the quaternionic Heisenberg group with its standard quaternionic
contact structure $\tilde{\Theta}$ and the sphere minus a point with its
standard structure $\tilde{\eta}$. In fact, by [10, Section 8.3] we have
$\Theta\ \overset{def}{=}\ \lambda\ \cdot(\mathcal{C}^{-1})^{*}\,\tilde{\eta}\
\cdot\bar{\lambda}\ =\ \frac{8}{\lvert
1+p^{\prime}\,\rvert^{2}}\,\tilde{\Theta}.$
where $\lambda\ ={\lvert 1+p\,\rvert}\,{(1+p)^{-1}}$ is a unit quaternion and
$\tilde{\eta}$ is the standard contact form on the sphere,
(2.1) $\tilde{\eta}\ =\ dq\cdot\bar{q}\ +\ dp\cdot\bar{p}\ -\ q\cdot
d\bar{q}-\ p\cdot d\bar{p}.$
###### Lemma 2.1.
The qc scalar curvature $\tilde{S}$ of the standard qc structure (2.1) on
$S^{4n+3}$ is
(2.2) $\tilde{S}\ =\ \frac{1}{2}(Q+2)(Q-6)=8n(n+2).$
###### Remark 2.2.
Notice that the standard qc contact form we consider here is twice the
3-Sasakian form on $S^{4n+3}$, which has qc scalar curvature equal to 16n(n+2)
[10].
###### Proof.
Let us introduce the functions
(2.3) $\displaystyle h$
$\displaystyle=\frac{1}{16}|1+p^{\prime}|^{2}=\frac{1}{16}\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right],\qquad(q^{\prime},p^{\prime})\in\Sigma\subset\mathbb{H}^{n}\times\mathbb{H},\quad
p^{\prime}=|q^{\prime}|^{2}+\omega^{\prime},$ and $\displaystyle\Phi$
$\displaystyle=\left({2h}\right)^{-(Q-2)/4}=8^{(Q-2)/4}\
{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4},$
so that now we have
$\Theta=\frac{1}{2h}\tilde{\Theta}=\Phi^{4/(Q-2)}\tilde{\Theta}.$
With the help of [10, Section 5.2] a small calculation shows that the sub-
laplacian of $h$ w.r.t. $\tilde{\Theta}$ is given by $\triangle
h=\frac{Q-6}{4}+\frac{Q+2}{4}|q^{\prime}|^{2}$ and thus $\Phi$ is a solution
of the qc Yamabe equation on the Heisenberg group $\Sigma$
(2.4) $\triangle\Phi=-K\,\Phi^{2^{*}-1},\qquad K=(Q-2)(Q-6)/8,$
where $\triangle$ is the sub-laplacian on the quaternionic Heisenberg group.
Denoting with $\mathcal{L}$ and $\tilde{\mathcal{L}}$ the conformal sub-
laplacians of $\Theta$ and $\tilde{\Theta}$, respectively, we have
$\Phi^{-1}\mathcal{L}(\Phi^{-1}u)=\Phi^{-2^{*}}\tilde{\mathcal{L}}u.$
We remind, cf. [1] and [11], that for a qc contact form $\Theta$ the conformal
sublaplacian is,
$\mathcal{L}=a\triangle_{\Theta}-S_{\Theta},\qquad a=4\frac{Q+2}{Q-2},$
where $\triangle_{\Theta}$ is the sub-laplacian associated to $\Theta$, i.e.,
$\triangle_{\Theta}u=tr(\nabla^{\Theta}du)$–the horizontal trace of the
Hessian of $u$, using the Biquard connection $\nabla^{\Theta}$ of $\Theta$,
and $S_{\Theta}$ is the qc scalar curvature of $\Theta$. Thus, letting
$u=\Phi$ we come to $\mathcal{L}(1)=\Phi^{1-2^{*}}\tilde{\mathcal{L}}\Phi$,
which shows $-S_{\Theta}=-4\frac{Q+2}{Q-2}K$. The latter is the same as that
of $\tilde{\eta}$ since the two structures are isomorphic via the
diffemorphism $\mathcal{C}$, or rather its extension, since we can consider
$\mathcal{C}$ as a quaternionic contact conformal transformation between the
whole sphere $S^{4n+3}$ and the compactification $\hat{\Sigma}\cup{\infty}$ of
the quaternionic Heisenberg group by adding the point at infinity, cf. [11,
Section 5.2]. ∎
We turn to the task of determining the first eigenvalue of the sub-laplacian
on $S^{4n+3}$. In fact, we shall need only the fact that the restriction of
every coordinate function is an eigenvalue. The proof of this fact can be seen
directly without any reference to the Biquard connection, but this will
require setting a lot of notation, so we prefer to use a result from [10].
###### Lemma 2.3.
If $\zeta$ is any of the (real) coordinate functions in
$\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$, then
(2.5)
$\tilde{\triangle}\zeta=-\lambda_{1}\zeta,\quad\lambda_{1}=\frac{\tilde{S}}{Q+2}={2n}$
for the horizontal trace of the Hessian, where $\tilde{\triangle}$ is the sub-
laplacian of the standard qc form $\tilde{\eta}$ of $S^{4n+3}$.
###### Proof.
It is enough to furnish a proof for the sub-laplacian on the 3-Sasakain sphere
since the two qc forms defer by a constant. We can see that every $\zeta$ of
the considered type is an eigenfunction by using [10, Corollary 6.24]. It will
be enough to see it for one coordinate function provided the sub-laplacian on
the sphere is rotation invariant. Thus, let us take $\zeta=t_{1}$. Notice that
$\zeta$ is quaternionic pluri-harmonic [10, Definition 6.7] since it is the
real part of the anti-regular function $t_{1}+ix_{1}-jy_{1}-kz_{1}$. So, its
restriction to the 3-Sasakain sphere is the real part of an anti-CRF function.
Therefore we apply [10, Corollary 6.24] which gives $tr(\nabla
d\zeta)=4\lambda n$ for the sub-laplacian of the 3-sasakain qc structure on
the sphere. Next, we compute $\lambda$, which can be found in [10, Theorem
6.20]. Using that the sphere is 3-Sasakian it follows the Reeb vector fields
are obtained from the outward pointing unit normal vector $N$ as follows,
$\xi_{1}=iN$, $\xi_{2}=jN$ and $\xi_{3}=kN$, where for a point on the sphere
we have $N(q)=q\in\mathbb{H}^{n+1}$. Therefore $\lambda=-t_{1}=-\zeta$. To see
this easier notice that only the first four coordinates of $N$ matter. So, if
we assume $n=0$ we have $iN=-x+it+ky-jz$, $jN=-y+iz+jt-kx$ and
$kN=-z-iy+jx+kt$, so we need to sum the _real dot product_ of these vectors
with $i$, $j$ and $k$, respectively, which gives $-t$. Thus, for the sub-
laplacian on the 3-Sasakian sphere we have
$tr(\nabla d\zeta)=-4n\zeta,$
where $\zeta$ is the restriction any of the coordinate functions of
$\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$. Since the qc contact form
$\tilde{\Theta}$ is twice the 3-Sasakain qc contact form on the sphere it
follows $\tilde{\triangle}$ is $1/2$ of the 3-Sasakain sub-laplacian. Thus
$\tilde{\triangle}=-2n\zeta,$
which shows $\lambda_{1}=2n=\frac{1}{2}(Q-6)={\tilde{S}}/{(Q+2)}$. ∎
We finish this section with a simple Lemma which will be used to relate the
various explicit constants. Its claim also follows from the conformal
invariance of the Yamabe equation, but we prefer to give a proof, which is
independent of the notion of qc scalar curvature. We recall, see [10, Chapter
8], that $Vol_{\eta}$ will denote the volume form determined by the qc form
$\eta$, thus
$Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$.
Also, for a qc form $\eta$ we let
$|\nabla^{\eta}F|^{2}=\sum_{\alpha=1}^{4n}|dF(e_{\alpha})|^{2}$ be the square
of the length of the horizontal gradient of a function $F$ taken with respect
to an orthonormal basis of the horizontal space $H=Ker\,\eta$ and the metric
determined by $\eta$.
###### Lemma 2.4.
Let $F\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$, cf. (3.1), be a
positive function with $\int_{\boldsymbol{G}}\
F^{2^{*}}Vol_{\tilde{\Theta}}=1$. Then we have
(2.6) $\int_{\boldsymbol{G}}\ a|\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}\ =\ \int_{S^{4n+3}}\
\left(a|\nabla^{\tilde{\eta}}g|^{2}+\tilde{S}g^{2}\right)\
Vol_{\tilde{\eta}},\qquad a=4(2^{*}-1),$
and
$\int_{\boldsymbol{G}}\ g^{2^{*}}Vol_{\tilde{\eta}}=1,$
where
(2.7) $g=\mathcal{C}^{*}(F\Phi^{-1}),$
and, as before, $\mathcal{C}:S^{4n+3}\rightarrow\Sigma$ is the Cayley
transform, $\Theta=\Phi^{4/(Q-2)}\tilde{\Theta}$, cf. (2.3).
###### Remark 2.5.
Notice that $Vol_{\tilde{\Theta}}=2^{-3}\,(2n)!\,dH$, where $dH$ is the
Lebesgue measure in $\mathbb{R}^{4n+3}$, which is a Haar measure on the group.
###### Proof.
It will be convenient for the remaining of this proof to denote by small
letters the pull-back by the Cayley transform of a function denoted with the
corresponding capital letter. Thus, $f=\mathcal{C}^{*}F=F\circ\mathcal{C}$,
$\phi=\mathcal{C}^{*}(\Phi)$ and $g=f\phi^{-1}$. By the conformality of the qc
structures on the group and the sphere we have
(2.8) $Vol_{\Theta}=\Phi^{2^{*}}Vol_{\tilde{\Theta}}$
By (2.8) we have
$F^{2^{*}}Vol_{\tilde{\Theta}}=f^{2^{*}}\phi^{-2^{*}}Vol_{\tilde{\eta}}$,
which motivates the definition (2.7) of the function $g$ which is defined on
the sphere and should be regarded as corresponding to the function $F$. Thus,
we have for example $F=G\Phi$. By definition we have
$\int_{\boldsymbol{G}}\ g^{2^{*}}Vol_{\tilde{\eta}}=1,$
so our next task is to see that the Yamabe integral is preserved
(2.9) $\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}\ =\ \int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+Kg^{2}\right)\ Vol_{\tilde{\eta}}.$
Here is where we shall exploit that a power of the conformal factor of the
Cayley transform is a solution of the Yamabe equation. Let
$\left<\nabla^{\Theta}\Phi,\nabla^{\Theta}G\right>=\sum_{a=1}^{4n}(e_{a}\Phi)\,(\textbf{e}_{a}G)$
where $\\{e_{1},\dots,e_{4n}\\}$ is an orthonormal basis of the horizontal
space $H$. Using the divergence formula from [10, Section 8.1] we find
$\int_{\boldsymbol{G}}\ |\tilde{\nabla}^{\Theta}F|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
|\nabla^{\tilde{\Theta}}(G\Phi)|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
\Big{(}G^{2}|\nabla^{\tilde{\Theta}}\Phi|^{2}+\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}+\left<\Phi\nabla^{\tilde{\Theta}}\Phi,\nabla^{\tilde{\Theta}}G^{2}\right>\Big{)}\
Vol_{\tilde{\Theta}}\\\ =\int_{\boldsymbol{G}}\
\Big{(}\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}-G^{2}\Phi\triangle_{\tilde{\Theta}}\Phi\
\Big{)}Vol_{\tilde{\Theta}}.$
Now, the Yamabe equation (2.4) gives
$\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
\left(\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}+KG^{2}\Phi^{2^{*}}\
\right)Vol_{\tilde{\Theta}}\\\ =\int_{S^{4n+3}}\
\left(\phi^{2-2^{*}}(|\nabla^{\tilde{\Theta}}G|\circ\mathcal{C})^{2}+Kg^{2}\
\right)Vol_{\tilde{\eta}}=\int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+Kg^{2}\ \right)Vol_{\tilde{\eta}},$
taking into account that $\mathcal{C}$ is a qc conformal map. Finally, a
glance at (2.4) and (2.2) shows $\tilde{S}/K=4(2^{*}-1)=(4(Q+2)/(Q-2)$ which
allows to put (2.9) in the form (2.6). ∎
## 3\. The best constant in the Folland-Stein inequality
In this section, following [6], we prove the main Theorem. It is important to
observe that a suitable adaptation of the method of concentration of
compactness due to P. L. Lions [16], [17] allows to prove that in any Carnot
group the Yamabe constant and optimal constant in the Folland-Stein inequality
is achieved in the space $\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$,
see [19] and [20]. Here
$\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})=\overline{C^{\infty}_{o}(\boldsymbol{G})}^{||\cdot||_{\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})}}.$
The space $\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$ is endowed with
the norm
(3.1) $||u||_{\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})}\ =\
||\,|\nabla u|\,||_{L^{2^{*}}(\boldsymbol{G})}.$
where $\nabla u$ is the horizontal gradient of $u$ and $|\nabla
u|^{2}=\sum_{a=1}^{4n}(e_{a}u)^{2}$ for an orthonormal basis
$\\{e_{1},\dots,e_{4n}\\}$ of horizontal left invariant vector fields.
In this regard an elementary, yet crucial observation, is that if $u$ is an
entire solution to the Yamabe equation, then such are also the two functions
(3.2) $\tau_{h}u\ \overset{def}{=}\ u\circ\tau_{h},\quad\quad\quad
h\in\boldsymbol{G},$
where $\tau_{h}:\boldsymbol{G}\to\boldsymbol{G}$ is the operator of left-
translation $\tau_{h}(g)=hg$, and
(3.3) $u_{\lambda}\ \overset{def}{=}\ \lambda^{(Q-2)/2}\
u\circ\delta_{\lambda},\quad\quad\quad\lambda>0.$
The Heisenberg dilations are defined by
$\delta_{\lambda}\left((q^{\prime},\omega^{\prime})\right)=\left((\lambda
q^{\prime},\lambda^{2}\omega^{\prime})\right),\qquad(q^{\prime},\omega^{\prime})\in\boldsymbol{G}$
It is also well known, [19] and [20], that there are smooth positive minimizer
of the Folland-Stein inequality on the quaternionic Heisenberg group
$\boldsymbol{G}$. These facts will be used without further notice on
regularity and existence.
We start with the ”new” key, see [2] and [6], allowing the ultimate solution
of the considered problem.
###### Lemma 3.1.
For every $v\in L^{1}(S^{4n+3})$ with $\int_{S^{4n+3}}v\ Vol_{\tilde{\eta}}=1$
there is a quaternionic contact conformal transformation $\psi$ such that
$\int_{S^{4n+3}}\psi\,v\ Vol_{\tilde{\eta}}=0.$
###### Proof.
Let $P\in S^{4n+3}$ be any point of the quaternionic sphere and $N$ be its
antipodal point. Let us consider the local coordinate system near $P$ defined
by the Cayley transform $\mathcal{C}_{N}$ from $N$. It is known that
$\mathcal{C}_{N}$ is a quaternionic contact conformal transformation between
$S^{4n+3}\setminus{N}$ and the quaternionic Heisenberg group. Notice that in
this coordinate system $P$ is mapped to the identity of the group. For every
$r$, $0<r<1$, let $\psi_{r,P}$ be the qc conformal transformation of the
sphere, which in the fixed coordinate chart is given on the group by a
dilation with center the identity by a factor $\delta_{r}$. If we select a
coordinate system in $\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$ so
that $P=(1,0)$ and $N=(-1,0)$ and then apply the formulas for the Cayley
transform from [10, Section 8.2] the formula for
$(q^{*},p^{*})=\psi_{r,P}(q,p)$ becomes
$\displaystyle q^{*}$
$\displaystyle=2r\left(1+r^{2}(1+p)^{-1}(1-p)\right)^{-1}\left(1+p\right)q$
$\displaystyle p^{*}$
$\displaystyle=\left(1+r^{2}(1+p)^{-1}(1-p)\right)^{-1}\left(1-r^{2}(1+p)^{-1}(1-p)\right),i.e,$
We can define then the map $\Psi:B\rightarrow\bar{B}$, where $B$ ( $\bar{B}$ )
is the open (closed) unit ball in $\mathbb{R}^{4n+4}$, by the formula
$\Psi(rP)=\int_{S^{4n+3}}\psi_{1-r,P}\,v\ Vol_{\tilde{\eta}}.$
Notice that $\Psi$ can be continuously extended to $\bar{B}$ since for any
point $P$ on the sphere, where $r=1$, we have $\psi_{1-r,P}(Q)\rightarrow P$
when $r\rightarrow 1$. In particular, $\Psi=id$ on $S^{4n+3}$. Since the
sphere is not a homotopy retract of the closed ball it follows that there are
$r$ and $P\in S^{4n+3}$ such that $\Psi(rP)=0$, i.e.,
$\int_{S^{4n+3}}\psi_{1-r,P}\,v\ Vol_{\tilde{\eta}}=0$. Thus,
$\psi=\psi_{1-r,P}$ has the required property. ∎
In the next step we prove that we can assume that the minimizer of the
Folland-Stein inequality satisfies the zero center of mass condition. A number
of well known invariance properties of the Yamabe functional will be
exploited.
###### Lemma 3.2.
Let $v$ be a smooth positive function on the sphere with
$\int_{S^{4n+3}}v^{2^{*}}\,Vol_{\tilde{\eta}}=1$. There is a smooth positive
function $u$ such that $\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla
u\rvert^{2}\ +\ \tilde{S}\,u^{2}\Bigr{)}\,Vol_{\tilde{\eta}}\
=\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla v\rvert^{2}\ +\
\tilde{S}\,v^{2}\Bigr{)}\,Vol_{\tilde{\eta}}$ and
$\int_{S^{4n+3}}u^{2^{*}}\,Vol_{\tilde{\eta}}=1$. In addition,
(3.4) $\int_{S^{4n+3}}P\,u^{2^{*}}(P)\,Vol_{\tilde{\eta}}=0,\qquad
P\in\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}.$
In particular, the Yamabe constant
(3.5) $\lambda(S^{4n+3},[\tilde{\eta}])\ =\
\inf\\{\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla v\rvert^{2}\ +\
\tilde{S}\,v^{2}\Bigr{)}Vol_{\tilde{\eta}}:\ \int_{S^{4n+3}}v^{2^{*}}\
Vol_{\tilde{\eta}}\ =\ 1,\ v>0\\}$
is achieved for a positive function $u$ with a zero center of mass, i.e., for
a function $u$ satisfying (3.4).
###### Proof.
By [10, Section 8.1],
$Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$ is a
volume form on a qc manifold with contact form $\eta$. Thus if $\eta$ is a qc
structure on the sphere which is qc conformal to the standard qc structure
$\tilde{\eta}$, $\eta=\phi^{4/(Q-2)}\tilde{\eta}$, then
$Vol_{\eta}=\phi^{2^{*}}Vol_{\tilde{\eta}}$. This allows to cast equation
(1.2) in the form
$\phi^{-1}v\,\mathcal{L}(\phi^{-1}v)\ Vol_{\eta}=v\mathcal{\tilde{L}}(v)\
Vol_{\tilde{\eta}}.$
Therefore, if we take a positive function $v$ on the sphere
$\int_{S^{4n+3}}v^{2^{*}}\,Vol_{\tilde{\eta}}=1$ and then consider the
function
(3.6) $u=\phi^{-1}(v\circ\psi^{-1}),$
where $\psi$ is the qc conformal map of Lemma 3.1,
$\eta\equiv(\psi^{-1})^{*}\tilde{\eta}$, and $\phi$ is the corresponding
conformal factor of $\psi$, we can see that $u$ achieves the claim of the
Lemma. ∎
We shall call a function $u$ on the sphere a well centered function when (3.4)
holds true. In the next step we show that a well centered minimizer has to be
constant.
###### Lemma 3.3.
If $u$ is a well centered local minimum of the problem (3.5), then $u\equiv
const$.
###### Proof.
Let $\zeta$ be a smooth function on the sphere $S^{4n+3}$. After applying the
divergence formula [10, Section 8] we obtain the formula
(3.7) $\Upsilon(\zeta u)=\int_{S^{4n+3}}\zeta^{2}\Bigl{(}4\frac{Q+2}{Q-2}\
\lvert\tilde{\nabla}u\rvert^{2}\ +\ \tilde{S}\,u^{2}\Bigr{)}\
Vol_{\tilde{\eta}}\ -\
4\frac{Q+2}{Q-2}\int_{S^{4n+3}}u^{2}\zeta{\tilde{\triangle}}\zeta\
Vol_{\tilde{\eta}}.$
This suggests to take as a test function $\zeta$ an eigenfunction of the sub-
laplacian ${\tilde{\triangle}}$ of the standard qc structure. In particular,
we can let $\zeta$ be any of the coordinate functions in
$\mathbb{H}^{n}\times\mathbb{H}$ in which case
${\tilde{\triangle}}\zeta=-\lambda_{1}\zeta$.
It will be useful to introduce the functional
$N(v)=\left(\int_{S^{4n+3}}v^{2^{*}}\ Vol_{\tilde{\eta}}\right)^{2/2^{*}}$ so
that
(3.8) $\lambda(S^{4n+3},[\tilde{\eta}])=\inf\\{\mathcal{E}(v):\,v\in
D\,(S^{4n+3})\\},\qquad\mathcal{E}(v)\overset{def}{=}\Upsilon(v)/N(v).$
Computing the second variation
$\delta^{2}\mathcal{E}(u)v=\frac{d^{2}}{dt^{2}}\mathcal{E}(u+tv)_{|_{t=0}}$ of
$\mathcal{E}(u)$ we see that the local minimum condition
$\delta^{2}\mathcal{E}(u)v\geq 0$ implies
$\Upsilon(v)-(2^{*}-1)\Upsilon(u)\int_{S^{4n+3}}u^{2^{*}-2}v^{2}\
Vol_{\tilde{\eta}}\ \geq 0$
for any function $v$ such that $\int_{S^{4n+3}}u^{2^{*}-1}v\
Vol_{\tilde{\eta}}=0$. Therefore, for $\zeta$ being any of the coordinate
functions in $\mathbb{H}^{n}\times\mathbb{H}$ we have
$\Upsilon(\zeta u)-(2^{*}-1)\Upsilon(u)\int_{S^{4n+3}}u^{2^{*}}\zeta^{2}\
Vol_{\tilde{\eta}}\ \geq 0,$
which after summation over all coordinate functions taking also into account
(3.7) gives
$\Upsilon(u)-(2^{*}-1)\Upsilon(u)-4\lambda_{1}(2^{*}-1)\int_{S^{4n+3}}u^{2}\
Vol_{\tilde{\eta}}\geq 0,$
which implies, recall $2^{*}-1=({Q+2})/({Q-2})$,
$0\leq 4(2^{*}-1)\left(2^{*}-2\right)\int_{S^{4n+3}}|\tilde{\nabla}u|^{2}\
Vol_{\tilde{\eta}}\\\ \leq\
\left(4\lambda_{1}(2^{*}-1)-\left(2^{*}-2\right)\tilde{S}\right)\int_{S^{4n+3}}u^{2^{*}}\
Vol_{\tilde{\eta}}.$
Thus, our task of showing that $u$ is constant will be achieved once we see
that
(3.9) $4\lambda_{1}(2^{*}-1)-\left(2^{*}-2\right)\tilde{S}\leq 0,\ \text{ i.e,
}\ \lambda_{1}\leq\tilde{S}/(Q+2).$
By Lemma 2.5 we have actually equality $\lambda_{1}={\tilde{S}}/{(Q+2)}$,
which completes the proof. It is worth observing that inequality (3.9) can be
written in the form
$\lambda_{1}\,a\leq(2^{*}-2)\,\tilde{S},$
where $a$ is the constant in front of the (sub-)laplacian in the conformal
(sub-)laplacian, i.e., $a=4\frac{Q+2}{Q-2}$ in our case. ∎
At this point the proof of our main Theorem 1.1 follows easily as follows.
###### Proof of Theorem 1.1.
Let $F$ be a minimizer (local minimum) of the Yamabe functional $\mathcal{E}$
on $\boldsymbol{G}$ and $g$ the corresponding function on the sphere defined
in Lemma 2.4. By Lemma 3.2 and (3.6) the function
$g_{0}=\phi^{-1}(g\circ\psi^{-1})$ will be well centered and a minimizer
(local minimum) of the Yamabe functional $\mathcal{E}$ on $S^{4n+3}$. The
latter claim uses also the fact that the map $v\mapsto u$ of equation (3.6) is
one-to-one and onto on the space of smooth positive functions on the sphere.
Now, from Lemma 3.3 we conclude that $g_{o}=const$. Looking back at the
corresponding functions on the group we see that
$F_{0}=\gamma\,{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4}$
for some $\gamma=const.>0.$ Furthermore, the proof of Lemma 3.1 shows that
$F_{0}$ is obtained from $F$ by a translation (3.2) and dilation (3.3).
Correspondingly, any positive minimizer (local maximum) of problem (3.11) is
given up to dilation or translation by the function
(3.10)
$F=\gamma\,{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4},\qquad\gamma=const.>0.$
Of course, translations (3.2) and dilations (3.3) do not change the value of
$\mathcal{E}$. Incidentally, this shows that any local minimum of the Yamabe
functional $\mathcal{E}$ on the sphere or the group has to be a global one.
We turn to the determination of the best constant. Let us define the constants
(3.11) $\displaystyle\Lambda_{\tilde{\Theta}}\ \overset{def}{=}\ inf\
\left\\{\underset{\boldsymbol{G}}{\int}|\nabla v|^{2}\ Vol_{\tilde{\Theta}}\
:\ v\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G}),\ v\geq 0,\
\underset{\boldsymbol{G}}{\int}|v|^{2^{*}}\ Vol_{\tilde{\Theta}}\ =\
1\right\\}$ and $\displaystyle\Lambda\overset{def}{=}\ inf\
\left\\{\underset{\boldsymbol{G}}{\int}|\nabla v|^{2}\ dH\ :\
v\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G}),\ v\geq 0,\
\underset{\boldsymbol{G}}{\int}|v|^{2^{*}}\ dH\ =\ 1\right\\}.$
Clearly, $\Lambda_{\tilde{\Theta}}\ {=}\ S_{\tilde{\Theta}}^{-2}$, where
$S_{\tilde{\Theta}}$ is the best constant in the $L^{2}$ Folland-Stein
inequality
(3.12) $\left(\int_{\boldsymbol{G}}\ |u|^{2^{*}}\
Vol_{\tilde{\Theta}}\right)^{1/2^{*}}\leq\ S_{\tilde{\Theta}}\
\left(\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}u|^{2}\
Vol_{\tilde{\Theta}}\right)^{1/2},$
while $\Lambda\ {=}\ S_{2}^{-2}$ is the best constant in the $L^{2}$ Folland-
Stein inequality (1.3) (taken with respect to the Lebesgue measure !). By
Remark 2.5 we have
$\Lambda_{\tilde{\Theta}}\ {=}\ \left[2^{-3}(2n)!\right]^{1/(2n+3)}\ \Lambda.$
Furthermore, by Lemma 3.3 and equations (2.6) and (2.7) with $g=const$, we
have
$\Lambda_{\tilde{\Theta}}\ =\ \frac{1}{S_{2}^{2}}=\frac{\int_{\boldsymbol{G}}\
|\nabla^{\tilde{\Theta}}F|^{2}\,Vol_{\tilde{\Theta}}}{\left[\int_{\boldsymbol{G}}|F|^{2^{*}}\,Vol_{\tilde{\Theta}}\right]^{2/2^{*}}}\\\
=\ \frac{\int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+\frac{\tilde{S}}{a}\,g^{2}\right)\,Vol_{\tilde{\eta}}}{\left[\int_{S^{4n+3}}|g|^{2^{*}}\,Vol_{\tilde{\eta}}\right]^{2/2^{*}}}\
=\ 4n(n+1)\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}.$
Here,
$\omega_{4n+3}=2\pi^{2n+2}/\Gamma(2n+2)=2\pi^{2n+2}/(2n+1)!$
is the volume of the unit sphere $S^{4n+3}\subset\mathbb{R}^{4n+4}$ and we
also took into account Remark 2.2 which shows that $Vol_{\tilde{\eta}}$ gives
$2^{2n+3}\left((2n)!\right)\omega_{4n+3}$ for the volume of $S^{4n+3}$. Thus,
$S_{\tilde{\Theta}}\ =\
\left(4n(n+1)\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}\right)^{-1/2}\
=\
\frac{\left[\left((2n)!\right)\omega_{4n+3}\right]^{-1/(4n+6)}}{2\sqrt{n(n+1)}},$
which completes the proof of part a).
b) The Yamabe constant of the sphere is calculated immediately by taking a
constant function in (3.8)
(3.13) $\lambda(S^{4n+3},[\tilde{\eta}])=a\,\Lambda_{\tilde{\Theta}},\qquad
a=4\frac{Q+2}{Q-2}=4\frac{n+2}{n+1}.$
This completes the proof of Theorem 1.1. ∎
###### Remark 3.4.
In view of the above Lemmas it follows that in the conformal class of the
standard qc structure on the sphere (or the quaternionic Heisenberg group)
there is an extremal qc contact form for problem (1.1) which is also qc-
Einstein, see [10, Definition 4.1], and has partial symmetry, see [7,
Definition 1.2], if viewed as a qc structure on the group. Thus, the above
precise constants and extremals can also be taken directly from [10, Theorem
1.1 and 1.2] or the result of [7]. However, the functions (3.10) depend on one
more arbitrary multiplicative parameter $\gamma$ since in the current paper we
are dealing with the functions realizing the infimum of (3.8) rather than with
the qc Yamabe equation with a fixed qc scalar curvature.
## References
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* [4] Cowling, M., Dooley, A. H., Korányi, A. & Ricci, F., _$H$ -type groups and Iwasawa decompositions_, Adv. Math., 87 (1991), 1–41.
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|
arxiv-papers
| 2010-09-15T17:56:45 |
2024-09-04T02:49:12.936112
|
{
"license": "Public Domain",
"authors": "Stefan Ivanov, Ivan Minchev and Dimiter Vassilev",
"submitter": "Dimiter Vassilev",
"url": "https://arxiv.org/abs/1009.2978"
}
|
1009.3308
|
# A pseudospectral quadrature method for Navier-Stokes equations on rotating
spheres
M. Ganesh Department of Mathematical and Computer Sciences, Colorado School
of Mines, Golden, CO 80401 mganesh@mines.edu , Q. T. Le Gia School of
Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052,
Australia qlegia@unsw.edu.au and I. H. Sloan School of Mathematics and
Statistics, University of New South Wales, Sydney, NSW 2052, Australia
i.sloan@unsw.edu.au
###### Abstract.
In this work, we describe, analyze, and implement a pseudospectral quadrature
method for a global computer modeling of the incompressible surface Navier-
Stokes equations on the rotating unit sphere. Our spectrally accurate
numerical error analysis is based on the Gevrey regularity of the solutions of
the Navier-Stokes equations on the sphere. The scheme is designed for
convenient application of fast evaluation techniques such as the fast Fourier
transform (FFT), and the implementation is based on a stable adaptive time
discretization.
###### Key words and phrases:
Navier-Stokes equations, unit sphere, vector spherical harmonics
###### 2000 Mathematics Subject Classification:
Primary 65M12; Secondary 76D05
## 1\. Introduction
In this paper we develop a pseudospectral quadrature method for the surface
Navier-Stokes partial differential equations (PDEs) on the rotating unit
sphere. Whereas the finite element method is best suited for handling non-
smooth processes, the spectral global basis computer models are very efficient
and perform extremely well for processes with smooth regularity. For example,
exponential convergence properties of the global Fourier basis spectral
Galerkin methods (without quadrature) for the Ginzburg-Landau and Navier-
Stokes PDEs on two dimensional periodic cells are based on the Gevrey
regularity of solutions of the PDEs [7, 20].
The complex three dimensional flows in the atmosphere and oceans are
considered to be accurately modeled by the Navier-Stokes PDEs of fluid
mechanics together with classical thermodynamics [22]. Difficulties in
computer modeling in these PDEs resulted in several simplified models for
which spectral approximations are well known [2, 15, 22]. A famous open
problem is to prove the global regularity for the three dimensional
incompressible Navier-Stokes PDEs [26]. However, the precise Gevrey regularity
of the unique solution of the (practically relevant) surface Navier-Stokes
PDEs on the rotating sphere was proved in [5]. (Because the Earth’s surface is
an approximate sphere, a standard surface model, to study global atmospheric
circulation on large planets, is the sphere.)
Consequently, a natural next step is to describe, analyze, and implement an
exponentially converging pseudospectral method for the Navier-Stokes PDEs on
the rotating sphere. In addition to the continuous model regularity results in
[5], this paper is also motivated by the recent work [12], where discrete
computer modeling of the Navier-Stokes PDEs on one-dimensional and toroidal
domains [8] was extended to the unit sphere.
This paper is concerned with both implementation of our algorithm and its
numerical analysis. The main numerical analysis contributions compared to
results in [5], for the continuous problem, and in [12], for a discrete
problem, are as follows.
For the spatially discrete pseudospectral quadrature Galerkin solutions of the
Navier-Stokes equations, we prove (i) the stability (that is, uniform
boundedness of approximate solutions, independent of their truncation
parameter $N$), see Theorem 4.1; and (ii) a spectrally accurate rate of
convergence [that is, $\mathcal{O}(N^{-2s})$ accuracy, with $s$ depending on
the smoothness of input data], see Theorem 4.2. We achieve these results by
first generalizing the main regularity result in [5] to complex valued times
(see Theorem 2.2 and 2.3), and then using this to prove the spectral rate of
convergence of the time derivative of a Stokes projection comparison function
(see Theorem 3.2). This time-derivative error result plays a crucial role in
the proof of spectral convergence of the approximate solutions (see the proof
of Theorem 4.2). We note that the main result in [12, page 978] establishes
only convergence of the semi-discrete Galerkin method without quadrature in
$L^{p}$ norms, but does not establish either stability or rate of convergence
of the scheme.
The rate of convergence results, supported by numerical experiments, formed
the core part of research on the Navier-Stokes equations on two dimensional
domains over the last few decades, see [8, 9, 27] and references therein.
There is a vast literature on numerical methods and analysis for the Navier-
Stokes PDE on bounded Euclidean domains (see [8, 9, 27] and references
therein), but their counterparts on closed manifolds are rarer (see [12] and
references therein). The implementation of the scheme in [12, page 978] is
based on a fixed time-step explicit Runge-Kutta method that has a small
stability region for the systems of ordinary differential equations arising
from the spatial discretization.
The outline of this paper is as follows. In the next section, we recall
various known preliminary results associated with the Navier-Stokes PDEs on
the unit sphere, in strong and weak form. In Section 3, we introduce essential
computational and numerical analysis tools required for the discretization and
analysis of the Navier-Stokes equations. In Section 4, we describe and prove
spectral accuracy of a pseudospectral quadrature method and give
implementation details required to apply the FFT and adaptive-in-time
simulation of the Navier-Stokes equations. In Section 5, we demonstrate
computationally the accuracy and applicability of the algorithm for well known
benchmark examples.
## 2\. Navier-Stokes equations on the rotating unit sphere
The surface Navier-Stokes equations (NSE) describing a tangential,
incompressible atmospheric stream on the rotating two-dimensional unit sphere
$S\subset\mathbb{R}^{3}$ can be written as [5, 16, 17, 19, 28]
(2.1) $\frac{\partial}{\partial t}{\bf u}+\boldsymbol{\mathrm{\nabla}}\,_{\bf
u}{\bf u}-\nu\boldsymbol{\mathrm{\Delta}}\,{\bf
u}+\boldsymbol{\mathrm{\omega}}\,\times{\bf
u}+\frac{1}{\rho}\boldsymbol{\mathrm{Grad}}\,p={\bf
f},\qquad\mathrm{Div}\,{\bf u}=0,\qquad{\bf u}|_{t=0}={\bf
u}_{0}\qquad\text{on}~{}~{}S.$
Here ${\bf u}={\bf u}(\widehat{{\bf x}},t)=\left(u_{1}(\widehat{{\bf
x}},t),u_{2}(\widehat{{\bf x}},t),u_{3}(\widehat{{\bf x}},t)\right)^{T}$ is
the unknown tangential divergence-free velocity field at $\widehat{{\bf x}}\in
S$ and $t\in[0,T]$, $p=p(\widehat{{\bf x}},t)$ is the unknown pressure. The
known components in (2.1) are the constant viscosity and density of the fluid,
respectively denoted by $\nu,\rho$, the normal vector field
$\boldsymbol{\mathrm{\omega}}\,=\boldsymbol{\mathrm{\omega}}\,(\widehat{{\bf
x}})=\omega(\widehat{{\bf x}})\widehat{{\bf x}}$ for the Coriolis acceleration
term, and the external flow driving vector field ${\bf f}={\bf
f}(\widehat{{\bf x}},t)$. The Coriolis function $\omega$ is given by
$\omega(\widehat{{\bf x}})=2\Omega\cos\theta$, where $\Omega$ is the angular
velocity of the rotating sphere, and $\theta$ is the angle between
$\widehat{{\bf x}}$ and the north pole. The vorticity of the flow associated
with the NSE (2.1), in the curvilinear coordinate system, is a normal vector
field, defined, for a fixed $t\geq 0$, by
(2.2) $\boldsymbol{\mathrm{Vort}}\,{\bf u}(\widehat{{\bf
x}},t)=\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf u}(\widehat{{\bf
x}},t)=\widehat{{\bf x}}\Delta\Psi(\widehat{{\bf
x}},t),\qquad\qquad\widehat{{\bf x}}\in S,$
for some scalar-valued vorticity stream function $\Psi$.
All spatial derivative operators in (2.1)-(2.2) are surface differential
operators, obtained by restricting the corresponding domain operators (defined
in a neighborhood of $S$) to the unit sphere, using standard differential
geometry concepts on closed manifolds in $\mathbb{R}^{3}$ [16, 17].
Using the fact that the outward unit normal at $\widehat{{\bf x}}\in S$ is
$\widehat{{\bf x}}$, the $\boldsymbol{\mathrm{Curl}}\,$ of a scalar function
$v$, of a normal vector field ${\bf w}=w\widehat{{\bf x}}$, and of a
tangential vector field ${\bf v}$ on $S$ are respectively defined by
(2.3) $\boldsymbol{\mathrm{Curl}}\,v=-\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Grad}}\,v,\qquad\boldsymbol{\mathrm{Curl}}\,{\bf
w}=-\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Grad}}\,w,\qquad\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}{\bf v}=-\widehat{{\bf x}}~{}\mathrm{Div}\,(\widehat{{\bf x}}\times{\bf
v}).$
The surface diffusion operator acting on tangential vector fields on $S$ is
denoted by $\boldsymbol{\mathrm{\Delta}}\,$ (known as the Laplace-Beltrami or
Laplace-de Rham operator) and is defined as
(2.4) $\boldsymbol{\mathrm{\Delta}}\,{\bf
v}=\boldsymbol{\mathrm{Grad}}\,\mathrm{Div}\,{\bf
v}-\boldsymbol{\mathrm{Curl}}\,\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}{\bf v}.$
The following relations connecting the above operators will be used throughout
the paper:
(2.5)
$\mathrm{Div}\,\boldsymbol{\mathrm{Curl}}\,v=0,\qquad\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}\boldsymbol{\mathrm{Curl}}\,v=-\widehat{{\bf x}}\Delta
v,\qquad\boldsymbol{\mathrm{\Delta}}\,\boldsymbol{\mathrm{Curl}}\,v=\boldsymbol{\mathrm{Curl}}\,\Delta
v,$ (2.6) $2\boldsymbol{\mathrm{\nabla}}\,_{\bf w}{\bf
v}=-\boldsymbol{\mathrm{Curl}}\,({\bf w}\times{\bf
v})+\boldsymbol{\mathrm{Grad}}\,({\bf w}\cdot{\bf v})-{\bf
v}\mathrm{Div}\,{\bf w}+{\bf w}\mathrm{Div}\,{\bf v}-{\bf
v}\times\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf w}-{\bf
w}\times\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf v}.$
In particular, for tangential divergence-free vector fields, such as the
solution ${\bf u}$ of the NSE, using (2.6), the nonlinear term in (2.1) can be
written as
(2.7) $\boldsymbol{\mathrm{\nabla}}\,_{\bf u}{\bf
u}=\boldsymbol{\mathrm{Grad}}\,\frac{|{\bf u}|^{2}}{2}-{\bf
u}\times\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf u}.$
### 2.1. A weak formulation
A standard technique for removing the scalar pressure field from the Navier-
Stokes equations is to multiply the first equation in (2.1) by test functions
${\bf v}$ from a space with elements having properties of the unknown velocity
field ${\bf u}$ (in particular, $\mathrm{Div}\,{\bf v}=0$) and then integrate
to obtain a weak formulation. (The unknown $p$, can then be computed by
solving a pressure Poisson equation, obtained by applying the surface
divergence operator in (2.1).)
To this end, we introduce the standard inner products on the space of all
square integrable (i) scalar functions on $S$, denoted by $L^{2}(S)$; and (ii)
tangential vector fields on $S$, denoted by $L^{2}(TS)$:
(2.8) $\displaystyle(v_{1},\;v_{2})$ $\displaystyle=$
$\displaystyle(v_{1},\;v_{2})_{L^{2}(S)}~{}~{}=\int_{S}v_{1}\overline{v_{2}}~{}dS,\quad\qquad
v_{2},v_{2}\in L^{2}(S),$ (2.9) $\displaystyle({\bf v}_{1},\;{\bf v}_{2})$
$\displaystyle=$ $\displaystyle({\bf v}_{1},\;{\bf
v}_{2})_{L^{2}(TS)}=\int_{S}{\bf v}_{1}\cdot\overline{{\bf
v}_{2}}~{}dS,\qquad{\bf u},{\bf v}\in L^{2}(TS),$
where $dS=\sin\theta d\theta d\phi$. Throughout the paper, the induced norm on
$L^{2}(TS)$ is denoted by $\|\cdot\|$ and for other inner product spaces, say
$X$ with inner product $(\cdot,\;\cdot)_{X}$, the associated norm is denoted
by $\|\cdot\|_{X}$. For example, for $s>0$, standard norms in the scalar and
vector valued functions Sobolev spaces $H^{s}(S)$ and $H^{s}(TS)$ are denoted
by $\|\cdot\|_{H^{s}(S)}$ and $\|\cdot\|_{H^{s}(TS)}$, respectively. Since
$H^{0}(TS)=L^{2}(TS)$, $\|\cdot\|_{H^{0}(TS)}=\|\cdot\|$.
We have the following identities for appropriate scalar and vector fields [16,
(2.4)-(2.6)]:
(2.10) $\displaystyle(\boldsymbol{\mathrm{Grad}}\,\psi,\;{\bf
v})=-(\psi,\;\mathrm{Div}\,{\bf v}),$
$\displaystyle\qquad(\boldsymbol{\mathrm{Curl}}\,\psi,\;{\bf
v})=(\psi,\;\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf v}),$ (2.11)
$\displaystyle(\boldsymbol{\mathrm{Curl}}\,\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}{\bf w},\;{\bf z})$ $\displaystyle=$
$\displaystyle(\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
w},\;\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf z}).$
In (2.10), the $L^{2}(TS)$ inner product is used on the left hand side and the
$L^{2}(S)$ inner product is used on the right hand side. Throughout the paper,
we identify a normal vector field ${\bf w}$ with a scalar field $w$ and hence
(2.12) $(\psi,\;{\bf w}):=(\psi,\;w)_{L^{2}(S)},\qquad{\bf w}=\widehat{{\bf
x}}w,\qquad\psi,w\in L^{2}(S).$
Using (2.5), smooth ($C^{\infty}$) tangential fields on $S$ can be decomposed
into two components, one in the space of all divergence-free fields and the
other through the Hodge decomposition theorem [1]:
(2.13) $C^{\infty}(TS)=C^{\infty}(TS;\boldsymbol{\mathrm{Grad}}\,)\oplus
C^{\infty}(TS;\boldsymbol{\mathrm{Curl}}\,),$
where
(2.14)
$C^{\infty}(TS;\boldsymbol{\mathrm{Grad}}\,)=\\{\boldsymbol{\mathrm{Grad}}\,\psi:\psi\in
C^{\infty}(S)\\},\;C^{\infty}(TS;\boldsymbol{\mathrm{Curl}}\,)=\\{\boldsymbol{\mathrm{Curl}}\,\psi:\psi\in
C^{\infty}(S)\\}.$
For $s\geq 0$, let $C^{\infty,s}(TS;\boldsymbol{\mathrm{Curl}}\,)$ denote the
closure of $C^{\infty}(TS;\boldsymbol{\mathrm{Curl}}\,)$ in the $H^{s}(TS)$
norm. In particular, following [5] we introduce a simpler notation
$\displaystyle H$ $\displaystyle=$ $\displaystyle\mbox{ closure of
}C^{\infty}(TS;\boldsymbol{\mathrm{Curl}}\,)\mbox{ in
}L^{2}(TS)\;=\;C^{\infty,0}(TS;\boldsymbol{\mathrm{Curl}}\,),$ $\displaystyle
V$ $\displaystyle=$ $\displaystyle\mbox{ closure of
}C^{\infty}(TS;\boldsymbol{\mathrm{Curl}}\,)\mbox{ in
}H^{1}(TS)\;=\;C^{\infty,1}(TS;\boldsymbol{\mathrm{Curl}}\,).$
Using the Gauss surface divergence theorem, for any scalar valued function $v$
on $S$ with $\boldsymbol{\mathrm{Grad}}\,v\in L^{2}(TS)$, using (2.10), we
have
(2.15) $(\boldsymbol{\mathrm{Grad}}\,v,\;{\bf
w})=\int_{S}\boldsymbol{\mathrm{Grad}}\,v\cdot\overline{{\bf
w}}~{}dS=-\int_{S}v\cdot\mathrm{Div}\,\overline{{\bf w}}~{}dS=0,\quad{\bf
w}\in V,$
and hence the unknown pressure can be eliminated from the first equation in
(2.1) through the weak formulation.
Following [16, Page 567], for the diffusion part of the NSE, we consider the
Stokes operator
(2.16)
$\boldsymbol{\mathrm{A}}=\boldsymbol{\mathrm{Curl}}\,\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}.$
Using (2.4) and (2.5), it is easy to see that the Stokes operator is the
restriction of the vector Laplace-de Rham operator
$-\boldsymbol{\mathrm{\Delta}}\,$ on $V$;
$\boldsymbol{\mathrm{A}}=-{\mathbf{P}}_{\boldsymbol{\mathrm{Curl}}\,}\boldsymbol{\mathrm{\Delta}}\,$,
where
${\mathbf{P}}_{\boldsymbol{\mathrm{Curl}}\,}:~{}L^{2}(TS)~{}\rightarrow~{}H$
is the orthogonal projection onto the divergence-free tangent space.
For each positive integer $L=1,2,\ldots$, the eigenvalue $\lambda_{L}$ and the
corresponding eigenvectors of the Stokes operator $\boldsymbol{\mathrm{A}}$
are given by
(2.17)
$\lambda_{L}=L(L+1),\qquad{\mathbf{Z}}_{L,m}(\theta,\varphi)=\lambda^{-1/2}_{L}\boldsymbol{\mathrm{Curl}}\,Y_{L,m}(\theta,\varphi),\quad
m=-L,\ldots,L,$
where $Y_{L,m}$ are the scalar orthonormal spherical harmonics of degree $L$,
defined by
(2.18)
$Y_{L,m}(\theta,\varphi)=\left[\frac{(2L+1)}{4\pi}\frac{(L-|m|)!}{(L+|m|)!}\right]^{1/2}P^{m}_{L}(\cos\theta)e^{im\varphi},\quad
m=-L,\ldots,L,$
with $P^{m}_{L}$ being the associated Legendre polynomials so that
$\overline{Y_{L,m}}=(-1)^{m}Y_{L,-m}$.
The spectral property
$\boldsymbol{\mathrm{A}}{\mathbf{Z}}_{L,m}=\lambda_{L}{\mathbf{Z}}_{L,m}$
follows from the fact that $Y_{L,m}$ are eigenfunctions of the scalar Laplace-
Beltrami operator $-\Delta$ with eigenvalues $\lambda_{L}$, the definition of
the Stokes operator $\boldsymbol{\mathrm{A}}$ in (2.16), ${\mathbf{Z}}_{L,m}$
in (2.17), (2.5), and (2.3). Since $\\{Y_{L,m}:L=0,1,\ldots;m=-L,\ldots,L\\}$
is an orthonormal basis for $L^{2}(S)$, it is easy to see that
$\\{{\mathbf{Z}}_{L,m}:L=1,\ldots;m=-L,\ldots,L\\}$ is an orthonormal basis
for $H$. Thus an arbitrary ${\bf v}\in H$ can be written as
(2.19) ${\bf v}=\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\widehat{{\bf
v}}_{L,m}{\mathbf{Z}}_{L,m},\qquad\widehat{{\bf v}}_{L,m}=\int_{S}{\bf
v}\cdot\overline{{\mathbf{Z}}_{L,m}}dS=({\bf v},\;{\mathbf{Z}}_{L,m}).$
We consider a subset of $H$,
(2.20) $\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2})=\left\\{{\bf v}\in H\;:{\bf
v}=\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\widehat{{\bf
v}}_{L,m}{\mathbf{Z}}_{L,m},\quad\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\lambda^{s}_{L}|\widehat{{\bf
v}}_{L,m}|^{2}<\infty\right\\},$
which is the divergence-free subset of the Sobolev space $H^{s}(TS)$. For
every ${\bf v}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2})$, we set
(2.21) $\|{\bf
v}\|_{H^{s}(TS)}=\left[\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\lambda^{s}_{L}|\widehat{{\bf
v}}_{L,m}|^{2}\right]^{1/2},$
and for ${\bf v}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2})$, we define
(2.22) $\boldsymbol{\mathrm{A}}^{s/2}{\bf
v}:=\quad\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\lambda^{s/2}_{L}\widehat{{\bf
v}}_{L,m}{\mathbf{Z}}_{L,m}\quad\in H.$
For a tangential vector field ${\bf v}$ on $S$, we define the Coriolis
operator $\boldsymbol{\mathrm{C}}$,
(2.23) $(\boldsymbol{\mathrm{C}}{\bf v})(\widehat{{\bf
x}})=\boldsymbol{\mathrm{\omega}}\,(\widehat{{\bf x}})\times{\bf
v}(\widehat{{\bf x}})=\omega(\widehat{{\bf x}})(\widehat{{\bf x}}\times{\bf
v}),\qquad\omega(\widehat{{\bf x}})=2\Omega\cos\theta.$
To treat the nonlinear term in (2.1), we consider the trilinear form $b$ on
$V\times V\times V$, defined as
(2.24) $b({\bf v},{\bf w},{\bf z})=(\boldsymbol{\mathrm{\nabla}}\,_{\bf v}{\bf
w},{\bf z})=\int_{S}\boldsymbol{\mathrm{\nabla}}\,_{\bf v}{\bf
w}\cdot{\overline{\bf z}}~{}dS,\qquad{\bf v},{\bf w},{\bf z}\in V.$
Using (2.6) and (2.10), for divergence free fields ${\bf v},{\bf w},{\bf z}$,
the trilinear form can be written as
(2.25) $b({\bf v},{\bf w},{\bf z})=\frac{1}{2}\int_{S}\left[-{\bf v}\times{\bf
w}\cdot\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\overline{\bf
z}}+\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf v}\times{\bf
w}\cdot{\overline{\bf z}}-{\bf
v}\times\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
w}\cdot{\overline{\bf z}}\right]~{}dS.$
Moreover [16, Lemma 2.1]
(2.26) $b({\bf v},{\bf w},{\bf w})=0,\qquad b({\bf v},{\bf z},{\bf w})=-b({\bf
v},{\bf w},{\bf z})\qquad{\bf v},{\bf w},{\bf z}\in V.$
Throughout the paper, the space $V$ is equipped with the norm
$\|\cdot\|^{2}_{V}=\left(\boldsymbol{\mathrm{A}}\cdot,\cdot\right)$.
Thus, using (2.4), (2.10), (2.16), and (2.25), a _weak solution_ of the
Navier-Stokes equations (2.1) is a vector field ${\bf u}\in L^{2}([0,T];V)$
with ${\bf u}(0)={\bf u}_{0}$ that satisfies the weak form
(2.27) $({\bf u}_{t},{\bf v})+b({\bf u},{\bf u},{\bf
v})+\nu(\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
u},\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
v})+(\boldsymbol{\mathrm{C}}{\bf u},{\bf v})=({\bf f},{\bf v}),\qquad{\bf
v}\in V.$
This weak formulation can be written in operator equation form on $V^{*}$, the
adjoint of $V$: Let ${\bf f}\in L^{2}([0,T];V^{*})$ and ${\bf u}_{0}\in H$.
Find a vector field ${\bf u}\in L^{2}([0,T];V)$, with ${\bf u}_{t}\in
L^{2}([0,T];V^{*})$ such that
(2.28) ${\bf u}_{t}+\nu\boldsymbol{\mathrm{A}}{\bf
u}+\boldsymbol{\mathrm{B}}({\bf u},{\bf u})+\boldsymbol{\mathrm{C}}{\bf
u}={\bf f},\qquad{\bf u}(0)={\bf u}_{0},$
where the bilinear form $\boldsymbol{\mathrm{B}}({\bf u},{\bf v})\in V^{*}$ is
defined by
(2.29) $(\boldsymbol{\mathrm{B}}({\bf u},{\bf v}),{\bf w})=b({\bf u},{\bf
v},{\bf w})\qquad{\bf w}\in V.$
In the subsequent error analysis, we need the following estimate for the
nonlinear term (see Lemma 6.1 in Appendix):
(2.30) $\|\boldsymbol{\mathrm{A}}^{-\delta}\boldsymbol{\mathrm{B}}({\bf
u},{\bf v})\|\leq\begin{cases}C\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf
u}\|\|{\bf v}\|\leq C\|\boldsymbol{\mathrm{A}}^{1/2}{\bf u}\|\|{\bf v}\|,&\cr
C\|{\bf u}\|\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf v}\|\leq C\|{\bf
u}\|\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
v}\|,&\end{cases}\quad\delta\in(1/2,1)\quad{\bf u},{\bf v}\in V.$
In (2.30), as throughout the paper, $C$ is a generic constant independent of
${\bf u}$ and ${\bf v}$, (and the discretization parameter $N$ introduced in
Section 3). From (2.30) we deduce the weak Lipschitz continuity property
(2.31) $\|\boldsymbol{\mathrm{A}}^{-\delta}(\boldsymbol{\mathrm{B}}({\bf
v},{\bf v})-\boldsymbol{\mathrm{B}}({\bf w},{\bf w}))\|\leq C\|{\bf v}-{\bf
w}\|,\quad\delta\in(1/2,1),\quad\text{if }\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
v}\|,\|\boldsymbol{\mathrm{A}}^{1/2}{\bf w}\|<C.$
The existence and uniqueness of the solution ${\bf u}\in L^{2}([0,T];V)$ of
the weak formulation (2.27) are discussed in [16, 17, 19]. A regular solution
of the Navier-Stokes equations (2.1) on $[0,T]$ is a tangential divergence-
free velocity field ${\bf u}$ that satisfies the equation obtained by
integrating in time the weak form (2.27), from $t_{0}$ to $t$, for almost
every $t_{0},t\in[0,T]$. In order to recall the existence, uniqueness, and
Gevrey regularity of the regular solution, we need a few more additional
details from [5]. These are also needed as tools for analyzing our
pseudospectral quadrature method.
### 2.2. Gevrey regularity of regular solution
The Gevrey class of functions of order $s>0$ and index $\sigma>0$, associated
with the Stokes operator defined in (2.16), is denoted by $G^{s/2}_{\sigma}$
and is defined as
(2.32) $G^{s/2}_{\sigma}:=\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2}e^{\sigma
A^{1/2}})\subset\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2}).$
Using (2.20), the Gevrey space
(2.33) $G^{s/2}_{\sigma}=\left\\{{\bf
v}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s/2})\ :{\bf
v}=\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\widehat{{\bf
v}}_{L,m}{\mathbf{Z}}_{L,m},\;\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\lambda^{s}_{L}e^{2\sigma\lambda^{1/2}_{L}}|\widehat{{\bf
v}}_{L,m}|^{2}<\infty\right\\}$
is a Hilbert space with respect to the inner product
(2.34) ${\left\langle{{\bf v}},{{\bf
w}}\right\rangle}_{G^{s/2}_{\sigma}}=\sum_{L=1}^{\infty}\sum_{m=-L}^{L}\lambda^{s}_{L}e^{2\sigma\lambda^{1/2}_{L}}\widehat{{\bf
v}}_{L,m}\overline{\widehat{{\bf w}}_{L,m}},\qquad{\bf v},{\bf w}\in
G^{s/2}_{\sigma}.$
First we recall the following result from [5, 29].
###### Theorem 2.1.
If ${\bf u}_{0}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2})$ and ${\bf
f}\in
L^{\infty}((0,\infty);\mathcal{D}(\boldsymbol{\mathrm{A}}^{s}e^{\sigma_{1}\boldsymbol{\mathrm{A}}^{1/2}}))$,
for some $s,\sigma_{1}>0$, then for all $t>0$ there exists a $T^{*}>0$,
depending only on $\nu,{\bf f}$, and $\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}_{0}\|_{L^{2}(TS)}$, such that the NSE (2.1) on $S$ have a unique regular
solution ${\bf u}(\cdot,t)$ and ${\bf u}(\cdot,t)\in G^{s+1/2}_{\sigma(t)}$,
where $\sigma(t)=\min\\{t,T^{*},\sigma_{1}\\}$.
In addition, from the assumption and proof of Theorem 2.1 [5, page 355],
(2.35)
$\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\sigma(t)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(t)\|^{2}\leq M_{0},\qquad t>0,$
where $M_{0}$ depends on $\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}(0)\|$,
$\sup_{t\geq 0}\|\boldsymbol{\mathrm{A}}^{s}{\bf f}(\cdot,t)\|$ and $\nu$ but
not on $t$. The bound in (2.35) is useful for establishing the quality of
approximation of the Stokes projection of ${\bf u}$ in the next section (see
Theorem 3.1). It is also convenient to have a similar bound for the time
derivative of the Stokes projections with $t$ in (2.35) replaced with certain
complex times $\zeta\in\mathbb{C}$, to prove the power of approximation of the
time derivative of the Stokes projection (see Theorem 3.2). To this end, we
consider the NSE extended to complex times $\zeta$,
(2.36) $\frac{d{\bf u}}{d\zeta}+\nu\boldsymbol{\mathrm{A}}{\bf
u}+\boldsymbol{\mathrm{B}}({\bf u},{\bf u})+\boldsymbol{\mathrm{C}}{\bf
u}={\bf f},\quad\mathrm{Div}\,{\bf u}=0,\quad{\bf u}(0)={\bf
u}_{0},\qquad\zeta\in\mathbb{C},\qquad\text{on}~{}~{}S,$
with standard complexification (see [9]) of all the spaces and operators
introduced earlier. In the next theorem, we extend arguments used in [9] for
the solution of the NSE on the plane to the case of the sphere. The arguments
differ in an essential way only for the nonlinear term.
###### Theorem 2.2.
Let ${\bf u}_{0}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2})$ and ${\bf
f}\in
C([0,T];\mathcal{D}(\boldsymbol{\mathrm{A}}^{s}e^{\sigma_{1}\boldsymbol{\mathrm{A}}^{1/2}}))$,
with $s~{}\geq~{}1/4$. Let the domain $\mathbb{T}$ be defined by
$\mathbb{T}:=\\{\zeta=re^{i\theta}:0\leq r\leq T;\ |\theta|\leq\pi/4\\}.$
We assume further that ${\bf f}(\cdot,\zeta)$ is analytic for
$\zeta\in\mathbb{T}$ and that
(2.37)
$K:=\sup\\{\|\boldsymbol{\mathrm{A}}^{s}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f}(\cdot,\zeta)\|^{2}:\zeta=re^{i\theta}\in\mathbb{T}\\}<\infty.$
Then there exists $T^{**}>0$ such that
(2.38)
$\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(\zeta)\|^{2}\leq M_{1},\qquad\zeta\in\mathbb{T}\mbox{ and }|\zeta|\leq
T^{**},$
where $M_{1}$ depends on $\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}(0)\|$ and
hence ${\bf u}(\cdot,\zeta)\in G^{s+1/2}_{\psi(r\cos\theta)}$, where
$\psi(x)~{}:=~{}\min\\{x,T^{**},\sigma_{1}\\}$.
###### Proof.
Let $\zeta=re^{i\theta}$ with $r>0$ and $|\theta|\leq\pi/4$, and let
${\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta):=\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(\zeta).$
The definition of $\psi$ gives $\frac{d}{dx}\psi:=\psi^{\prime}(x)\leq 1$, and
$\psi\leq\sigma_{1}$, thus for fixed $\theta$ we find
(2.39) $\displaystyle\frac{d}{dr}{\bf u}_{\boldsymbol{\mathrm{A}}}(\zeta)$
$\displaystyle=$
$\displaystyle\psi^{\prime}(r\cos\theta)\cos\theta\boldsymbol{\mathrm{A}}^{s+1}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(\zeta)$
$\displaystyle+e^{i\theta}\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}\frac{d{\bf
u}}{d\zeta}(\zeta).$
Using (2.39) in
$\displaystyle\frac{1}{2}\frac{d}{dr}\left\|{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta)\right\|^{2}=\Re\left(\frac{d}{dr}{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta),{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta)\right),$
where $\Re(\cdot)$ denotes the real-part function, we get
(2.40) $\displaystyle\frac{1}{2}\frac{d}{dr}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta)\|^{2}$ $\displaystyle=$
$\displaystyle\psi^{\prime}(r\cos\theta)\cos\theta\;\Re\;(\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta),{\bf u}_{\boldsymbol{\mathrm{A}}}(\zeta))$
$\displaystyle+\Re\;e^{i\theta}(\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}\frac{d{\bf
u}}{d\zeta}(\zeta),{\bf u}_{\boldsymbol{\mathrm{A}}}(\zeta)).$
Using (2.36) for the last term in (2.40) together with the fact that
$(\boldsymbol{\mathrm{A}}^{s}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}\boldsymbol{\mathrm{C}}{\bf
u},\boldsymbol{\mathrm{A}}^{s+1}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u})=0$
(see [5, Lemma 1]), we find
(2.41) $\displaystyle\frac{1}{2}\frac{d}{dr}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta)\|^{2}+\nu\cos\theta\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta)\|^{2}$ $\displaystyle=$
$\displaystyle\psi^{\prime}(r\cos\theta)\cos\theta(\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}},{\bf
u}_{\boldsymbol{\mathrm{A}}})-\Re\;e^{i\theta}(\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}\boldsymbol{\mathrm{B}}({\bf
u},{\bf u}),{\bf u}_{\boldsymbol{\mathrm{A}}})$
$\displaystyle+\Re\;e^{i\theta}(\boldsymbol{\mathrm{A}}^{s}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f},\boldsymbol{\mathrm{A}}^{1/2}{\bf u}_{\boldsymbol{\mathrm{A}}}),$
where in (2.41) and below we write ${\bf u}_{\boldsymbol{\mathrm{A}}}={\bf
u}_{\boldsymbol{\mathrm{A}}}(\zeta),{\bf u}={\bf u}(\zeta),{\bf f}={\bf
f}(\zeta),\psi=\psi(r\cos\theta)$. From [5, Lemma 2], we have (with
$p=\max\\{2-s,7/4\\}=7/4$, since $s\geq 1/4$),
(2.42)
$|(\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}\boldsymbol{\mathrm{B}}({\bf
u},{\bf u}),{\bf u}_{\boldsymbol{\mathrm{A}}})|\leq C\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{3-p}\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{p}.$
Applying $\psi^{\prime}\leq 1$, the Cauchy-Schwarz inequality, (2.42) and
Young’s inequality ($ab\leq a^{q}/q+b^{q^{\prime}}/q^{\prime}$ with
$1/q+1/q^{\prime}=1$) with $q=2$ and $q=2/(2-p)$ in (2.41), we get
$\displaystyle\frac{1}{2}\frac{d}{dr}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}+\nu\cos\theta\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}$ $\displaystyle\leq$
$\displaystyle\cos\theta\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|\|{\bf u}_{\boldsymbol{\mathrm{A}}}\|+C\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{3-p}\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{p}+\|\boldsymbol{\mathrm{A}}^{s}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f}\|\|\boldsymbol{\mathrm{A}}^{1/2}{\bf u}_{\boldsymbol{\mathrm{A}}}\|$
$\displaystyle\leq$
$\displaystyle\frac{\nu\cos\theta}{4}\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}+\frac{\cos\theta}{\nu}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}$
$\displaystyle+C^{\frac{2}{2-p}}\frac{(2-p)}{p}\left(\frac{p}{\nu\cos\theta}\right)^{\frac{p}{2-p}}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{\frac{2(3-p)}{2-p}}+\frac{\nu\cos\theta}{2}\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}$
$\displaystyle+\frac{1}{\nu\cos\theta}\|\boldsymbol{\mathrm{A}}^{s}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f}\|^{2}+\frac{\nu\cos\theta}{4}\|\boldsymbol{\mathrm{A}}^{1/2}{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}.$
Therefore,
(2.43) $\displaystyle\frac{d}{dr}\|{\bf u}_{\boldsymbol{\mathrm{A}}}\|^{2}$
$\displaystyle\leq$ $\displaystyle\frac{2\cos\theta}{\nu}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2}+C\left(\frac{1}{\nu\cos\theta}\right)^{\frac{p}{2-p}}\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{\frac{2(3-p)}{2-p}}+\frac{2}{\nu\cos\theta}\|\boldsymbol{\mathrm{A}}^{s}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f}\|^{2}$ $\displaystyle\leq$ $\displaystyle\frac{2\cos\theta}{\nu}(1+\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2})+C\left(\frac{1}{\nu\cos\theta}\right)^{\frac{p}{2-p}}(1+\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2})^{\frac{3-p}{2-p}}+$
$\displaystyle\frac{2}{\nu\cos\theta}\|\boldsymbol{\mathrm{A}}^{s}e^{\psi\boldsymbol{\mathrm{A}}^{1/2}}{\bf
f}\|^{2}.$
Using $|\theta|\leq\pi/4$, and $\frac{3-p}{2-p}=5$, in (2.43), we get
(2.44) $\frac{d}{dr}\|{\bf u}_{\boldsymbol{\mathrm{A}}}\|^{2}\leq C(1+\|{\bf
u}_{\boldsymbol{\mathrm{A}}}\|^{2})^{5}+\frac{2\sqrt{2}}{\nu}K.$
With $|\theta|\leq\pi/4$ fixed, let
$y(r)=1+\|{\bf u}_{\boldsymbol{\mathrm{A}}}\|^{2}.$
Then on using (2.44) and $y\geq 1$ we obtain
$\frac{d}{dr}y\leq Cy^{5},$
and hence on integrating the inequality we find that
$y(r)\leq 2y(0)$
provided that
$0\leq
r\leq\frac{15}{64C}\frac{1}{(y(0))^{4}}=\frac{15}{64C}\frac{1}{(1+\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}(0)\|^{2})^{4}}.$
By setting
$T^{**}:=\frac{15}{64C}\frac{1}{(1+\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}(0)\|^{2})^{4}},\quad M_{1}=1+2\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}(0)\|^{2},$
we deduce that (2.38) holds for $0\leq r\leq T^{**}$. ∎
We can extend the bound (2.38) to a larger domain which contains the interval
$[0,T]$ using the following property of the NSE solution on the sphere [29]:
(2.45) $\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}(\cdot,t)\|\leq M_{2}\mbox{
for all }t\in[0,T],$
where the constant $M_{2}$ depends only on
$\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}_{0}\|$, $\sup_{0\leq t\leq
T}\|\boldsymbol{\mathrm{A}}^{s}{\bf f}(\cdot,t)\|$ and $\nu$ but not on $T$.
###### Theorem 2.3.
Suppose ${\bf u}_{0}$ and ${\bf f}$ satisfy all the conditions in the domain
$\mathbb{T}$ as in Theorem 2.2. Then
(2.46)
$\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(\zeta)\|^{2}\leq M_{3},\qquad\zeta\in\mathbb{T}\mbox{ and }|\mbox{Im
}\zeta|\leq T^{**}/\sqrt{2},$
where $M_{3}:=1+2M_{2}^{2}$, $T^{**}$ depends on $M_{2}$, and hence ${\bf
u}(\cdot,\zeta)\in G^{s+1/2}_{\psi(r\cos\theta)}$ for all $\zeta\in\mathbb{T}$
with $|\mbox{Im }\zeta|\leq T^{**}/\sqrt{2}$.
###### Proof.
We proceed as in the proof of Theorem 2.2 to obtain the ordinary differential
equation
$\frac{d}{dr}y\leq Cy^{5},$
where
$y(r)=1+\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(re^{i\theta})\|^{2}$. On integrating the inequality we find that
$y(r)\leq 2y(0)$
provided that
$0\leq r\leq\frac{15}{64C}\frac{1}{(1+\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}(0)\|^{2})^{4}}.$
We define
$T_{1}(\rho):=\frac{15}{64C}\frac{1}{(1+\rho^{2})^{4}},\quad\rho\geq 0.$
If $T_{1}(\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}(0)\|)\geq T$ we have
finished the proof. Otherwise, we let
$T^{**}=T_{1}(M_{2}),$
where $M_{2}$ is given in (2.45). For $\zeta=re^{i\theta}\in\mathbb{T}$,
$0\leq r\leq T^{**}$,
(2.47)
$\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(re^{i\theta})\|^{2}\leq 1+2\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf
u}(0)\|^{2}.$
Consequently, (2.46) holds for $0\leq r\leq T^{**}$ with
$M_{3}=1+2\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf u}(0)\|^{2}$.
Next we consider the case $\zeta=T^{**}+re^{i\theta}$, with $r\in[0,T^{**}]$.
We define, for $|\theta|\leq\pi/4$,
${\bf v}(re^{i\theta}):={\bf u}(T^{**}+re^{i\theta}),\qquad r\in[0,T^{**}].$
Using $\|\boldsymbol{\mathrm{A}}^{s+1/2}{\bf v}(0)\|\leq M_{2}$ we can apply
the previous arguments to obtain (2.46) (with ${\bf u}$ replaced with ${\bf
v}$) for $0\leq r\leq T^{**}$. We complete the proof and obtain the bound
(2.46) by repeating the last argument $n$ times, where $n=\lceil
T/T^{**}\rceil$. ∎
## 3\. Finite dimensional spaces and Stokes projections
Throughout the remainder of the paper, with $s$ and $\sigma_{1}$ as in Theorem
2.1 and Theorem 2.2, we assume that
(3.1) ${\bf u}_{0}\in\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2}),\qquad{\bf
f}\in
C([0,T];\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2}e^{\sigma_{1}\boldsymbol{\mathrm{A}}^{1/2}})),\qquad
s\geq 1/4.$
Natural finite dimensional spaces (depending on a parameter $N>0$) in which to
seek approximations to ${\bf u}(t)$ are
(3.2) $V_{N}:=\mbox{ span
}\\{{\mathbf{Z}}_{L,m}:L=1,\ldots,N;~{}~{}m=-L,\ldots,L\\}.$
The dimension of $V_{N}$ is $N^{2}+2N$. Let ${\mathbf{\Pi}}_{N}:H\rightarrow
V_{N}$ be the orthogonal projection with respect to the $L^{2}(TS)$ inner
product defined by
(3.3) ${\mathbf{\Pi}}_{N}({\bf v})=\sum_{L=1}^{N}\sum_{m=-L}^{L}\widehat{{\bf
v}}_{L,m}{\mathbf{Z}}_{L,m}.$
###### Lemma 3.1.
Let $\alpha>0$ be given. If ${\bf
v}\in{\mathcal{D}}(\boldsymbol{\mathrm{A}}^{\alpha/2})$ then
(3.4) $\|{\bf v}-{\mathbf{\Pi}}_{N}{\bf v}\|\leq N^{-\alpha}\|{\bf
v}\|_{H^{\alpha}(TS)}.$
###### Proof.
Using (2.17), (2.19), and (2.21) we get
$\displaystyle\|{\bf v}-{\mathbf{\Pi}}_{N}{\bf
v}\|^{2}=\sum_{L=N+1}^{\infty}\sum_{m=-L}^{L}|\widehat{{\bf v}}_{L,m}|^{2}$
$\displaystyle\leq
N^{-2\alpha}\sum_{L=N+1}^{\infty}\sum_{m=-L}^{L}\lambda^{\alpha}_{L}|\widehat{{\bf
v}}_{L,m}|^{2}$ $\displaystyle\leq N^{-2\alpha}\|{\bf
v}\|^{2}_{H^{\alpha}(TS)}.$
∎
In particular, using (3.1), we get
(3.5) $\|{\bf f}-{\mathbf{\Pi}}_{N}{\bf f}\|\leq N^{-(2s+1)}\|{\bf
f}\|_{H^{2s+1}(TS)},\qquad t\in[0,T].$
For computer implementation, the Fourier coefficients in (3.3) and all
Galerkin type integrals in computational schemes for the NSE need to be
approximated by cubature rules on the sphere, leading to a pseudospectral
method. To this end, for a continuous scalar field $\psi$ on $S$, we consider
a Gauss-rectangle quadrature sum $Q_{M}(\psi)$ with quadrature points
$\\{\widehat{\boldsymbol{\xi}}_{p,q}=\boldsymbol{p}(\theta_{p},\phi_{q})\\}$
and positive weights ${w_{p}}$ of the form
(3.6)
$Q_{M}(\psi):=\frac{2\pi}{M}\sum_{q=1}^{M}\sum_{p=1}^{M/2}w_{p}\psi(\widehat{\boldsymbol{\xi}}_{p,q})=\frac{2\pi}{M}\sum_{q=1}^{M}\sum_{p=1}^{M/2}w_{p}\psi(\theta_{p},\phi_{q}),$
where $M\geq 2$ is an even integer, $w_{p}$ and $\cos\theta_{p}$ for $\quad
p=1,\ldots,M/2$ are the Gauss-Legendre weights and nodes on $[-1,1]$ and
$\phi_{q}=2q\pi/M$, $q=1,\ldots,M$. The rule (3.6) is exact when $\psi$ is a
polynomial of degree $M-1$ on $S$, that is,
$Q_{M}\psi=\int_{S}\psi\;dS,\qquad\psi\in\mathcal{P}_{M-1}.$
Hence corresponding to (2.8) and (2.9), we define discrete inner products for
scalar and vector fields on the unit sphere as
(3.7) $(v_{1},\;v_{2})_{M}=Q_{M}(v_{1}v_{2}),\qquad\qquad({\bf v}_{1},\;{\bf
v}_{2})_{M}=Q_{M}({\bf v}_{1}\cdot{\bf v}_{2}).$
The choice of $M$ is very important; we choose $M$ such that all Galerkin
integrals with polynomial terms in our scheme are evaluated exactly. In
particular, with the unknown tangential divergence-free velocity field sought
in the polynomial space $V_{N}$, and knowing that the NSE nonlinearity is
quadratic, we choose $M$ such that
(3.8) $(\boldsymbol{\mathrm{B}}({\bf v},{\bf w}),{\bf
z})=(\boldsymbol{\mathrm{B}}({\bf v},{\bf w}),{\bf z})_{M},\qquad{\bf v},{\bf
w},{\bf z}\in V_{N}.$
This holds, for example, if $M=3N+2$. We define a computable counterpart of
(3.3), using ${\mathbf{L}}_{N}:H\cap C(TS)\rightarrow V_{N}$, a discrete
orthogonal projection with respect to the $M^{2}/2$ point discrete inner
product, as
(3.9) ${\mathbf{L}}_{N}({\bf v})=\sum_{L=1}^{N}\sum_{m=-L}^{L}\widehat{{\bf
v}}_{L,m,M}{\mathbf{Z}}_{L,m},\qquad\widehat{{\bf v}}_{L,m,M}=Q_{M}({\bf
v}\cdot\overline{{\mathbf{Z}}_{L,m}})=({\bf v},\;{\mathbf{Z}}_{L,m})_{M}.$
With $M$ chosen to satisfy (3.8), it is easy to see that
(3.10) ${\mathbf{L}}_{N}({\bf v})={\mathbf{\Pi}}_{N}({\bf v})={\bf
v},\qquad{\bf v}\in V_{N},$
and for ${\bf v}\in H\cap C(TS)$ and ${\bf w}\in H\cap C^{k}(TS)$,
(3.11) $\|{\mathbf{L}}_{N}({\bf v})\|\leq C\|{\bf
v}\|_{\infty},\qquad\qquad\|{\bf w}-{\mathbf{L}}_{N}({\bf w})\|\leq
CN^{-k}\|{\bf w}\|_{C^{k}(TS)},$
where the last two inequalities follow from simple arguments used in Theorem
13 and Lemma 14 of [24]. In particular, since
$\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2})\subset H\cap C^{2s}(TS)$, for an
integer $2s$, using (3.1)
(3.12) $\|{\bf f}-{\mathbf{L}}_{N}({\bf f})\|\leq CN^{-2s}\|{\bf
f}\|_{C^{2s}(TS)},\qquad t\in[0,T].$
Next we consider the Stokes projection in $V_{N}$ of the exact unique regular
solution ${\bf u}(t):={\bf u}(.,t)$ of (2.1). For each fixed $t$, the Stokes
projection $\widetilde{{\bf u}}_{N}\in V_{N}$ of ${\bf u}$ is defined by
(3.13) $(\boldsymbol{\mathrm{A}}\widetilde{{\bf u}}_{N},{\bf
v})=(\boldsymbol{\mathrm{A}}{\bf u},{\bf v}),\qquad{\bf v}\in V_{N}.$
Since
${\mathbf{\Pi}}_{N}\boldsymbol{\mathrm{A}}=\boldsymbol{\mathrm{A}}{\mathbf{\Pi}}_{N}$,
it follows that $\widetilde{{\bf u}}_{N}={\mathbf{\Pi}}_{N}({\bf u})$.
Following standard techniques in finite element analysis, the Stokes
projection of ${\bf u}$ plays an important role as a comparison function in
the main analysis in the next section.
###### Theorem 3.1.
Let ${\bf u}_{0}$ and ${\bf f}$ satisfy (3.1). Then
(3.14) $\|{\bf u}-\widetilde{{\bf u}}_{N}\|\leq
C\lambda^{-s-1/2}_{N+1}e^{-\sigma(t)\lambda^{1/2}_{N+1}}\leq
CN^{-2s-1}e^{-\sigma(t)N},\qquad t\in[0,T],$
where $\sigma(t)$ is as in Theorem 2.1.
###### Proof.
Using the fact that $\widetilde{{\bf u}}_{N}={\mathbf{\Pi}}_{N}{\bf u}$, we
have
$\displaystyle\|{\bf u}-\widetilde{{\bf u}}_{N}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{L>N}\sum_{|m|\leq L}|\widehat{{\bf u}}_{L,m}|^{2}$
$\displaystyle\leq$
$\displaystyle\lambda^{-2s-1}_{N+1}e^{-2\sigma(t)\lambda^{1/2}_{N+1}}\sum_{L>N}\sum_{|m|\leq
L}\lambda^{2s+1}_{L}e^{2\sigma(t)\lambda^{1/2}_{L}}|\widehat{{\bf
u}}_{L,m}|^{2}$ $\displaystyle\leq$
$\displaystyle\lambda^{-2s-1}_{N+1}e^{-2\sigma(t)\lambda^{1/2}_{N+1}}\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\sigma(t)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(t)\|^{2}$ $\displaystyle\leq$ $\displaystyle
C\lambda^{-2s-1}_{N+1}e^{-2\sigma(t)\lambda^{1/2}_{N+1}}$
where in the last step we used (2.35). The last inequality in (3.14) follows
from the fact that $N^{2}\leq\lambda_{N+1}=(N+1)(N+2)$. ∎
###### Theorem 3.2.
Let ${\bf u}_{0}$, ${\bf f}$ satisfy (3.1). We assume further that ${\bf f}$
is analytic in $\mathbb{T}$ and (2.37) holds. Then for $t\in(0,T)$,
(3.15) $\left\|\frac{d}{dt}\left({\bf u}-\widetilde{{\bf
u}}_{N}\right)\right\|\leq
C\lambda^{-s-1/2}_{N+1}e^{-\psi_{1}(t)\lambda^{1/2}_{N+1}}\leq
CN^{-2s-1}e^{-\psi_{1}(t)N},$
where $\psi_{1}(t)=\min\\{(1-1/\sqrt{2})t,T^{**},\sigma_{1}\\}$, and $T^{**}$
is as in Theorem 2.3.
###### Proof.
Let $t\in(0,T)$ be fixed. Let ${\bf p}_{N}(\zeta)=({\bf u}-\widetilde{{\bf
u}}_{N})(\zeta)$ be the standard complexification of ${\bf u}-\widetilde{{\bf
u}}_{N}$ at $\zeta=re^{i\theta}$. Using Theorem 2.3 and the Cauchy integral
formula,
$\frac{d{\bf p}_{N}(t)}{dt}=\frac{1}{2\pi i}\int_{\Gamma}\frac{{\bf
p}_{N}(\zeta)}{(t-\zeta)^{2}}d\zeta,$
where for $t>0$, $\Gamma$ is a circle in the $\zeta$ plane with center $(t,0)$
and radius
$\min\\{t/\sqrt{2},T^{**}/\sqrt{2},T-t\\}$, a condition that ensures that
$\zeta=re^{i\theta}\in\Gamma$ lies in the region $\mathbb{T}$ with $|\mbox{Im
}\zeta|\leq T^{**}/\sqrt{2}$. Using the fact that $\widetilde{{\bf
u}}_{N}={\mathbf{\Pi}}_{N}{\bf u}$, for $\zeta=re^{i\theta}\in\Gamma$ we have,
from Theorem 2.3,
(3.16) $\displaystyle\|{\bf p}_{N}(\zeta)\|$ $\displaystyle=$
$\displaystyle\|{\bf u}(\zeta)-\widetilde{{\bf u}}_{N}(\zeta)\|^{2}$
$\displaystyle=$ $\displaystyle\sum_{L>N}\sum_{|m|\leq L}|\widehat{{\bf
u}}_{L,m}|^{2}$ $\displaystyle\leq$
$\displaystyle\lambda^{-2s-1}_{N+1}e^{-2\psi(r\cos\theta)\lambda^{1/2}_{N+1}}\sum_{L>N}\sum_{|m|\leq
L}\lambda^{2s+1}_{L}e^{2\psi(r\cos\theta)\lambda^{1/2}_{L}}|\widehat{{\bf
u}}_{L,m}|^{2}$ $\displaystyle\leq$
$\displaystyle\lambda^{-2s-1}_{N+1}e^{-2\psi(r\cos\theta)\lambda^{1/2}_{N+1}}\|\boldsymbol{\mathrm{A}}^{s+1/2}e^{\psi(r\cos\theta)\boldsymbol{\mathrm{A}}^{1/2}}{\bf
u}(\zeta)\|^{2}$ $\displaystyle\leq$ $\displaystyle
C\lambda^{-2s-1}_{N+1}e^{-2\psi(r\cos\theta)\lambda^{1/2}_{N+1}}$
$\displaystyle\leq$ $\displaystyle CN^{-2(2s+1)}e^{-2\psi(r\cos\theta)N}.$
For $\zeta=re^{i\theta}\in\Gamma$ it is easily seen that
$r\cos\theta\geq(1-1/\sqrt{2})t$, and hence that
(3.17)
$\psi(r\cos\theta)\geq\min\\{(1-1/\sqrt{2})t,T^{**},\sigma_{1}\\}=:\psi_{1}(t).$
On using (3.16) in (3.14) we get
$\left\|\frac{d}{dt}\left({\bf u}-\widetilde{{\bf
u}}_{N}\right)(t)\right\|\leq\frac{1}{2\pi}\int_{\Gamma}\frac{\|{\bf
p}_{N}(\zeta)\|}{|t-\zeta|^{2}}d\zeta\leq CN^{-2s-1}e^{-\psi_{1}(t)N}.$
∎
## 4\. A pseudospectral quadrature method
We are now ready to describe, analyze, and implement a spectrally accurate
scheme to compute approximate solutions of the NSE (2.1) in $V_{N}$, through
its weak formulation (2.27). The task is then to compute ${\bf
u}_{N}(\cdot,t)\in V_{N}$ for $t\in[0,T]$ with ${\bf u}_{N}(0,\widehat{{\bf
x}})={\mathbf{L}}_{N}{\bf u}_{0}(\widehat{{\bf x}})$, $\widehat{{\bf x}}\in
S$, satisfying the (spatially) discrete system of ordinary differential
equations
(4.1) $\frac{d}{dt}\left({\bf u}_{N},{\bf v}\right)_{M}+b({\bf u}_{N},{\bf
u}_{N},{\bf v})_{M}+\nu(\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
u}_{N},\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf
v})_{M}+(\boldsymbol{\mathrm{C}}{\bf u}_{N},{\bf v})_{M}=({\bf f},{\bf
v})_{M},$
for all ${\bf v}\in V_{N}$ and prove that the scheme is spectrally accurate
with respect to the parameter $N$ (that is, converges with rate determined by
the smoothness of the given data), and demonstrate the theory with numerical
experiments.
Using (3.2), the exactness properties of the discrete inner product, (3.8),
(2.10), (2.16), (2.23), and (2.29), the system (4.1) can be written as
(4.2) $\displaystyle\left(\frac{d}{dt}{\bf
u}_{N}+\nu\boldsymbol{\mathrm{A}}{\bf u}_{N}+\boldsymbol{\mathrm{B}}({\bf
u}_{N},{\bf u}_{N})+\boldsymbol{\mathrm{C}}{\bf
u}_{N},{\mathbf{Z}}_{L,m}\right)$ $\displaystyle=$ $\displaystyle({\bf
f},{\mathbf{Z}}_{L,m})_{M},$ $\displaystyle L=1,\cdots,N;~{}~{}m=-L,\cdots,L.$
### 4.1. Stability and convergence analysis
First we establish the stability of the approximate solution ${\bf u}_{N}$ of
(4.24). That is, similar to [6, Proposition 9.1], we prove that
$\max_{t\in[0,T]}\|{\bf u}_{N}\|_{V}$ is uniformly bounded with the bound
depending only on the initial data, forcing function, and the viscosity term
in (4.24).
###### Theorem 4.1.
Let ${\bf u}_{0}$ and ${\bf f}$ satisfy (3.1). Let $N\geq 1$ be an integer.
Let ${\bf u}_{N}$ be the solution of the pseudospectral quadrature Galerkin
system (4.1). Then there exists a constant $C$ depending on $\nu,\|{\bf
u}_{0}\|_{V}$ and $\|{\bf f}\|_{\infty}:=\max_{t\in[0,T]}\|{\bf
f}(t)\|_{C(TS)}$ so that
$\max_{t\in[0,T]}\|{\bf u}_{N}\|_{V}\leq C.$
###### Proof.
The proof follows by repeating the arguments described in the first four pages
of [6, Section 9] (proving [6, Proposition 9.1]), provided that we establish
[6, Inequalities (9.3) and (9.13)] for our system (4.1) on the spherical
surface with additional Coriolis term and quadrature approximations.
Using (2.26), (2.29) and the exactness of the quadrature rule (given by
(3.7)-(3.8)), we have $\left(\boldsymbol{\mathrm{B}}({\bf u}_{N},{\bf
u}_{N}),{\bf u}_{N}\right)_{M}=0$. Using (4.27), the symmetry of the
coefficients of ${\bf u}_{N}$ in (4.24) and the exactness of the quadrature,
we get
(4.3) $(\boldsymbol{\mathrm{C}}{\bf u}_{N},{\bf
u}_{N})_{M}=(\boldsymbol{\mathrm{C}}{\bf u}_{N},{\bf u}_{N})=(-2\Omega
i)\sum_{L=1}^{N}\lambda^{-1}_{L}\sum_{|m|\leq L}m|\alpha_{L,m}|^{2}=0.$
By taking ${\bf v}$ to be ${\bf u}_{N}$ in (4.1) and using $\|{\bf
u}_{N}\|^{2}=({\bf u}_{N},{\bf u}_{N})_{M}$, $\|{\bf
u}_{N}\|_{V}^{2}=(\boldsymbol{\mathrm{A}}{\bf u}_{N},{\bf u}_{N})_{M}$, (4.3),
Young’s inequality and the fact that all the eigenvalues $\lambda_{J}$ of
$\boldsymbol{\mathrm{A}}$ (corresponding to eigenvectors in ${\bf u}_{N}$)
satisfy $\lambda_{J}\geq\lambda_{1}=2,J=1,\cdots,N$, we obtain
$\frac{1}{2}\frac{d}{dt}\|{\bf u}_{N}\|^{2}+\nu\|{\bf u}_{N}\|_{V}^{2}=({\bf
f},{\bf u}_{N})_{M}\leq\|{\bf f}\|_{\infty}\|{\bf u}_{N}\|\leq\frac{\|{\bf
f}\|^{2}_{\infty}}{4\nu}+\nu\|{\bf u}_{N}\|^{2}\leq\frac{\|{\bf
f}\|^{2}_{\infty}}{4\nu}+\frac{\nu}{2}\|{\bf u}_{N}\|_{V}^{2},$
Hence, for our discrete system (4.1), we obtain [6, Inequality (9.3)]:
(4.4) $\frac{d}{dt}\|{\bf u}_{N}\|^{2}+\nu\|{\bf
u}_{N}\|_{V}^{2}\leq\frac{\|{\bf f}\|^{2}_{\infty}}{\nu\lambda_{1}}.$
Again using (4.27), the exactness of the quadrature rule, eigenfunction
properties of $\boldsymbol{\mathrm{A}}$, and the symmetry of the coefficients
of ${\bf u}_{N}$ in (4.24), we get
(4.5) $(\boldsymbol{\mathrm{C}}{\bf u}_{N},\boldsymbol{\mathrm{A}}{\bf
u}_{N})=(-2\Omega i)\sum_{L=1}^{N}\sum_{|m|\leq L}m|\alpha_{L,m}|^{2}=0,$
By taking ${\bf v}$ to be $\boldsymbol{\mathrm{A}}{\bf u}_{N}$ in equation
(4.1), and using $\|\boldsymbol{\mathrm{A}}{\bf
u}_{N}\|^{2}=(\boldsymbol{\mathrm{A}}{\bf u}_{N},\boldsymbol{\mathrm{A}}{\bf
u}_{N})_{M}$, (3.8), and (4.3), we obtain
(4.6) $\frac{1}{2}\frac{d}{dt}\|{\bf
u}_{N}\|^{2}_{V}+\nu\|\boldsymbol{\mathrm{A}}{\bf u}_{N}\|^{2}+b({\bf
u}_{N},{\bf u}_{N},\boldsymbol{\mathrm{A}}{\bf u}_{N})=({\bf
f},\boldsymbol{\mathrm{A}}{\bf u}_{N})_{M}.$
Using [16, Lemma 3.1], $b({\bf u}_{N},{\bf u}_{N},\boldsymbol{\mathrm{A}}{\bf
u}_{N})=0$. The term $({\bf f},\boldsymbol{\mathrm{A}}{\bf u}_{N})_{M}$ can be
estimated by using the exactness of the quadrature and Young’s inequality:
$({\bf f},\boldsymbol{\mathrm{A}}{\bf u}_{N})_{M}\leq\|{\bf
f}\|_{\infty}\|\boldsymbol{\mathrm{A}}{\bf u}_{N}\|\leq\frac{\|{\bf
f}\|^{2}_{\infty}}{2\nu}+\frac{\nu}{2}\|\boldsymbol{\mathrm{A}}{\bf
u}_{N}\|^{2}$
Hence we obtain a stronger version of [6, Inequality (9.13)] for our
quadrature discrete scheme (4.1):
$\frac{d}{dt}\|{\bf u}_{N}\|^{2}_{V}+\nu\|\boldsymbol{\mathrm{A}}{\bf
u}_{N}\|^{2}\leq\frac{\|{\bf f}\|^{2}_{\infty}}{\nu\lambda_{1}}.$
Thus, the result follows from arguments in [6, Page 74-77]. ∎
Next, using Theorem 3.1, 3.2, and 4.1, we prove the spectral convergence of
the solution ${\bf u}_{N}$ of (4.2) to the solution of ${\bf u}$ of (2.27).
###### Theorem 4.2.
Let ${\bf u}_{0}$, ${\bf f}$ satisfy (2.37), (3.1). Then there exists a
$T^{\\#}>0$, depending only on $\nu,{\bf f}$, ${\bf u}_{0}$ and the uniform
bound in (2.45) (and hence there exists
$0<\mu(t)<\min\\{t,T^{\\#},\sigma_{1}\\}$) such that for all $t\in(0,T)$,
(4.7) $\|{\bf u}-{\bf u}_{N}\|\leq
C\left[N^{-2s-1}e^{-\mu(t)N}+\left\|{\mathbf{\Pi}}_{N}{\bf
f}-{\mathbf{L}}_{N}{\bf f}\right\|\right].$
In particular, with $2s$ being an integer
(4.8) $\|{\bf u}-{\bf u}_{N}\|\leq CN^{-2s}.$
###### Proof.
Let ${\bf w}_{N}=\widetilde{{\bf u}}_{N}-{\bf u}_{N}$, where the comparison
function $\widetilde{{\bf u}}_{N}$ is the solution of (3.13). Since ${\bf
u}-{\bf u}_{N}={\bf p}_{N}+{\bf w}_{N}$, where ${\bf p}_{N}={\bf
u}-\widetilde{{\bf u}}_{N}$, in view of Theorem 3.1 and 3.2, existence of
$T^{\\#}$ and $\mu(t)$ follows and it is sufficient to show that $\|{\bf
w}_{N}\|\leq C\left[N^{-2s-1}e^{-\mu(t)N}+\left\|{\mathbf{\Pi}}_{N}{\bf
f}-{\mathbf{L}}_{N}{\bf f}\right\|\right]$. For any ${\bf v}\in V_{N}$, using
(2.28), (3.13), and (4.2),
$\displaystyle(({\bf w}_{N})_{t},{\bf v})+\nu(\boldsymbol{\mathrm{A}}{\bf
w}_{N},{\bf v})+(\boldsymbol{\mathrm{C}}{\bf w}_{N},{\bf v})$
$\displaystyle=(({\widetilde{{\bf u}}_{N}})_{t},{\bf
v})+\nu(\boldsymbol{\mathrm{A}}{\widetilde{{\bf u}}_{N}},{\bf
v})+(\boldsymbol{\mathrm{C}}{\widetilde{{\bf u}}}_{N},{\bf v})-(({\bf
u}_{N})_{t},{\bf v})-\nu(\boldsymbol{\mathrm{A}}{\bf u}_{N},{\bf
v})-(\boldsymbol{\mathrm{C}}{\bf u}_{N},{\bf v})$
$\displaystyle=(({\widetilde{{\bf u}}_{N}})_{t},{\bf
v})+\nu(\boldsymbol{\mathrm{A}}{\widetilde{{\bf u}}_{N}},{\bf
v})+(\boldsymbol{\mathrm{C}}{\widetilde{{\bf u}}}_{N},{\bf v})-({\bf f},{\bf
v})_{M}+(\boldsymbol{\mathrm{B}}({\bf u}_{N},{\bf u}_{N}),{\bf v})$
$\displaystyle=(({\widetilde{{\bf u}}_{N}})_{t},{\bf
v})+\nu(\boldsymbol{\mathrm{A}}{\bf u},{\bf
v})+(\boldsymbol{\mathrm{C}}{\widetilde{{\bf u}}}_{N},{\bf v})-({\bf f},{\bf
v})_{M}+(\boldsymbol{\mathrm{B}}({\bf u}_{N},{\bf u}_{N}),{\bf v})$
$\displaystyle=(({\widetilde{{\bf u}}_{N}}-{\bf u})_{t},{\bf v})+({\bf f},{\bf
v})-({\bf f},{\bf v})_{M}+(\boldsymbol{\mathrm{C}}{\widetilde{{\bf
u}}_{N}}-\boldsymbol{\mathrm{C}}{\bf u},{\bf v})+(\boldsymbol{\mathrm{B}}({\bf
u}_{N},{\bf u}_{N})-\boldsymbol{\mathrm{B}}({\bf u},{\bf u}),{\bf v}).$
Using the orthogonal projection ${\mathbf{\Pi}}_{N}$ in (3.2), we can write
this relation in functional form as
$\frac{d{\bf
w}_{N}}{dt}=(-\nu\boldsymbol{\mathrm{A}}-\boldsymbol{\mathrm{C}}){\bf
w}_{N}-\mathbf{{\mathbf{\Pi}}_{N}}\left[({\bf
p}_{N})_{t}+\boldsymbol{\mathrm{C}}{\bf p}_{N}\right]+{\mathbf{\Pi}}_{N}{\bf
f}-{\mathbf{L}}_{N}{\bf
f}+{\mathbf{\Pi}}_{N}\left[\boldsymbol{\mathrm{B}}({\bf u}_{N},{\bf
u}_{N})-\boldsymbol{\mathrm{B}}({\bf u},{\bf u})\right].$
Integrating with respect to $t$ and using ${\bf w}_{N}(0)=0$, we have
(4.9) $\displaystyle{\bf w}_{N}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}e^{-(t-s)(\nu\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}})}\left[-{\mathbf{\Pi}}_{N}(\frac{d}{ds}{\bf
p}_{N}+\boldsymbol{\mathrm{C}}{\bf p}_{N})+{\mathbf{\Pi}}_{N}{\bf
f}-{\mathbf{L}}_{N}{\bf f}\right](s)\;ds$
$\displaystyle+\int_{0}^{t}e^{-(t-s)(\nu\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}})}{\mathbf{\Pi}}_{N}\left[\boldsymbol{\mathrm{B}}({\bf
u}_{N},{\bf u}_{N})-\boldsymbol{\mathrm{B}}({\bf u},{\bf u})\right](s)\;ds.$
Let
(4.10)
$R_{N}(\epsilon,t-s)=\|\nu^{\epsilon}{\mathbf{\Pi}}_{N}\boldsymbol{\mathrm{A}}^{\epsilon}e^{-(t-s)(\nu\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}})}\|.$
On using (2.31), with $\delta\in(1/2,1)$, and the uniform boundedness of the
orthogonal projection ${\mathbf{\Pi}}_{N}$, we get
(4.11)
$\displaystyle\|e^{-(t-s)(\nu\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}})}{\mathbf{\Pi}}_{N}\left[\boldsymbol{\mathrm{B}}({\bf
u}_{N},{\bf u}_{N})-\boldsymbol{\mathrm{B}}({\bf u},{\bf u})\right]\|$
$\displaystyle\leq\frac{R_{N}(\delta,t-s)}{\nu^{\delta}}\|\boldsymbol{\mathrm{A}}^{-\delta}{\mathbf{\Pi}}_{N}(\boldsymbol{\mathrm{B}}({\bf
u}_{N},{\bf u}_{N})-\boldsymbol{\mathrm{B}}({\bf u},{\bf u}))\|\leq
CR_{N}(\delta,t-s)\|{\bf u}_{N}-{\bf u}\|.$
Taking norms and using $\|{\bf u}_{N}-{\bf u}\|\leq\|{\bf w}_{N}\|+\|{\bf
p}_{N}\|$ together with (4.10), and (4.1) in (4.9), we obtain
$\displaystyle\|{\bf w}_{N}(t)\|$ $\displaystyle\leq$
$\displaystyle\left[\|\frac{d}{dt}{\bf p}_{N}\|+\|\boldsymbol{\mathrm{C}}{\bf
p}_{N}\|+\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\mathbf{L}}_{N}{\bf
f}\right\|\right]\int_{0}^{t}R_{N}(0,t-s)\;ds$
$\displaystyle+C\int_{0}^{t}R_{N}(\delta,t-s)(\|{\bf p}_{N}(s)\|+\|{\bf
w}_{N}(s)\|)ds.$
Using Gronwall’s inequality, we obtain for each $t\in[0,T]$,
(4.12) $\displaystyle\|{\bf w}_{N}(t)\|$ $\displaystyle\leq$ $\displaystyle
C\left\\{\left[\|\frac{d}{dt}{\bf p}_{N}\|+\|\boldsymbol{\mathrm{C}}{\bf
p}_{N}\|+\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\mathbf{L}}_{N}{\bf
f}\right\|\right]\int_{0}^{t}R_{N}(0,t-s)\;ds\right.$
$\displaystyle\qquad\left.+\|{\bf p}_{N}\|\
\int_{0}^{t}R_{N}(\delta,t-s)\;ds\right\\}.$
For $\epsilon\in[0,1]$, we have to bound
(4.13)
$\int_{0}^{t}R_{N}(\epsilon,t-s)\;ds=\int_{0}^{t}R_{N}(\epsilon,r)\;dr.$
Using also (4.10) and (4.27),
$R_{N}(\epsilon,r)\leq\max_{1\leq L\leq N;|m|\leq
L}|(\nu\lambda_{L})^{\epsilon}e^{-\nu\lambda_{L}r}e^{-2\Omega
irm\lambda^{-1/2}_{L}}|\leq\max_{z\in[\nu\lambda_{1},\nu\lambda_{N}]}z^{\epsilon}e^{-rz}.$
Thus
(4.14)
$R_{N}(r)\leq\begin{cases}(\nu\lambda_{N})^{\epsilon}e^{-\nu\lambda_{N}r}&\mbox{
if
}r\leq\epsilon/(\nu\lambda_{N}),\cr\epsilon^{\epsilon}e^{-\epsilon}r^{-\epsilon}&\mbox{
if }\epsilon/(\nu\lambda_{N})\leq
r\leq\epsilon/(\nu\lambda_{1}),\cr(\nu\lambda_{1})^{\epsilon}e^{-\nu\lambda_{1}r}&\mbox{
if }r\geq\epsilon/(\nu\lambda_{1}).\cr\end{cases}$
With
(4.15) $I_{1}=[0,\epsilon/(\nu\lambda_{N})]\cap[0,t],\quad
I_{2}=[\epsilon/(\nu\lambda_{N}),\epsilon/(\nu\lambda_{1})]\cap[0,t],\quad
I_{3}=[\epsilon/(\nu\lambda_{1}),t]\cap[0,t],$
the interval of integration in (4.13) can be subdivided into these three
intervals. In particular, using (4.14) and (4.15), we get
(4.16)
$\int_{I_{1}}R_{N}(r)dr\leq\int_{I_{1}}(\nu\lambda_{N})^{\epsilon}e^{-\nu\lambda_{N}r}dr\leq\frac{(1-e^{-\epsilon})}{(\nu\lambda_{N})^{1-\epsilon}}\leq
C,$ (4.17)
$\int_{I_{2}}R_{N}(r)dr\leq\int_{I_{2}}\epsilon^{\epsilon}e^{-\epsilon}r^{-\epsilon}dr\leq\frac{\epsilon
e^{-\epsilon}}{(1-\epsilon)}\left[\left(\frac{1}{\nu\lambda_{1}}\right)^{(1-\epsilon)}-\left(\frac{1}{\nu\lambda_{N}}\right)^{(1-\epsilon)}\right]\leq
C,$
and
(4.18)
$\int_{I_{3}}R_{N}(r)dr\leq\int_{I_{3}}(\nu\lambda_{1})^{\epsilon}e^{\nu\lambda_{1}r}dr\leq\frac{(e^{-\epsilon}-e^{-\nu\lambda_{1}t})}{(\nu\lambda_{1})^{1-\epsilon}}\leq
C.$
Using (4.16)- (4.18) in (4.13), we get
(4.19) $\int_{0}^{t}R_{N}(\epsilon,t-s)\;ds\leq C,\qquad\epsilon\in[0,1].$
Substituting this in (4.12), we get
(4.20) $\|{\bf w}_{N}(t)\|\leq C\left[\|\frac{d}{dt}{\bf
p}_{N}\|+\|\boldsymbol{\mathrm{C}}{\bf p}_{N}\|+\|{\bf p}_{N}\|\
+\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\mathbf{L}}_{N}{\bf f}\right\|\right].$
The term $\|\boldsymbol{\mathrm{C}}{\bf p}_{N}\|$ in (4.20) can be simplified
using (2.23) and the fact that ${\bf p}_{N}$ is tangential (and hence
$\widehat{{\bf x}}\cdot{\bf p}_{N}(\widehat{{\bf x}})=0$),
$\left[\widehat{{\bf x}}\times{\bf p}_{N}(\widehat{{\bf
x}})\right]\cdot\left[\overline{(\widehat{{\bf x}}\times{\bf
p}_{N}(\widehat{{\bf x}})}\right]=\left[\widehat{{\bf x}}\cdot\widehat{{\bf
x}}\right]\left[{\bf p}_{N}(\widehat{{\bf x}})\cdot\overline{{\bf
p}_{N}(\widehat{{\bf x}})}\right]=\|{\bf p}_{N}\|^{2},$
and hence
(4.21) $\|{\bf w}_{N}(t)\|\leq C\left[\|\frac{d}{dt}{\bf p}_{N}\|+\|{\bf
p}_{N}\|\ +\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\mathbf{L}}_{N}{\bf
f}\right\|\right].$
Hence from Theorem 3.1 and 3.2, for all $t\in(0,T)$ we have
(4.22) $\|{\bf u}-{\bf u}_{N}\|\leq
C\left[N^{-2s-1}e^{-\mu(t)N}+\left\|{\mathbf{\Pi}}_{N}{\bf
f}-{\mathbf{L}}_{N}{\bf f}\right\|\right].$
In particular, using (3.5) and (3.12) with $2s$ being an integer, we get
(4.23) $\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\mathbf{L}}_{N}{\bf
f}\right\|\leq\left\|{\mathbf{\Pi}}_{N}{\bf f}-{\bf f}\right\|+\left\|{\bf
f}-{\mathbf{L}}_{N}{\bf f}\right\|\leq CN^{-2s}.$
Now the result (4.8) follows from (4.22) and (4.23). ∎
### 4.2. Adaptive and fast implementation of the pseudospectral method
Having established spectrally accurate convergence of the spatially discrete
scheme (4.2), for implementation of the scheme (4.2) to simulate stable and
accurate solutions of (2.1) and compare with benchmark random flow simulations
in the literature, we need to discretize the time derivative operator
$\frac{d}{dt}$ in (4.2). Further, at each discrete time step we develop a
(FFT-based) fast evaluation technique to set up the resulting fully discrete
nonlinear system with spatial $\mathcal{O}(N^{4})$ complexity. First we
consider discretization of $\frac{d}{dt}$ in (4.2).
In order to the make (4.2) fully discrete in space and time, and hence compute
the $N^{2}+2N$ unknown time-dependent coefficients in the representation of
the tangential divergence-free approximate real velocity vector field
(4.24) ${\bf u}_{N}(\widehat{{\bf x}},t):=\sum_{L=1}^{N}\sum_{|m|\leq
L}\alpha_{L,m}(t){\mathbf{Z}}_{L,m}(\widehat{{\bf
x}}),\quad\alpha_{L,m}=\overline{\alpha}_{L,-m},\quad\alpha_{L,m}(0)=({\bf
u}_{0},{\mathbf{Z}}_{L,m})_{M},$
for $\widehat{{\bf x}}\in S$ and $t\geq 0$, the standard fixed-time-step
backward-Euler (or Crank-Nicolson) Galerkin approach could be used in (4.1),
leading to a first-order (or a second-order, respectively) in time non-
adaptive scheme [13]. However, due to the complicated unknown flow behavior of
the NSE solutions, when the initial states are random, it is more efficient
instead to integrate (4.1) using a combination of multi-order integration
formulas that allow adaptive choice of time step, leading to computation of
solutions with a specified accuracy in time. In this paper we follow the
latter approach.
For implementation purposes, we first need to substitute (4.24) in (4.2). We
write the resulting $N^{2}+2N$-dimensional system of ordinary differential
equations (ODE), for the unknown $N^{2}+2N$ time dependent coefficients of
${\bf u}_{N}(\widehat{{\bf x}},t)$ in (4.24), as
(4.25)
$\frac{d}{dt}\boldsymbol{\alpha}(t)=\mathbf{F}(t,\boldsymbol{\alpha}(t)).$
It is well known that such nonlinear ODE systems are stiff, and hence it is
important to use time discretization techniques with a large stability region
[14]. Further, it is important to use high-order implicit formulas whenever
possible, but the high-order discretization formulas are appropriate only at
those time steps where the unknown exact solution is smooth.
For practical problems such as the Navier-Stokes equations (with initial
random state) where the necessary spatial discretization (at each time step)
is expensive, it is important to optimize computing time by simulating only up
to a required accuracy by choosing adaptive discrete time steps. Unlike the
adaptive spatial mesh for elliptic PDEs based on the a posteriori estimates,
adaptive time steps can be computed by comparing numerical solutions obtained
using two distinct order formulas [14] and hence simulation using multi-order
formulas is appropriate.
In particular, for practical realization of large stiff nonlinear ODE systems,
multi-order implicit backward differentiation formulas (BDF) and their
generalizations such as the numerical differential formulas (NDF) are most
appropriate. The implicit NDF formula of order $p$ (NDFp) with a parameter
$\kappa_{p}$ (so that $\kappa_{p}=0$ corresponds to BDFp) for the system
(4.25), with $\boldsymbol{\alpha}_{n}\approx\boldsymbol{\alpha}(t_{n})$ and
$\nabla^{m}$ denoting the $m$-th Newton backward difference operator, is
(4.26)
$\sum_{m=1}^{p}\frac{1}{m}\nabla^{m}\boldsymbol{\alpha}_{n+1}=h\mathbf{F}(t_{n+1},\boldsymbol{\alpha}_{n+1})+\kappa_{p}\nabla^{p+1}\boldsymbol{\alpha}_{n+1}\sum_{j=1}^{p}\frac{1}{j}.$
It is well known [14] that BDFp (and hence NDFp) are unstable for $p>6$, and
for $p=6$ the stability region is small and hence not practically useful in
our case. Further the celebrated Dahlquist barrier [14] implies that BDFp (and
hence NDFp) cannot be absolutely stable [that is, $A(\alpha)$-stable with
$\alpha=90\,^{\circ}$] for $p>2$.
Following details in [25], for simulation of (4.25) we use multi-order NDFp
with $p=1,2,3,4,5$ (and respective
$\kappa_{p}=-0.1850,-1/9,-0.0823,-0.0415,0$) and these are
$A(\alpha_{p})$-stable, with respective
$\alpha_{p}=90^{\circ},90^{\circ},80^{\circ},66^{\circ},51^{\circ}$. For
$p=1,2,3,4,5$, NDFp is more accurate than BDFp, however NDFp has slightly
smaller stability angle compared BDFp only for $p=3,4$ (with respective
$\alpha_{p}=86^{\circ},73^{\circ}$) and the same stability angle for
$p=1,2,5$.
For each fixed time discretization step, the computational cost is dominated
by evaluation of $\mathbf{F}(t_{n+1},\boldsymbol{\alpha}_{n+1})$ in (4.26) and
it is important to have an efficient method to set up the spatial part of the
nonlinear system (4.2). Using the spectral properties of the Stokes operator
$\boldsymbol{\mathrm{A}}$ given by (2.17), the linear second term in (4.2) is
trivial to evaluate using the diagonal matrix consisting of the eigenvalues of
$\boldsymbol{\mathrm{A}}$.
The Coriolis term can be evaluated similarly using the identity [5, Equation
(24)]
(4.27)
$(\boldsymbol{\mathrm{C}}\;\boldsymbol{\mathrm{Curl}}\,Y_{J,k},{\mathbf{Z}}_{L,m})=(2\Omega\cos\theta\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Curl}}\,Y_{J,k},{\mathbf{Z}}_{L,m})=-2\Omega
i\frac{m}{\lambda^{1/2}_{L}}\delta_{L,J}\delta_{k,m}.$
For the first component in the nonlinear third term in (4.2), we use (2.7) to
write
(4.28)
$\displaystyle\boldsymbol{\mathrm{B}}({\mathbf{Z}}_{R,s},{\mathbf{Z}}_{J,k})$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{\lambda_{R}\lambda_{J}}}{\mathbf{P}}_{\boldsymbol{\mathrm{Curl}}\,}(\Delta
Y_{R,s}\boldsymbol{\mathrm{Grad}}\,Y_{J,k}),$ (4.29)
$\displaystyle\left(\boldsymbol{\mathrm{B}}({\mathbf{Z}}_{R,s},{\mathbf{Z}}_{J,k}),{\mathbf{Z}}_{L,m}\right)$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\lambda_{R}}{\lambda_{J}\lambda_{L}}}\left(Y_{R,s}\boldsymbol{\mathrm{Grad}}\,Y_{J,k},\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Grad}}\,Y_{L,m}\right).$
It is convenient to write $\boldsymbol{\mathrm{Grad}}\,Y_{J,k}$ and
$\widehat{{\bf x}}\times\boldsymbol{\mathrm{Grad}}\,Y_{L,m}$ in terms of
expressions similar to those in (2.18). Such explicit representations are also
useful for the efficient evaluation of the $N^{2}$ Fourier coefficients $({\bf
f},{\mathbf{Z}}_{L,m})_{M}$ of the source term in (4.2), and eventually for
the computation of the vorticity field.
In order to express the tangential (and normal, needed for computing the
approximate vorticity from ${\bf u}_{N}$) vector harmonics as a linear
combination of the scalar harmonics (2.18), we first recall, from the
classical quantum mechanics literature (see, for example, [30]), the covariant
spherical basis vectors
(4.30) ${\bf e}_{+1}=-\frac{1}{\sqrt{2}}([1,0,0]^{T}+i[0,1,0]^{T}),\quad{\bf
e}_{0}=[0,0,1]^{T},\quad{\bf
e}_{-1}=\frac{1}{\sqrt{2}}([1,0,0]^{T}-i[0,1,0]^{T}),$
and the Clebsch-Gordan coefficients
(4.31)
$C^{j,m}_{j_{1},m_{1},j_{2},m_{2}}:=(-1)^{(m+j_{1}-j_{2})}\sqrt{2j+1}\left(\begin{array}[]{lll}j_{1}&j_{2}&j\\\
m_{1}&m_{2}&-m\end{array}\right),$
where $\left(\begin{array}[]{lll}a&b&c\\\
\alpha&\beta&\gamma\end{array}\right)$ are the Wigner 3j-symbols given, for
example, by the Racah formula,
$\displaystyle\left(\begin{array}[]{lll}a&b&c\\\
\alpha&\beta&\gamma\end{array}\right)$
$\displaystyle=(-1)^{(a-b-\gamma)}\sqrt{T(abc)}\sqrt{(a+\alpha)!(a-\alpha)!(b+\beta)!(b-\beta)!(c+\gamma)!(c-\gamma)!}\times$
$\displaystyle\sum_{t}\frac{(-1)^{t}}{t!(c-b+t+\alpha)!(c-a+t-\beta)!(a+b-c-t)!(a-t-\alpha)!(b-t+\beta)!},$
where the sum is over all integers $t$ for which the factorials in the
denominator all have nonnegative arguments. In particular, the number of terms
in the sum is $1+\min\\{a\pm\alpha,b\pm\beta,c\pm\gamma,a+b-c,b+c-a,c+a-b\\}$.
The triangle coefficient $T(abc)$ is defined by
$T(abc)=\left[\frac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!}\right].$
Below, we require $C^{j,m}_{j_{1},m_{1},j_{2},m_{2}}$ only for some
$j_{2},m_{2}\in\\{-1,0,1\\}$, and using various symmetry and other known
properties (such as $C^{j,m}_{j_{1},m_{1},j_{2},m_{2}}=0$ unless the
conditions $|j_{1}-j_{2}|\leq j\leq j_{1}+j_{2}$ and $m_{1}+m_{2}=m$ hold) of
Wigner 3j-symbols, these coefficients can be efficiently pre-computed and
stored.
In our computation, we used the following basis functions for the tangential
vector fields: $(i)~{}\boldsymbol{\mathrm{Grad}}\,Y_{L,m},(ii)~{}\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Grad}}\,Y_{J,k}$. For the vorticity components of
${\bf u}_{N}$, in addition we used
$(iii)~{}\boldsymbol{\mathrm{Vort}}\,{\mathbf{Z}}_{J,m}=\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf
x}}}\times{\mathbf{Z}}_{J,m}=\lambda_{J}^{1/2}\widehat{{\bf
x}}Y_{J,m}=-\lambda_{J}^{-1/2}\widehat{{\bf x}}\Delta Y_{J,m}$. In particular,
using (4.24) our approximation to the vorticity in (2.2), for a fixed $t\geq
0$ and $\widehat{{\bf x}}\in S$, is
(4.33) $\boldsymbol{\mathrm{Vort}}\,{\bf u}_{N}(\widehat{{\bf
x}},t)=\boldsymbol{\mathrm{Curl}}_{\widehat{{\bf x}}}{\bf u}_{N}(\widehat{{\bf
x}})=\widehat{{\bf x}}\Delta\Psi_{N}(\widehat{{\bf x}}),$
where
$\Psi_{N}(\widehat{{\bf x}},t)=-\sum_{L=1}^{N}\sum_{|m|\leq
L}\lambda_{L}^{-1/2}\alpha_{L,m}(t)Y_{L,m}(\widehat{{\bf x}}).$
To facilitate easy application of fast transforms to evaluate these functions
at the $M=\mathcal{O}(N^{2})$ quadrature points
$\\{\widehat{\boldsymbol{\xi}}_{p,q}=\boldsymbol{p}(\theta_{p},\phi_{q})\\}$,
we represent these three types of fields first as a linear combination of the
covariant vectors in (4.30):
(4.34) $\displaystyle\boldsymbol{\mathrm{Grad}}\,Y_{L,m}$ $\displaystyle=$
$\displaystyle B_{+1,L,m}{\bf e}_{+1}+B_{0,L,m}{\bf e}_{0}+B_{-1,L,m}{\bf
e}_{-1},$ (4.35) $\displaystyle(\widehat{{\bf
x}}\times\boldsymbol{\mathrm{Grad}}\,Y_{J,k})$ $\displaystyle=$ $\displaystyle
D_{+1,J,k}{\bf e}_{+1}+D_{0,J,k}{\bf e}_{0}+D_{-1,J,k}{\bf e}_{-1},$
With $c_{L}=(L+1)\sqrt{\frac{L}{2L+1}},d_{L}=L\sqrt{\frac{L+1}{2L+1}}$, these
coefficients are explicitly given by
$\displaystyle B_{+1,L,m}$ $\displaystyle=$
$\displaystyle\left\\{c_{L}C^{L,m}_{L-1,m-1,1,1}P^{m-1}_{L-1}(\cos\theta)+d_{L}C^{L,m}_{L+1,m-1,1,1}P^{m-1}_{L+1}(\cos\theta)\right\\}e^{i(m-1)\varphi}$
$\displaystyle B_{0,L,m}$ $\displaystyle=$
$\displaystyle\left\\{c_{L}C^{L,m}_{L-1,m,1,0}P^{m}_{L-1}(\cos\theta)+d_{L}C^{L,m}_{L+1,m,1,0}P^{m}_{L+1}(\cos\theta)\right\\}e^{im\varphi}$
$\displaystyle B_{-1,L,m}$ $\displaystyle=$
$\displaystyle\left\\{c_{L}C^{L,m}_{L-1,m+1,1,-1}P^{m+1}_{L-1}(\cos\theta)+d_{L}C^{L,m}_{L+1,m+1,1,-1}P^{m+1}_{L+1}(\cos\theta)\right\\}e^{i(m+1)\varphi}.$
$\displaystyle D_{+1,J,k}$ $\displaystyle=$ $\displaystyle
i\sqrt{\lambda_{J}}C^{J,k}_{J,k-1,1,1}P^{k-1}_{J}(\cos\theta)e^{i(k-1)\varphi},$
$\displaystyle D_{0,J,k}$ $\displaystyle=$ $\displaystyle
i\sqrt{\lambda_{J}}C^{J,k}_{J,k,1,0}P^{k}_{J}(\cos\theta)e^{ik\varphi},$
$\displaystyle D_{-1,J,k}$ $\displaystyle=$ $\displaystyle
i\sqrt{\lambda_{J}}C^{J,k}_{J,k+1,1,-1}P^{k+1}_{J}(\cos\theta)e^{i(k+1)\varphi}.$
Noting (i) the complex azimuthal exponential terms
$e^{ik\varphi},e^{im\varphi}$ in (2.18) and (4.34)-(4.35) (via the above
representations for $B$ and $D$) for $|k|\leq J,|m|\leq L;1\leq L,J\leq N$,
and (ii) the need to evaluate several $\mathcal{O}(N^{2})$ sums, of the form
in (4.24) and (4.28)-(4.28), at the equally spaced $\mathcal{O}(N)$ azimuthal
quadrature points (see (3.6)), we reduce the complexity by $\mathcal{O}(N)$ in
each of these sums, at each adaptive-time step (described below), by using the
FFT for setting up the nonlinear system (4.2), similar to the approach in [4].
In our numerical experiments (see Section 5) for adaptive-time simulation of a
flow induced by random initial states, we observed that such an efficient FFT
based implementation reduced the (non-FFT code) computing time substantially
for the case $N=100$, to simulate from $t=0$ to $t=60$.
In addition, by using the fast Legendre/spherical transforms along the
latitudinal direction (obtained, for example, by modifying the NFFT algorithm
in [21] for evaluation of the Legendre functions in the above terms at
$\mathcal{O}(N)$ non-uniform latitudinal quadrature points), we could reduce
the complexity by $\mathcal{O}(N^{2})$. We did not use the fast
Legendre/spherical transforms in our implementation due to the spectral
convergence of the scheme and the fact that $N\leq 100$ in our simulations.
(In these complexity counts, we ignored $\mathcal{O}(\log N)$ and
$\mathcal{O}(\log^{2}N)$ terms.)
## 5\. Numerical Experiments
We demonstrate the fully discrete pseudospectral quadrature algorithm by
simulating (i) a known solution test case with low to high frequency modes and
(ii) a benchmark example [8, page 305] in which the unknown velocity and
vorticity fields are generated by a random initial state.
The first test example is useful to demonstrate that the pseudospectral
quadrature algorithm reproduces any number of high frequency modes in the
solution (provided $V_{N}$ contains all these modes), with computational error
dominated only by the chosen accuracy for the adaptive time evolution for the
ordinary differential system (4.25).
### 5.1. Example 1.
Our test case first example is (2.1) with
(5.1) ${\bf u}|_{t=0}(\widehat{{\bf x}})={\bf u}_{0}(\widehat{{\bf
x}})=g(0)[{\mathbf{W}}_{1}(\widehat{{\bf x}})-{\mathbf{W}}_{2}(\widehat{{\bf
x}})],$ (5.2) $g(t)=\nu
e^{-t}[\sin(5t)+\cos(10t)],\quad{\mathbf{W}}_{1}={\mathbf{Z}}_{1,0}+2\Re({\mathbf{Z}}_{1,1}),\;{\mathbf{W}}_{2}={\mathbf{Z}}_{2,0}+2\Re({\mathbf{Z}}_{2,1}+{\mathbf{Z}}_{2,2}),$
where ${\mathbf{Z}}_{L,m}$ is given by (2.17), $\Re(\cdot)$ denotes the real-
part function, and the external force ${\bf f}(\widehat{{\bf x}},t)$ in (2.1)
is chosen so that
(5.3) ${\bf u}(\widehat{{\bf x}},t)=tg(t)\sum_{L=1}^{N_{0}}\left[{\bf
Z}_{L,0}+2\sum_{m=1}^{L}\Re({\mathbf{Z}}_{L,m})\right](\widehat{{\bf
x}})+g(t){\mathbf{W}}_{1}(\widehat{{\bf
x}})+(t-1)g(t){\mathbf{W}}_{2}(\widehat{{\bf x}}),$
is the exact tangential divergence-free velocity field, solving the NSE (2.1).
The exact test field (5.2)-(5.3) has high oscillations both in space and time,
and exponentially decays in time. Note the dependence on a parameter $N_{0}$,
the maximum order of the spherical harmonics in the exact solution.
In our calculation of the approximate solution ${\bf u}_{N}$, we chose
$N=N_{0}$, so that all frequencies of the exact solution can be recovered. The
solution (5.3) is then used to validate our algorithm and code by numerical
adaptive time-integration of (4.1), for various values of $N=N_{0}$. In
particular, for a fixed integration tolerance error, we demonstrate in Figure
1 that all $N$ modes in (5.3) can indeed be recovered by the approximate
solution ${\bf u}_{N}$, within the chosen error tolerance, for all
$N=N_{0}=70,80,90,100$.
Figure 1. $\|{\bf u}-{\bf u}_{N}\|$ for Example 1 with a fixed time-
integration error, $N=N_{0}=70,80,90,100$.
### 5.2. Example 2.
Having established the validity of our algorithm for a simple known solution,
we use the same code to simulate unknown velocity and vorticity fields
generated by (2.1) with random initial velocity field as in [8, 12] with
angular velocity of the rotation $\Omega=1$ and $\nu^{-1}=10,000$ in (2.1).
The initial flow and external force in this benchmark example satisfy the main
assumption (3.1) for any $s,\sigma_{1}>0$ and hence, as proved in Theorem 4.2,
the approximate solution ${\bf u}_{N}$ is spectrally accurate and converges
super-algebraically with order given by (4.8) for any $s>0$. As mentioned in
Section 1, this is the main advantage of the present paper over the recent
paper [12], where such spectrally accurate convergence results are neither
discussed nor proved. On the other hand, convergence results for two-
dimensional problems on a Euclidean plane, supported by numerical experiments,
have formed a core part of research on the NSE over the last few decades, see
[8, 9, 27] and references therein.
The random initial tangential divergence-free velocity field, having
properties similar to those considered in [8, page 305] and [12, page 988]
(but not exactly same as in [8, 12], due to randomness), is a smooth function
${\bf u}_{0}={\bf v}\in G^{s/2}_{\sigma}$, a Gevrey class of order $s$ and
index $\sigma$, see (2.33), for any $s,\sigma>0$, with Fourier coefficients
$\widehat{{\bf v}}_{L,m},~{}~{}L=1,2,\cdots,|m|\leq L$, defined by
(5.4) $\widehat{{\bf
v}}_{L,m}=\begin{cases}a_{L}\exp(i\phi_{m})&L=1,\ldots,20;~{}m=0,\ldots,L,\cr
a_{L}(-1)^{m}\exp(-i\phi_{m})&L=1,\ldots,20;~{}m=-L,\cdots,-1,\cr
0,&L>20;~{}m=-L,\ldots,L,\end{cases}$
where $\phi_{0}=0$, and $\phi_{m}\in(0,2\pi)$ are random numbers for $m>0$,
and $a_{L}=b_{L}/||\mathbf{b}||$, with $\mathbf{b}\in\mathbb{R}^{20}$ having
components $b_{L}=2/\left[L+(\nu L)^{2.5}\right],~{}L=1,\ldots,20$. The
vorticity stream function $\Psi$ (see (2.2)) of
$\boldsymbol{\mathrm{Vort}}\,{\bf u}(\cdot,0)$ in Figure 3 demonstrates the
randomness of the field at time $t=0$.
The external force field ${\bf f}={\bf f}(\widehat{{\bf x}},t)$ in (2.1) for
our simulation is motivated by that considered in [8, page 305] and is exactly
same as that in [12, page 988]. The source term ${\bf f}$ is a decaying
tangential divergence-free field which, for any $s,\sigma>0$, belongs to
$C([0,T];\mathcal{D}(\boldsymbol{\mathrm{A}}^{s+1/2}e^{\sigma\boldsymbol{\mathrm{A}}^{1/2}}))$
(for any $T>0$) with the only non-zero Fourier coefficient being
$\widehat{{\bf f}(t)}_{3,0}$. The Fourier coefficient $\widehat{{\bf
f}(t)}_{3,0}$ is a continuous function in time and is defined by
$\widehat{{\bf f}(t)}_{3,0}=\begin{cases}1,&0\leq t\leq 10,\cr\cos(\pi
t/5)\exp(-(t-10)/5)&t>10.\cr\end{cases}$
The impact of the external force on the numerical velocity and its complement
(generating the approximate inertial manifold) is demonstrated in Figure 3
with the time evolution of the velocity field matching that of the external
force, leading to little change in evolution process of the velocity as the
external force gets smaller and smaller.
We chose a fixed relative error tolerance to be of accuracy at least
$\mathcal{O}(10^{-3})$, for adaptive time-integration solving the
$N^{2}+2N$-dimensional system ordinary differential equations (4.1) in time,
using backward differential formulas with variable order (one to five) and
variable adaptive time step sizes that meet the fixed error tolerance.
As discussed in the next subsection, the high-frequency components of the
solution turn out to be unreliable as $t$ increases, [8], presumably because
of the time discretization error. We therefore retained only the frequency
components up to some order $N_{1}\leq N$, correspondingly, we define an
additional approximation of ${\bf u}_{N}$, defined in (4.1)-(4.24), by
(5.5) ${\bf u}_{N_{1};N}(\cdot,t):={\mathbf{\Pi}}_{N_{1}}{\bf
u}_{N}(\cdot,t),\qquad N_{1}\leq N.$
As in Theorem 4.2, for the spatially discrete pseudospectral quadrature
method, assuming (3.1) and exact time integration of (4.1), and hence using
(2.35), (3.4), and (4.8), we get spectral convergence:
$\|{\bf u}-{\bf u}_{N_{1};N}\|\leq\left\|{\bf u}-{\mathbf{\Pi}}_{N_{1}}{\bf
u}\right\|+\left\|{\mathbf{\Pi}}_{N_{1}}\left({\bf u}-{\bf
u}_{N}\right)\right\|\leq C\left[N_{1}^{-2(s+1)}+N^{-2s}\right]\leq
CN_{1}^{-2s}.$
Our simulated approximate velocity fields
${\bf U}_{75}(t):={\bf u}_{75;100}(t),\quad\mbox{ for }t=10,20,30,40,50,60,$
are in Figure 6–9 and the associated vorticity stream function $\Psi_{75}(t)$
of $\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}(t)$ (computed using (4.33)) are
in Figure 12–15. These figures demonstrate that the initial random flow with
several smaller structures evolve into regular flow with larger structures,
similar to those observed in [8, page 307]. The choice of $N_{1}$ will be
discussed in the next subsection.
### 5.3. Energy spectrum of the solution
If ${\bf u}$ (and hence the number of modes in ${\bf u}$) is unknown, as it is
in the case in the benchmark test Example 2, and if ${\bf u}_{N}$ does not
contain all of the modes in ${\bf u}$, the higher modes of ${\bf u}_{N}$ are
usually less accurate than the chosen practical error tolerance and hence can
even violate important physical properties of ${\bf u}$ (because of the error
time-integration tolerance being not very small). In such cases, it is
important to choose $N_{1}<N$, depending on certain known physical properties
of ${\bf u}$.
For a fixed time $t$, the $L$-th mode energy spectrum of a tangential
divergence-free flow ${\bf u}$ on the sphere is defined by
(5.6) $E(L)=E({\bf u},L)=\sum_{|m|\leq L}|\beta_{L,m}(t)|^{2},\qquad\qquad{\bf
u}(\widehat{{\bf x}},t):=\sum_{L=1}^{\infty}\sum_{|m|\leq
L}\beta_{L,m}(t){\mathbf{Z}}_{L,m}(\widehat{{\bf x}}).$
Although the analytical form of the flow in Example 2 is not known, several
investigations have been carried out for such fields with initial spectrum of
the $L$-th mode decaying with order $L^{-1}$ or $L^{-2}$. In particular, it is
well known (see [8]), for this benchmark test case (on periodic two
dimensional geometries), that the energy spectrum of the velocity has a power-
law inertial range and an exponential decay (dissipation range) for wave
numbers larger than the Kraichnan’s dissipation wave number. Further, several
random smaller structures built in the initial random vorticity evolve into
regular flow with larger structures.
With ${\bf u}$ being the unique solution of the NSE (2.1), let us decompose
${\bf u}=\widetilde{{\bf u}}_{N_{1}}+{\bf w}_{N_{1}}$, where $\widetilde{{\bf
u}}_{N_{1}}={\mathbf{\Pi}}_{N_{1}}({\bf u})$ contains all modes lower or equal
$N_{1}$ and ${\bf w}_{N_{1}}:={\bf u}-{\mathbf{\Pi}}_{N_{1}}({\bf u})$
contains all higher modes. The existence of a relation between ${\bf
w}_{N_{1}}$ and $\widetilde{{\bf u}}_{N_{1}}$ of a form ${\bf
w}_{N_{1}}=\Phi(\widetilde{{\bf u}}_{N_{1}})$ was established in [28]. The
graph of $\Phi$ is known as the inertial manifold of (2.1). For computational
purposes, the higher modes can be computed efficiently using an approximate
inertial manifold:
(5.7) $\widetilde{\Phi}_{\widetilde{N_{1}}}(\widetilde{{\bf
u}}_{N_{1}})=(\nu\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}})^{-1}\left({\mathbf{\Pi}}_{\widetilde{N_{1}}}-{\mathbf{\Pi}}_{N_{1}}\right)\left[{\bf
f}-\boldsymbol{\mathrm{B}}\left(\widetilde{{\bf u}}_{N_{1}},\widetilde{{\bf
u}}_{N_{1}}\right)\right],~{}~{}\widetilde{N_{1}}>N_{1}.$
This well known approximation (without the Coriolis term) was introduced in
[10, 11] for nonlinear dissipative systems, including the NSE on domains and
we choose $\widetilde{N_{1}}=2N_{1}$.
For the benchmark test case, in the dissipation range, even
$L^{4}E(\widetilde{\Phi}_{2N_{1}}(\widetilde{{\bf u}}_{N_{1}}),L)$ decays
exponentially to zero, further justifying restriction of the infinite
dimensional range after certain values of $N_{1}$. As discussed in [8, page
280, 307] (and repeated in [12, page 991]), such a faster decay (one order
higher than the $L^{-3}$ decay known is Kraichnan theory of turbulence) is
expected due to the Reynolds number considered in [8, 12] being much small
than $25,000$. (Extensive study in [3, 23] shows that the turbulence theory
decay can occur only when the Reynolds number is of the order $25,000$.)
Briefly (without repeating technical details in [8]), using the viscosity term
$\nu$ in (2.1), the Reynolds number is $\mathcal{O}(\nu^{-1})$ with the order
constant $r$ given by the product of the mean fluid velocity and the
characteristic length-scale. With $\nu^{-1}=10,000$ and the constant rotation
rate $\Omega=1$, for the Coriolis parameter in (2.23), the decay of energy
spectrum of a numerical velocity and exponential decay of its approximate
complement in Figure 16-17 is well supported by extensive simulations in [3,
23, 8], highlighting further the benchmark applicability of our algorithm,
extending periodic domain results in [3, 23, 8] to a practically relevant
rotating sphere case with Coriolis effect.
Further, the simulated results are substantiated by the well known exponential
decay of $E(\widetilde{\Phi}_{2N_{1}}(\widetilde{{\bf u}}_{N_{1}}),L)$,
observed in Figures 16-17. Finally, the exponential decay of the energy
spectrum show that $N=100,N_{1}=3N/4$ is sufficient to understand the flow
behavior for this benchmark example with our algorithm using adaptive variable
order and variable time-step highly stable backward differentiation formulas
with a practically useful relative error tolerance $\mathcal{O}(10^{-3})$.
Figure 2. Initial random vorticity stream function $\Psi$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf u}$ at $t=0$.
|
---|---
Time evolution of ${\bf U}_{75}(t)$ and $f(t)$. | Time evolution of $\widetilde{\Phi}_{150}({\bf U}_{75}(t))$ and ${\bf f}(t)$.
Figure 3. Impact of the external force on a numerical velocity and its
complement.
Figure 4. Numerical velocity ${\bf U}_{75}(t)$, at $t=10$.
Figure 5. Numerical velocity ${\bf U}_{75}(t)$, at $t=20$.
Figure 6. Numerical velocity ${\bf U}_{75}(t)$, at $t=30$.
Figure 7. Numerical velocity ${\bf U}_{75}(t)$, at $t=40$.
Figure 8. Numerical velocity ${\bf U}_{75}(t)$, at $t=50$.
Figure 9. Numerical velocity ${\bf U}_{75}(t)$, at $t=60$.
Figure 10. Vorticity stream function $\Psi_{75}$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=10$.
Figure 11. Vorticity stream function $\Psi_{75}$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=20$.
Figure 12. Vorticity stream function $\Psi_{75}$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=30$.
Figure 13. Vorticity stream function $\Psi_{75}$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=40$.
Figure 14. Vorticity stream function $\Psi_{75}$ of
$\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=50$.
Figure 15. Vorticity stream function $\Psi_{75}$ of $\boldsymbol{\mathrm{Vort}}\,{\bf U}_{75}$ at $t=60$. |
---|---
Energy spectrum $E(L)$ of ${\bf U}_{75}(t)$. | Energy spectrum $E(L)$ of $\widetilde{\Phi}_{150}({\bf U}_{75}(t))$.
Figure 16. Energy spectra of velocity ${\bf U}_{75}(t)$ and $\Phi_{150}({\bf U}_{75}(t))$. |
---|---
Energy spectrum $L^{4}*E(L)$ of ${\bf U}_{75}(t)$ | Energy spectrum $L^{4}*E(L)$ of $\widetilde{\Phi}_{150}({\bf U}_{75}(t))$.
Figure 17. Scaled energy spectra $L^{4}*E(L)$ of ${\bf U}_{75}(t)$ and
$\widetilde{\Phi}_{150}({\bf U}_{75}(t))$.
## 6\. Appendix
In this section, we generalize certain known domain case estimates, that are
fundamental for the NSE analysis, to the spherical surface case and hence
prove (2.30). Following [16, Page 574], for ${\bf u}\in C^{\infty}(TS)$, we
extend ${\bf u}$ to the spherical layer $S\times I$,
$I=(r_{1},r_{2}),0<r_{1}<1<r_{2}<\infty$ by the formula
(6.1) $\widetilde{{\bf u}}(\widehat{{\bf x}})=\varphi(|\widehat{{\bf x}}|){\bf
u}(\widehat{{\bf x}}/|\widehat{{\bf x}}|),$
where $\varphi(t)\in C_{0}^{\infty}(I)$, $\varphi(t)\geq 0,~{}t\in I$, and
$\varphi(1)=1$. We have
$\int_{S\times I}|{\widetilde{{\bf
u}}}|^{p}=\int_{r_{1}}^{r_{2}}\varphi^{p}(r)\int_{S}|{\bf u}|^{p}dS$
In other words
(6.2) $\|{\widetilde{{\bf u}}}\|_{L^{p}(S\times I)}=c\|{\bf u}\|_{L^{p}(TS)},$
where $c=c(\varphi,r_{1},r_{2})$. Suppose ${\bf u},{\bf v},{\bf w}$ are
extended from $S$ to the spherical layer $S\times I$ by (6.1). Then from [16,
Lemma 4.3] we have
(6.3) $b({\bf u},{\bf v},{\bf w})=cb(\widetilde{{\bf u}},\widetilde{{\bf
v}},\widetilde{{\bf w}}),$
On the sphere $S$, we have the following version of Sobolev embedding
inequality [1]
(6.4) $\|{\bf u}\|_{L^{q}(TS)}\leq C\|{\bf u}\|_{H^{s}(TS)},\qquad
s<1,\quad\frac{1}{q}=\frac{1}{2}-\frac{s}{2}.$
The following nonlinearity estimate is an adaptation of [6, Proposition 6.1]
for $S$.
###### Proposition 6.1.
Let $s_{1},s_{2},s_{3}\geq 0$ be real numbers, and we assume that
$s_{1}+s_{2}+s_{3}\geq 1$ and
$(s_{1},s_{2},s_{3})\neq(0,0,1),(0,1,0),(1,0,0)$. Then there exists a constant
depending on $s_{1},s_{2},s_{3}$ such that
$|b({\bf u},{\bf v},{\bf w})|\leq C\|{\bf u}\|_{H^{s_{1}}(TS)}\|{\bf
v}\|_{H^{s_{2}+1}(TS)}\|{\bf w}\|_{H^{s_{3}}(TS)}$
or in the extrapolated form, writing $H^{s}(TS)=H^{s}_{TS}$, for all ${\bf
u},{\bf v},{\bf w}\in C^{\infty}(TS)$
$\displaystyle|b({\bf u},{\bf v},{\bf w})|\leq C$ $\displaystyle\|{\bf
u}\|^{1+[s_{1}]-s_{1}}_{H^{[s_{1}]}_{TS}}\|{\bf
u}\|^{s_{1}-[s_{1}]}_{H^{[s_{1}]+1}_{TS}}\|{\bf
v}\|^{1+[s_{2}]-s_{2}}_{H^{[s_{2}]+1}_{TS}}\|{\bf
v}\|^{s_{2}-[s_{2}]}_{H^{[s_{2}]+2}_{TS}}\|{\bf
w}\|^{1+[s_{3}]-s_{3}}_{H^{[s_{3}]}_{TS}}\|{\bf
w}\|^{s_{3}-[s_{3}]}_{H^{[s_{3}]+1}_{TS}}.$
###### Proof.
Let ${\bf u},{\bf v},{\bf w}\in C^{\infty}(TS)$ and $\widetilde{{\bf
u}},\widetilde{{\bf v}},\widetilde{{\bf w}}$ be their corresponding extension
to the spherical layer $\widetilde{\Omega}:=S\times I$. Let us consider first
the case $s_{i}<1$ for $i=1,2,3$. Define associated constants
$q_{1},q_{2},q_{2},q_{4}$ so that $\sum_{i=1}^{4}\frac{1}{q_{i}}=1$ and
$\frac{1}{q_{i}}=\frac{1}{2}-\frac{s_{i}}{2}$ for $i=1,2,3$. Then, by Hölder’s
inequality
$\displaystyle|b({\widetilde{{\bf u}}},{\widetilde{{\bf v}}},{\widetilde{{\bf
w}}})|$ $\displaystyle=$
$\displaystyle\left|\sum_{i,j=1}^{3}\int_{\widetilde{\Omega}}\widetilde{u}_{j}\frac{\partial\widetilde{v}_{i}}{\partial
x_{j}}\widetilde{w}_{i}1\right|\leq\|{\widetilde{{\bf
u}}}\|_{L^{q_{1}}(\Omega)}\|\nabla{\widetilde{{\bf
v}}}\|_{L^{q_{2}}(\Omega)}\|{\widetilde{{\bf
w}}}\|_{L^{q_{3}}(\Omega)}\|1\|_{L^{q_{4}}(\Omega)}$
Restricting to the sphere by (6.3) and (6.2) then using the Sobolev embedding
theorem on the sphere (6.4) we have
$\displaystyle|b({\bf u},{\bf v},{\bf w})|$ $\displaystyle\leq$ $\displaystyle
C\|{\bf u}\|_{L^{q_{1}}_{TS}}\|\boldsymbol{\mathrm{Grad}}\,{\bf
v}\|_{L^{q_{2}}_{TS}}\|{\bf w}\|_{L^{q_{3}}_{TS}}\leq C\|{\bf
u}\|_{H^{s_{1}}_{TS}}\|\boldsymbol{\mathrm{Grad}}\,{\bf
v}\|_{H^{s_{2}}_{TS}}\|{\bf w}\|_{H^{s_{3}}_{TS}}.$
∎
###### Lemma 6.1.
Let $\delta\in(1/2,1)$ be given and ${\bf u},{\bf v}\in V$. Then there exists
$C$, independent of ${\bf u}$ and ${\bf v}$, such that
$\|\boldsymbol{\mathrm{A}}^{-\delta}\boldsymbol{\mathrm{B}}({\bf u},{\bf
v})\|\leq C\left\\{\begin{array}[]{cc}\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf
u}\|\|{\bf v}\|&\\\ \|{\bf u}\|\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf
v}\|&\\\ \end{array},\right.\qquad{\bf u},{\bf v}\in V.$
###### Proof.
Let ${\bf u},{\bf v},{\bf w}\in V$. Hence from (2.26), and by using
Proposition 6.1 with $s_{1}=0$, $s_{2}=2\delta-1>0$ and $s_{3}=2-2\delta>0$,
$|b({\bf u},{\bf v},\boldsymbol{\mathrm{A}}^{-\delta}{\bf w})|=|b({\bf
u},\boldsymbol{\mathrm{A}}^{-\delta}{\bf w},{\bf v})|\leq C\|{\bf
u}\|\|\boldsymbol{\mathrm{A}}^{-\delta}{\bf w}\|_{H^{2\delta}(TS)}\|{\bf
v}\|_{H^{2-2\delta}(TS)}.$
Since $\|\boldsymbol{\mathrm{A}}^{-\delta}{\bf
w}\|_{H^{2\delta}(TS)}=\|\boldsymbol{\mathrm{A}}^{\delta}\boldsymbol{\mathrm{A}}^{-\delta}{\bf
w}\|=\|{\bf w}\|$, we get
$|b({\bf u},{\bf v},\boldsymbol{\mathrm{A}}^{-\delta}{\bf w})|\leq C\|{\bf
u}\|\|{\bf w}\|\|{\bf v}\|_{H^{2-2\delta}(TS)},\qquad{\bf w}\in V.$
Since the inequality is true for all ${\bf w}\in V$, we obtain the first bound
$\|\boldsymbol{\mathrm{A}}^{-\delta}\boldsymbol{\mathrm{B}}({\bf u},{\bf
v})\|\leq C\|{\bf u}\|\|{\bf v}\|_{H^{2-2\delta}(TS)}=C\|{\bf
u}\|\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf v}\|.$
To obtain the second bound, we again use (2.26) and Proposition 6.1 but with
$s_{1}=2-2\delta$, $s_{2}=2\delta-1$ and $s_{3}=0$,
$\displaystyle|b({\bf u},{\bf v},\boldsymbol{\mathrm{A}}^{-\delta}{\bf
w})|=|b({\bf u},\boldsymbol{\mathrm{A}}^{-\delta}{\bf w},{\bf v})|$
$\displaystyle\leq$ $\displaystyle C\|{\bf
u}\|_{H^{2-2\delta}(TS)}\|\boldsymbol{\mathrm{A}}^{-\delta}{\bf
w}\|_{H^{2\delta}(TS)}\|{\bf v}\|$ $\displaystyle=$ $\displaystyle
C\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf u}\|\|{\bf w}\|\|{\bf
v}\|,\qquad{\bf w}\in V.$
Since the inequality is true for all ${\bf w}\in H$, we obtain
$\|\boldsymbol{\mathrm{A}}^{-\delta}\boldsymbol{\mathrm{B}}({\bf u},{\bf
v})\|\leq C\|\boldsymbol{\mathrm{A}}^{1-\delta}{\bf u}\|\|{\bf v}\|.$
∎
#### Acknowledgments:
The support of the Australian Research Council under its Discovery and Centre
of Excellence programs is gratefully acknowledged. The authors thank
Professors M. Farge and E. S. Titi [29] for valuable discussions.
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|
arxiv-papers
| 2010-09-17T01:30:01 |
2024-09-04T02:49:12.952009
|
{
"license": "Public Domain",
"authors": "M. Ganesh, Q. T. Le Gia, I. H. Sloan",
"submitter": "Quoc Thong Le Gia",
"url": "https://arxiv.org/abs/1009.3308"
}
|
1009.3727
|
# Analytical three-dimensional bright solitons and soliton-pairs in Bose-
Einstein condensates with time-space modulation
Zhenya Yan1,2 and Chao Hang2,3 1Key Laboratory of Mathematics Mechanization,
Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing
100080, China
2Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo
Interdisciplinar, Lisboa 1649-003, Portugal
3Department of Physics, East China Normal University, Shanghai 200062, China
###### Abstract
We provide analytical three-dimensional bright multi-soliton solutions to the
(3+1)-dimensional Gross-Pitaevskii (GP) equation with time and space-dependent
potential, time-dependent nonlinearity, and gain/loss. The zigzag propagation
trace and the breathing behavior of solitons are observed. Different shapes of
bright solitons and fascinating interactions between two solitons can be
achieved with different parameters. The obtained results may raise the
possibility of relative experiments and potential applications.
###### pacs:
05.45.Yv, 03.75.Lm, 42.65.Tg
## I Introduction
Solitons describe a class of fascinating nonlinear wave propagation phenomena
appearing as a result of balance between nonlinearity and dispersion or
diffraction properties of the medium under nonlinear excitations, which leads
to undistorted propagation over extended distance Haus . One of the most
important physically relevant realizations of solitons is provided by the
matter-wave solitons in Bose-Einstein condensed atomic gas bec1 . Based on the
successful experimental realization and theoretical analysis of Bose-Einstein
condensations (BECs) in weakly interacting atomic gases bec1 , matter-wave
dark solitons mws , vortices vor , bright solitons 3dsol , gap solitons gaps ,
and soliton chains chains have been observed and studied. These studies have
stimulated a large amount of research activities, which enable the extension
of linear atom optics to nonlinear atom optics bec2 .
The realization of higher-dimensional matter-wave solitons in BECs is still a
challengeable topic because those solutions are usually unstable for (2+1)-D
and (3+1)-D constant-coefficient nonlinear Schrödinger (NLS) equation due to
the weak and strong collapse Sulem . However, different situations are
observed in BECs with temporally or spatially modulated parameters. Alteration
of atomic scattering length achieved by Feshbach resonance feshbach has been
used to dynamically stabilize higher-dimensional bright solitons stab1 while
periodic external potentials achieved by optical lattice has been used to
generate and control higher-dimensional gap solitons stab2 . 1D periodic wave
solutions are also predicted in BECs with time-space varying parameters 1DNLS
. Moreover, the bright solitons 3bec and periodic wave solutions yanc were
obtained in spinor BECs governed by a system of three coupled mean-field
equations.
In this work, we present a detailed study on dynamics of analytical 3D bright
matter-wave single solitons and soliton-pairs in BECs with time-space
modulation. We note that 3D periodic wave solutions have been studied in the
generalized NLS equation very recently belic ; yank . However, the authors did
not study the soliton pair solutions and their interaction properties. By
using the similarity 1DNLS ; yank ; simi and bilinear transformations bi , we
can achieve different shapes of bright solitons and fascinating interactions
between two solitons. In addition, the experimental possibilities for
observability are discussed and the stability of solitons is illustrated
numerically.
The paper is organized as follows. In the next section, the model under study
is introduced. In Sec. III, the methods for solving the model equation are
introduced. A relationship between the model and a practical system is
established. In Sec. IV, we give the expressions of the bright solitons and
the soliton-pairs. The interactions between two solitons are also
investigated. In the last section, the case of the dark solitons is discussed
and the outcomes are summarized.
## II The GP model
The dynamics of a weakly interacting Bose gas at zero temperature is well
described by the (3+1)-D GP model with time-space modulation bec1
$\displaystyle\begin{array}[]{l}\displaystyle
i\hbar\frac{\partial\Psi}{\partial
t}\\!=\\!\left[-\frac{\hbar^{2}}{2m}\nabla^{2}\\!+V_{\rm ext}(t,{\bf
r})+G(t)|\Psi|^{2}\right]\\!\\!\Psi+i\,\Gamma(t)\Psi,\end{array}\vspace{-0.1in}$
(2)
where $\nabla=(\partial_{x},\partial_{y},\partial_{z})$, ${\bf r}=(x,y,z)$,
$\Psi\equiv\Psi(t,{\bf r})$ denotes the order parameter with
$N=\int|\Psi|^{2}d{\bf r}$ being the number of atoms in the condensate,
$G(t)=4\pi\hbar^{2}a_{s}(t)/m$ is the interaction function with $a_{s}(t)$
being the $s$-wave scattering length modulated by a Feshbach resonance, and
$\Gamma(t)$ is the gain/loss term, which is phenomenologically incorporated to
account for the interaction of atomic or thermal clouds. We note that the
dissipative dynamics originating from the interaction between the radial and
axial degrees of freedom has also been studied recently dissip . Here the
potential is chosen as a harmonic trap $V_{\rm ext}(t,{\bf r})=(m/2)({\bf
r}-{\bf e}(t))\omega^{2}(t)({\bf r}-{\bf e}(t))$ with $\omega(t)={\rm
diag}(\omega_{x}(t),\omega_{y}(t),\omega_{z}(t))$ being a diagonal matrix of
the trap frequencies in three directions and ${\bf e}(t)=$($e_{1}(t)$,
$e_{2}(t)$, $e_{3}(t)$) corresponding to its center.
Using the suitably scales and variables: ${\bf r}=a_{z}{\bf r}^{\prime},\
t=\tilde{\omega}_{z}^{-1}t^{\prime},\ {\bf e}(t)=a_{z}{\bf e}^{\prime}(t)$,
$\Psi=\sqrt{N/a_{z}^{3}}\psi$, $a_{z}=[\hbar/(m\tilde{\omega}_{z})]^{1/2}$,
and $\tilde{\omega}_{z}=\int\omega_{z}(t)dt$, we arrive at the dimensionless
GP equation in the (3+1)-D space after dropping the primes
$\displaystyle\begin{array}[]{l}\displaystyle i\frac{\partial\psi}{\partial
t}=\left[-\frac{1}{2}\nabla^{2}+v(t,{\bf
r})+g(t)|\psi|^{2}\right]\psi+i\,\gamma(t)\psi,\end{array}$ (4)
where $g(t)=4\pi N^{2}a_{s}(t)/a_{z}^{4}$,
$\gamma(t)=\Gamma(t)/(\hbar\tilde{\omega}_{z})$, and
$\displaystyle\begin{array}[]{l}\displaystyle v(t,{\bf r})=\frac{1}{2}({\bf
r}-{\bf e}(t)){\bf\alpha}^{2}(t)({\bf r}-{\bf e}(t))\end{array}$ (6)
with $\alpha(t)={\rm
diag}(\alpha_{1}(t),\alpha_{2}(t),\alpha_{3}(t))=\tilde{\omega}_{z}^{-1}\omega(t)$.
Eq. (4) is associated with $\delta\mathcal{L}/\delta\psi^{*}=0$ in which the
Lagrangian density can be written as
$\displaystyle\begin{array}[]{l}\mathcal{L}=i(\psi\psi^{*}_{t}-\psi^{*}\psi_{t})+|\nabla\psi|^{2}-g(t)|\psi|^{4}\vspace{0.1in}\cr\qquad\quad-2[v(t,{\bf
r})+i\gamma(t)]|\psi|^{2}.\end{array}$ (9)
## III Similarity solutions
Here we focus on the spatially localized bright solitons and soliton-paris for
which $\lim_{|{\bf r}|\rightarrow\infty}\psi(t,{\bf r})=0$. Our first
objective is to reduce Eq. (4) to the tractable NLS equation
$\displaystyle\begin{array}[]{l}\displaystyle
i\frac{\partial\Phi(\tau,\xi)}{\partial\tau}=-\frac{1}{2}\frac{\partial^{2}\Phi(\tau,\xi)}{\partial\xi^{2}}+\mathcal{G}|\Phi(\tau,\xi)|^{2}\Phi(\tau,\xi)\end{array}$
(11)
using a proper similarity transformation, where $\tau\equiv\tau(t)$ and
$\xi\equiv\xi(t,{\bf r})$ are both the unknown variables, and $\mathcal{G}$ is
a constant. We explore the attractive nonlinearity, i.e. $\mathcal{G}=-1$,
resulting in the bright multi-soliton solutions. The case $\mathcal{G}=1$
resulting in the dark multi-soliton solutions does not pose new challenges and
will be discussed in the last section. Using the similarity transformation
1DNLS ; yank ; simi
$\displaystyle\psi(t,{\bf r})=\rho(t)e^{i\varphi(t,{\bf
r})}\Phi(\tau(t),\xi(t,{\bf r})),$ (12)
and requiring $\Phi(\tau(t),\xi(t,{\bf r}))$ to satisfy Eq. (11) and
$\psi(t,{\bf r})$ to be the solution of Eq. (4), we find a set of equations
$\displaystyle\begin{array}[]{l}\nabla^{2}\xi=0,\ \
\xi_{t}+\nabla\xi\cdot\nabla\varphi=0,\ \
\tau_{t}=|\nabla\xi|^{2},\end{array}\quad$ (13b)
$\displaystyle\begin{array}[]{l}\displaystyle v(t,{\bf
r})=-\frac{1}{2}|\nabla\varphi|^{2}-\varphi_{t},\ \
g(t)=\mathcal{G}\rho^{-2}|\nabla\xi|^{2},\end{array}\quad$ (13d)
$\displaystyle\begin{array}[]{l}\displaystyle\gamma(t)=\frac{1}{2}\nabla^{2}\varphi+\rho_{t}/\rho.\end{array}\quad$
(13f)
Here for the harmonic trapping potential $v(t,{\bf r})$ given by Eq. (6),
after some algebra it follows from system (13) that the similarity variables
can be expressed as
$\displaystyle\begin{array}[]{l}\displaystyle\xi(t,{\bf
r})\\!=\beta(t)\cdot{\bf
r}\\!+\\!\int^{t}_{0}\beta(s)\cdot\sigma(s)ds,\vspace{0.1in}\cr\displaystyle\tau(t)\\!=\int^{t}_{0}|\beta(s)|^{2}ds,\end{array}$
(16)
where $\beta(t)=(\beta_{1}(t),\beta_{2}(t),\beta_{3}(t))$ denotes the vector
of the inverse spatial widths of the localized solutions along $x,y,z$
directions, respectively, and
$\sigma(t)=(\sigma_{1}(t),\sigma_{2}(t),\sigma_{3}(t))$ with
$\sigma_{j}=\beta_{j}\int^{t}_{0}e_{j}\alpha_{j}^{2}\beta_{j}^{-1}dt$ relating
to the velocity of the solitons. Moreover the nontrivial phase has the
quadratic form
$\displaystyle\begin{array}[]{l}\displaystyle\varphi(t,{\bf
r})=-\frac{1}{2}{\bf r}A(t){\bf r}+\sigma\cdot{\bf
r}-\frac{1}{2}\int^{t}_{0}({\bf e}\alpha^{2}{\bf
e}+|\sigma|^{2})dt,\end{array}$ (18)
where $A(t)={\rm
diag}(\dot{\beta}_{1}/\beta_{1},\dot{\beta}_{2}/\beta_{2},\dot{\beta}_{3}/\beta_{3})$.
The additional relations between $\alpha_{j}(t)$ and
$\beta_{j}(t)=1/\nu_{j}(t)$ result in the Mathieu equations
$\displaystyle\ddot{\nu}_{j}(t)+\alpha_{j}^{2}(t)\nu_{j}(t)=0,\quad(j=1,2,3).$
(19)
Finally, the function $\rho(t)$ modulating the amplitude of solution $\psi$
and nonlinearity $g(t)$ can be also found by
$\displaystyle\begin{array}[]{l}\rho(t)=\displaystyle\rho_{0}[\beta_{1}(t)\beta_{2}(t)\beta_{3}(t)]^{1/2}\exp\left[\int_{0}^{t}\gamma(s)ds\right],\vspace{0.1in}\cr
g(t)=-\rho^{-2}(t)|\beta(t)|^{2},\end{array}$ (22)
which depend on both $\beta_{j}(t)$ and gain/loss coefficient $\gamma(t)$ with
$\rho_{0}$ being a non-zero parameter. Note that for the given $\beta_{j}(t)$,
the nonlinearity $g(t)$ must attenuate (grow) exponentially in the gain (loss)
medium $\gamma(t)>0\ (<0)$.
For the given $\alpha_{j}(t)$, one can, in principle, obtain corresponding
$\beta_{j}(t)$ (or equivalently for the given $\beta_{j}(t)$ one can obtain
$\alpha_{j}(t)$) based on Eq.(19). Furthermore, the bright $N$-soliton
solutions of Eq.(11) can be obtained using the bilinear transformation bi :
$\Phi_{N}=P^{(N)}(\tau,\xi)/Q^{(N)}(\tau,\xi)$. Here, $P^{(N)}$ and $Q^{(N)}$
satisfy $(iD_{\tau}+1/2D_{\xi}^{2})P^{(N)}\cdot Q^{(N)}=0$ and
$D_{\xi}^{2}Q^{(N)}\cdot Q^{(N)}=2|P^{(N)}|^{2}$ with $D_{t}$ and $D_{\xi}$
being the bilinear operators and
$P^{(N)}=\sum_{j=1}^{N}\epsilon^{2j-1}P_{2j-1}(\tau,\xi)$ and
$Q^{(N)}=1+\sum_{j=1}^{N}\epsilon^{2j}Q_{2j}(\tau,\xi)$. Thus, by choosing
$\beta_{j}(t)$ and $\gamma(t)$, we can generate $v(t,{\bf r})$ and $g(t)$ for
which the generic bright $N$-soliton solutions of Eq. (4) can be found from
Eq. (11) on the basis of Eq. (12). We will use this analytical result to
construct the exact bright $N$-soliton solutions with many interesting
nontrivial features.
For the convenience of analyzing different dynamical regimes described by the
given model, we specify the magnitude of main physical parameters, which are
feasible in experiments. We consider a condensed sodium sample trapped in the
state $|3S_{1/2},F=1,m_{F}=-1\rangle$, which has the scattering length
$a_{s}=2.75$ nm csys . The other parameters can be taken as $N=1.2\times
10^{6}$ and $\tilde{\omega}_{z}=(2\pi)\times 21$ Hz, which leads to
$a_{z}=4.55$ $\mu$m and $\sqrt{N/a_{z}^{3}}=1.13\times 10^{2}$ $\mu$m-3/2. To
make sure the frequencies $\alpha_{j}(t)$ and nonlinearity $g(t)$ are bounded
for realistic cases, we choose $\beta_{j}(t)$ and the gain/loss coefficient
$\gamma(t)$ as the periodic functions
$\displaystyle\beta(t)={\rm dn}(t,m){\bf b},\quad\gamma(t)=\gamma_{0}\,{\rm
cn}(t,n),\ \ \ \gamma_{0}\in\mathbb{R}$ (23)
where ${\bf b}=(b_{1},b_{2},b_{3})$ is a real constant vector describing the
inverse of the width of the potential and the frequency, $m\in[0,\ 1)$ and
$n\in[0,\ 1]$ are the modules of Jacobi elliptic functions. It is easy to see
that $\gamma(t)=\gamma_{0}\,{\rm sech}(t)>0$ corresponds to the dissipative
case when $n=1$ and $\gamma_{0}>0$ (we will focus on this condition next). In
practical systems, the modulations of $\beta_{j}(t)$, $g(t)$ and $\gamma(t)$
depend on the use of the optical lattice and Feshbach-resonance techniques,
i.e. we can achieve $\Gamma(t)$ and $a_{s}(t)$ by exerting particular time-
dependent optical field and magnetic field.
It follows from Eqs. (19) and (23) that $\alpha_{j}(t)$ is given by
$\displaystyle\alpha_{j}^{2}(t)=m^{2}\left[(2-m^{2})\,{\rm sd}^{2}(t,m)-{\rm
nd}^{2}(t,m)\right].$ (24)
Figure 1 shows the curves of $\alpha_{j}(t)$, $g(t)$, and $\gamma(t)$ vs $t$.
For simplicity, we take $b_{j}=1$, i.e.
$\alpha_{1}(t)=\alpha_{2}(t)=\alpha_{3}(t)$ corresponding to the isotropic
potential, and consider that the center of the potential locates at the origin
($e_{j}=0$). We can also change $b_{j}$ to get an anisotropic potential and
use nonzero $e_{j}$ to obtain moving bright solitons as those discussed in 1D
case 1DNLS .
Figure 1: (color online). Curves of $\alpha_{j}(t)$, $g(t)$, and $\gamma(t)$
given by Eqs. (23) and (24) vs $t$ for $\rho_{0}=6.0$, $\gamma_{0}=m=0.1$, and
$b_{j}=n=1.0$ ($j=1$, 2, 3).
## IV Bright solitons and soliton pairs
Based on the discussions in the previous section, we arrive at the fundamental
3D time-varying bright solitons
$\displaystyle\begin{array}[]{ll}\psi_{1}(t,{\bf r})\\!=r_{1}\rho(t)\,{\rm
sech}\big{[}r_{1}(\xi(t,{\bf
r})\\!-\\!s_{1}\tau(t))\\!-\\!\ln|2r_{1}|\big{]}e^{i\theta},\vspace{-0.05in}\end{array}$
(26)
where $\theta=s_{1}\xi(t,{\bf
r})+(r_{1}^{2}-s_{1}^{2})/2\tau(t)+\varphi(t,{\bf r})$ with
$r_{1},s_{1}\in\mathbb{R}$, and $\rho(t)$, $\varphi(t,{\bf r})$, $\xi(t,{\bf
r})$, and $\tau(t)$ are given by Eqs. (16), (18) and (22).
Figure 2 exhibits the dynamics of the time-varying bright soliton (26). A
breathing behavior is also evident, which can be managed by $\beta_{j}(t)$,
$\gamma_{j}(t)$, and $\rho(t)$. For the case $m=0$, we have $\beta_{j}=b_{j}$
and $\alpha_{j}=0$, in which the travelling-wave bright soliton is obtained.
In experiments, it can be simply realized for the zero linear potential. The
bright soliton propagates in a zigzag trace for $m=0.1$ [see Fig.2(a)]. An
important feature is that while $m\rightarrow 1$ ($\neq 1$) resulting in the
larger period of $\beta_{j}(t)$ given by Eq.(23), the amplitude of the soliton
close to the corners attenuates rapidly so that a soliton chain is generated
[see Fig.2(b)]. In experiments, it can be realized by taking
$\omega_{x}=\omega_{y}=\omega_{z}=m[(2-m^{2}){\rm sd}^{2}(t,m)-{\rm
nd}^{2}(t,m)]^{1/2}\tilde{\omega}_{z}$.
Figure 2: (color online). Propagations (left) and contour plots (right) of
density for the bright soliton (26) in $(t,\,\zeta\equiv{\bf b}\cdot{\bf
r})$-space for $r_{1}=0.5,\ s_{1}=\rho_{0}=2.0$, and $\gamma_{0}=0.01$. The
others are the same as Fig.1. (a) The breathing bright soliton propagating in
a zigzag trace for $m=0.1$. In the given sodium sample, the maximum density
and width of the soliton are about $1.28$ $\mu$m-3 and $12.13$ $\mu$m. The
period is about $53.1$ ms. (b) The bright soliton chain for $m=0.9$. The
maximum density and width of the soliton are about $1.28$ $\mu$m-3 and $9.1$
$\mu$m. The period is about $91.0$ ms.
The interaction of the bright solitons plays an important role in the study of
BECs. Here, we will also study the interaction between two bright solitons.
The analytical 3D time-varying bright soliton pairs read
$\displaystyle\begin{array}[]{l}\psi_{2}(t,{\bf r})=\rho(t)e^{i\varphi(t,{\bf
r})}P(t,{\bf r})/Q(t,{\bf r}),\end{array}$ (28)
where $P(t,{\bf r})$ and $Q(t,{\bf r})$ can be expressed as the series of
exponential functions of $(t,{\bf r})$
$\displaystyle\begin{array}[]{l}\displaystyle P(t,{\bf
r})=\sum_{j=1}^{2}\delta_{j}e^{\eta_{j}}+\sum_{j,k=1,j\not=k}^{2}\lambda_{jk}e^{\eta_{j}+\eta_{j}^{*}+\eta_{k}},\vspace{0.1in}\cr\displaystyle
Q(t,{\bf r})=1+\sum_{j,k=1}^{2}\Lambda_{jk}e^{\eta_{j}+\eta_{k}^{*}}+\Omega
e^{\eta_{1}+\eta_{2}+\eta_{1}^{*}+\eta_{2}^{*}},\end{array}$ (31)
with $\eta_{j}=\mu_{j}\xi(t,{\bf r})+\frac{i}{2}\mu_{j}^{2}\tau(t),\
\mu_{j}=r_{j}+is_{j}\ (r_{j},s_{j},\delta_{j}\in\mathbb{R})$,
$\Lambda_{jk}=\delta_{j}\delta_{k}^{*}(\mu_{j}+\mu_{k}^{*})^{-2},\
\lambda_{jk}=(\mu_{k}-\mu_{j})\big{[}\delta_{k}\Lambda_{jk}(\mu_{k}+\mu_{k}^{*})^{-1}-\delta_{j}\Lambda_{kk}(\mu_{j}+\mu_{k}^{*})^{-1}\big{]}\
(j\not=k)$, and
$\Omega=(|\delta_{1}\delta_{2}|)^{-1}|\mu_{1}-\mu_{2}|^{2}\big{(}\Lambda_{11}\Lambda_{22}\sqrt{\Lambda_{12}\Lambda_{21}}-\Lambda_{12}\Lambda_{21}\sqrt{\Lambda_{11}\Lambda_{22}}\big{)}$.
The dynamics of the 3D time-varying bright two-soliton solutions (28) is
exhibited in Figure 3. Under the different parameters, we exhibit three cases
for two weak zigzag solitons without interaction [see Fig.3(a)], two strong
zigzag solitons with interaction [see Fig.3(b)], and strong-weak zigzag
solitons with interaction [see Fig.3(c)]. Notice that similar with the bright
solitons shown in Fig.2(b), for the case $m\rightarrow 1$ ($m\neq 1$), the
amplitudes of the soliton-pairs close to the corners will almost decrease to
zero so that panel (a) will degenerate to two parallel soliton chains while
panels (b) and (c) will degenerate to the $><$-shaped soliton chains. The
experimental realization of the dynamics regimes for the two-soliton solutions
is similar with that for the bright one-soliton solutions.
We stress that the important feature that distinguishes our solutions from the
reported in the literature mws ; vor ; 3dsol ; chains is the appearance of
the time- and space-dependent functions in both the phase and the amplitude
and which strongly affect the form and the behavior of bright solitons and
their interactions.
Figure 3: (color online). Propagations (left ) and contour plots (right) for
collisions between bright two solitons (28) in $(t,\,\zeta\equiv{\bf
b}\cdot{\bf r})$-space for $\gamma_{0}=0.01,\ \delta_{1,2}=\rho_{0}=1.0$, and
$m=0.6$. The others are the same as Fig.1. (a) Two zigzag solitons without
interaction for $r_{1}=1.0,\ r_{2}=1.2$ and $s_{1,2}=0$. In the given sodium
sample, the maximum density and width of the left (right) soliton are about
$1.8$ $\mu$m-3 and $7.6$ $\mu$m ($1.5$ $\mu$m-3 and $9.1$ $\mu$m). (b) Two
strong zigzag solitons with interaction for $r_{1,2}=s_{1}=-s_{2}=1.2$. The
maximum density and width of the left (right) soliton are about $6.2$ $\mu$m-3
and $3.0$ $\mu$m ($17.5$ $\mu$m-3 and $2.4$ $\mu$m) (c) Two strong-weak zigzag
solitons with interaction for $r_{1}=1.2,\ r_{2}=1.5,\ s_{1}=1.05$ and
$s_{2}=0$. The maximum density and width of the left (right) soliton are about
$6.2$ $\mu$m-3 and $4.2$ $\mu$m ($13.1$ $\mu$m-3 and $3.8$ $\mu$m). The period
of the zigzag oscillation is about $26.5$ ms in all panels.
In order to check the stability of the time-varying bright soliton (26), we
make numerical simulations of Eq. (4) with the initial conditions given by Eq.
(26) and different values of $m$. We find that the bright solitons are very
stable for $m$ being small (e.g. $m=0.1$) [see Fig.4(a)]. With the increase of
$m$, the bright solitons become unstable [see Fig.4(b)]. This is because large
$m$ results in stronger oscillations of $\beta_{j}(t)$, which affect the
coefficients of Eq. (4) and the behavior of the solutions.
Figure 4: (color online). Numerical simulations of bright soliton from Fig. 2
vs $\zeta$ at $t=0$, 10, and 20. The initial conditions are given by Eq. (26)
with $s_{1}=0.1$, $m=0.1$ in panel (a) and $m=0.9$ in panel (b). The other
parameters are the same as those used in Fig. 2.
## V Discussions and Conclusions
For completeness, we consider the repulsive nonlinearity in Eq. (11), i.e.
$\mathcal{G}=-1$. In this case, the equation admits 3D dark soliton solutions
in the form with a nontrial phase
$\displaystyle\psi(t,{\bf r})=\big{\\{}iv+k\tanh[k(\xi(t,{\bf
r})-v\tau(t))]\big{\\}}e^{-i\mu\tau(t)},$ (32)
where $\mu$ is the chemical potential, $v=\sqrt{\mu-k^{2}}$, and $k$ is a free
parameter satisfying $k^{2}<\mu$.
In summary, we have analytically constructed the novel 3D time-varying bright
multi-soliton solutions for the (3+1)-D GP equation with time-space
modulation. We focus on the bounded potential, nonlinearity, and gain/loss
case to analyze the dynamics of the breathing and the zigzag propagation trace
of the obtained solitons. Different shapes of the one-soliton solutions and
the fascinating interactions between soliton-pairs were achieved. The
stability of bright solitons have been checked numerically. The method we
present here can be extended to study the higher-dimensional bright soliton
solutions of other nonlinear systems and their various interaction properties.
The model (4) can also be extended to describe 3D nonlinear optical media with
varying coefficients belic after the transformation $z\leftrightarrow t$. The
results we obtained may raise the possibility of relative experiments and
potential applications.
###### Acknowledgements.
The work of Z.Y. was supported by FCT under Grant No. SFRH/BPD/41367/2007 and
the NSFC60821002/F02. The work of C.H. was supported by FCT under Grant No.
SFRH/BPD/36385/2007.
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|
arxiv-papers
| 2010-09-20T08:11:43 |
2024-09-04T02:49:12.970007
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhenya Yan, Chao Hang",
"submitter": "Z Yan",
"url": "https://arxiv.org/abs/1009.3727"
}
|
1009.3794
|
Derived equivalences for Cohen-Macaulay Auslander algebras
Shengyong Pan
###### Abstract
Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. We
prove that, if $A$ and $B$ are derived equivalent, then their Cohen-Macaulay
Auslander algebras are also derived equivalent.
2000 AMS Classification: 16G10, 18G05, 16S50; 16P10, 20C05, 18E10.
Key words: Cohen-Macaulay finite type, Cohen-Macaulay Auslander algebras,
derived equivalence.
## 1 Introduction
Triangulated categories and derived categories were introduced by Grothendieck
and Verdier [27]. Today, they have widely been used in many branches:
algebraic geometry, stable homotopy theory, representation theory, etc. In the
representation theory of algebras, we will restrict our attention to the
equivalences of derived categories, that is, derived equivalences. Derived
equivalences have been shown to preserve many invariants and provide new
connection. For instance, Hochschild homology and cohomology [26], finiteness
of finitistic dimension [23] have been shown to be invariant under derived
equivalences. Moreover, derived equivalences are related to cluster categories
and cluster tilting objects [5]. As is known, Rickard’s Morita theory for
derived categories leaves something to be desired, though, as for some pairs
of rings, or algebras, it is currently difficult, sometimes even impossible to
verify whether there exists a tilting complex. It is of interest to construct
a new derived equivalence from given one by finding a suitable tilting
complex. Rickard [25, 26] used tensor products and trivial extensions to get
new derived equivalences. In the recent years, Hu and Xi have provided various
techniques to construct new derived equivalences. In [15] they established an
amazing connection between derived equivalences and Auslander-Reiten sequences
via BB-tilting modules, and obtained derived equivalences from Auslander-
Reiten triangles. In [17] they constructed new derived equivalences between
$\Phi$-Auslander-Yoneda algebras from a given almost $\nu$-stable equivalence.
In [17, Corollary 3.13] Hu and Xi proved that, if two representation finite
self-injective Artin algebras are derived equivalent, then their Auslander
algebras are derived equivalent. In this paper, we generalize their result and
prove that, if two Cohen-Macaulay finite Gorenstein Artin algebras are derived
equivalent, then their Cohen-Macaulay Auslander algebras are also derived
equivalent.
This paper is organized as follows. In Section 2, we review some facts on
derived categories and derived equivalences. In Section 3, we state and prove
our main result.
## 2 Preliminaries
In this section, we shall recall some definitions and notations on derived
categories and derived equivalences.
Let $\mathscr{A}$ be an abelian category. For two morphisms
$\alpha:X\rightarrow Y$ and $\beta:Y\rightarrow Z$, their composition is
denoted by $\alpha\beta$. An object $X\in\mathscr{A}$ is called a additive
generator for $\mathscr{A}$ if ${\rm add\,}(X)=\mathscr{A}$, where ${\rm
add\,}(X)$ is the additive subcategory of $\mathscr{A}$ consisting of all
direct summands of finite direct sums of the copies of $X$. A complex
$X^{\bullet}=(X^{i},d_{X}^{i})$ over $\mathscr{A}$ is a sequence of objects
$X^{i}$ and morphisms $d_{X}^{i}$ in $\mathscr{A}$ of the form:
$\cdots\rightarrow X^{i}\stackrel{{\scriptstyle
d^{i}}}{{\rightarrow}}X^{i+1}\stackrel{{\scriptstyle
d^{i+1}}}{{\rightarrow}}X^{i+1}\rightarrow\cdots$, such that $d^{i}d^{i+1}=0$
for all $i\in\mathbb{Z}$. If $X^{\bullet}=(X^{i},d_{X}^{i})$ and
$Y^{\bullet}=(Y^{i},d_{Y}^{i})$ are two complexes, then a morphism
$f^{\bullet}:X^{\bullet}\rightarrow Y^{\bullet}$ is a sequence of morphisms
$f^{i}:X^{i}\rightarrow Y^{i}$ of $\mathscr{A}$ such that
$d^{i}_{X}f^{i+1}=f^{i}d^{i}_{Y}$ for all $i\in\mathbb{Z}$. The map
$f^{\bullet}$ is called a chain map between $X^{\bullet}$ and $Y^{\bullet}$.
The category of complexes over $\mathscr{A}$ with chain maps is denoted by
${\rm C}(\mathscr{A})$. The homotopy category of complexes over $\mathscr{A}$
is denoted by ${\rm K}(\mathscr{A})$ and the derived category of complexes is
denoted by $\rm D\,(\mathscr{A})$.
Let $R$ be a commutative Artin ring. And let $A$ be an Artin $R$-algebra. We
denote by $A$-mod the category of finitely generated left $A$-modules. The
full subcategory of $A$-mod consisting of projective modules is denoted by
${}_{A}\mathcal{P}$. Recall that a homomorphism $f:X\rightarrow Y$ of
$A$-modules is called a radical map provided that for any $A$-module $Z$ and
homomorphisms $g:Y\rightarrow Z$ and $h:Z\rightarrow X$, the composition $hfg$
is not an isomorphism. A complex of $A$-modules is called a radical complex if
its differential maps are radical maps. Let ${\rm K^{b}}(A)$ denote the
homotopy category of bounded complexes of $A$-modules. We denote by ${\rm
D^{b}}(A)$ by the bounded derived category of $A$-mod.
The fundamental theory on derived equivalences has been established. Rickard
[24] gave a Morita theory for derived categories in the following theorem.
###### Theorem 2.1
[24, Therem 6.4] Let $A$ and $B$ be rings. The following conditions are
equivalent.
$(i)$ ${\rm D^{b}}(A\text{-}{\rm Mod})$ and ${\rm D^{b}}(B\text{-}{\rm Mod})$
are equivalent as triangulated categories.
$(ii)$ ${\rm K}^{-}(A\mbox{{\rm-Proj}})$ and ${\rm K}^{-}(B\mbox{{\rm-Proj}})$
are equivalent as triangulated categories.
$(iii)$ ${\rm K^{b}}(A\mbox{{\rm-Proj}})$ and ${\rm K^{b}}(B\mbox{{\rm-
Proj}})$ are equivalent as triangulated categories.
$(iv)$ ${\rm K^{b}}(_{A}\mathcal{P})$ and ${\rm K^{b}}(_{B}\mathcal{P})$ are
equivalent as triangulated categories.
$(v)$ $B$ is isomorphic to ${\rm End\,}_{{\rm D^{b}}(A)}(T^{\bullet})$ for
some complex $T^{\bullet}$ in ${\rm K^{b}}(_{A}\mathcal{P})$ satisfying
$(1)$ ${\rm Hom\,}_{{\rm D^{b}}(A)}(T^{\bullet},T^{\bullet}[n])=0$ for all
$n\neq 0$.
$(2)$ ${\rm add\,}(T^{\bullet})$, the category of direct summands of finite
direct sums of copies of $T^{\bullet}$, generates ${\rm
K^{b}}(_{A}\mathcal{P})$ as a triangulated category.
Here $A$-Proj is the subcategory of $A$-Mod consisting of all projective
$A$-modules.
Remarks. (1) The rings $A$ and $B$ are said to be derived equivalent if $A$
and $B$ satisfy the conditions of the above theorem. The complex $T^{\bullet}$
in Theorem 2.1 is called a tilting complex for $A$.
(2) By [24, Corollary 8.3], two Artin $R$-algebras $A$ and $B$ are said to be
derived equivalent if their derived categories ${\rm D^{b}}(A)$ and ${\rm
D^{b}}(B)$ are equivalent as triangulated categories. By Theorem 2.1, Artin
algebras $A$ and $B$ are derived equivalent if and only if $B$ is isomorphic
to the endomorphism algebra of a tilting complex $T^{\bullet}$. If
$T^{\bullet}$ is a tilting complex for $A$, then there is an equivalence
$F:{\rm D^{b}}(A)\rightarrow{\rm D^{b}}(B)$ that sends $T^{\bullet}$ to $B$.
On the other hand, for each derived equivalence $F:{\rm
D^{b}}(A)\rightarrow{\rm D^{b}}(B)$, there is an associated tilting complex
$T^{\bullet}$ for $A$ such that $F(T^{\bullet})$ is isomorphic to $B$ in ${\rm
D^{b}}(B)$.
## 3 Derived equivalences for Cohen-Macaulay Auslander Algebras
In this section, we shall prove the main result of this paper. First, let us
recall the definition of Cohen-Macaulay Auslander algebras.
### 3.1 Cohen-Macaulay Auslander algebras
Let $A$ be an Artin algebra. Recall that $A$ is of finite representation type
provided that there are only finitely many indecomposable finitely generated
$A$-modules up to isomorphism. If an $A$-module $X$ satisfies ${\rm
Ext}_{A}^{i}(X,A)=0$ for $i>0$, then $X$ is said to be a Cohen-Macaulay
$A$-module. Denote by ${}_{A}\mathcal{X}$ the category of Cohen-Macaulay
$A$-modules. It is easy to see that if $A$ is a self-injective algebra, then
${}_{A}\mathcal{X}=A$-mod. By a ${\rm Hom\,}_{A}(-,X)$-exact sequence
$Y^{\bullet}=(Y^{i},d^{i})$, we mean that the sequence $Y^{\bullet}$ itself is
exact, and that ${\rm Hom\,}_{A}(Y^{\bullet},X)$ remains to be exact. An
$A$-module $X$ is said to be Gorenstein projective if there is a ${\rm
Hom\,}_{A}(-,Q)$-exact sequence
$\cdots\rightarrow P^{-1}\stackrel{{\scriptstyle
d^{-1}}}{{\rightarrow}}P^{0}\stackrel{{\scriptstyle
d^{0}}}{{\rightarrow}}P^{1}\stackrel{{\scriptstyle
d^{1}}}{{\rightarrow}}\cdots$
such that $X\simeq Imd^{0}$, where $P^{i}$ (for each $i$) and $Q$ are
projective $A$-modules. Denote by $A$-Gproj the subcategory of $A$-mod
consisting of Gorenstein projective $A$-modules. Note that Gorenstein
projective modules are Cohen-Macaulay $A$-modules. Following [3, Example
8.4(2)] an Artin algebra $A$ is said to be of Cohen-Macaulay finite type
provided that there are only finitely many indecomposable finitely generated
Gorenstein projective $A$-modules up to isomorphism. It is easy to see that
algebras of finite representation type are of Cohen-Macaulay finite type.
Suppose that $A$ is of Cohen-Macaulay finite type. In other words, $A$-Gproj
has an additive generator $M$, that is, ${\rm add\,}(M)=A$-Gproj.
###### Definition 3.1
[6] Suppose that an Artin algebra $A$ is of Cohen-Macaulay finite type. Let
$M$ be an additive generator in $A$-Gproj. We call $\Lambda={\rm End\,}(M)$ a
Cohen-Macaulay Auslander algebra of $A$.
Remark. For a Cohen-Macaulay finite algebra $A$, its Cohen-Macaulay Auslander
algebra is unique up to Morita equivalences.
Example. Let $A=k[x]/(x^{2})$ and consider the Artin algebra
$T_{2}(A)=\left(\begin{array}[]{cc}A&A\\\ 0&A\end{array}\right).$
Then $T_{2}(A)$ is a $1$-Gorenstein Artin algebra of Cohen-Macaulay type [7]
or [10]. $T_{2}(A)$ has indecomposable Gorenstein projective modules [4,
p.101]:
$M_{1}=\left(\begin{array}[]{cc}k\\\
0\end{array}\right),M_{2}=\left(\begin{array}[]{cc}A\\\
0\end{array}\right),M_{3}=\left(\begin{array}[]{cc}A\\\
A\end{array}\right),M_{4}=\left(\begin{array}[]{cc}k\\\
k\end{array}\right),M_{5}=\left(\begin{array}[]{cc}A\\\ k\end{array}\right).$
Set $M=\oplus_{1\leq i\leq 5}M_{i}$. Then Cohen-Macaulay Auslander algebra
${\rm End\,}_{T_{2}(A)}(M)$ of $T_{2}(A)$ is given by the following quiver and
relations $xy=0=v\alpha-yu=\alpha z=\alpha\beta\gamma$ [8].
12345$y$$v$$x$$u$$\alpha$$z$$\beta$$\gamma$
### 3.2 The proof of the main result
We shall give the proof of the main result of this paper.
Suppose $A$ and $B$ are Artin algebras. Let $F:{\rm
D^{b}}(A)\longrightarrow{\rm D^{b}}(B)$ be a derived equivalence and let
$P^{\bullet}$ be the tilting complex associated to $F$. Without loss of
generality, we assume that $P^{\bullet}$ is a radical complex of the following
form
$0\rightarrow P^{-n}\rightarrow P^{-n+1}\rightarrow\cdots\rightarrow
P^{-1}\rightarrow P^{0}\rightarrow 0.$
Then we have the following fact.
###### Lemma 3.2
[15, lemma 2.1] Let $F:{\rm D^{b}}(A)\longrightarrow{\rm D^{b}}(B)$ be a
derived equivalence between Artin algebras $A$ and $B$. Then we have a tilting
complex $\bar{P}^{\bullet}$ for $B$ associated to the quasi-inverse of $F$ of
the form
$0\rightarrow\bar{P}^{0}\rightarrow\bar{P}^{1}\rightarrow\cdots\rightarrow\bar{P}^{n-1}\rightarrow\bar{P}^{n}\rightarrow
0,$
with the differential being radical maps.
Suppose that $X^{\bullet}$ is a complex of $A$-modules. We define the
following truncations:
$\tau_{\geq 1}(X^{\bullet}):\cdots\rightarrow 0\rightarrow 0\rightarrow
X^{1}\rightarrow X^{2}\rightarrow\cdots$,
$\tau_{\leq 0}(X^{\bullet}):\cdots\rightarrow X^{-1}\rightarrow
X^{0}\rightarrow 0\rightarrow 0\cdots$.
Using the properties of Cohen-Macaulay $A$-modules, we can prove the following
lemma.
###### Lemma 3.3
Let $F:{\rm D^{b}}(A)\longrightarrow{\rm D^{b}}(B)$ be a derived equivalence
between Artin algebras $A$ and $B$, and let $G$ be the quasi-inverse of $F$.
Suppose that $P^{\bullet}$ and $\bar{P}^{\bullet}$ are the tilting complexes
associated to $F$ and $G$, respectively. Then
$(i)$ For $X\in_{A}\mathcal{X}$, the complex $F(X)$ is isomorphic in ${\rm
D^{b}}(B)$ to a radical complex $\bar{P}^{\bullet}_{X}$ of the form
$0\rightarrow\bar{P}_{X}^{0}\rightarrow\bar{P}_{X}^{1}\rightarrow\cdots\rightarrow\bar{P}_{X}^{n-1}\rightarrow\bar{P}_{X}^{n}\rightarrow
0$
with $\bar{P}_{X}^{0}\in_{B}\mathcal{X}$ and $\bar{P}_{X}^{i}$ projective
$B$-modules for $1\leq i\leq n$.
$(ii)$ For $Y\in_{B}\mathcal{X}$, the complex $G(Y)$ is isomorphic in ${\rm
D^{b}}(A)$ to a radical complex $P^{\bullet}_{Y}$ of the form
$0\rightarrow P_{Y}^{-n}\rightarrow P_{Y}^{-n+1}\rightarrow\cdots\rightarrow
P_{Y}^{-1}\rightarrow P_{Y}^{0}\rightarrow 0$
with $P_{Y}^{-n}\in_{A}\mathcal{X}$ and $P_{Y}^{i}$ projective $A$-modules for
$-n+1\leq i\leq 0$.
Proof. We just show the first case. The proof of ($ii$) is similar to that of
($i$).
($i$) For $X\in_{A}\mathcal{X}$, by [16, Lemma 3.1], we see that the complex
$F(X)$ is isomorphic in ${\rm D^{b}}(B)$ to a complex $\bar{P}^{\bullet}_{X}$
of the form
$0\rightarrow\bar{P}_{X}^{0}\rightarrow\bar{P}_{X}^{1}\rightarrow\cdots\rightarrow\bar{P}_{X}^{n-1}\rightarrow\bar{P}_{X}^{n}\rightarrow
0,$
with $\bar{P}_{X}^{i}$ projective $B$-modules for $i>0$. We only need to show
that $\bar{P}_{X}^{0}$ is in ${}_{B}\mathcal{X}$. It suffices to prove that
${\rm End\,}^{i}_{B}(\bar{P}_{X}^{0},B)=0$ for $i\geq 1$. Indeed, there exists
a distinguished triangle
$\bar{P}^{+}_{X}\rightarrow\bar{P}^{\bullet}_{X}\rightarrow\bar{P}^{0}_{X}\rightarrow\bar{P}^{+}_{X}[1]$
in ${\rm K^{b}}(B)$, where $\bar{P}^{+}_{X}$ denotes the complex $\tau_{\geq
1}(\bar{P}_{X}^{\bullet})$. For each $i\in\mathbb{Z}$, applying the functor
${\rm Hom\,}_{{\rm D^{b}}(B)}(-,B[i])$ to the above distinguished triangle, we
get an exact sequence
$\displaystyle\cdots\rightarrow{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{+}_{X}[1],B[i])\rightarrow{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},B[i])\rightarrow{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet}_{X},B[i])$ $\displaystyle\rightarrow{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{+}_{X},B[i])\rightarrow\cdots.$
On the other hand, ${\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{+}_{X},B[i])\simeq{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{+}_{X},B[i])=0$ for $i\geq 0$. By [23, lemma 2.1] and
${\rm End\,}^{i}_{A}(X,A)=0$ for $i\geq 1$, we get ${\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet}_{X},B[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(A)}(X,P^{\bullet}[i])=0$ for all $i\geq 1$. Consequently, we get ${\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},B[i])=0$ for all $i\geq 1$ by the
above exact sequence. Therefore,
${\rm End\,}^{i}_{B}(\bar{P}_{X}^{0},B)\simeq{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},B[i])=0,\;\;\;\text{for}\;\;\;i\geq 1.$
This implies that $\bar{P}_{X}^{0}\in_{B}\mathcal{X}$. $\square$
Now we give a lemma, which is useful in the following argument.
###### Lemma 3.4
Let $A$ be an Artin algebra and $f:X\rightarrow Y$ a homomorphism of
$A$-modules with $X,Y\in_{A}\mathcal{X}$. Suppose $Q^{\bullet}$ is a complex
in ${\rm K^{b}}(_{A}\mathcal{P})$. If $f$ factors through $Q^{\bullet}$ in
${\rm D^{b}}(A)$, then $f$ factors through a projective $A$-module.
Proof. There is a distinguished triangle
$\tau_{\leq 0}(Q^{\bullet})\rightarrow\tau_{\geq
1}(Q^{\bullet})\stackrel{{\scriptstyle
a}}{{\rightarrow}}Q^{\bullet}\stackrel{{\scriptstyle
b}}{{\rightarrow}}\tau_{\leq 0}(Q^{\bullet})[1]\quad\text{in}\quad{\rm
D^{b}}(A).$
Suppose that $f=gh$, where $g:X\rightarrow Q^{\bullet}$ and
$h:Q^{\bullet}\rightarrow Y$. Since ${\rm Hom\,}_{{\rm D^{b}}(A)}(\tau_{\geq
1}(Q^{\bullet}),Y)\simeq{\rm Hom\,}_{{\rm K^{b}}(A)}(\tau_{\geq
1}(Q^{\bullet}),Y)=0$, it follows that $ah=0$. Then there is a map
$x:\tau_{\leq 0}(Q^{\bullet})[1]\rightarrow Y$, such that $h=bx$. Thus, we get
$f=gbx$. Now, it is sufficient to show that $f$ factors through $\tau_{\leq
0}(Q^{\bullet})$. Consider the following distinguished triangle
$Q^{0}\stackrel{{\scriptstyle c}}{{\rightarrow}}\tau_{\leq
0}(Q^{\bullet})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\tau_{\leq-1}(Q^{\bullet})\rightarrow
Q^{0}[1]\quad\text{in}\quad{\rm D^{b}}(A).$
Note that ${\rm Ext}_{A}^{i}(X,A)=0$ for $i\geq 1$. By [23, Lemma 2.1], we
have ${\rm Hom\,}_{{\rm D^{b}}(A)}(X,\tau_{\leq-1}(Q^{\bullet}))=0$. Thus, we
get $gbd=0$. Then there is a morphism $u:X\rightarrow Q^{0}$ such that
$gb=uc$. Consequently, $f=ucx$, which implies that $f$ factors through a
projective $A$-module $Q^{0}$. $\square$
Choose an $A$-module $X\in_{A}\mathcal{X}$, by Lemma 3.3, we know that $F(X)$
is isomorphic to a radical complex of the form
$0\rightarrow\bar{P}_{X}^{0}\rightarrow\bar{P}_{X}^{1}\rightarrow\cdots\rightarrow\bar{P}_{X}^{n-1}\rightarrow\bar{P}_{X}^{n}\rightarrow
0$
such that $\bar{P}_{X}^{0}\in_{B}\mathcal{X}$ and $\bar{P}_{X}^{i}$ are
projective $B$-modules for $1\leq i\leq n$. In the following, we try to define
a functor
$\underline{F}:\underline{{}_{A}\mathcal{X}}\rightarrow\underline{{}_{B}\mathcal{X}}$.
###### Proposition 3.5
Let $F:{\rm D^{b}}(A)\longrightarrow{\rm D^{b}}(B)$ be a derived equivalence.
Then there is an additive functor
$\underline{F}:\underline{{}_{A}\mathcal{X}}\rightarrow\underline{{}_{B}\mathcal{X}}$
sending $X$ to $\bar{P}_{X}^{0}$, such that the following diagram
$\textstyle{\underline{{}_{A}\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
can}}$$\scriptstyle{\underline{F}}$$\textstyle{{\rm D^{b}}(A)/{\rm
K^{b}}(_{A}\mathcal{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{F}}$$\textstyle{\underline{{}_{B}\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
can}}$$\textstyle{{\rm D^{b}}(B)/{\rm K^{b}}(_{B}\mathcal{P})}$
is commutative up to natural isomorphism.
Proof. The idea of the proof is similar to that of [15, Proposition 3.4]. For
convenience, we give the details here.
For each $f:X\rightarrow Y$ in ${}_{A}\mathcal{X}$, we denote by
$\underline{f}$ the image of $f$ in $\underline{{}_{A}\mathcal{X}}$. By Lemma
3.3, we have a distinguished triangle
$\bar{P}^{+}_{X}\stackrel{{\scriptstyle
i_{X}}}{{\rightarrow}}F(X)\stackrel{{\scriptstyle
j_{X}}}{{\rightarrow}}\bar{P}^{0}_{X}\stackrel{{\scriptstyle
m_{X}}}{{\rightarrow}}\bar{P}^{+}_{X}[1]\quad\text{in}\quad{\rm D^{b}}(B).$
Moreover, for each $f:X\rightarrow Y$ in ${}_{A}\mathcal{X}$, there is a
commutative diagram
$\textstyle{\bar{P}^{+}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{X}}$$\scriptstyle{\alpha_{f}}$$\textstyle{F(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{X}}$$\scriptstyle{F(f)}$$\textstyle{\bar{P}^{0}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{X}}$$\scriptstyle{\beta_{f}}$$\textstyle{\bar{P}^{+}_{X}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{f[1]}}$$\textstyle{\bar{P}^{+}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y}}$$\textstyle{F(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{Y}}$$\textstyle{\bar{P}^{0}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{Y}}$$\textstyle{\bar{P}^{+}_{Y}[1].}$
Since ${\rm Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{+}_{X},\bar{P}^{0}_{Y})\simeq{\rm
Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{+}_{X},\bar{P}^{0}_{Y})=0$, it follows that
$i_{X}F(f)j_{Y}=0$. Then there exists a homomorphism
$\alpha_{f}:\bar{P}^{+}_{X}\rightarrow\bar{P}^{+}_{Y}$. Note that $B$-mod is
fully embedding into ${\rm D^{b}}(B)$, hence $\beta_{f}$ is a morphism of
$B$-modules. If there is another morphism $\beta^{\prime}_{f}$ such that
$j_{X}\beta^{\prime}_{f}=F(f)j_{Y}$, then
$j_{X}(\beta_{f}-\beta^{\prime}_{f})=0$. Thus $\beta_{f}-\beta^{\prime}_{f}$
factors through $\bar{P}^{+}_{X}[1]$, which implies that
$\beta_{f}-\beta^{\prime}_{f}$ factors through a projective $B$-module by
Lemma 3.4. Therefore, the morphism $\bar{\beta_{f}}$ is uniquely determined by
$f$.
Let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be morphisms in
${}_{A}\mathcal{X}$. Then we have $F(fg)j_{Z}=j_{X}\beta_{fg}$ and
$F(fg)j_{Z}=j_{X}\beta_{f}\beta_{g}$. By the uniqueness of
$\underline{\beta_{fg}}$, we have
$\underline{\beta_{fg}}=\underline{\beta_{f}}$ $\underline{\beta_{g}}$.
Moreover, if $f$ factors through a projective $A$-module, then $\beta_{f}$
also factors through a projective $B$-module.
For each $X\in_{A}\mathcal{X}$, we define $\underline{F}(X):=\bar{P}_{X}^{0}$.
Set $\underline{F}(\underline{f})=\underline{\beta_{f}}$, for each
$\underline{f}\in{\rm Hom\,}_{\underline{\mathcal{X}_{A}}}(X,Y)$. Then $F$ is
well-defined and an additive functor.
To complete the proof of the lemma, it remains to show that
$j_{X}:F(X)\rightarrow\underline{F}(X)$ is a natural isomorphism in ${\rm
D^{b}}(B)/{\rm K^{b}}(_{B}\mathcal{P})$. Since $\bar{P}^{+}_{X}$ is in ${\rm
K^{b}}(_{B}\mathcal{P})$, then
$j_{X}:F(X)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\underline{F}(X)$ in
${\rm D^{b}}(B)/{\rm K^{b}}(_{B}\mathcal{P})$. The following commutative
diagram
$\textstyle{F(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{X}}$$\scriptstyle{F(f)}$$\textstyle{\underline{F}(X)=\bar{P}^{0}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{f}}$$\textstyle{F(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{Y}}$$\textstyle{\underline{F}(Y)=\bar{P}^{0}_{Y}}$
shows that $j_{X}:F(X)\rightarrow\underline{F}(X)$ is a natural isomorphism in
${\rm D^{b}}(B)/{\rm K^{b}}(_{B}\mathcal{P})$. $\square$
The following lemma is quoted from [15] which will be used frequently.
###### Lemma 3.6
[15, Lemma 2.2] Let $R$ be an arbitrary ring, and let $R$-Mod be the category
of all left $A$-modules. Suppose $X^{\bullet}$ is a bounded above complex and
$Y^{\bullet}$ is a bounded below complex over $R$-Mod. Let $m$ be an integer.
If $X^{i}$ is projective for all $i>m$ and $Y^{j}=0$ for all $j<m$, then ${\rm
Hom\,}_{{\rm K}(R\text{-}{\rm Mod})}(X^{\bullet},Y^{\bullet})\simeq{\rm
Hom\,}_{\rm D\,(R\text{-}Mod)}(X^{\bullet},Y^{\bullet})$.
Let $A$ be an Artin algebra and let $X$ be in ${}_{A}\mathcal{X}$ which is not
a projective $A$-module. Set $\Lambda={\rm End\,}_{A}(A\oplus X)$,
$N=B\oplus\underline{F}(X)$ and $\Gamma={\rm End\,}_{B}(N)$. Let
$\bar{T}^{\bullet}$ be the complex
$\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X}$. Then $\bar{T}^{\bullet}$ is in
${\rm K^{b}}({\rm add\,}_{B}N)$.
The proof of the following lemma is different from [17, Lemma 3.6], and in
fact extends Hu and Xi’s original methods for the self-injective case.
###### Lemma 3.7
Keep the notations above. We have the following statements.
$(1)$ ${\rm Hom\,}_{{\rm K^{b}}({\rm
add\,}_{B}N)}(\bar{T}^{\bullet},\bar{T}^{\bullet}[i])=0$ for $i\neq 0$.
$(2)$ ${\rm add\,}\bar{T}^{\bullet}$ generates ${\rm K^{b}}({\rm add\,}_{B}N)$
as a triangulated category.
Proof. (1) Decompose the complex $\bar{T}^{\bullet}$ as
$\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X}$. Then we have the following
isomorphisms
$\displaystyle{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{T}^{\bullet},\bar{T}^{\bullet}[i])\simeq{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X},(\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X})[i])\simeq{\rm
Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{\bullet},\bar{P}^{\bullet}[i])\oplus$
$\displaystyle{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])\oplus{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\oplus{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i]).$
The proof falls naturally into three parts.
(a) Since $\bar{P}^{\bullet}$ is a tilting complex over $B$, we have ${\rm
Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{\bullet},\bar{P}^{\bullet}[i])=0$ for all
$i\neq 0$. Furthermore,
${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(A)}(A,X[i])=0\quad\text{for all}\quad i\neq 0.$
(b) We claim that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])=0$ for $i\neq 0$.
Indeed, applying the functors ${\rm Hom\,}_{{\rm
K}(B)}(-,\bar{P}^{\bullet}[i])$ and ${\rm Hom\,}_{{\rm
D^{b}}(B)}(-,\bar{P}^{\bullet}[i])$ to the distinguished triangle
$\bar{P}^{+}_{X}\rightarrow\bar{P}^{\bullet}_{X}\rightarrow\bar{P}^{0}_{X}\rightarrow\bar{P}^{+}_{X}[1]$
in ${\rm K^{b}}(B)$, we obtain the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{+}_{X}[1],\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(A)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{+}_{X}[1],\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(\bar{P}^{0}_{X},\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$
Note that
${\rm Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\simeq{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet}).$
Indeed, since $P^{\bullet}$ is a bounded complex and Lemma 3.6, it suffices to
show that for the complex $\bar{P}^{\bullet}$ of length $2$ of the form
$0\rightarrow\bar{P}^{0}\rightarrow\bar{P}^{1}\rightarrow 0$, we get
${\rm Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\simeq{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet}).$
In this case, we have a distinguished triangle
$\bar{P}^{1}[-1]\rightarrow\bar{P}^{\bullet}\rightarrow\bar{P}^{0}\rightarrow\bar{P}^{1}\quad\text{in}\quad{\rm
K^{b}}(B).$
Applying the functors ${\rm Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{0}_{X},-)$ and
${\rm Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},-)$ to the distinguished
triangle
$\bar{P}^{1}[-1]\rightarrow\bar{P}^{\bullet}\rightarrow\bar{P}^{0}\rightarrow\bar{P}^{1}$,
we obtain the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1}).}$
Since ${\rm Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])=0$ and
${\rm Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}^{1}[1])=0$, we conclude
that that, for $i\neq 0$, we have
${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(A)}(X,A[i])=0.$
(c) We claim that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i\neq 0$.
Indeed, it follows that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i<0$ by
Lemma 3.6. It suffices to show that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i>0$. Note
that there is a distinguished triangle
$(\star)\quad\quad\bar{P}^{+}_{X}\rightarrow\bar{P}^{\bullet}_{X}\rightarrow\bar{P}^{0}_{X}\rightarrow\bar{P}^{+}_{X}[1]\quad\text{in}\quad{\rm
K^{b}}(B),\text{\;where\;}\bar{P}^{+}_{X}\text{\;denotes the
complex\;}\tau_{\geq 1}(\bar{P}_{X}^{\bullet}).$
Applying the functor ${\rm Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{\bullet}_{X},-)$
to ($\star$), we get a long exact sequence
$\cdots\rightarrow{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\rightarrow{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])\rightarrow{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X}[i])\rightarrow\cdots(\star\star).$
From the distinguished triangle
$\bar{P}^{+}_{X}\rightarrow\bar{P}^{\bullet}_{X}\rightarrow\bar{P}^{0}_{X}\rightarrow\bar{P}^{+}_{X}[1]$,
we conclude that $H^{i}(G(\bar{P}^{+}_{X}))=0$ for $i>1$ and
$G(\bar{P}^{+}_{X})$ is a radical complex $Q^{\bullet}_{X}$ of the form
$\cdots\rightarrow Q^{-1}_{X}\rightarrow Q^{0}_{X}\rightarrow
Q^{1}_{X}\rightarrow 0.$
Applying the functors ${\rm Hom\,}_{{\rm K^{b}}(B)}(-,\bar{P}^{+}_{X}[i])$ and
${\rm Hom\,}_{{\rm K^{b}}(B)}(-,\bar{P}^{+}_{X}[i])$ to ($\star$) again, we
have the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{+}_{X}[1],\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{+}_{X}[1],\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{0}_{X},\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+}[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$
Therefore,
$\displaystyle{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\simeq{\rm Hom\,}_{{\rm
D^{b}}(A)}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{+}_{X}[i]))$
$\displaystyle\simeq{\rm Hom\,}_{{\rm D^{b}}(A)}(X,G(\bar{P}^{+}_{X})[i]).$
By [23, lemma 2.1], it follows that ${\rm Hom\,}_{{\rm
D^{b}}(A)}(X,G(\bar{P}^{+}_{X})[i])=0$ for all $i>1$. Consequently, ${\rm
Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])=0$ for
$i>1$. Since ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X}[i])=0$ for $i>0$ by shifting,
it follows that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i>1$ by the
long exact sequence $(\star\star)$. It remains to prove that ${\rm
Hom\,}_{{\rm K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[1])=0$. To
get ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[1])=0$, it suffices to
show that the map
$(\maltese)\quad\quad{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\rightarrow{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\quad\text{is \;
surjective}.$
From the above argument, we have the following commutative diagram
$\textstyle{Q^{\bullet}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\simeq}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{M(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{Q^{\bullet}_{X}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{G(\bar{P}^{+}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\bar{P}^{\bullet}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\bar{P}^{0}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\bar{P}^{+}_{X})[1]}$
in ${\rm D^{b}}(A)$, where all the vertical maps are isomorphisms, and the
morphism $a$ is chosen in ${\rm K^{b}}(A)$ such that the first square is
commutative. Applying the functor ${\rm Hom\,}_{{\rm K^{b}}(A)}(X,-)$ to the
first horizontal distinguished triangle, we get an exact sequence
${\rm Hom\,}_{{\rm K^{b}}(A)}(X,M(a))\rightarrow{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,Q^{\bullet}_{X}[1])\rightarrow 0,\quad\text{since}\quad{\rm
Hom\,}_{{\rm K^{b}}(A)}(X,X[1])=0.$
We have the following formulas.
$\displaystyle{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\stackrel{{\scriptstyle(\ast)}}{{\simeq}}{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\simeq{\rm
Hom\,}_{{\rm D^{b}}(A)}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{0}_{X}))$
$\displaystyle\simeq{\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))$
and
$\displaystyle{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\stackrel{{\scriptstyle(\ast\ast)}}{{\simeq}}{\rm
Hom\,}_{{\rm D^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\simeq{\rm
Hom\,}_{{\rm D^{b}}(A)}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{+}_{X})[1])$
$\displaystyle\simeq{\rm Hom\,}_{{\rm D^{b}}(A)}(X,Q^{\bullet}_{X}[1]).$
The isomorphisms $(\ast)$ and $(\ast\ast)$ are deduced by Lemma 3.6. Then we
have the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,M(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm D^{b}}(A)}(X,Q^{+}[1]).}$
From the above diagram, to show the map ($\maltese$) is surjective, it is
sufficient to show the map
${\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))\rightarrow{\rm Hom\,}_{{\rm
D^{b}}(A)}(X,Q^{\bullet}_{X}[1])\quad\text{is \; surjective}.$
Applying the functor ${\rm Hom\,}(X,-)$ and ${\rm Hom\,}(X,-)$ to the
distinguished triangle $Q^{\bullet}\rightarrow X\rightarrow M(a)\rightarrow
Q^{\bullet}[1]$, we get the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,M(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(A)}(X,Q_{X}^{\bullet}[1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,M(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,Q_{X}^{\bullet}[1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm D^{b}}(A)}(X,X[1]).}$
Thus, to get the map
${\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))\rightarrow{\rm Hom\,}_{{\rm
D^{b}}(A)}(X,Q^{\bullet}_{X}[1])\quad\text{is\; surjective},$
it suffices to show the following isomorphisms
$(i)\quad{\rm Hom\,}_{{\rm D^{b}}(A)}(X,Q^{\bullet}_{X}[1])\simeq{\rm
Hom\,}_{{\rm K^{b}}(A)}(X,Q^{\bullet}_{X}[1])$
and
$(ii)\quad{\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))\simeq{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,M(a)).$
Firstly, we show that
$(i)\quad{\rm Hom\,}_{{\rm D^{b}}(A)}(X,Q^{\bullet}_{X}[1])\simeq{\rm
Hom\,}_{{\rm K^{b}}(A)}(X,Q^{\bullet}_{X}[1]).$
Indeed, it suffices to show that for the complex $Q^{\bullet}_{X}$ of the form
$0\rightarrow Q_{X}^{-1}\rightarrow Q_{X}^{0}\rightarrow 0$, we get ($i$).
There is a distinguished triangle
$(\clubsuit)\quad\quad Q_{X}^{-1}\rightarrow Q_{X}^{0}\rightarrow
Q_{X}^{\bullet}\rightarrow Q_{X}^{-1}[1]\quad\text{in}\quad{\rm K^{b}}(A).$
Applying the functors ${\rm Hom\,}_{{\rm K^{b}}(A)}(X,-)$, ${\rm Hom\,}_{{\rm
D^{b}}(A)}(X,-)$ to ($\clubsuit$), we obtain the following commutative diagram
$\textstyle{{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,Q_{X}^{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(A)}(X,Q_{X}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(A)}(X,Q_{X}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
K^{b}}(A)}(X,P^{-1}[1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,Q_{X}^{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,Q_{X}^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm
D^{b}}(A)}(X,Q_{X}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom\,}_{{\rm D^{b}}(A)}(X,P^{-1}[1]).}$
Since ${\rm End\,}^{i}_{A}(X,A)=0$ for $i\geq 1$, it follows that ${\rm
Hom\,}_{{\rm D^{b}}(A)}(X,P^{-1}[1])=0$. Moreover,
${\rm Hom\,}_{{\rm K^{b}}(A)}(X,P^{-1}[1])=0$. We thus get ${\rm Hom\,}_{{\rm
D^{b}}(A)}(X,Q^{\bullet}_{X}[1])\simeq{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,Q^{\bullet}_{X}[1])$. Next, we prove that
$(ii)\quad{\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))\simeq{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,M(a)).$
Indeed, there exists a distinguished triangle
$(\spadesuit)\quad\quad M(a)^{0}\rightarrow M(a)\rightarrow
M(a)^{-}\rightarrow M(a)^{0}[1]\quad\text{in}\quad{\rm K^{b}}(A),$
where $M(a)^{-}$ denotes the truncated complex $\tau_{\leq-1}(M(a))$. Applying
the homological functors ${\rm Hom\,}_{{\rm K^{b}}(A)}(X,-)$ and ${\rm
Hom\,}_{{\rm D^{b}}(A)}(X,-)$ to ($\spadesuit$), we obtain the following
commutative diagram
$\textstyle{{}_{{\rm
K^{b}}(A)}(X,M(a)^{-}[-1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{}_{{\rm
K^{b}}(A)}(X,M(a)^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{}_{{\rm
K^{b}}(A)}(X,M(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{{\rm
K^{b}}(A)}(X,M(a)^{-})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{{\rm
D^{b}}(A)}(X,M(a)^{-}[-1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{{\rm
D^{b}}(A)}(X,M(a)^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{{\rm
D^{b}}(A)}(X,M(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{{\rm
D^{b}}(A)}(X,M(a)^{-}).}$
According to ($i$), we have ${\rm Hom\,}_{{\rm
K^{b}}(A)}(X,M(a)^{-})\simeq{\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a)^{-})=0$.
Therefore, ${\rm Hom\,}_{{\rm D^{b}}(A)}(X,M(a))\simeq{\rm Hom\,}_{{\rm
K^{b}}(A)}(X,M(a))$.
From the above argument, we have shown that ${\rm Hom\,}_{{\rm
K^{b}}(B)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i\neq 0$.
Since ${\rm K^{b}}(B)$ is a full subcategory of ${\rm K^{b}}({\rm
add\,}_{B}N)$, it follows that ${\rm Hom\,}_{{\rm K^{b}}({\rm
add\,}_{B}N)}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$ for $i\neq
0$.
(2) Since $\bar{P}^{\bullet}$ is a tilting complex for $B$, we see that ${\rm
add\,}\bar{P}^{\bullet}$ generates ${\rm K^{b}}({\rm add\,}_{B}B)$ as
triangulated category. All the terms of $\bar{P}^{+}_{X}$ are in ${\rm
add\,}_{B}B$. From the distinguished triangle
$\bar{P}^{+}_{X}\rightarrow\bar{P}^{\bullet}_{X}\rightarrow\bar{P}^{0}_{X}\rightarrow\bar{P}^{+}_{X}[1],$
it follows that $\bar{P}^{0}_{X}$ is in the triangulated subcategory generated
by ${\rm add\,}(\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X})$. Therefore,
${\rm add\,}\bar{T}^{\bullet}$ generates ${\rm K^{b}}({\rm add\,}_{B}N)$ as a
triangulated category. $\square$
###### Proposition 3.8
The complex ${\rm Hom\,}(N,\bar{T}^{\bullet})$ is a tiling complex over
$\Gamma$ with the endomorphism ${\rm End\,}({\rm
Hom\,}(N,\bar{T}^{\bullet}))\simeq\Lambda$. In particular, Artin algebras
$\Lambda$ and $\Gamma$ are derived equivalent associated with the tilting
complex ${\rm Hom\,}(N,\bar{T}^{\bullet})$.
Proof. We have an equivalence of categories
${\rm Hom\,}_{B}(N,-):{\rm
add\,}_{B}N\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}_{\Gamma}\mathcal{P}.$
We thus get an equivalence of triangulated categories induced by ${\rm
Hom\,}_{B}(N,-)$ as follows
${\rm K^{b}}({\rm
add\,}_{B}N)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
K^{b}}(_{\Gamma}\mathcal{P}).$
Then ${\rm Hom\,}(N,\bar{T}^{\bullet})\in{\rm K^{b}}(_{\Gamma}\mathcal{P})$.
By the Lemma 3.7, we see that ${\rm add\,}{\rm Hom\,}(N,\bar{T}^{\bullet})$
generates ${\rm K^{b}}(_{\Gamma}\mathcal{P})$ as a triangulated category, and
${\rm End\,}({\rm Hom\,}(N,\bar{T}^{\bullet}))\simeq{\rm
End\,}(\bar{T}^{\bullet})\simeq\Lambda$. $\square$
We have the following lemma, its proof is due to Happel [10, Lemma 4.4].
###### Lemma 3.9
Suppose that ${\rm inj.dim\,}_{A}A<\infty$. Then the following statements are
equivalent.
$(i)$ ${\rm proj.dim\,}_{A}(\rm D\,(A_{A}))<\infty$.
$(ii)$ For $X\in_{A}\mathcal{X}$, there exists an exact sequence $0\rightarrow
X\rightarrow P\rightarrow X^{{}^{\prime}}\rightarrow 0$, with
$X^{{}^{\prime}}\in_{A}\mathcal{X}$ and $P$ a projective $A$-module.
$(iii)$ If $X\in_{A}\mathcal{X}$ satisfies ${\rm inj.dim\,}_{A}X<\infty$, then
$X$ is a projective $A$-module.
Recall that an Artin algebra $A$ is called Gorenstein if the regular module A
has finite injective dimension on both sides. If $A$ is a Gorenstein algebra,
then it follows from Lemma $7.2.8$ that ${}_{A}\mathcal{X}=A$-Gproj, and that
${}_{A}\mathcal{X}$ is a Frobenius category and its category
$\underline{{}_{A}\mathcal{X}}$ is a triangulated category.
###### Proposition 3.10
Let $A$ and $B$ be Gorenstein Artin algebras. Suppose that $F$ is a derived
equivalence between $A$ and $B$. Then we have the following statements.
$(1)$ There is an equivalence
$\underline{F}:\underline{{}_{A}\mathcal{X}}\rightarrow\underline{{}_{B}\mathcal{X}}$.
$(2)$ If $A$ and $B$ are finite dimensional algebras over a field $k$, then
there exist bimodules ${}_{A}M_{B}$ and ${}_{B}L_{A}$ such that the pair of
functors
${}_{A}M_{B}\otimes-:A\text{-}mod\rightarrow
B\text{-}mod,\;_{B}L_{A}\otimes-:B\text{-}mod\rightarrow A\text{-}mod$
induces an equivalence of triangulated categories
$\underline{{}_{A}\mathcal{X}}$ and $\underline{{}_{B}\mathcal{X}}$.
Proof. We refer to [10, Theorem 4.6] and [18, Theorem 5.4] for the proofs of
(1) and (2), respectively. $\square$
Our main result in this chapter is the following theorem.
###### Theorem 3.11
Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. If
$A$ and $B$ are derived equivalent, then the Cohen-Macaulay Auslander algebras
$\Lambda$ and $\Gamma$ of $A$ and $B$ are also derived equivalent.
Proof. In fact, if Artin algebras $A$ and $B$ are derived equivalent, then $A$
is Gorenstein if and only if $B$ is Gorenstein. By Proposition 3.10 or [3,
Theorem 8.11], if Gorenstein Artin algebras $A$ and $B$ are derived
equivalent, then $A$ is of Cohen-Macaulay finite type if and only if $B$ is.
Let $F:{\rm D^{b}}(A)\longrightarrow{\rm D^{b}}(B)$ be a derived equivalence.
Set $\Lambda={\rm End\,}(A\oplus X)$ with $X=\oplus_{0\leq i\leq m}X_{i}$,
where each $X_{i}$ is indecomposable non-projective Gorenstein projective
$A$-module. Then $\Lambda$ is the Cohen-Macaulay Auslander algebra of $A$. By
Proposition 3.10, it follows that $Y_{i}=\underline{F}(X_{i})$ is the
indecomposable non-projective Gorenstein projective $B$-module. Set
$Y=\oplus_{0\leq i\leq m}Y_{i}$. Then $\Gamma={\rm End\,}(B\oplus Y)$ is the
Cohen-Macaulay Auslander algebra of $B$. Let $N$ be the $B$-module $(B\oplus
Y)$ and let $\bar{T}^{\bullet}$ be the complex $F(A\oplus X)$. Thus, we
construct a tilting complex ${\rm Hom\,}(N,\bar{T}^{\bullet})$. The result
follows from Proposition 3.8. $\square$
Remark. Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite
type. According to a result of Liu and Xi [20, Theorem 1.1], we see that, if
$A$ and $B$ are stably equivalent of Morita type, then the Cohen-Macaulay
Auslander algebras of $A$ and $B$ are also stably equivalent of Morita type.
As a corollary of Theorem 3.11, we re-obtain the following result of Hu and Xi
[17] since self-injective Artin algebras of finite representation type are
Gorenstein Artin algebras of Cohen-Macaulay finite type.
###### Corollary 3.12
[17, Corollary 3.13] Suppose that $A$ and $B$ are self-injective Artin
algebras of finite representation type. If $A$ and $B$ are derived equivalent,
then the Auslander algebras of $A$ and $B$ are also derived equivalent.
Acknowledgements. The author would like to thank his supervisor Professor
Changchang Xi. He is grateful to him for his guidance, patience and kindness.
## References
* [1] H. Abe and M. Hoshino, Gorenstein orders associated with modules. Comm. Algebra 38 (2010), 165-180.
* [2] M. Auslander, I. Reiten and S. O. Smalø, Representation Thoery of Artin Algebras. Cambridge University Press, 1995.
* [3] A. Beligiannis, Cohen-Macaulay modules, (co)torsion pairs and virtually Goren- stein algebras. J. Algebra 288(2005), 137-211.
* [4] A. Beligiannis and I. Reiten, Homological and Homotopical Aspects of Torsion Theories. Mem. Amer. Math. Soc., 188(2007).
* [5] A. Buan, R. Marsh, M, Reineke and I. Reiten, Tilting theory and cluster combinatorics. Adv. Math. 204(2006), 572-618.
* [6] X. W. Chen, Gorenstein homological algebra of Artin algebras, Postdoctoral Report, USTC, 2010.
* [7] R. M. Fossum, P. A. Griffith and I. Riten, Trivial Extensions of Abelian Categories. Springer Lecture Notes 456, Heidelberg 1975.
* [8] N. Gao and P. Zhang, Gorenstein derived categories. J. Algebra 323(2010), 2041-2057.
* [9] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press, Cambridge. 1988.
* [10] D. Happel, On Gorenstein algebras. In: Representation theory of finite groups and finitedimensional algebras (Proc. Conf. at Bielefeld, 1991), 389-404, Progress in Math., vol. 95, Birkh$\ddot{a}$user, Basel, 1991.
* [11] M. Hoshino and Y. Kato, Tilting complexes defined by idempotents. Comm. Algebra 30(2002), 83-100.
* [12] M. Hoshino and Y. Kato, Tilting complexes associated with a sequence of idempotents. J. Algebra 183(2003), 105-124.
* [13] W. Hu, On iterated almost $\nu$-stable derived equivalences. Preprint, available at : http://arxiv.org/abs/0811.0926v3, 2008.
* [14] W. Hu, S. Koenig and C. C. Xi, Derived eqivalences from cohomological approximations, and mutations of perforated Yoneda algebras. Preprint, available at : http://arxiv.org/abs/arXiv:1102.2790, 2011.
* [15] W. Hu and C. C. Xi, $\mathcal{D}$-split sequences and derived equivalences. To appear in Adv. Math. (2011).
* [16] W. Hu and C. C. Xi, Derived equivalences and stable equivalences of Morita type, I. Preprint, available at : http://math.bnu.edu.cn/ ccxi/Papers/Articles/xihu-3.pdf, 2007, to appear in Nagoya Math. J.
* [17] W. Hu and C. C. Xi, Derived equivalences for $\Phi$-Auslander-Yoneda algebras. Preprint, available at : http://math.bnu.edu.cn/ ccxi/Papers/Articles/xihu-4.pdf, 2009.
* [18] Y. Kato, On derived equivalent coherent rings. Comm. Algebra 30(2002), 4437-4454.
* [19] Y. M. Liu and C. C. Xi, Constructions of stable equivalences of Morita type for finite-dimensional algebras. II. _Math. Z._ 251(2005), no.1, 21-39.
* [20] Y. M. Liu and C. C. Xi, Constructions of stable equivalences of Morita type for finite dimensional algebras III. J. London Math. Soc. 76(2007), 567-585.
* [21] A. Neeman, Triangulated Categories. Annals of Mathematics Studies 148, Princeton University Press, Princeton and Oxford, 2001.
* [22] S. Y. Pan, Derived equivalences for $\Phi$-Cohen Macaulay Auslander-Yoneda algebras. In preparation.
* [23] S. Y. Pan and C. C. Xi, Finiteness of finitistic dimension is invariant under derived equivalences. J. Algebra 322(2009), 21-24.
* [24] J. Rickard, Morita theory for derived categories. J. London Math. Soc. 39 (1989), 436-456.
* [25] J. Rickard, Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989), 303-317.
* [26] J. Rickard, Derived equivalences as derived functors. J. London Math. Soc. 43 (1991), 37-48.
* [27] J. Verdier, Catégories dérivées, état 0. Lecture Notes in Math. 569(1977), Springer, Berlin, 262-311.
Shengyong Pan
Department of Mathematics,
Beijing Jiaotong University, Beijing 100044,
People’s Republic of China
School of Mathematical Sciences,
Beijing Normal University, Beijing 100875,
People’s Republic of China
E-mail:shypan@bjtu.edu.cn
|
arxiv-papers
| 2010-09-20T12:35:42 |
2024-09-04T02:49:12.977686
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shengyong Pan",
"submitter": "Shengyong Pan",
"url": "https://arxiv.org/abs/1009.3794"
}
|
1009.3933
|
# Kernel-phase in Fizeau Interferometry
Frantz Martinache11affiliation: Subaru Telescope, Hilo HI
###### Abstract
The detection of high contrast companions at small angular separation appears
feasible in conventional direct images using the self-calibration properties
of interferometric observable quantities. The friendly notion of closure-
phase, which is key to the recent observational successes of non-redundant
aperture masking interferometry used with Adaptive Optics, appears to be one
example of a wide family of observable quantities that are not contaminated by
phase-noise. In the high-Strehl regime, soon to be available thanks to the
coming generation of extreme Adaptive Optics systems on ground based
telescopes, and already available from space, closure-phase like information
can be extracted from any direct image, even taken with a redundant aperture.
These new phase-noise immune observable quantities, called kernel-phases, are
determined a-priori from the knowledge of the geometry of the pupil only. Re-
analysis of archive data acquired with the Hubble Space Telescope NICMOS
instrument, using this new kernel-phase algorithm demonstrates the power of
the method as it clearly detects and locates with milli-arcsecond precision a
known companion to a star at angular separation less than the diffraction
limit.
###### Subject headings:
techniques: high angular resolution, image processing, interferometric;
methods: data analysis; stars: low-mass, close binaries
## 1\. Phase in the Fourier Plane
Only two parameters essentially determine whether a source is detectable
during an observation: its brightness at the wavelength $\lambda$ of interest
and the angular resolution necessary to separate the source or feature from
its direct environment. The angular resolution is ultimately constrained by
the diffraction of the telescope, and astronomers usually follow the rule of
thumb known as the Rayleigh criterion, stating that to be resolved, two
sources need to be separated by $1.22\lambda/D$, where $D$ is the diameter of
the telescope used, to design their observations.
The development of optical interferometry has however made this criterion
obsolete: thanks to the exquisite level of calibration it permits,
interferometry indeed makes it possible to detect sources or constraint the
extent of features around objects at separations significantly smaller than
the diffraction limit. Even at the scale of one single telescope, the results
obtained with the technique known as non-redundant masking (NRM)
interferometry, first, seeing-limited (Haniff et al., 1987; Readhead et al.,
1988) and more recently used with Adaptive Optics (AO) systems (Tuthill et
al., 2000, 2006; Lloyd et al., 2006; Ireland et al., 2008; Kraus et al., 2008;
Martinache et al., 2009) demonstrate the relevance of this technique for the
detection of structures at small angular separation, that would not be
accessible from conventional AO images (Rajagopal et al., 2004).
Even if it only uses one single telescope, in order to reach this level of
resolution, one however needs to accept that the familiar product called
“image” may not necessarily constitute the best final data-product. Instead,
when interested in high-angular resolution properties of partially resolved
objects, it is convenient to derive information not from the image itself, but
from its Fourier-transform counterpart. This information, known as complex
visibility, is extracted from the Fourier-plane, calibrated and then tested
against a model of the observed object.
In optical interferometry, this approach is often mandatory: the paucity of
apertures ($N\sim 2-5$) and baselines make the content of a direct (Fizeau)
image of limited value. Information rich images can be reconstructed after
extraction of the complex visibility function from the $u,v$ plane, but only
with a large ($N>10$) number of apertures like in radio-interferometry, or
after using image synthesis. The optical image reconstruction known as pupil
densification that is used in hypertelescopes (Labeyrie, 1996) does provide an
alternative, but again, only becomes compelling if a large number of apertures
is used (Labeyrie et al., 2008). But even when an image can be reconstructed
from optical interferometry measurements, e.g. the images of the binary
Capella by Baldwin et al. (1996), the intensity map of the surface of Altair
by Monnier et al. (2007) or the spectacular images of the disk eclipsing
$\epsilon$ Aurigae (Kloppenborg et al., 2010), quantitative characteristics of
the sources can only be deduced from the fit of the interferometric data by
parametric models. In the case of a marginally resolved binary star, precise
measurements of angular separation, orientation and contrast, with confidence
intervals, deduced from a model-fit of complex visibilities carry much more
scientific value than an image of “blurry blobs”.
Visibilities in the Fourier-plane are complex numbers, whose amplitude and
phase are usually considered separately. This paper focuses on the treatment
of the phase and ignores the amplitude. In general, the power contained at
given spatial frequency is the result of the coherent sum of $R$ random
phasors, with $R$ a scalar coding the redundancy of the spatial frequency. In
the presence of residual optical path differences (OPD), this coherent sum of
$R$ random phasors loses the phase information and results in the formation of
speckles in seeing-limited images with a visible/IR telescope. NRM-
interferometry solves this problem, by discarding light with a pupil mask
designed so that each baseline is unique ($R=1$), which makes the extraction
of the phase possible.
The phases alone, being corrupted by residual OPDs, are of restricted
interest. It is however possible to combine them to form what is known as
closure-phase (Jennison, 1958), that is the sum of three phases measured by
baselines forming a closed triangle. This remarkable interferometric quantity
(cf. the introduction to closure phase by Monnier (2000)) exhibits a
compelling property: it rejects all residual pupil-plane phase errors.
Moreover, because it is determined from the analysis of the final science
detector and not on a separate arm wavefront sensor, it is also immune to non-
common path errors between the wavefront sensor and the science camera.
Once extracted and calibrated, the closure-phases can then be compared to a
parametric model, for instance of a binary star, to confirm or infirm the
presence of a companion around a given source, while uncertainties provide
contrast detection (i.e. sensitivity) limits. This approach was successfully
used by (Lloyd et al., 2006; Martinache et al., 2007; Ireland et al., 2008;
Kraus et al., 2008; Martinache et al., 2009), who typically report sensitivity
of 5-6 magnitudes in the near infrared at separations ranging from 0.5 to 4
$\lambda/D$.
This paper aims at generalizing the notion of closure-phase, and shows that
closure-phase like quantities, i.e. sharing the same property of independence
to pupil-plane phase errors, can be constructed even in the case of redundant
apertures.
## 2\. Kernel-phase
### 2.1. Linear model
Whether contiguous (i.e. single-dish) or not (i.e. interferometric), the 2D
pupil of an imaging system can be discretized into a finite collection of $N$
elementary sub-apertures. One of these elementary sub-apertures taken as zero-
phase reference, the pupil-plane phase of a coherent point-like light source
can be written as a $N-1$-component vector $\varphi$. Given that the image, or
interferogram, of this source is sufficiently sampled, then in the Fourier
plane (a.k.a. $(u,v)$-plane in interferometry) one will be able to sample up
to $M$ phases, where $M$ is a function of the pupil geometry only. For a non-
redundant array made of $N$ elementary sub-apertures, the number of sampled
$(u,v)$ phases is maximum $M=(^{N}_{2})$. The same number of sub-apertures
organized in a redundant array, for instance following a regular grid,
produces significantly less distinct $(u,v)$ sample points as each point
receives the contribution of several pairs of sub-apertures.
In most cases, since each point receives the sum of several random phasors,
both phase and amplitude are lost and cannot be simply retrieved: this results
in the formation of speckles. However, if the Strehl is high enough, the
complex amplitude associated to the instrumental phase in one point of the
pupil, $\varphi_{k}$, can be approximated by $e^{i\varphi_{k}}\approx
1+i\varphi_{k}$. Direct application of the approach is therefore for now,
restricted to space-borne diffraction-limited optical and mid-IR telescopes
like HST (cf. Section 3), but should also prove relevant to the upcoming
generation of extreme AO systems.
Given that the proposed approximation holds, while observing a point source,
the unknown (instrumental) phase distribution in the pupil $\varphi$ can be
related to the phases $\Phi$ measured in the Fourier plane with a single
linear operator. To build an intuitive understanding of this relation, let us
consider the following scenarios:
* •
If the phase is constant across the entire pupil, then none of the baselines
formed by any pair of elementary sub-apertures does record a phase difference,
and the phase in the Fourier plane is zero everywhere.
* •
If a phase offset $\delta_{\phi}$ is added to one single sub-aperture, then
each baseline involving this aperture records a phase difference, which is
exactly $\pm\delta_{\phi}$. Figure 1 represents several such scenarios.
* •
If the pupil-plane phase vector $\varphi$ is completely random, each of the
$M$ samples in the Fourier-plane is then the average of $R$ phase differences
on the pupil, where $R$ is the redundancy of the considered baseline.
To reproduce this behavior, the following linear model will be used:
$\Phi=\mathbf{R^{-1}}\cdot\mathbf{A}\cdot\varphi,$ (1)
where $\Phi$ represents the $M$-component Fourier plane phase vector,
$\mathbf{R}$ a $M\times M$ diagonal matrix whose diagonal elements code the
redundancy of the baselines, and $\mathbf{A}$ represents a $M\times(N-1)$
transfer matrix, whose properties form the core of the discussion of this
work. To be complete, the model should also include the phase information
intrinsic to the observed source, represented by the term $\Phi_{O}$ that
simply adds on top of the instrumental phase. One can then multiply both sides
of the equation by the matrix $\mathbf{R}$ so that it becomes:
$\mathbf{R}\cdot\Phi=\mathbf{A}\cdot\varphi+\mathbf{R}\cdot\Phi_{O}.$ (2)
Figure 1.— Iterative process for the determination of the transfer matrix
$\mathbf{A}$. The top row shows the sub-aperture of the full 2D pupil
(circular aperture with 30% central obscuration) where a phase offset is
applied (three cases are represented). The bottom row shows the resulting
distribution of phase in the Fourier plane. The overlaid dashed-line circle in
the bottom row marks the cutoff spatial frequency of the transfer function.
While $\mathbf{R}$ and $\mathbf{A}$ could have been merged into one single
operator, there are intentionally kept distinct. The rationale for this choice
so is that the left hand side of Eq. 2, i.e. the measurements, can be acquired
by reading directly the imaginary part of the complex visibility. Given that
the next (quadratic) term in the Taylor expansion of $e^{i\varphi}$ being
real, this makes the approximation valid to the third order in phase. This
also makes $\mathbf{A}$ of striking aspect as it is then exclusively filled
with values 0, 1 or -1.
If the matrix $\mathbf{A}$ were invertible, then the analysis of one unique
focal plane image of a single star (case corresponding to Eq. 1) would be
sufficient to determine the instrumental phase $\varphi$ as seen from the
detector, and drive an AO system and/or delay lines. Except for the special
case of a non-redundant aperture, the problem is however known to be
degenerate, despite the larger number of measures than unknowns
($M>N-1$).111The use of this model for wavefront sensing purposes will be the
object of another paper.
As demonstrated by the successes of NRM-interferometry, a complete
characterization of the wavefront is not essential if one can determine
observable quantities that are pupil-phase independent. The closure relations
used in interferometry can be related to the operator $\mathbf{A}$: these
relations are simply linear combinations (modelized by an operator
$\mathbf{K}$) of rows of $\mathbf{A}$ that produce 0, forming something known
as the left null space of A:
$\mathbf{K}\cdot\mathbf{A}=\mathbf{0}.$ (3)
For a non-redundant array, each closure relation will fill a row of
$\mathbf{K}$ with mostly zeroes, except in three positions corresponding to
the baselines forming a closing triangle, that will contain 1 or -1. These
relations are however not the only possible ones, and less trivial
combinations, involving more than three rows at a time, can be constructed.
The total number of independent relations however remains unchanged and is
only imposed by the geometry of the array.
Although not impossible, finding the operator $\mathbf{K}$ “by hand” (i.e.
finding a basis for the left null-space of $\mathbf{A}$) for a redundant
aperture is a tedious task, as the matrix $\mathbf{A}$ can get quite large. A
very efficient way to do this is to calcutate the singular value decomposition
(SVD) of $\mathbf{A}^{T}$. The SVD algorithm (Press et al., 2002), allows to
decompose the now $(N-1)\times M$ matrix $\mathbf{A}^{T}$ as the product of a
$(N-1)\times M$ column-orthogonal matrix $\mathbf{U}$, a $M\times M$ diagonal
matrix $\mathbf{W}$ with positive or zero elements (the so-called singular
values) and the transpose of an $M\times M$ orthogonal matrix $\mathbf{V}$:
$\mathbf{A}^{T}=\mathbf{U}\cdot\mathbf{W}\cdot\mathbf{V}^{T}.$ (4)
One relevant property of the SVD is that it explicitly constructs orthonormal
bases for both the null-space and the range of the matrix $\mathbf{A}^{T}$. Of
particular interest here, are the columns of $\mathbf{V}$ that correspond to
singular values equal to zero: these vectors form an orthonormal base for the
null-space, also refered to as Kernel of $\mathbf{A}^{T}$, that is exactly
what is needed to fill in the rows of $\mathbf{K}$.
If the observed target is not perfectly symmetric, and exhibits actual phase
information (i.e. $\Phi_{O}\neq 0$, see for instance Monnier (2000)), Eq. 2 is
required. Multiplying it with the left side operator $\mathbf{K}$ leads to a
new series of new phase-like quantities that are not contaminated by
instrumental phase, generalizing the notion of closure-phase (Baldwin et al.,
1986) on which NRM-interferometry from the ground (Tuthill et al., 2000) and
from space (Sivaramakrishnan et al., 2009) rely entirely.
While not as immediately tangible as the notion of closure-phase, this
proposed generalization, hereafter refered to as kernel-phase (or Ker-phase)
since it relates to the kernel of the matrix $\mathbf{A}^{T}$, exhibits a
unique advantage over the classical closure-phase: it is not restricted to
non-redundant apertures and makes it possible to extract phase-residual immune
information from images acquired from telescopes of arbitrary pupil geometry.
Figure 2.— Model for the geometry of the HST/NICMOS pupil and location of the
sample points for the determination of the Ker-phase relations. The 156 sample
points of the pupil fall on a regular square grid with a step of 1/16th of a
pupil diameter that do not intersect with the central obstruction or the
spider arms.
This has some obvious advantages over the restrictive non-redundant case:
* •
throughput: non-redundant aperture masks exhibit a typical 5-15% throughput
(Martinache et al., 2007), and photon noise of the companion one tries to
detect may be the dominant source of noise. Given that it benefits from the
same phase-noise cancelling properties as closure-phase, for a given readout
noise and exposure time, kernel-phase on an unmasked aperture offers an
immediate boost in sensitivity (or dynamic range) on faint sources.
* •
number of observable quantities: a common non-redundant aperture mask design
exhibits nine sub-apertures, therefore forming $(^{9}_{2})=36$ baselines and
$(^{8}_{2})=28$ independent closure phases (Monnier, 2000). More independent
kernel-phases can be extracted from the Fourier transform of a full-aperture
image, which will provide a better characterization of the target.
Another incidental advantage is that, being a product of the SVD, all the
kernel-phase relations contained in $\mathbf{K}$ form an orthonormal basis,
and therefore do not introduce correlation in the data. A consequence is that
manipulating Ker-phases does not require to keep track of the covariance
matrix used for closure-phases in masking interferometry, which simplifies
their interpretation.
### 2.2. Calibration
In discretizing the pupil into a finite number of sub-apertures, one important
assumption is made: the phase (or more generally, the complex amplitude of the
electric field) is assumed to be uniform within each sub-aperture. Yet even
for a space borne telescope, in the absence of atmosphere, this is only an
idealization as small scale structures like polishing imperfections of the
primary mirror for instance, will impact, to some extent, the value of the
Ker-phases. This issue is not proper to Ker-phases and also affects closure-
phases. Thus, unless perfect (i.e. single-mode) spatial filtering is performed
within each sub-aperture of a non-redundant array, the closure phase on a
point source is not exactly be zero.
This effect can somewhat be mitigated by substracting from the Ker-phases of a
science target, the Ker-phase signal measured on a point source observed in
identical conditions. NRM-interferometry results reported in Martinache et al.
(2009) for instance, make extensive use of this kind of calibration: from the
ground, this approach is very powerful as it makes it possible to calibrate
other sources of systematic errors like the effect of broadband filters which
smear out the Fourier plane and differential atmospheric refraction. From
space, this may not be as essential depending on the science goal: if the Ker-
phases obtained on a binary system are non-calibrated, then they will simply
contain a systematic error term that will limit the achievable contrast.
## 3\. Kernel phase analysis of HST/NICMOS data
While the kernel-phase approach may prove difficult to apply to ground based
observations until extreme Adaptive Optics become available, it can readily be
applied to diffraction-limited observations made from space. It is tested here
on a series of non-coronagraphic narrow band images acquired with the Near-
Infrared Camera and Multiobject Spectrometer (NICMOS) onboard the Hubble Space
Telescope. Two datasets acquired with the NICMOS1 in the F190N filter on two
distinct objects are used: the first target is a calibration star, SAO 179809,
which was observed in 1998; the second is is the high-proper motion star GJ
164, around which a companion was astrometrically discovered and whose
existence was confirmed after PSF modelization and substraction of these
NICMOS1 images by Pravdo et al. (2004). This latter target is an ideal
benchmark: given its expected $<10:1$ luminosity contrast, one should expect a
strong, unambiguous Ker-phase signal.
Moreover, ground-based infrared aperture masking interferometry measurements
reported by Martinache et al. (2009) combined with the astrometry have lead to
strong constraints on the orbit of the companion around the primary. The
location of GJ164 B measured from the Ker-phase analysis of the data can be
compared to the orbit prediction.
Figure 2 shows the model of the pupil used for this exercise. The HST pupil
exhibits a 30 % central obscuration as well as 90∘ spider arms (actual
dimensions were taken from the NIC1 configuration file in the TinyTim PSF
simulation package for HST). The phase across the pupil is discretized into a
156 elementary sub-aperture array, whose locations fall on a regular square
grid of step of $1/16^{th}$ of the outer pupil diameter. The phase sample is
assembled into a 155-component ($N-1$) vector $\varphi$.
Figure 3.— From left to right: example of a narrow band (F190N) NICMOS1 image
used for this work, visualized with a non-linear brightness scale; The Fourier
transform of this image. The 366 sample points for the phase in the Fourier
domain are overlaid; A comparison of histograms of the 288 Ker-phases
calculated using the relations identified in Section 2.1. By design, the Ker-
phases calculated from images of a single star are expected to be zero within
uncertainty: the corresponding histogram (gray curve) confirms this
expectation. In comparison, the Ker-phase histogram of the binary (dark curve)
appears significantly larger. The same GJ164 Ker-phases plotted against the
model of a binary star that best fits the data achieve to convince of the
presence of a companion in the data.
These 156 pupil phase samples map in the $(u,v)$-plane onto a square grid of
366 distinct elements 222Note for reference, that a non-redundant array of 156
sub-apertures would produce exactly 12,090 distinct $(u,v)$ points.. The
resulting $(u,v)$-sampling is illustrated in Fig. 3. For this analysis,
$\mathbf{A}$ (cf. Eq. 1) is therefore a $155\times 366$ rectangular matrix,
whose SVD reveals that 78 singular values are non-zero, leaving $366-78=288$
Ker-phase relations.
The GJ164 data consists of a total of 80 frames, acquired at average Julian
Date 2453049.3 (February 14, 2004 UT). Each image is a non-saturated 32 second
exposure, and the target was acquired in a total of 10 distinct dither
positions. Note that this dataset does not include images on a point-source
and therefore, the Ker-phases calculated from this dataset are non-calibrated.
Images corresponding to one dither position were simply coadded forming a
final total of ten 250-second exposure images, and assembled into a datacube.
The SAO 179809 data consists of four distinct 20-second exposure frames
assembled into a separate datacube.
For both datacubes, the images were then centered with sub-pixel accuracy and
windowed by a super-Gaussian function as described by Kraus et al. (2008) to
limit sensitivity to readout noise. The window size is about $25\lambda/D$ in
diameter, which is significantly larger than the field of view in which this
technique is relevant.
After this preparatory stage, the images are simply Fourier-transformed (cf.
second panel of Fig. 3), and the signal $\mathbf{R}\cdot\Phi$ is directly
measured for each of the 366 $(u,v)$ points by sampling the imaginary part of
the local complex visibility. The uncertainty associated to the measurement of
each phase is estimated from the dispersion of the signal in the direct
neighborhood of the $(u,v)$ point.
The $(u,v)$ signals are then assembled into Ker-phases using the relations
gathered in the rows of $\mathbf{K}$ and uncertainties are propagated. The
procedure is repeated for each of the frames within each datacube. The final
retained series of 288 Ker-phases is the weighted average for all frames.
Because the Ker-phase relations are designed to produce quantities independent
from pupil phase errors, a point source is expected to exhibit zero signal
within uncertainty. Despite the small number statistics (four frames acquired
on SAO 179809), the Ker-phase of the calibrator do average to zero (with a
$19.7^{\circ}$ standard deviation), while the binary exhibits a large signal
amplitude ($>100^{\circ}$) in comparison with the uncertainty of individual
Ker-phases ($\sim 2^{\circ}$). The third panel of Fig. 3 compares the Ker-
phase histograms of both datasets.
To further investigate the GJ 164 data, a parametric model of the
$(u,v)$-plane phase $\Phi_{O}$ for a binary star is needed. The parameters
are: the angular separation, the position angle of the secondary relative to
the primary and the luminosity contrast ratio. The model phase $\Phi_{O}$ is
then multiplied by the diagonal matrix $\mathbf{R}$, and finally, transformed
into model Ker-phases using the relations established during the SVD.
Figure 4.— To determine confidence intervals for the parameters of the
binary, a likelihood analysis comparable to the one presented by for closure-
phase was performed. These panels show the three projections of the likelihood
function in the region of best fit. Except for the expected correlation
between angular separation and contrast ratio for a detection within 1
$\lambda/D$, the solution is unambiguous and well constrained, demonstrating
the elegance of the Ker-phase approach.
The agreement between the data and the model is very good (cf. panel 4 of Fig.
3), considering the large number of measurements (288) adjuted by only three
parameters. The uncertainties on the Ker-phases, determined from the scatter
of the data however lead to a best fit reduced $\chi^{2}$ larger than one. A
global error term ($10^{\circ}$) is then added in quadrature to the
uncertainty to account for a systematic error in the non-calibrated Ker-phase
and produce a final reduced $\chi^{2}=1$.
One can then proceed with determining the uncertainty on the parameters of the
model fit, by close examination of the likelihood function, very much like
what is described in (Martinache et al., 2009). The three panels of Fig. 4
show the evolution of this function in the parameter space region near the
best solution. Just like with closure-phase data, at angular separations less
than 1 $\lambda/D$, contrast and separation appear to be correlated.
The uncertainty on each parameter of the model-fit is determined after
marginalization of the likelihood function over the other two parameters.
Despite the noted correlation, the constraint on the parameters appears
satisfactory, and the best fit (cf. Table 1) lies well within one $\sigma$ of
the position predicted from the orbital parameters determined from NRM
interferometry from the ground. It also matches the location reported by
Pravdo et al. (2004), after substraction of a simulated PSF from the same
data, only with a constraint on the position angle improved by a factor of 10.
Table 1KER-PHASE DETECTION OF GJ164B IN NICMOS DATA COMPARED TO PREDICTION FROM ORBITAL PARAMETERS Parameter | Ker-phase fit | Prediction
---|---|---
Sep. (mas) | 88.5 $\pm$ | 3.6 | 88.2
P.A. (degrees) | 100.6 $\pm$ | 0.3 | 100.4
Contrast | 9.1 $\pm$ | 1.2 |
From its (H-K) color index, Martinache et al. (2009) were able to conclude
that GJ 164 B is of spectral type later than M8.5, while the primary is well
characterized as a M4.5 dwarf. One of the most prominent spectral features for
M dwarfs is the broad absorption band of water at 1.8 $\mu m$, getting deeper
with later types (Jones et al., 1994; Leggett et al., 2001). The $\sim 5:1$
contrast ratio quoted in the NRM paper was determined over the full Ks filter
(bandwidth 2.0-2.3 $\mu m$). A careful examination of the spectral sequence by
(Jones et al., 1994) reveals that for this combination of spectral types, the
luminosity of GJ 164B relative to GJ164A seen in the NICMOS F190N filter is
expected to drop by 30 to 40 % due to the water absorbtion band. The $9:1$
contrast determined from the Ker-phase model (cf Table 1) in this narrow
filter reflects this evolution. The analysis of this GJ164 data demonstrates
the validity of the Ker-phase approach, by positively detecting a companion
whose existence was known beforehand. This $<10:1$ contrast detection was
however expected to be easy, despite the small angular separation of the
detection (0.6 $\lambda/D$).
Figure 5.— Level of confidence in the detection of a companion from the
analysis of the HST/NICMOS data with the Ker-phase algorithm. A darker color
indicates a region of lower confidence level. Three levels are highlighted:
the 90%, 99% and 99.9% confidence levels. At angular separation 0.5
$\lambda/D$ (i.e. 80 mas at $\lambda=1.9\mu m$), a contrast limit better than
50:1 is possible at the 99% confidence level.
Typical NICMOS1 datasets on a given target usually consist of four frames
only. The SAO 179809 dataset is then a representative example and the
statistics of its Ker-phase ($\sigma=19.7^{\circ}$) can be used in a Monte-
Carlo simulation to determine contrast detection limits.
Because the sampling of the $(u,v)$-plane exhibits no gap, the sensitivity
does not depend on the the position angle relative to the central star. One
can however expect it to be a function of angular separation. A total of
10,000 simulations were performed per point in the angular separation/contrast
plane to produce the sensitivity map displayed in Fig. 5. The map highlights
the 90, 99 and 99.9 % confidence level detection thresholds.
The technique looks promising: for such a dataset, at 0.5 $\lambda/D$, a 50:1
contrast detection appears possible at the 99 % confidence level. The
sensitivity increases and peaks at 180 mas, which unsurprisingly corresponds
to the location of the first zero of the diffraction for the centrally
obstructed telescope (about 1.1 $\lambda/D$), and reaches $\sim$200:1.
## 4\. Conclusion
Classical closure-phase appears to be one special case of a wider family of
observable quantities that are immune to phase noise and non-common path
errors. In the high Strehl regime, it was demonstrated that closure-phase like
quantities, called Ker-phases, can be extracted from focal plane images, and
provide high quality “interferometric grade” information on a source, even
when the pupil is redundant. The Ker-phase technique was successfully applied
to a series of archive NICMOS images, clearly detecting a 10:1 contrast
companion at a separation of $0.5\lambda/D$. Non-calibrated Ker-phase appears
sensitive to the presence of 200:1 contrast companion at angular separation
$1\lambda/D$. Re-analysis of other comparable NICMOS datasets with this
technique might very well lead to the detection of previously undetected
objects in the direct neighborhood of nearby stars.
Unlike closure-phases, which are extremely robust to large wavefront errors,
the use of Ker-phases is however for now restricted to the high-Strehl regime,
and will only become relevant to ground based observations, when extreme AO
systems become available. There is nevertheless hope to be able to extend the
application of Ker-phases to not-so-well corrected AO images, using additional
differential techniques. One possibility, consists in using integral field
spectroscopy, to follow in the Fourier plane, the evolution of the complex
visibilities as a function of wavelength. With enough resolution and spectral
coverage, this indeed allows to identify the phasors contributing to the power
contained at one spatial frequency.
The singular value decomposition of the transfer matrix used to create Ker-
phase relations can also be used to produce a pseudo inverse to the matrix,
and in some cases, allows to inverse the relation linking the $(u,v)$ phases
to the pupil phases. This means that under certain conditions, a single
monochromatic focal plane image can also be useful for wavefront sensing
purposes. This is particularly interesting since the measurement is happening
at the level of the final science detector, which therefore allows to
calibrate non-common path errors. The application of the formalism to
wavefront sensing will be the object of a future publication.
The author thanks Michael J. Ireland and Olivier Guyon for the useful
discussions of the ideas presented in this work.
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|
arxiv-papers
| 2010-09-20T20:00:07 |
2024-09-04T02:49:12.986029
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Frantz Martinache",
"submitter": "Frantz Martinache",
"url": "https://arxiv.org/abs/1009.3933"
}
|
1009.4275
|
# Stability and preconditioning for a hybrid approximation on the sphere
Q. T. Le Gia School of Mathematics and Statistics, University of New South
Wales, Sydney, NSW 2052, Australia qlegia@unsw.edu.au , Ian H. Sloan School
of Mathematics and Statistics, University of New South Wales, Sydney, NSW
2052, Australia i.sloan@unsw.edu.au and Andrew J. Wathen Mathematical
Institute, Oxford University, 24-29 St Giles, Oxford OX1 3LB, UK
wathen@maths.ox.ac.uk
###### Abstract.
This paper proposes a new preconditioning scheme for a linear system with a
saddle-point structure arising from a hybrid approximation scheme on the
sphere, an approximation scheme that combines (local) spherical radial basis
functions and (global) spherical polynomials. Making use of a recently derived
inf-sup condition [13] and the Brezzi stability and convergence theorem for
this approximation scheme, we show that the linear system can be optimally
preconditioned with a suitable block-diagonal preconditioner. Numerical
experiments with a non-uniform distribution of data points support the
theoretical conclusions.
###### Key words and phrases:
radial basis functions, approximation on the sphere, saddle-point systems,
iterative methods, preconditioning
###### 1991 Mathematics Subject Classification:
65F10, 65N22, 65F50
## 1\. Introduction
Amongst approaches for scattered data approximation on the sphere, the hybrid
interpolation scheme of von Golitschek & Light [6] and Sloan & Sommariva [12],
which employs both radial basis functions and spherical polynomials, seems an
attractive method, especially when the data is concentrated in some regions
(such as over mountain ranges and trenches on the Earth’s surface), yet
relatively sparse in other regions. The underlying idea is that radial basis
functions can give good approximation for rapidly varying data over short
distances, whereas the polynomial component can more effectively represent
smooth variations on a global scale. The radial basis functions are centered
at data points which are supposed given, and the linear combination of radial
basis functions is constrained to be orthogonal, in a natural sense, to the
finite dimensional space of polynomials.
However, the hybrid scheme poses difficulties in implementation, compared with
a pure radial basis function approximation, when the number of centers is
large. In the case of a pure radial basis function approximation with a
(strictly) positive definite kernel, the resulting linear system has a matrix
that is positive definite, allowing an iterative solution by the conjugate
gradient method, and preconditioning by, for example, the additive Schwarz
method, see [7]. For the hybrid scheme, in contrast, the linear equations for
the relevant expansion coefficients have the saddle-point form, see [12],
(1) $\left[\begin{array}[]{cc}A&Q\\\
Q^{T}&0\end{array}\right]\left[\begin{array}[]{c}{{\mathbb{\alpha}}}\\\
{{\mathbb{\beta}}}\end{array}\right]=\left[\begin{array}[]{c}{{\bf f}}_{X}\\\
{\bf{0}}\end{array}\right],$
where $A\in\mathbb{R}^{N\times N}$ is a positive definite matrix arising from
the radial basis function part of the function approximation, and
$Q\in\mathbb{R}^{N\times M}$ is a matrix of spherical harmonics evaluated at
the data centers, with $M\leq N$. (The matrices $A$ and $Q$ are defined
properly in Section 2). The saddle-point structure means that the overall
matrix is not positive definite, and that the conjugate gradient method is no
longer a suitable iterative solver. More fundamentally, because the matrix has
both positive and negative eigenvalues, the problem of constructing a good
preconditioner becomes more delicate. For a thorough review of strategies and
challenges for the numerical solution of saddle-point problems, see [2, 5].
In this manuscript we concentrate on the stability of the saddle-point
formulation of this hybrid scheme, and devise and validate a rapid
preconditioned iterative solution method for the solution of the equations
from the approximation. We make use of the Brezzi stability and convergence
theorem well known in the context of mixed finite elements, along with the new
inf-sup condition of [13] to establish convergence of the approximation
scheme; and then use the inf-sup condition to obtain an optimal
preconditioner.
A leading contender amongst the solution methods for equations with the
structure (1) (see [2]) is the block preconditioning method of [8] employing
an approximation to the Schur complement
$S:=Q^{T}A^{-1}Q.$
The use of Schur complement approximations in preconditioners is by now well
established (see for example [5]) in the setting of mixed finite element
methods. In particular, Verfürth [14] showed that for the mixed finite element
approximation of the Stokes flow problem the Schur complement is spectrally
equivalent to the identity operator (or to the mass matrix or $L_{2}$
projection matrix in the finite element setting), making this a suitable
approximation to the Schur complement. Verfürth’s proof makes essential use of
the Babuska-Brezzi or inf-sup condition (Assumption 2.1 in [14]), see [3, 4].
In the present setting, the Schur complement turns out to be spectrally
equivalent not to the identity operator/matrix, but rather to a specific
diagonal but non-constant matrix. This spectral equivalence, a main result of
the paper, is stated in Theorem 4. The key ingredient here, in analogy with
the known inf-sup condition for the Stokes flow problem, is the inf-sup
stability condition recently established by Sloan and Wendland [13] for the
hybrid approximation problem.
An approximate solver for the primal operator (the radial basis function
interpolation matrix in this case) is also required. This could be provided,
for example, by the domain decomposition method of Le Gia, Sloan and Tran [7],
or by any other preconditioner for the pure radial basis function problem. The
resulting block diagonal preconditioner is symmetric and positive definite,
hence the preconditioned MINRES method ([10],[5]) is applicable to the full
problem (which is symmetric but not positive definite).
In Section 2 we formulate the hybrid approximation scheme and establish
notation. Then in Section 3 we describe the inf-sup condition of [13], and use
the Brezzi theorem to establish stability and convergence of the scheme. In
Section 4 we turn to preconditioning, establishing there the main spectral
equivalence result. In Section 5 numerical results are presented (using the
primal preconditioner of [7]) and we conclude in Section 6.
## 2\. Problem formulation
Let $X=X_{N}=\\{{\bf x}_{1},{\bf x}_{2},\ldots,{\bf x}_{N}\\}$ be a set of $N$
distinct points on the sphere ${\mathbb{S}}^{d}$ in ${\mathbb{R}}^{d+1}$.
Using these as centers we define a radial basis function approximation space
${\mathcal{X}}_{X_{N}}={\mathcal{X}}_{N}:=\\{\sum_{i=1}^{N}\alpha_{i}\phi(\cdot,{\bf
x}_{i}):\alpha_{1},\ldots,\alpha_{N}\in{\mathbb{R}}\\}$
with a suitable kernel function $\phi$. The kernel is assumed to be (strictly)
positive definite, that is
$\sum_{i=1}^{N}\sum_{j=1}^{N}\alpha_{i}\phi(\mathbf{x}_{i},\mathbf{x}_{j})\alpha_{j}\geq
0$
for every set of points
$X_{N}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\\}\in{\mathbb{S}}^{d}$ and for
all $N\in{\mathbb{N}}$, with equality for distinct points $\mathbf{x}_{j}$
only if $\alpha_{1}=\alpha_{2}=\ldots=\alpha_{N}=0.$
The native space $\mathcal{N}_{\phi}$ is defined as the completion under the
inner product
(2) $\langle\sum_{i}\alpha_{i}\phi(\cdot,{\bf
x}_{i}),\sum_{j}\alpha_{j}^{\prime}\phi(\cdot,{\bf
x}_{j})\rangle_{\phi}=\sum_{i}\sum_{j}\alpha_{i}\alpha_{j}^{\prime}\phi({\bf
x}_{i},{\bf x}_{j})$
of the linear space
$F_{\phi}:=\\{\sum_{j=1}^{N}\alpha_{j}\phi(\cdot,{\bf
x}_{j}),\alpha_{j}\in{\mathbb{R}},{\bf
x}_{j}\in{\mathbb{S}}^{d},j=1,\ldots,N,N\in{\mathbb{N}}\\},$
where we insist that the points ${\bf x}_{j}$ are distinct. The norm is as
usual defined by
$\|\cdot\|_{\phi}=\langle\cdot,\cdot\rangle_{\phi}^{1/2}.$
It is well-known that $\mathcal{N}_{\phi}$ is a reproducing kernel Hilbert
space (see [1]) with the reproducing kernel $\phi(\cdot,\cdot)$. That is
$\displaystyle\phi({\bf x},{\bf y})$ $\displaystyle=$ $\displaystyle\phi({\bf
y},{\bf x}),\quad{\bf x},{\bf y}\in{\mathbb{S}}^{d}$
$\displaystyle\phi(\cdot,{\bf y})$ $\displaystyle\in$
$\displaystyle\mathcal{N}_{\phi},\quad{\bf y}\in{\mathbb{S}}^{d}$
and for $f\in\mathcal{N}_{\phi}$
(3) $\langle f,\phi(\cdot,{\bf y})\rangle_{\phi}=f({\bf y}),\quad{\bf
y}\in{\mathbb{S}}^{d}.$
The kernel function $\phi$ needs to be positive definite in order that the
inner product $\langle\cdot,\cdot\rangle_{\phi}$ satisfy the positivity axiom
for an inner product. Equivalently, the matrices $A_{X}$ defined by
(4) $\left(A_{X}\right)_{i,j}:=\phi({\bf x}_{i},{\bf x}_{j}),\quad
i,j=1,\ldots,N$
are positive definite as well as symmetric for every $X$ and every
$N\in{\mathbb{N}}$.
Taking now a fixed $N\in{\mathbb{N}}$ and a fixed set
$X_{N}\subset{\mathbb{S}}^{d}$, we may define the usual radial basis function
interpolant to a continuous function $f$ on ${\mathbb{S}}^{d}$ by
$f_{N}({\bf x})=\sum_{j=1}^{N}\alpha_{j}\phi({\bf x},{\bf x}_{j}),$
where $\alpha_{1},\ldots,\alpha_{N}$ are such that
$f_{N}({\bf x}_{i})=f({\bf x}_{i}),\quad i=1,\ldots,N,$
which we may write as
$\sum_{j=1}^{N}\phi(\mathbf{x}_{i},\mathbf{x}_{j})\alpha_{j}=f(\mathbf{x}_{i})\textrm{
for }i=1,\ldots,N.$
That is, the vector ${\mathbb{\alpha}}=(\alpha_{1},\ldots,\alpha_{N})^{T}$ of
coefficients satisfies
(5) $A_{X}{\mathbb{\alpha}}={\bf f}_{X},$
where
(6) ${\bf f}_{X}:=(f({\bf x}_{1}),\ldots,f({\bf x}_{N}))^{T}.$
The hybrid approximation scheme of von Golitschek & Light [6] and Sloan &
Sommariva, see [13], employs not only the radial basis functions, but also
spherical polynomials of total degree up to some conveniently chosen $L\geq
0.$ We define
${\mathcal{P}}_{L}=\hbox{span}\\{Y_{\ell,k}:\,k=1,\ldots,M(d,\ell),\,\ell=0,\ldots,L\\},$
where $Y_{\ell,k}$ is a spherical harmonic of degree $\ell$, that is, the
restriction to ${\mathbb{S}}^{d}$ of a homogeneous harmonic polynomial in
${\mathbb{R}}^{d+1}$ of degree $\ell$, and $M(d,\ell)$ is the dimension of the
space spanned by the spherical harmonics of degree $\ell$. Then
${\mathcal{P}}_{L}$ is the set of spherical polynomials of degree $\leq L$. We
shall assume that $\\{Y_{\ell,k}:\,k=1,\ldots,M(d,\ell),\,\ell=0,1,\ldots\\}$
is an orthonormal set with respect to the usual $L_{2}$ inner product, that is
$\int_{{\mathbb{S}}^{d}}Y_{\ell,k}({\bf x})Y_{\ell^{\prime},k^{\prime}}({\bf
x})d\omega({\bf x})=\delta_{\ell,\ell^{\prime}}\delta_{k,k^{\prime}},$
where $d\omega({\bf x})$ denotes surface measure on ${\mathbb{S}}^{d}$. Then
it is well known that
$\\{Y_{\ell,k}:\,k=1,\ldots,M(d,\ell),\,\ell=0,1,\ldots\\}$ is a complete
orthonormal basis for $L_{2}({\mathbb{S}}^{d})$.
For a given function $f$, the hybrid approximation scheme is then to find
(7) $u_{N,L}({\bf x})=\sum_{j=1}^{N}\alpha_{j}\phi({\bf x},{\bf
x}_{j})\in{\mathcal{X}}_{N}$
and
(8) $p_{N,L}({\bf
x})=\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\beta_{l,k}Y_{\ell,k}({\bf
x})\in{\mathcal{P}}_{L}$
such that
(9) $u_{N,L}(\mathbf{x}_{i})+p_{N,L}(\mathbf{x}_{i})=f(\mathbf{x}_{i}),$
or equivalently,
(10) $\sum_{j=1}^{N}\alpha_{j}\phi({\bf x}_{i},{\bf
x}_{j})+\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\beta_{\ell,k}Y_{\ell,k}({\bf
x}_{i})=f({\bf x}_{i}),\quad i=1,\ldots,N,$
which is to be solved subject to the side condition
(11) $\sum_{j=1}^{N}\alpha_{j}q({\bf x}_{j})=0\quad\forall
q\in{\mathcal{P}}_{L}.$
The condition (11) is equivalent, via (3), to $\langle
q,u_{N,L}\rangle_{\phi}=0$ for all $q\in{\mathcal{P}}_{L}$, forcing the radial
basis function component to be $\mathcal{N}_{\phi}$-orthogonal to
${\mathcal{P}}_{L}$. It also ensures that the defining linear system is square
and symmetric.
The conditions (10),(11) can also be seen to be those which derive from the
solution of the constrained optimization problem
$\min_{u_{N,L}\in{\mathcal{X}}_{N}}\;\,\frac{1}{2}\|u_{N,L}-f\|_{\phi}^{2}\;\;\mbox{subject
to}\;\;\langle Y_{\ell,k},u_{N,L}\rangle_{\phi}=0$
for all $\ell=0,\ldots,L,k=1,\ldots,{M(d,\ell)}$, the coefficients
$\beta_{\ell,k}$ being the Lagrange multipliers in the Lagrangian
$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\|u_{N,L}-f\|_{\phi}^{2}+\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\beta_{\ell,k}\langle
Y_{\ell,k},u_{N,L}\rangle_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\|u_{N,L}-f\|_{\phi}^{2}+\langle
p_{N,L},u_{N,L}\rangle_{\phi}.$
This is therefore another way of expressing the hybrid approximation problem.
By choosing the spherical harmonic functions as the basis for
${\mathcal{P}}_{L}$ in (11), we can write (10),(11) as a so-called ‘saddle-
point’ linear system of equations
(12) $\left[\begin{array}[]{cc}A_{X}&Q_{X,L}\\\
Q_{X,L}^{T}&0\end{array}\right]\left[\begin{array}[]{c}{{\mathbb{\alpha}}}\\\
{{\mathbb{\beta}}}\end{array}\right]=\left[\begin{array}[]{c}{{\bf f}}_{X}\\\
{\bf{0}}\end{array}\right],$
where $A_{X}$ is defined by (4), ${\mathbb{\alpha}}$ is the vector of
coefficients $\alpha_{j}$ as defined above, ${\mathbb{\beta}}$ is a vector
containing the coefficients $\beta_{\ell,k}$ for
$k=1,\ldots,M(d,\ell),\,\ell=0,\ldots,L$, and $Q_{X,L}$ is the $N\times M$
matrix defined by
(13) $(Q_{X,L})_{i,\ell k}:=Y_{\ell,k}({\bf x}_{i}),\quad
i=1,\ldots,N,\;k=1,\ldots,M(d,\ell),\;\ell=0,\ldots,L,$
and
$M:=\sum_{\ell=0}M(d,\ell)=\mbox{dim }({\mathcal{P}}_{L}).$
In the present application we need to prescribe more precisely the nature of
the kernel $\phi(\mathbf{x},\mathbf{y})$. In the first place we shall assume
that it is zonal, meaning that
$\phi(\mathbf{x},\mathbf{y})=\Phi(\mathbf{x}\cdot\mathbf{y})$
for some function $\Phi\in C[-1,1]$, where $\mathbf{x}\cdot\mathbf{y}$ denotes
the Euclidean inner product in $\mathbb{R}^{d+1}$. More precisely, we shall
assume that $\phi(\mathbf{x},\mathbf{y})$ has an expansion of the form
(14)
$\phi(\mathbf{x},\mathbf{y})=\sum_{\ell=0}^{\infty}\sum_{k=1}^{M(d,\ell)}a_{\ell}Y_{\ell,k}(\mathbf{x})Y_{\ell,k}(\mathbf{y}),$
with $a_{\ell}>0$ for all $\ell\geq 0$. That the expansion is zonal follows
from the addition theorem for spherical harmonics,
$\sum_{k=1}^{M(d,\ell)}Y_{\ell,k}({\bf x})Y_{\ell,k}({\bf
y})=\frac{M(d,\ell)}{\omega_{d}}P_{\ell}(d+1,{\bf x}\cdot{\bf y}),$
where $P_{\ell}(d+1,z)$ is the Legendre polynomial of degree $\ell$ in
dimension $d+1$ normalized to $P_{\ell}(d+1,1)=1$, and $\omega_{d}$ is the
total surface measure of ${\mathbb{S}}^{d}$,
$\omega_{d}=\int_{{\mathbb{S}}^{d}}d\omega({\bf x}).$
In this situation it is well known that the inner product in
$\mathcal{N}_{\phi}$ can be written as
(15) $\langle
u,v\rangle_{\phi}=\sum_{\ell=0}^{\infty}\sum_{k=1}^{M(d,\ell)}\dfrac{\widehat{u}_{\ell,k}\widehat{v}_{\ell,k}}{a_{\ell}},$
where
$\widehat{u}_{\ell,k}=\int_{{\mathbb{S}}^{d}}u(\mathbf{x})Y_{\ell,k}(\mathbf{x})d\omega(\mathbf{x}).$
Indeed, as a special case of (15) we find
$\langle
f,\phi(\cdot,\mathbf{y})\rangle_{\phi}=\sum_{\ell=0}^{\infty}\sum_{k=1}^{M(d,\ell)}\dfrac{\widehat{f}_{\ell,k}a_{\ell}Y_{\ell,k}}{a_{\ell}}=\sum_{\ell=0}^{\infty}\sum_{k=1}^{M(d,\ell)}\widehat{f}_{\ell,k}Y_{\ell,k}(\mathbf{y})=f({\bf
y}),$
thus verifying the reproducing kernel property (3).
If we further assume that for large $\ell$
(16) $a_{\ell}\sim(\ell+1)^{-2s},$
then it follows from (15) and (16) that the native space $\mathcal{N}_{\phi}$
is equivalent to the Sobolev space $H^{s}({\mathbb{S}}^{d})$ with inner
product
(17) $\langle
u,v\rangle_{H^{s}}=\sum_{\ell=0}^{\infty}\sum_{k=1}^{M(d,\ell)}(\ell+1)^{2s}\widehat{u}_{\ell,k}\widehat{v}_{\ell,k}.$
Technicalities aside, we remark that the essential difficulty in analysing the
hybrid approximation is that ${\mathcal{X}}_{N}$ and ${\mathcal{P}}_{L}$ are
both subsets of $\mathcal{N}_{\phi}$, and that in an appropriate sense both
can approximate $\mathcal{N}_{\phi}$ as $N$ or $L$ tend to $\infty$. We need
the inf-sup condition, now to be introduced, to allow both subspaces to
coexist comfortably within the one approximation.
## 3\. Inf-sup condition, and the Brezzi theorem
Typically, uniqueness of the solution and optimal error estimates for saddle-
point problems follow from so-called inf-sup conditions together with
appropriate coercivity of the primal operator. For us an essential tool will
be the following inf-sup theorem proved in [13]. In this theorem $h_{X}$, for
a given point set $X=X_{N}\subset{\mathbb{S}}^{d}$, is the mesh norm, defined
by
$h_{X}:=\sup_{\mathbf{x}\in{\mathbb{S}}^{d}}\inf_{\mathbf{x}_{j}\in
X}\rm{cos}^{-1}(\mathbf{x}\cdot\mathbf{x}_{j}).$
In words, $h_{X}$ is the maximum geodesic distance from a point on
${\mathbb{S}}^{d}$ to the nearest point of $X$.
###### Theorem 1.
Let $\phi$ be a kernel satisfying (14) and (16) for some $s>d/2$. There exist
constants $\gamma>0$ and $\tau>0$ depending only on $d$ and $s$ such that for
all $L\geq 1$ and all
$X_{N}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\\}\subset{\mathbb{S}}^{d}$
satisfying $h_{X}\leq\tau/L$ the following inequality holds:
$\inf_{p\in{\mathcal{P}}_{L}\setminus\\{0\\}}\sup_{v\in{\mathcal{X}}_{N}\setminus\\{0\\}}\dfrac{\langle
p,v\rangle_{\phi}}{\|v\|_{\phi}\|p\|_{\phi}}\geq\gamma.$
To use this to prove stability we start with the following well-known theorem
from Brezzi [4], which is at the heart of most analyses of mixed finite
elements.
###### Theorem 2.
Let $H$ and $J$ be real Hilbert spaces, $a(\xi_{1},\xi_{2})$ be a continuous
bilinear form on $H\times H$ and $b(\psi,\xi)$ be a continuous bilinear form
on $J\times H$. Let $\\{H_{N}:N\in{\mathbb{N}}\\}$ and
$\\{J_{L}:L\in{\mathbb{N}}\\}$ be sequences of subspaces of $H$ and $J$
respectively. Set
$K=\\{\eta\in H:b(\theta,\eta)=0\;\;\forall\theta\in
J\\},\;\;K_{N,L}=\\{\eta\in H_{N}:b(\theta,\eta)=0\;\;\forall\theta\in
J_{L}\\}.$
If
(18) $\exists\gamma_{0}>0\;\text{ such that }\quad
a(\eta,\eta)\geq\gamma_{0}\|\eta\|_{H}^{2}\quad\forall\eta\in K\cup K_{N,L},$
and
$\exists\gamma_{1}>0\;\text{ such that }\quad\sup_{\eta\in
H\setminus\\{0\\}}\frac{b(\theta,\eta)}{\|\eta\|_{H}}\geq\gamma_{1}\|\theta\|_{J}\quad\
\forall\theta\in J$ (19) $\text{ and }\sup_{\eta\in
H_{N}\setminus\\{0\\}}\frac{b(\theta,\eta)}{\|\eta\|_{H}}\geq\gamma_{1}\|\theta\|_{J}\quad\
\forall\theta\in J_{L},$
then for every $\ell_{1}\in H^{\prime}$ and $\ell_{2}\in J^{\prime}$ and every
$N,L>0$ the discrete problem of finding $\xi_{N,L}\in H_{N}$ and
$\psi_{N,L}\in J_{L}$ such that
$\displaystyle a(\xi_{N,L},\eta)+b(\psi_{N,L},\eta)$ $\displaystyle=$
$\displaystyle\langle\ell_{1},\eta\rangle\quad\forall\eta\in H_{N}$
$\displaystyle b(\theta,\xi_{N,L})$ $\displaystyle=$
$\displaystyle\langle\ell_{2},\theta\rangle\quad\forall\theta\in J_{L}$
has a unique solution, and there exists a constant
$C=C(\gamma_{0},\gamma_{1})>0$ such that
$\|\xi-\xi_{N,L}\|_{H}+\|\psi-\psi_{N,L}\|_{J}\leq
C\left(\inf_{\widehat{\xi}_{N}\in
H_{N}}\|\xi-\widehat{\xi}_{N}\|_{H}+\inf_{\widehat{\psi}_{L}\in
J_{L}}\|\psi-\widehat{\psi}_{L}\|_{J}\right)$
where $\xi\in H$ and $\psi\in J$ are defined by
$\displaystyle a(\xi,\eta)+b(\psi,\eta)$ $\displaystyle=$
$\displaystyle\langle\ell_{1},\xi\rangle\quad\forall\eta\in H,$ $\displaystyle
b(\theta,\xi)$ $\displaystyle=$
$\displaystyle\langle\ell_{2},\theta\rangle\quad\forall\theta\in J.$
To apply this theorem, we first observe that the hybrid approximation with its
defining equations (10) and (11) can be written, using the reproducing kernel
property (3), as the problem of finding $u_{N,L}\in{\mathcal{X}}_{N}$ and
$p_{N,L}\in{\mathcal{P}}_{L}$ such that
(20) $\langle u_{N,L},\eta\rangle_{\phi}+\langle
p_{N,L},\eta\rangle_{\phi}=\langle
f,\eta\rangle_{\phi}\quad\forall\eta\in{\mathcal{X}}_{N},$ (21) $\langle
q,u_{N,L}\rangle_{\phi}=0\qquad\forall q\in{\mathcal{P}}_{L}.$
To use Theorem 2 we take $H=\mathcal{N}_{\phi}$, $J={\mathcal{P}}_{L}$,
$H_{N}={\mathcal{X}}_{N}$ and $J_{L}={\mathcal{P}}_{L}$, with the inner
product on $\mathcal{N}_{\phi}$ being defined by (2), and the bilinear forms
$a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ both equal to the $\mathcal{N}_{\phi}$
inner product. The coercivity condition (18) is trivially satisfied on the
whole space $H=\mathcal{N}_{\phi}$ with $\gamma_{0}=1$ since
$a(u,u)=\langle u,u\rangle_{\phi}=\|u\|_{\phi}^{2}.$
The existence of a constant $\gamma_{1}$ independent of $N$ and $L$ satisfying
(19) is ensured by Theorem 1, provided $h_{X}\leq\tau/L$. The first part of
Theorem 2 then confirms that the solution of the system (20) and (21) exists
and is unique provided $h_{X}\leq\tau/L$. The last part of that theorem
defines the comparison quantities: it defines $u_{L}\in\mathcal{N}_{\phi}$ and
$p_{L}\in{\mathcal{P}}_{L}$ such that
$\langle u_{L},\eta\rangle_{\phi}+\langle p_{L},\eta\rangle_{\phi}=\langle
f,\eta\rangle_{\phi}\quad\forall\eta\in\mathcal{N}_{\phi},$ $\langle
q,u_{L}\rangle_{\phi}=0\qquad\forall q\in{\mathcal{P}}_{L}.$
The second equation says that $u_{L}$ is orthogonal to the space
${\mathcal{P}}_{L}$. In principle the orthogonality is with respect to the
$\mathcal{N}_{\phi}$ inner product, but because of the zonal property of the
kernel it is easy to see that this is the same as the $L_{2}$ orthogonal
projection. Indeed, from (15) we have
$\langle q,u_{L}\rangle_{\phi}=0\,\forall
q\in{\mathcal{P}}_{L}\implies(\widehat{u}_{L})_{\ell,k}=0\mbox{ for
}\ell\in[0,L]\implies\langle q,u_{L}\rangle_{L_{2}}=0\,\forall
q\in{\mathcal{P}}_{L}.$
The first of the latter set of equations then becomes, on specialising the
choice of $\eta$ to $q\in{\mathcal{P}}_{L}$,
$\langle p_{L},q\rangle_{\phi}=\langle f,q\rangle_{\phi}\quad\forall
q\in{\mathcal{P}}_{L},$
thus $p_{L}$ is the orthogonal projection of $f$ on the subspace
${\mathcal{P}}_{L}$ with respect to either the $L_{2}$ or the
$\mathcal{N}_{\phi}$ inner products. We write this orthogonal projection as
$P_{L}f$. Now we can write $p_{L}=P_{L}f$, and $u_{L}=f-p_{L}=f-P_{L}f$.
Theorem 2 with $\xi=u_{L}$ and $\psi=p_{L}$ now gives the following
convergence result, recovering a result obtained by a direct argument in [13].
Note that even though we have taken $J={\mathcal{P}}_{L}$ in the theorem, the
constants $\gamma_{0}$ and $\gamma_{1}$ do not depend on $L$, and hence
neither does $C$.
###### Theorem 3.
Let $\phi$ be a kernel satisfying (14) and (16) for some $s>d/2$. There exist
constants $C>0$ and $\tau>0$ depending only on $d$ and $s$ such that for all
$L\geq 1$ and all
$X=X_{N}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\\}\subset{\mathbb{S}}^{d}$
satisfying $h_{X}\leq\tau/L$ the solutions of (7),(8) and (9) satisfy
(22) $\|f-u_{N,L}-p_{N,L}\|_{\phi}\leq
C\inf_{\widehat{\xi}_{N}\in{\mathcal{X}}_{N}}\|(f-P_{L}f)-\widehat{\xi}_{N}\|_{\phi}.$
Explicit error bounds in the $L_{2}$ norm for $f\in H^{s}$ and $f\in H^{2s}$
can then be obtained as in [13].
## 4\. Preconditioning
Now we turn our attention to the linear algebra aspects of the hybrid
approximation described in Section 2.
We have noted already that the hybrid approximation can be written as the
linear system (12), with $A_{X},Q_{X,L}$ and ${\bf f}$ defined by (4),(13),
and (6).
The solution of saddle-point linear systems such as (1) has received much
attention in recent years - see [2] for an overview of possible approaches. In
particular, it was shown in [8] that a suitable preconditioner for the saddle
point system
(23) $\left[\begin{array}[]{cc}A&Q\\\ Q^{T}&0\end{array}\right]$
with positive definite $A$ is
(24) $\left[\begin{array}[]{cc}A&0\\\ 0&S\end{array}\right],$
where
$S=Q^{T}A^{-1}Q$
is the Schur complement. This is because of the remarkable fact that the
product of (23) by the inverse of (24) is a diagonalisable matrix with just
three distinct eigenvalues, namely $1,(1\pm\sqrt{5})/2$. Thus an appropriate
Krylov subspace iteration such as MINRES (see [10]) will converge in just
three iterations. While this is generally not a practical preconditioner, an
approximate preconditioner of the form
(25) $\left[\begin{array}[]{cc}\widehat{A}&0\\\
0&\widehat{S}\end{array}\right],$
where $\widehat{A}$ is a preconditioner for the problem (5) involving only
$A$, and $\widehat{S}$ is a suitable Schur complement approximation, will lead
to rapid convergence.
In the present work we shall assume that an approximate preconditioner
$\widehat{A}$ for $A=A_{X}$ is already available; one example would be the
domain decomposition preconditioner from [7] \- this is the one we employ in
the numerical results presented in the next section. Our interest here is in
finding an appropriate approximation to the Schur complement
$S_{X}=Q^{T}_{X,L}A_{X}^{-1}Q_{X,L}$. We shall see that this is handed to us
by the inf-sup result in Theorem 1.
That inf-sup condition can be stated as
$\sup_{v\in{\mathcal{X}}_{N}\setminus\\{0\\}}\frac{\langle
p,v\rangle_{\phi}}{\|v\|_{\phi}}\geq\gamma_{1}\|p\|_{\phi}\quad\ \forall
p\in{\mathcal{P}}_{L},$
provided $h_{X}\leq\tau/L$. With the help of the Cauchy-Schwarz inequality,
this can be strengthened to a two-sided inequality,
(26) $\|p\|_{\phi}=\sup_{v\in\mathcal{N}_{\phi}\setminus\\{0\\}}\frac{\langle
p,v\rangle_{\phi}}{\|v\|_{\phi}}\geq\sup_{v\in{\mathcal{X}}_{N}\setminus\\{0\\}}\frac{\langle
p,v\rangle_{\phi}}{\|v\|_{\phi}}\geq\gamma_{1}\|p\|_{\phi}\quad\ \forall
p\in{\mathcal{P}}_{L},$
provided $h_{X}\leq\tau/L$. To find an equivalent matrix expression, we write
$p\in{\mathcal{P}}_{L}$ and $v\in{\mathcal{X}}_{N}$ as
$p=\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\beta_{\ell,k}Y_{\ell,k},\qquad
v=\sum_{i=1}^{N}\alpha_{i}\phi(\cdot,\mathbf{x}_{i}).$
With the help of the reproducing property (3), we find
$\displaystyle\langle p,v\rangle_{\phi}$
$\displaystyle=\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\sum_{i=1}^{N}\beta_{\ell,k}\alpha_{i}\langle
Y_{\ell,k},\phi(\cdot,\mathbf{x}_{i})\rangle_{\phi}$
$\displaystyle=\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\sum_{i=1}^{N}\beta_{\ell,k}\alpha_{i}Y_{\ell,k}(\mathbf{x}_{i})={\mathbb{\beta}}^{T}Q^{T}_{X,L}{\mathbb{\alpha}},$
and
$\|v\|_{\phi}=\langle
v,v\rangle_{\phi}^{1/2}=(\sum_{i=1}^{N}\sum_{j=1}^{N}\alpha_{i}\alpha_{j}\phi(\mathbf{x}_{i},\mathbf{x}_{j}))^{1/2}=({\mathbb{\alpha}}^{T}A_{X}{\mathbb{\alpha}})^{1/2}.$
Also, with the aid of (15) we obtain
$\|p\|_{\phi}=\left(\sum_{\ell=0}^{L}\sum_{k=1}^{M(d,\ell)}\dfrac{\beta_{\ell,k}^{2}}{a_{\ell}}\right)^{{\frac{1}{2}}}=\left({\mathbb{\beta}}^{T}\Lambda_{L}{\mathbb{\beta}}\right)^{\frac{1}{2}},$
where $\Lambda_{L}$ is the $M\times M$ diagonal matrix given by
(27) $(\Lambda_{L})_{\ell
k,\ell^{\prime}k^{\prime}}=\delta_{\ell\ell^{\prime}}\delta_{kk^{\prime}}/a_{\ell}.$
Thus in matrix terms (26) can be written as
(28)
$({\mathbb{\beta}}^{T}\Lambda_{L}{\mathbb{\beta}})^{{\frac{1}{2}}}\geq\sup_{{\mathbb{\alpha}}\in{\mathbb{R}}^{N}\setminus\\{0\\}}\frac{{\mathbb{\beta}}^{T}Q^{T}_{X,L}{\mathbb{\alpha}}}{({\mathbb{\alpha}}^{T}A_{X}{\mathbb{\alpha}})^{{\frac{1}{2}}}}\geq\gamma_{1}({\mathbb{\beta}}^{T}\Lambda_{L}{\mathbb{\beta}})^{{\frac{1}{2}}}\quad\forall{\mathbb{\beta}}\in{\mathbb{R}}^{M}.$
Because $A_{X}$ is symmetric and positive definite, the central term can be
simplified by the substitution ${\mathbb{\alpha}}=A_{X}^{-{\frac{1}{2}}}{\bf
a}$, making it
$\sup_{{\bf
a}\in{\mathbb{R}}^{N}\setminus\\{0\\}}\frac{{\mathbb{\beta}}^{T}Q_{X,L}^{T}A_{X}^{-{\frac{1}{2}}}{\bf
a}}{({\bf a}^{T}{\bf
a})^{{\frac{1}{2}}}}\,=\,({\mathbb{\beta}}^{T}Q_{X,L}^{T}A_{X}^{-1}Q_{X,L}{\mathbb{\beta}})^{\frac{1}{2}},$
with the last step following because the supremum over ${\bf a}$ is clearly
achieved by ${\bf
a}=({\mathbb{\beta}}^{T}Q^{T}_{X,L}A_{X}^{-{\frac{1}{2}}})^{T}$. Thus in
matrix terms (26) can be expressed as
(29)
${\mathbb{\beta}}^{T}\Lambda_{L}{\mathbb{\beta}}\geq{\mathbb{\beta}}^{T}Q^{T}_{X,L}A^{-1}_{X}Q_{X,L}{\mathbb{\beta}}\geq\gamma_{1}^{2}{\mathbb{\beta}}^{T}\Lambda_{L}{\mathbb{\beta}}\quad\forall{\mathbb{\beta}}\in{\mathbb{R}}^{M}.$
Through the above arguments we have established the following theorem.
###### Theorem 4.
Let $\phi$ be a kernel satisfying (14) and (16) for some $s>d/2$, and let
$A_{X}$ and $Q_{X,L}$ be given by (4) and (13). For all $L\geq 1$ and
$h_{X}\leq\tau/L$, where $\tau$ is as in Theorem 1, the Schur complement
$S_{X}=Q_{X,L}^{T}A_{X}^{-1}Q_{X,L}$ is spectrally equivalent to the diagonal
matrix $\Lambda_{L}$ given by (27).
It follows from the theorem that our practical recommendation for the hybrid
problem is a preconditioner of the form
(30) $\left[\begin{array}[]{cc}\widehat{A}&0\\\
0&\Lambda_{L}\end{array}\right],$
where $\widehat{A}$ is an approximation to $A$, and $\Lambda_{L}$ is defined
by (27).
## 5\. Numerical examples
We will use the following kernel
$\phi(\mathbf{x},\mathbf{y})=\psi(|\mathbf{x}-\mathbf{y}|)=\psi(\sqrt{2-2\mathbf{x}\cdot\mathbf{y}}),\quad\mathbf{x},\mathbf{y}\in{\mathbb{S}}^{2},$
where the radial basis function $\psi(r)$ is one of the three choices
$\psi_{0}(r)=(1-r)^{2}_{+},\quad\psi_{1}(r)=(1-r)^{4}_{+}(4r+1),\quad\psi_{2}(r)=(1-r)^{6}_{+}(35r^{2}+18r+3)$
with $(x)_{+}=x$ for $x\geq 0$ and $0$ otherwise. Note that $\psi_{0}\in
C^{0}({\mathbb{R}}^{3}),\psi_{1}\in C^{2}({\mathbb{R}}^{3})$ and $\psi_{2}\in
C^{4}({\mathbb{R}}^{3})$ and each is positive definite (see [15]). We comment
that we have not here employed any scaling of the compactly supported RBFs.
Using functions with smaller support would improve matrix conditioning, but
because it would also reduce approximation accuracy we have chosen not to use
any scaling. It is shown theoretically in [9] and verified numerically in
Figure 1 that the coefficients $a_{\ell}$ in the expansion of the kernel
$\phi$ defined from $\psi_{1}$ are of order $(1+\ell)^{-5}$. This is
consistent with (16) but because the constants in the equivalence are large we
have chosen to work directly with the Fourier-Legendre coefficients
$a_{\ell}$, which being $1$-dimensional integrals are easily evaluated
numerically. For the sphere ${\mathbb{S}}^{2}$, we have
$a_{\ell}=2\pi\int_{-1}^{1}\Phi(t)P_{\ell}(3,t)dt,\quad\mbox{ where
}\Phi(t)=\psi(\sqrt{2-2t}).$
Figure 1. Numerical values of $(\ell+1)^{5}a_{\ell}$
For the preconditioner, as described in [7], the matrix $A$ is preconditioned
using a domain decomposition technique. First, given a fixed parameter
$0<\nu<\pi$, an appropriate set of centers
$\\{\mathbf{p}_{1},\ldots,\mathbf{p}_{J}\\}\subset X$ is chosen over the whole
sphere so that
$\min_{i\neq j}\cos^{-1}(\mathbf{p}_{i}\cdot\mathbf{p}_{j})\geq\nu.$
Second, with another fixed parameter $0<\mu<\pi/3$, we decompose the point set
$X$ into a collection of smaller sets $X_{j}$, for $j=1,\ldots,J$, defined by
$X_{j}:=\\{\mathbf{x}\in
X:\cos^{-1}(\mathbf{x}\cdot\mathbf{p}_{j})\leq\mu\\}.$
The sets $X_{j}$ with cardinality $m_{j}$, for $j=1,\ldots,J$, may overlap
each other and must satisfy $\cup_{j=1}^{J}X_{j}=X$. The restriction operator
from ${\mathbb{R}}^{N}$ to ${\mathbb{R}}^{m_{j}}$ is denoted by $R_{j}$ and
the extension operator from ${\mathbb{R}}^{m_{j}}$ to ${\mathbb{R}}^{N}$ is
$R^{T}_{j}$. Given a vector $\mathbf{r}\in{\mathbb{R}}^{N}$, the action of the
preconditioner $\widehat{A}$ is given by
$\widehat{A}^{-1}\mathbf{r}=\sum_{j=1}^{J}R^{T}_{j}(A_{j})^{-1}R_{j}\mathbf{r}$
where the matrix $A_{j}$ is the restriction of the full matrix $A$ on the
subdomain $X_{j}$ (see [7] for more details).
By Theorem 4 the diagonal matrix $\Lambda_{L}$ defined by (27) is spectrally
equivalent to $S_{X}=Q^{T}_{X,L}A^{-1}_{X}Q_{X,L}$. The block diagonal
preconditioner for (23) is therefore the matrix (30).
The results here are for interpolation of the function
$f(x,y,z)=\exp(x+y+z)+[0.01-x^{2}-y^{2}-(z-1)^{2}]^{2}_{+},$
consisting of a smooth first term and a second term whose support is a cap of
Euclidean radius $0.1$.
Using each of the kernel functions $\phi$ obtained from the radial basis
functions $\psi_{m},m=0,1,2$, we employ $N=2000$, $4000$, $8000$, $16000$ and
$32000$ points, and maximum polynomial degree $L=0,5,10,15,20,25$. In each
case, a thousand of the points were generated in a cap about the $z$ axis
subtending an angle of $0.1$ radians at the origin and the remaining points
distributed outside this cap. The Saff-Kuijlars equal area algorithm described
in [11] was used to generate these points in the following manner. Firstly the
Saff-Kuijlars points are generated on the whole sphere and those in the cap
region are discarded. Then a similar equal area construction only for the cap
is used to generate $1000$ points in this region.
The number of MINRES iterations and the CPU time in seconds required for
convergence to a residual norm tolerance of $10^{-9}$ are tabulated for the
unpreconditioned case in Table 1 and for the preconditioner introduced here in
Table 2. The computer code is written in Fortran 90, compiled with the Intel
compiler and run on a single core of an SGI Altix XE320 with two Intel Xeon
X5472 CPUs.
$m$ | $N$ | 2000 | 4000 | 8000 | 16000 | 32000
---|---|---|---|---|---|---
0 | $L=0$ | 207 ( 14) | 228 ( 62) | 273 ( 297) | 294 ( 1229) | 367 ( 6277)
| $L=5$ | 813 ( 57) | 1116 (304) | 1531 (1662) | 1996 ( 8297) | 2535 (43513)
| $L=10$ | 1053 ( 77) | 1488 (413) | 2240 (2463) | 3016 (12573) | 4001 (67005)
| $L=15$ | 1080 ( 85) | 1492 (498) | 2208 (2475) | 3037 (12788) | 4270 (71728)
| $L=20$ | 1126 ( 99) | 1466 (464) | 2032 (2334) | 2777 (17227) | 3832 (66025)
| $L=25$ | 1262 (123) | 1467 (499) | 1976 (2382) | 2562 (11069) | 3461 (58451)
1 | $L=0$ | 1041 ( 75) | 790 ( 218) | 695 ( 755) | 912 ( 3954) | 995 (17436)
| $L=5$ | 2654 (193) | 4293 (1194) | 3564 (3889) | 3172 (13736) | 2321 (40754)
| $L=10$ | 3647 (280) | 4370 (1237) | 4047 (4488) | 5335 (42445) | 3381 (59228)
| $L=15$ | 3267 (265) | 4308 (1281) | 4021 (4594) | 3792 (16908) | 2808 (48999)
| $L=20$ | 2883 (264) | 2923 ( 935) | 3329 (6906) | 3977 (17987) | $>$24 hours
| $L=25$ | 2659 (265) | 2837 (1006) | 3295 (4048) | 2756 (12735) | 3461 (61280)
2 | $L=0$ | 1079 ( 82) | 1578 (457) | 1903 ( 2164) | 2743 (12233) | 2671 (48177)
| $L=5$ | 1978 (152) | 2594 (757) | 14424 (16476) | $>$24 hours | $>$40 hours
| $L=10$ | 2205 (176) | 3130 (929) | 13569 (15603) | $>$24 hours | $>$40 hours
| $L=15$ | 2402 (205) | 2803 (873) | 11970 (14053) | $>$24 hours | $>$40 hours
| $L=20$ | 1796 (169) | 1869 (624) | 8710 (10687) | 13551 (63016) | $>$40 hours
| $L=25$ | 1615 (168) | 1758 (629) | 6417 ( 8239) | 10299 (48420) | $>$40 hours
Table 1. MINRES iteration count (CPU time) without preconditioning $m$ | $N$ | 2000 | 4000 | 8000 | 16000 | 32000
---|---|---|---|---|---|---
0 | $L=0$ | 31 ( 6) | 39 (18) | 30 ( 51) | 29 (225) | 39 ( 959)
| $L=5$ | 59 (11) | 71 (31) | 58 ( 96) | 62 (465) | 75 (1802)
| $L=10$ | 70 (13) | 88 (39) | 70 (116) | 71 (532) | 89 (2120)
| $L=15$ | 76 (15) | 93 (43) | 76 (128) | 76 (572) | 96 (2279)
| $L=20$ | 83 (17) | 98 (48) | 80 (138) | 82 (780) | 99 (2393)
| $L=25$ | 95 (20) | 97 (50) | 80 (142) | 84 (649) | 104 (2505)
1 | $L=0$ | 43 ( 8) | 75 (34) | 35 ( 66) | 29 (227) | 47 (1161)
| $L=5$ | 76 (15) | 128 (57) | 83 (138) | 74 (557) | 105 (2559)
| $L=10$ | 91 (18) | 148 (67) | 98 (163) | 94 (705) | 136 (3296)
| $L=15$ | 98 (19) | 168 (78) | 96 (163) | 100 (758) | 148 (3615)
| $L=20$ | 107 (22) | 170 (83) | 103 (180) | 103 (788) | 153 (3754)
| $L=25$ | 106 (23) | 174 (89) | 103 (185) | 113 (870) | 161 (3972)
2 | $L=0$ | 64 (13) | 149 ( 61) | 46 ( 81) | 30 ( 243) | 61 (1419)
| $L=5$ | 95 (19) | 157 ( 70) | 88 (151) | 103 ( 797) | 140 (3380)
| $L=10$ | 95 (19) | 171 ( 71) | 102 (176) | 111 ( 861) | 146 (3425)
| $L=15$ | 112 (23) | 187 (123) | 113 (199) | 118 ( 923) | 165 (3614)
| $L=20$ | 119 (25) | 197 ( 86) | 115 (207) | 133 (1052) | 196 (4301)
| $L=25$ | 125 (28) | 201 ( 91) | 119 (220) | 131 (1046) | 203 (4433)
Table 2. MINRES iteration count(CPU time) with preconditioning $\widehat{A}$
and $\widehat{S}$
The preconditioning is seen to be effective: as anticipated from the theory
above, the number of iterations remains approximately constant over all
choices of $N$ for each degree $L$. Indeed, for each $N$ aside from the simple
case $L=0$ (in which the radial basis function approximation matrix is only
supplemented by one row and one column), the iteration counts grow only slowly
with increasing $L$.
In order to determine the descriptiveness of the bound (29) we have also
computed the generalised eigenvalues $\lambda_{i}$ of the pencil
$Q_{X,L}^{T}A_{X}^{-1}Q_{X,L}-\lambda\Lambda_{L}$ for the case $N=4000$
points, for $m=0,1$ and for $L=5,10,15,20,25$. Note that the generalised
eigenvalues are exactly the eigenvalues of
$\Lambda^{-1}_{L}(Q_{X,L}^{T}A_{X}^{-1}Q_{X,L})$. The minimum and maximum
computed eigenvalues are given in Table 3. It is noticeable how close the
largest eigenvalue is to the analytical upper bound of $1$ and that, although
the lowest eigenvalue does decrease for larger $L$, it remains reasonably
close to $1$. (Note that for fixed $X$ and increasing $L$ the inf-sup
condition must eventually break down.)
$m$ | $L$ | 5 | 10 | 15 | 20 | 25
---|---|---|---|---|---|---
0 | $\lambda_{\min}$ | 0.9987434 | 0.9899326 | 0.9623012 | 0.9068357 | 0.8348191
| $\lambda_{\max}$ | 0.9997653 | 0.9997658 | 0.9997674 | 0.9997753 | 0.9998099
1 | $\lambda_{\min}$ | 0.9999955 | 0.9999125 | 0.9993989 | 0.9973949 | 0.9908182
| $\lambda_{\max}$ | 0.9999986 | 0.9999986 | 0.9999986 | 0.9999986 | 0.9999989
Table 3. Extreme eigenvalues of
$Q_{X,L}^{T}A_{X}^{-1}Q_{X,L}-\lambda\Lambda_{L}$ for $N=4000$
## 6\. Conclusions
By employing a recent inf-sup stability result of Sloan and Wendland, we have
derived an effective preconditioned iterative solver for the hybrid radial
basis function and spherical polynomial approximation scheme of Sloan and
Sommariva. The preconditioner requires only a good approximation for the
radial basis function interpolation problem and a simple diagonal scaling
matrix for the Schur complement based on the Fourier-Legendre coefficients of
the kernel function used in the radial basis function interpolant. We have
established theoretically that the preconditioner for the Schur complement is
optimal.
Acknowledgements. The support of the Australian Research Council is gratefully
acknowledged.
## References
* [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
* [2] M. Benzi, G. H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica 14 (2005), 1–137.
* [3] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, R.A.I.R.O, R-2 (1974), 129–151.
* [4] F. Brezzi, New applications of mixed finite element methods, Proc Int. Congress of Mathematicians, Berkeley, CA (1986), 1335–1347.
* [5] H. C. Elman, D. J. Silvester, A. J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2005.
* [6] M. von Golitschek, W. Light, Interpolation by polynomials and radial basis functions on spheres, Constr. Approx. 17 (2001), 1–18.
* [7] Q. T. Le Gia, I. H. Sloan, T. Tran, Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere, Math.Comp. 78 (2009), 79–101.
* [8] M. F. Murphy, G. H. Golub, A. J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput. 21 (2000), 1969–1972.
* [9] F. J. Narcowich, J. D. Ward, Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33 (2002), 1393–1410.
* [10] C. C. Paige, M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), 617–629.
* [11] E. B. Saff, A. B. J. Kuijlaars, Distributing many points on a sphere, Math. Intell. 19 (1997), 5–11.
* [12] I. H. Sloan, A. Sommariva, Approximation on the sphere using radial basis functions plus polynomials, Adv. Comput. Math. 29 (2008), 147–177.
* [13] I. H. Sloan, H. Wendland, Inf-sup condition for spherical polynomials and radial basis functions on spheres, Math. Comp. 78 (2009), 1319–1331.
* [14] R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. Numer. Anal. 4 (1984), 441–455.
* [15] H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, UK, 2005.
|
arxiv-papers
| 2010-09-22T05:46:24 |
2024-09-04T02:49:12.998464
|
{
"license": "Public Domain",
"authors": "Q. T. Le Gia, Ian H. Sloan, Andrew J. Wathen",
"submitter": "Quoc Thong Le Gia",
"url": "https://arxiv.org/abs/1009.4275"
}
|
1009.4397
|
# Improving quantum entanglement through single-qubit operations
Xiang-Bin Wang Department of Physics and State Key Laboratory of Low-
Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China
Advanced Science Institute, RIKEN, Wako-shi, Saitama, 351-0198, Japan Zong-
Wen Yu Department of Physics and the Key Laboratory of Atomic and
Nanosciences, Ministry of Education, Tsinghua University, Beijing 100084,
China Jia-Zhong Hu Department of Physics and the Key Laboratory of Atomic
and Nanosciences, Ministry of Education, Tsinghua University, Beijing 100084,
China Franco Nori Advanced Science Institute, RIKEN, Wako-shi, Saitama,
351-0198, Japan Physics Department,The University of Michigan, Ann Arbor,
Michigan 48109-1040, USA
###### Abstract
We show that the entanglement of a $2\times 2$ bipartite state can be improved
and maximized probabilistically through single-qubit operations only. An
experiment is proposed and it is numerically simulated.
###### pacs:
03.65.Ud, 03.67.Ac
Introduction.— Quantum entanglement plays a central role in quantum
information and also in the foundations of quantum physics. Thus, it has been
extensively studied (see, e.g., amico ; horodecki ; pv ; Yutin ; sci ; nor ).
One important topic here is how to improve quantum entanglement of a bipartite
quantum state bennett . As is well known, quantum entanglement can be improved
through entanglement purification bennett where a bipartite state is first
transformed to a Werner state and then two-qubit operations at each sides are
needed to improve the quantum entanglement probabilistically.
In this letter, we shall present a theorem (Theorem 2) to maximize the
entanglement of a two-qubit mixed state through single-qubit operations only.
The theorem can be used to efficiently improve the quantum entanglement of a
mixed state without the difficult 2-qubit operations. Explicitly, given a two-
qubit mixed state $\rho_{\textrm{in}}=\rho_{12}$, by taking local (non-trace-
preserving Italy ) maps on qubit 1 and qubit 2 separately, what is the
maximally achievable entanglement of the normalized outcome state, and what
are the specific maps needed on each qubits.
To make a clear picture of our work we consider the following example with a
pure state $\rho_{\textrm{in}}=|\chi\rangle\langle\chi|$ and
$|\chi\rangle=a|00\rangle+b|11\rangle$ first. Take the following specific non-
trace-preserving map on the first qubit:
$\varepsilon\otimes I(\rho_{{\rm in}})=\hat{M}(\tilde{a},\tilde{b})\otimes
I\cdot\rho_{{\rm in}}\cdot\hat{M}(\tilde{a},\tilde{b})\otimes I,$ (1)
where $\hat{M}(\tilde{a},\tilde{b})=\tilde{a}|0\rangle\langle
0|+\tilde{b}|1\rangle\langle 1|$, and $|\tilde{a}|^{2}+|\tilde{b}|^{2}=1$. We
have
$\varepsilon\otimes I(\rho_{{\rm
in}})=\gamma|\chi^{\prime}\rangle\langle\chi^{\prime}|,$ (2)
and $\gamma=\sqrt{|a\tilde{a}|^{2}+|b\tilde{b}|^{2}}$,
$|\chi^{\prime}\rangle=\frac{a\tilde{a}}{\gamma}|00\rangle+\frac{b\tilde{b}}{\gamma}|11\rangle$.
The entanglement concurrence of the outcome state is
$C(|\chi^{\prime}\rangle\langle\chi^{\prime}|)=\frac{2|a\tilde{a}b\tilde{b}|}{|a\tilde{a}|^{2}+|b\tilde{b}|^{2}}.$
(3)
Setting $|\tilde{a}|=|b|$ and $|\tilde{b}|=|a|$, we shall obtain the maximum
output entangled state
$|\phi^{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ state (up to a
normalization factor). Physically, the map $\hat{M}(\tilde{a},\tilde{b})$ can
be easily realized. For example qip , one can use a polarization-dependent
attenuator, with transmittance proportional to $\tilde{a}$ for a horizontally
polarized photon (state $|0\rangle$) and transmittance proportional to
$\tilde{b}$ for a vertically polarized photon (state $|1\rangle$). Once we
find a photon at the outcome port of the attenuator, the initial state
$|\chi\rangle\langle\chi|$ has been mapped to the outcome state
$|\chi^{\prime}\rangle\langle\chi^{\prime}|$.
Outline of our work.— Our goal is to look for the largest achievable
entanglement through local operations, i.e., among all physical maps
$\varepsilon\otimes\varepsilon^{\prime}:\rho_{\textrm{out}}=\varepsilon\otimes\varepsilon^{\prime}(\rho_{\textrm{in}})$,
which map gives out the largest entanglement of the outcome state
$\rho_{\textrm{out}}$. Most generally, any local map
$\varepsilon\otimes\varepsilon^{\prime}$ can be represented in the form of
Kruss operators Italy :
$\rho_{\textrm{out}}=\sum_{i}{\Gamma_{i}\otimes\Gamma_{i}^{\prime}\cdot\rho_{\textrm{in}}\cdot(\Gamma_{i}\otimes\Gamma_{i}^{\prime})^{\dagger}}=\sum_{i}p_{i}\rho_{i},$
(4)
where
$p_{i}\rho_{i}=\Gamma_{i}\otimes\Gamma_{i}^{\prime}\rho_{\textrm{in}}(\Gamma_{i}\otimes\Gamma_{i}^{\prime})^{\dagger}$.
Denote $C(\rho)$ as the entanglement concurrence Wooters of state $\rho$.
Suppose $C(\rho_{m})$ is the largest among all $\\{C(\rho_{i})\\}$. Obviously,
$C(\rho_{\textrm{out}})\;\leq\;\sum_{i}p_{i}C(\rho_{i})\;\leq\;C(\rho_{m}).$
(5)
Therefore, to find the largest entanglement concurrence of the outcome state
among all local maps, we only need to seek it in the following special class
of maps:
$\rho_{\textrm{out}}=\mathcal{Q}\otimes\mathcal{Q^{\prime}}\rho_{\textrm{in}}(\mathcal{Q}\otimes\mathcal{Q^{\prime}})^{\dagger}$
where $\mathcal{Q},\mathcal{Q}^{\prime}$ are $2\times 2$ positive matrices.
According to the singular-value decomposition, the positive matrix
$\mathcal{Q}$($\mathcal{Q^{\prime}}$) can be decomposed into
$\mathcal{Q}=U_{1}DU$($\mathcal{Q^{\prime}}=U_{1}^{\prime}D^{\prime}U^{\prime}$)
where $U_{1},U$($U_{1}^{\prime},U^{\prime}$) are unitary matrices and
$D$($D^{\prime}$) is a positive-definite diagonal matrix. Since a unitary
transformation plays no role in the entanglement, we only need to consider the
positive matrices in the form of $DU$ ($D^{\prime}U^{\prime}$).
As shown in Lemma 2, any two-qubit state $\rho_{\textrm{in}}$ can be generated
from the maximally-entangled state $|\phi^{+}\rangle$ acted by a one-sided map
$I\otimes\varepsilon^{\prime}$, i.e.,
$\rho_{\textrm{in}}=I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)$.
Therefore, we can start with entanglement evolution and maximization under
non-trace-preserving one-sided maps and then apply the result to the general
problem of improving and maximizing quantum entanglement through single-qubit
operations.
Entanglement evolution and maximization under non-trace-preserving maps.— A
non-trace-preserving one-sided map $I\otimes\varepsilon^{\prime}$ is fully
characterized by
$\rho_{\varepsilon^{\prime}}=I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)$
liusky . We assume
$I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)=f\rho_{\varepsilon^{\prime}},\;I\otimes\varepsilon^{\prime}(|\psi\rangle\langle\psi|)=f^{\prime}\rho_{\psi}\;,$
(6)
where $f={\rm
tr}\left[I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)\right]$,
$f^{\prime}={\rm
tr}\left[I\otimes\varepsilon^{\prime}(|\psi\rangle\langle\psi|)\right]$.
A $2\times 2$ pure state $|\chi\rangle=a|00\rangle+b|11\rangle$ can be
rewritten in the form $|\chi\rangle\langle\chi|=2\hat{M}(a,b)\otimes
I(|\phi^{+}\rangle\langle\phi^{+}|)\hat{M}(a,b)\otimes I$. From Eq. (6) we
have
$I\otimes\varepsilon^{\prime}(|\chi\rangle\langle\chi|)=2f\hat{M}(a,b)\otimes
I\rho_{\varepsilon^{\prime}}\hat{M}(a,b)\otimes I=f^{\prime}\rho_{\chi}$. We
emphasize here that even though $\rho_{\varepsilon^{\prime}}$ is normalized,
the operator $2\hat{M}(a,b)\otimes
I\rho_{\varepsilon^{\prime}}\hat{M}(a,b)\otimes I$ is not necessarily
normalized. Define the following function $C$ of an arbitrary non-negative
definite $4\times 4$ matrix (operator) $N$
$C(N)=\rm{max}\\{0,\sqrt{\xi}_{1}-\sqrt{\xi}_{2}-\sqrt{\xi}_{3}-\sqrt{\xi}_{4}\\},$
(7)
where $\\{\xi_{i}\\}$ are the eigenvalues of $N\cdot\tilde{N}$, in descending
order, with
$\tilde{N}=\sigma_{y}\otimes\sigma_{y}N^{*}\sigma_{y}\otimes\sigma_{y}$, and
$N^{*}$ is the complex conjugate of $N$. If $N$ is a density matrix of a
$2\times 2$ system, $C(N)$ is just the entanglement concurrence of the system
Wooters . With this definition of $C$, we can summarize the major result,
equation (5) in Ref. 1 as:
Lemma 1. Given any density matrix $\rho_{\varepsilon^{\prime}}$, if
$N=2\hat{M}(a,b)\otimes
I\rho_{\varepsilon^{\prime}}\hat{M}^{\dagger}(a,b)\otimes I$, then
$C(N)\;=\;C(|\chi\rangle\langle\chi|)\cdot
C(\rho_{\varepsilon^{\prime}})\;=\;2|ab|C(\rho_{\varepsilon^{\prime}}).$ (8)
However, this is not the entanglement concurrence of $\rho_{\chi}$ because $N$
is not necessarily normalized, even though $\rho_{\varepsilon^{\prime}}$ is.
Now denote $N=g\rho_{\chi}$, and $g={\rm tr}N$. According to the definition of
$C$ and $\rho_{\psi}$ in Eq. (6),
$C(\rho_{\chi})\ =\ C(N)/g\ =\ 2|ab|C(\rho_{\varepsilon^{\prime}})/g$ (9)
where $g={\rm tr}N=2{\rm tr}[\hat{M}(a,b)\otimes
I\rho_{\varepsilon^{\prime}}\hat{M}^{\dagger}(a,b)\otimes I]$. To avoid
meaningless results, we assume $C(\rho_{\varepsilon}^{\prime})>0$ throughout
this paper. Assume that the density matrix of the first qubit of
$\rho_{\varepsilon^{\prime}}$ is
$\rho_{0}=\rm{tr}_{2}\rho_{\varepsilon^{\prime}}=\left(\begin{array}[]{cc}c_{1}&\alpha\\\
\alpha^{*}&c_{2}\end{array}\right)=K_{0}$, where $\rm{tr}_{2}$ is the partial
trace over the subspace of the second qubit and $c_{1}=\langle 0|{\rm
tr}_{2}\rho_{\varepsilon^{\prime}}|0\rangle,\;c_{2}=\langle 1|{\rm
tr}_{2}\rho_{\varepsilon^{\prime}}|1\rangle.$ Consequently,
$g=2{\rm tr}[M(a,b)\rho_{0}M^{\dagger}(a,b)]=2|a|^{2}c_{1}+2|b|^{2}c_{2}.$
(10)
Therefore, the value of output entanglement
$C(\rho_{\chi})=\frac{2|ab|C(\rho_{\varepsilon^{\prime}})}{(|a|^{2}c_{1}+|b|^{2}c_{2})},$
(11)
is maximized when $|a|=\sqrt{c_{2}},\;|b|=\sqrt{c_{1}}$, with the maximum
value
$C(\rho_{\chi})=\frac{C(\rho_{\varepsilon^{\prime}})}{2\sqrt{c_{1}c_{2}}}.$
(12)
More generally, the initial pure state can be
$|\psi\rangle=I\otimes U|\chi\rangle=\sqrt{2}\hat{M}(a,b)\otimes
U|\phi\rangle,$ (13)
where $U$ is an arbitrary unitary operator. Given the fact that $U^{*}\otimes
U|\phi^{+}\rangle=|\phi^{+}\rangle$ for any unitary $U$, we have
$\sqrt{2}\hat{M}(a,b)\otimes
U|\phi^{+}\rangle=\sqrt{2}\hat{M}(a,b)U^{T}\otimes I|\phi^{+}\rangle.$ (14)
In such a case, we obtain
$C(\rho_{\psi})=2|ab|\cdot C(\rho_{\varepsilon^{\prime}})/g^{\prime}$ (15)
and $g^{\prime}={\rm tr}[\hat{M}(a,b)U^{T}\otimes
I\rho_{\varepsilon^{\prime}}U^{*}\hat{M}(a,b)\otimes I]$. To maximize
$C(\rho_{\psi})$, we first fix $U$ and maximize it with $a,b$. Assume
$U^{T}K_{0}U^{*}=\left(\begin{array}[]{cc}c_{1}^{\prime}&\alpha^{\prime}\\\
\alpha^{\prime*}&c_{2}^{\prime}\end{array}\right).$ The largest value for
$C(\rho_{\psi})$ is
${C(\rho_{\varepsilon^{\prime}})}/{2\sqrt{c_{1}^{\prime}c_{2}^{\prime}}}$, as
shown already. To maximize the value over all $U$, we only need to minimize
$c_{1}^{\prime}c_{2}^{\prime}$. Since $U$ is unitary,
$\det(U^{T}K_{0}U^{*})=\det K_{0}$. Therefore
$c_{1}^{\prime}c_{2}^{\prime}=\det K_{0}+|\alpha^{\prime}|^{2}$, which is
minimized when $\alpha^{\prime}=0$. Namely, $C(\rho_{\psi})$ is maximized when
$U^{T}K_{0}U^{*}$ is diagonalized and
$\sqrt{a}=c_{2}^{\prime},~{}\sqrt{b}=c_{1}^{\prime}$, i.e.,
$\hat{M}(a,b)U^{T}\left({\rm
tr}_{2}\rho_{\varepsilon^{\prime}}\right)U^{*}\hat{M}(a,b)={\rm
diag}[1/2,1/2]$. We obtain
Theorem 1. Denote $\mathcal{Q}$ to be a $2\times 2$ positive-definite matrix.
Given the inseparable two-qubit density matrix
$\rho_{\textrm{in}}=\rho_{\varepsilon^{\prime}}=I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)$,
the entanglement of the normalized density matrix $\rho_{1}=\mathcal{Q}\otimes
I\rho_{\textrm{in}}(\mathcal{Q}\otimes I)^{\dagger}$ maximizes when
$\mathcal{Q}\left({\rm
tr_{2}}\rho_{\textrm{in}}\right){\mathcal{Q}}^{\dagger}={\rm diag}[1/2,1/2]$
and the entanglement concurrence is:
$C_{M}=\frac{C(\rho_{\varepsilon^{\prime}})}{2\sqrt{\det\left[{\rm
tr}_{2}\rho_{\varepsilon^{\prime}}\right]}}.$ (16)
Improving and maximizing quantum entanglement through single-qubit
operations.— To apply our theorem, we need the following lemma:
Lemma 2. Given any $2\times 2$ bipartite mixed state $\rho_{12}$, there exists
a map $\varepsilon^{\prime}$ such that
$\rho_{\textrm{in}}=I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)$.
Note that map $\varepsilon^{\prime}$ here is in general non-trace-preserving.
Since any two-qubit density matrix $\rho_{\textrm{in}}$ can be decomposed into
the mixture of a few pure states, say
$\rho_{\textrm{in}}=\sum_{i}\lambda_{i}|\psi_{i}\rangle\langle\psi_{i}|$.
Obviously, for any bipartite pure state $|\psi_{i}\rangle$, there always
exists a positive operator $\hat{M}_{i}^{\prime}$ such that
$|\psi_{i}\rangle=I\otimes\hat{M}_{i}^{\prime}|\phi^{+}\rangle$. Therefore, we
have
$\rho_{\textrm{in}}=\sum_{i}\lambda_{i}I\otimes\hat{M}_{i}^{\prime}|\phi^{+}\rangle\langle\phi^{+}|I\otimes\hat{M}_{i}^{\prime\dagger}$.
Denoting
$I\otimes\varepsilon^{\prime}(|\phi^{+}\rangle\langle\phi^{+}|)=\sum_{i}I\otimes\sqrt{\lambda}_{i}\hat{M}_{i}^{\prime}|\phi^{+}\rangle\langle\phi^{+}|I\otimes\sqrt{\lambda}_{i}\hat{M}_{i}^{\prime\dagger}$
completes the proof.
With Theorem 1 and Lemma 2, we can improve the quantum entanglement of any
2-qubit state $\rho_{\textrm{in}}$ (here and after, the 2-qibit states are
normalized) step by step, with single-qubit operations only. Denote
$K_{1}={\rm tr}_{2}\rho_{\textrm{in}}$, if $\det K_{1}<1/4$, we construct
$\hat{M}_{1}(\tilde{a}_{1},\tilde{b}_{1})$ and local unitary $U_{1}$ such that
$\hat{M}_{1}U_{1}K_{1}U_{1}^{\dagger}\hat{M}_{1}^{\dagger}={\rm
diag[1/2,1/2]}$. The local operation on qubit 1 transforms state
$\rho_{\textrm{in}}$ into the outcome state $\rho_{1}=\hat{M}_{1}U_{1}\otimes
I\rho_{\textrm{in}}U_{1}^{\dagger}\hat{M}_{1}^{\dagger}\otimes I$. According
to Theorem 1, the entanglement concurrence of the outcome state is
$C(\rho_{1})=C(\rho_{\textrm{in}})/(2\sqrt{\det{K_{1}}})>C(\rho_{\textrm{in}})$.
The normalized density matrix of qubit 1 is $\textrm{tr}_{2}\rho_{1}={\rm
diag}[1/2,1/2]$ now, but in general the normalized density operator of qubit 2
is not ${\rm diag}[1/2,1/2]$ now. Using Lemma 2, we know $\rho_{1}$ can be
written in the form of $\rho_{1}=\varepsilon\otimes
I(|\phi^{+}\rangle\langle\phi^{+}|)$. We can now apply Theorem 1 again to
further improve the quantum entanglement through operation on qubit 2. Denote
$K_{1}^{\prime}={\rm{tr}_{1}}\rho_{1}$. If $\det(K_{1}^{\prime})<1/4$, we
construct new operators $\hat{M}_{1}^{\prime}$ and $U_{1}^{\prime}$ such that
the density matrix of qubit 2 is ${\rm diag}[1/2,1/2]$ after the operation,
i.e., $\hat{M}_{1}^{\prime}U_{1}^{\prime}({\rm
tr}_{1}\rho_{1})U_{1}^{{}^{\prime}\dagger}\hat{M}_{1}^{{}^{\prime}\dagger}={\rm
diag}[1/2,1/2]$. The operation on qubit 2 leads to a new outcome state
$\rho_{1}^{\prime}=I\otimes\hat{M}_{1}^{\prime}U_{1}^{\prime}\rho_{1}I\otimes
U_{1}^{{}^{\prime}\dagger}\hat{M}_{1}^{{}^{\prime}\dagger}$. The operation on
qubit 2 improves the entanglement concurrence to
$C(\rho_{1}^{\prime})=C(\rho_{1})/(2\sqrt{\det K_{1}^{\prime}})>C(\rho_{1})$.
After the non-trace-preserving operation above on qubit 2, we have
$K_{1}^{\prime}=\textrm{tr}_{1}\rho_{1}^{\prime}={\rm diag}[1/2,1/2]$, but in
general the density matrix of qubit 1 is not ${\rm diag}[1/2,1/2]$, i.e.
$K_{2}=\textrm{tr}_{2}\rho_{1}^{\prime}\not={\rm diag}[1/2,1/2]$ now. We can
construct new operators $\hat{M}_{2}$ and $U_{2}$ to improve the entanglement
of $\rho_{1}^{\prime}$. The process will continue step by step until the
determinant of two reduced density matrices are all equal to $1/4$ after many
steps of iterations. Since the entanglement concurrence of a two-qubit state
can never be greater than 1 and the entanglement always increases during the
iteration process above, there must exist a limit value of the entanglement in
the process say, after many steps of iterations, the process gives out the
largest entanglement concurrence. This also means that after many steps of
iterations, the process always produces a two-qubit state where the reduced
density matrices of each qubit are ${\rm diag}[1/2,1/2]$ simultaneously.
Therefore the process that transfers the reduced density matrix of qubit 1 and
the reduced density matrix of qubit 2 into ${\rm diag}[1/2,1/2]$
simultaneously always exists and can be written in the following form:
$\rho_{f}=\prod_{k=1}^{\infty}{\hat{M}_{k}U_{k}\otimes\hat{M}_{k}^{\prime}U_{k}^{\prime}}\cdot\rho_{\textrm{in}}\cdot\prod_{k=1}^{\infty}{U_{k}^{\dagger}\hat{M}_{k}^{\dagger}\otimes
U_{k}^{{}^{\prime}\dagger}\hat{M}_{k}^{{}^{\prime}\dagger}}.$ (17)
At the same time, the final state $\rho_{f}$ satisfies the following
condition:
${\rm tr_{1}}\rho_{f}={\rm tr_{2}}\rho_{f}=I/2.$ (18)
As an example, consider the imperfect entangled state
$\tilde{\rho}=0.1|\varphi^{\prime}\rangle\langle\varphi^{\prime}|+0.12|\varphi^{\prime\prime}\rangle\langle\varphi^{\prime\prime}|+0.78|\phi^{+}\rangle\langle\phi^{+}|,$
(19)
where $|\varphi^{\prime}\rangle=\hat{R}(\theta_{1})\otimes I|00\rangle$,
$|\varphi^{\prime\prime}\rangle=\hat{R}(\theta_{2})\otimes I|11\rangle$ with
$\theta_{1}=\pi/5,\theta_{2}=-3\pi/10$ and
$\hat{R}(\theta)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{array}\right)$. The entanglement increase through 7
steps of iteration is shown in Fig. 1.
Figure 1: (color online) The concurrence $C$ versus number of iterations of
single-qubit operation.
The remaining task is to show that, starting from the same state
$\rho_{\textrm{in}}$, all final states satisfying Eq. (18) have the same value
for entanglement concurrence.
Lemma 3. If state $\rho_{f}$ satisfies Eq. (18), then state
$\rho_{f}^{\prime}=U\otimes
U^{\prime}\rho_{f}U^{\dagger}\otimes{U^{\prime}}^{\dagger}$ also satisfies Eq.
(18). Here $U,U^{\prime}$ are any two unitary operators.
This conclusion is obvious since a unity density operator remains to be unity
after any local unitary transformation.
Assume we have two different final states $\rho_{u},\rho_{v}$ obtained by
using different processes from the same initial state, and they satisfy Eq.
(18). Suppose
$\rho_{u}=\prod_{k=1}^{\infty}{\hat{\mathcal{M}}_{k}\mathcal{U}_{k}\otimes\hat{\mathcal{M}}_{k}^{\prime}\mathcal{U}_{k}^{\prime}}\cdot\rho_{\textrm{in}}\cdot\prod_{k=1}^{\infty}{\mathcal{U}_{k}^{\dagger}\hat{\mathcal{M}}_{k}^{\dagger}\otimes\mathcal{U}_{k}^{{}^{\prime}\dagger}\hat{\mathcal{M}}_{k}^{{}^{\prime}\dagger}},$
$\rho_{v}=\prod_{k=1}^{\infty}{\hat{\mathcal{N}}_{k}\mathcal{V}_{k}\otimes\hat{\mathcal{N}}_{k}^{\prime}\mathcal{V}_{k}^{\prime}}\cdot\rho_{\textrm{in}}\cdot\prod_{k=1}^{\infty}{\mathcal{V}_{k}^{\dagger}\hat{\mathcal{N}}_{k}^{\dagger}\otimes\mathcal{V}_{k}^{{}^{\prime}\dagger}\hat{\mathcal{N}}_{k}^{{}^{\prime}\dagger}},$
where
$\hat{\mathcal{M}}_{k},\hat{\mathcal{M}}_{k}^{\prime},\hat{\mathcal{N}}_{k},\hat{\mathcal{N}}_{k}^{\prime}$
are projective operators and
$\mathcal{U}_{k},\mathcal{U}_{k}^{\prime},\mathcal{V}_{k},\mathcal{V}_{k}^{\prime}$
are unitary operators. By using singular-value decomposition, We have
$\rho_{v}=\tilde{\mathcal{W}}\mathcal{P}\mathcal{W}\otimes\tilde{\mathcal{W}}^{\prime}\mathcal{P}^{\prime}\mathcal{W}^{\prime}\cdot\rho_{u}\cdot\mathcal{W}^{\dagger}\mathcal{P}^{\dagger}\tilde{\mathcal{W}}^{\dagger}\otimes{\mathcal{W}^{\prime\dagger}}{\mathcal{P}^{\prime}}^{\dagger}{\tilde{\mathcal{W}}^{\prime\dagger}},$
(20)
where
$\tilde{\mathcal{W}},\mathcal{W},\tilde{\mathcal{W}}^{\prime},\mathcal{W}^{\prime}$
are unitary operators and $\mathcal{P},\mathcal{P}^{\prime}$ are projective
operators defined in Eq. (1). Denote
$\tilde{\rho}_{w}=\tilde{\mathcal{W}}^{\dagger}\otimes{\tilde{\mathcal{W}}^{\prime\dagger}}\rho_{v}\tilde{\mathcal{W}}\otimes\tilde{\mathcal{W}}^{\prime}$
and
${\rho}_{w}=\mathcal{W}\otimes\mathcal{W}^{\prime}\rho_{u}\mathcal{W}^{\dagger}\otimes{\mathcal{W}^{\prime\dagger}}$.
We have
$\tilde{\rho}_{w}=\mathcal{P}\otimes\mathcal{P}^{\prime}\rho_{w}\mathcal{P}^{\dagger}\otimes{\mathcal{P}^{\prime\dagger}}.$
(21)
According to Lemma 3, we know that $\tilde{\rho}_{w},\rho_{w}$ satisfy Eq.
(18). Thus $\mathcal{P}$ and $\mathcal{P}^{\prime}$ must be either identity or
$\textrm{diag}[1,i]$. This indicates that $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ are unitary therefore the entanglement concurrence of
$\tilde{\rho}_{w}$ and $\rho_{w}$ must be same. We now obtain the major result
of this letter:
Theorem 2. Given any inseparable two-qubit initial state $\rho_{\textrm{in}}$,
the entanglement concurrence can be improved through single-qubit operations
provided that the reduced density matrix of any one qubit is not ${\rm
diag[1/2,1/2]}$. Among all out-come states
$\\{\rho_{\textrm{out}}|\rho_{\textrm{out}}=\varepsilon\otimes\varepsilon^{\prime}(\rho_{\textrm{in}})\\}$
through positive-definite local maps, the state
$\rho_{\textrm{out}}=\mathcal{Q}\otimes\mathcal{Q^{\prime}}\rho_{\textrm{in}}\mathcal{Q}^{\dagger}\otimes\mathcal{Q^{\prime}}^{\dagger}$
has the largest entanglement concurrence if the density matrices of each qubit
of the outcome state are $I/2$. The corresponding local maps at each side are
simply positive-definite matrices $\mathcal{Q},\mathcal{Q^{\prime}}$ which can
be constructed by Eq. (17), i.e.,
$\mathcal{Q}=\prod_{k=1}^{\infty}{\hat{M}_{k}U_{k}},\mathcal{Q^{\prime}}=\prod_{k=1}^{\infty}{\hat{M}_{k}^{\prime}U_{k}^{\prime}}$
where ${\hat{M}_{k}U_{k}}$ ($\hat{M}_{k}^{\prime}U_{k}^{\prime}$) diagonalize
the state of the first (second) qubit into the form of $I/2$ at the
corresponding step. Specifically, $\textrm{tr}_{2}[(\hat{M}_{k}U_{k}\otimes
I)\rho_{k-1}^{\prime}(\hat{M}_{k}U_{k}\otimes I)^{\dagger}]=I/2$ and
$\textrm{tr}_{1}[(I\otimes\hat{M}_{k}^{\prime}U_{k}^{\prime})\rho_{k}(I\otimes\hat{M}_{k}^{\prime}U_{k}^{\prime})^{\dagger}]=I/2$
where
$\rho_{k}^{\prime}=\prod_{i=1}^{k}{(\hat{M}_{i}U_{i}\otimes\hat{M}_{i}^{\prime}U_{i}^{\prime})\rho_{\textrm{in}}(\hat{M}_{i}U_{i}\otimes\hat{M}_{i}^{\prime}U_{i}^{\prime})^{\dagger}}$
and $\rho_{k}=\hat{M}_{k}U_{k}\otimes
I\rho_{k-1}^{\prime}(\hat{M}_{k}U_{k})^{\dagger}\otimes I$.
Remark: In the theorem, we have presented a mathematical way to construct
$\mathcal{Q},\mathcal{Q^{\prime}}$ by iteration. We emphasize that, in
applying our theorem in a real experiment, one can compute
$\mathcal{Q},\mathcal{Q^{\prime}}$ and then realize the physical process in
only one step.
Proposed experiment and numerical simulation.— We propose to test Theorem 2
with the initial state $\tilde{\rho}$ as defined in Eq. (19). With many
iterations, we have
$\mathcal{Q}=\prod_{k}\hat{M}_{k}U_{k}=\left(\begin{array}[]{cc}0.5875&-0.8090\\\
0.0130&0.0095\end{array}\right)$ and
$\mathcal{Q^{\prime}}=\prod_{k}{\hat{M}^{\prime}}_{k}U_{k}^{\prime}=\left(\begin{array}[]{cc}0.0106&-0.0145\\\
0.8091&0.5874\end{array}\right)$. Then we find that the entanglement
concurrence of the final state
$\mathcal{Q}\otimes\mathcal{Q}^{\prime}\tilde{\rho}\mathcal{Q}^{\dagger}\otimes{\mathcal{Q}^{\prime}}^{\dagger}$
is 0.8858. Changing matrices $\mathcal{Q}$ and $\mathcal{Q^{\prime}}$, the
outcome entanglement is always smaller than 0.8858. Numerical results are
presented in Fig.(2) and Fig.(3).
Figure 2: (color online) The concurrence $C$ versus $\theta_{1}$ and
$\theta_{2}$. Here we have
$\mathcal{Q}=\hat{M}(a_{1},b_{1})U(\theta_{1}),\mathcal{Q^{\prime}}=\hat{M}(a_{2},b_{2})U(\theta_{2})$
with $a_{1}=0.99987,a_{2}=0.01797,U(\theta)=\hat{R}(\theta)\sigma_{z}$ and
$\sigma_{z}$ is the Pauli-$z$ matrix. The peak point indicates the maximum
concurrence 0.8858 with $\theta_{1}=0.9427$ and $\theta_{2}=0.9428$.
Figure 3: (color online) The concurrence $C$ versus $\theta_{1}$ [in (a)] and
$\theta_{2}$ [in (b)]. Here we have
$\mathcal{Q}=\hat{M}(a_{1},b_{1})U(\theta_{1}),\mathcal{Q^{\prime}}=\hat{M}(a_{2},b_{2})U(\theta_{2})$
where $U(\theta)=\hat{R}(\theta)\sigma_{z}$. We set
$a_{1}=0.99987,a_{2}=0.01797$ [in (a)] and $a_{2}=0.01797,\theta_{1}=0.9427$
[in (b)]. The peak points indicate the maximum concurrence 0.8858 with
$\theta_{1}=0.9427$ [in (a)] and $\theta_{2}=0.9428$ [in (b)].
Concluding remark.— In summary, we have presented explicit results on
probabilistically improving and maximizing the quantum entanglement of a mixed
state through single-qubit operations only. Testing schemes are proposed with
numerical simulations. The local operator maximize the outcome entanglement
concurrence and can be constructed numerically by iteration. It is interesting
to construct the operators directly from the initial $\rho_{\textrm{in}}$
analytically.
###### Acknowledgements.
XBW is supported by the National Natural Science Foundation of China under
Grant No. 60725416, the National Fundamental Research Programs of China Grant
No. 2007CB807900 and 2007CB807901, and China Hi-Tech Program Grant No.
2006AA01Z420. FN acknowledges partial support from the NSA, LPS, ARO, AFOSR,
DARPA, NSF Grant No. 0726909, JSPS-RFBR Contract No. 09-02-92114, Grant-in-Aid
for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the
JSPS-FIRST Funding Program.
## References
* (1) L. Amico et al., Rev. Mod. Phys. 80, 517 (2008)
* (2) R. Horodecki et al., Rev. Mod. Phys. 81, 865 (2009)
* (3) M.B. Plenio, V. Vedral, Contemp. Phys. 39, 431 (1998).
* (4) T. Yu, J.H. Eberly, Science 323, 598 (2009).
* (5) O. Jimenez Farias et al., Science, 324, 1414, (2009).
* (6) J. Ma, X. Wang, C. P. Sun, F. Nori, Phys. Reports, in press (2011); arXiv:1011.2978v2
* (7) C.H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993); M. Zukowski et al., Phys. Rev. Lett. 71, 4287 (1993); J.W. Pan et al., Phys. Rev. Lett. 80, 3891 (1998).
* (8) See e.g., I. Bongioanni et al., arXiv: 1008.5334v1
* (9) M. Tiersch et al., Quantum Inf. Process. 8, 523 (2009).
* (10) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
* (11) A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972).
* (12) T. Konrad et al., Nature Physics 4, 99 (2008).
|
arxiv-papers
| 2010-09-22T15:57:19 |
2024-09-04T02:49:13.007583
|
{
"license": "Public Domain",
"authors": "X.-B. Wang, Z.-W. Yu, J.-Z. Hu, and F. Nori",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1009.4397"
}
|
1009.4489
|
###### Abstract
We exploit the symmetry concepts developed in the companion review of this
article to introduce a stochastic version of link reversal symmetry, which
leads to an improved understanding of the reciprocity of directed networks. We
apply our formalism to the international trade network and show that a strong
embedding in economic space determines particular symmetries of the network,
while the observed evolution of reciprocity is consistent with a symmetry
breaking taking place in production space. Our results show that networks can
be strongly affected by symmetry-breaking phenomena occurring in embedding
spaces, and that stochastic network symmetries can successfully suggest, or
rule out, possible underlying mechanisms.
###### keywords:
complexity; networks; symmetry breaking; World Trade Web
10.3390/—— xx Received: xx / Accepted: xx / Published: xx Complex Networks and
Symmetry II: Reciprocity and Evolution of World Trade Franco Ruzzenenti 1,⋆,
Diego Garlaschelli 2 and Riccardo Basosi 3 E-Mail: ruzzenenti@unisi.it; Tel.:
+39-0577-234240; Fax: +39-0577-234239.
## 1 Introduction
In this paper, we take full advantage to the symmetry concepts developed in
the companion review symmetry1 in order to study in great detail two
applications of stochastic symmetry in networks. First, we discuss link
reversal symmetry in directed networks and introduce its stochastic variant to
highlight the connections with the important property of reciprocity. Second,
we consider the problem of (stochastic) symmetry breaking in spatially
embedded networks (where the embedding space is in general not necessarily
Euclidean or geographic, but specified by different variables). Both
applications will be discussed in the context of an empirical analysis of a
particular network: the World Trade Web, defined by the international trade
relationships existing among all world countries.
The investigation of symmetry naturally leads to the fascinating problem of
symmetry breaking. ‘All scientific applications of symmetry are based on the
principle that _identical causes produce identical effects_ ’ 1 . That is to
say, the symmetry of the effect must be at least that of the cause, or in a
mathematical jargon: the order of the _symmetry group_ of the effect must be
at least equivalent to that of the cause. Nevertheless, for a qualitatively
new phenomenon to occur, symmetry cannot be conserved. Pierre Curie was the
first in modern science who highlighted the relevance of spontaneous symmetry
breaking in various phenomena 2 . His studies of criticality in phase
transitions overcame the boundaries of solid state physics and posed a
suitable analytical framework for further studies, in different fields.
Prigogine himself, a renowned and illustrious precursor of complex systems
research, referred to Curie’s contribution in order to elucidate the meaning
of symmetry breaking in dissipative structures: ‘We see therefore, that the
appearance of a periodic reaction is a time-symmetry breaking process exactly
as ferromagnetism is a space-symmetry breaking one’ 3 . Two main viewpoints on
symmetry breaking developed in science: one concerned with space symmetry and
one with time symmetry. Symmetry breaking can be indeed approached from two
different sides: from a time-scale perspective, as we see the phenomenon as a
dynamic system, or from a spatial perspective, as we focus on changes in the
system’s space. With the former approach we tend to consider changes that are
endogenous to the system, while space is taken as homogeneous all the way
through; with the latter, the system is embedded in some space and changes are
considered exogenous. However, whereas in physics space-symmetry breaking has
been a prominent research subject, in the field of complex nonlinear systems
time-symmetry breaking has received much more attention. The following passage
by Mainzer illustrates on what basis time symmetry breaking became a major
topic in the science of complexity: ‘Thus, bifurcation mathematically only
means the emergence of new solutions of equations at critical values.
Actually, bifurcation and symmetry breaking is a purely mathematical
consequence of the theory of nonlinear differential equations. But,
bifurcations of final states as solutions of differential equations correspond
to qualitative changes of dynamical systems and the emergence of new phenomena
in nature and society [..]’4 .
In what follows, spatial symmetry breaking will be approached in the framework
of complex networks. Symmetries relevant to networks can be either ‘internal’,
if they involve purely topological quantities, or ‘external’, if they are
defined with respect to additional properties such as positions in some
embedding space. In the latter case, symmetries relative to the external space
can be reflected in some topological property displayed by the network
symmetry1 . In this sense, spatial symmetry breaking has so far received
little attention in the field of network theory, despite the latter developed
considerably in recent years guidosbook ; largescalestructure ;
dynamicalprocessesoncomplexnetworks ; ecologicalnetworks ; internet ;
networksincellbiology ; adaptivenetworks . On the other hand, specific
analyses of processes that are well described within a network framework
suggest that spatial symmetry breaking can occur with respect to some
embedding space and manifest itself in major structural changes at a
topological level, as happened in the evolution of vascular systems in living
beings west , of river basins banavar , and of production networks in modern
economies 7 ; 8 . In the present paper, we exploit our review of network
symmetries symmetry1 to start investigating this problem. The paper is
organized as follows. In Section 2 we will introduce link reversal symmetry
and we define a new stochastic variant of it. In Section 3 we will then
investigate how stochastic link reversal symmetry, together with other
symmetries discussed in referencsymmetry1 , enable to achieve an improved
understanding of network structure in a specific case, i.e. the problem of
_reciprocity_ in directed networks. We will highlight how different measures
of reciprocity capture different symmetry properties of a network. This will
help us disentangle distinct possible mechanisms explaining the observed
reciprocity structure of real networks. As a particular application, in
Section 5 we will consider the evolution of reciprocity in the World Trade
Web. We will also emphasize the role of spatial embedding, which relates the
topology of the network to underlying geographical coordinates and economic
variables. We will advance heuristic explanations for the observed evolution
of reciprocity in the World Trade Web in terms of symmetry breaking phenomena
due to changes in the underlying economic structure. These analyses highlights
the idea that complex networks are not phenomena _per se_ , but maps of
physical phenomena that are immersed in physical space—or any other space,
depending on the variables determining the system’s dynamics. Symmetry
breaking can occur in some geographical, economic, or different space, and be
mirrored in the topological space the network belongs to. In real imperfect
systems, stochastic symmetry is able to capture spatial patterns that are
undetected by exact symmetries. Interactions between the underlying system’s
‘spaces’ is an intriguing challenge for network theory and pertains the study
of network dynamics.
## 2 Exact and Stochastic Link Reversal Symmetry
In this section we make use of the notion of _stochastic graph symmetries_ we
introduced in Reference symmetry1 to define a new graph invariance, i.e. the
stochastic version of _link reversal symmetry_. To this end, we first briefly
recall the concept of graph ensembles and equiprobability symmetry1 , and then
discuss link reversal symmetry, first in its exact version and finally in its
stochastic variant.
### 2.1 Graph Ensembles and Stochastic Symmetries
In Reference symmetry1 we introduced the concept of graph ensembles as
collections of graphs with specified properties and probability. Each graph
$G$ in a statistical ensemble of graphs has an associated occurrence
probability $P(G)$ satisfying
$\sum_{G}P(G)=1$ (1)
Two graphs $G_{1}$ and $G_{2}$ such that $P(G_{1})=P(G_{2})$ are said to be
_equiprobable_ in the ensemble considered. The probability $P(G)$ can have
different forms depending on the structure of the ensemble under
consideration. In what follows, we will make use of _(grand)canonical_
ensembles, and in particular maximally random graphs with specified
constraints symmetry1 . Such ensembles are defined by specifying the expected
value (ensemble average) of a chosen set of topological properties (the
constraints), and are maximally random otherwise. This means that the
probability $P(G)$ must maximize the entropy of the ensemble subject to the
enforced constraints. The constraints will be denoted as a collection
$\\{c_{1},\dots,c_{K}\\}$ of $K$ topological properties, and the Lagrange
multipliers involved in the constrained maximization problem will be denoted
as the conjugate parameters $\\{\theta_{1},\dots,\theta_{K}\\}$. The
probability $P(G)$ will depend on such parameters, and its explicit form is
$P(G)=\frac{e^{-H(G)}}{Z}$ (2)
where $H(G)$ is the _graph Hamiltonian_
$H(G)\equiv\sum_{a=1}^{K}\theta_{a}c_{a}(G)$ (3)
and $Z$ is the _partition function_
$Z\equiv\sum_{G}e^{-H(G)}$ (4)
The expected value of a topological property $X$, which evaluates to $X(G)$ on
the particular graph $G$, is
$\langle X(\theta_{1},\dots,\theta_{K})\rangle\equiv\sum_{G}P(G)X(G)$ (5)
The values of the parameters $\\{\theta_{1},\dots,\theta_{K}\\}$ are such that
the expected values $\\{\langle c_{1}\rangle,\dots,\langle c_{K}\rangle\\}$ of
the constraints match the specified values. In particular, if the ensemble is
meant as a null model symmetry1 of a real network $G^{*}$, the expected
values of the constraints will have to match the empirical values of the
properties $\\{c_{1}(G^{*}),\dots,c_{K}(G^{*})\\}$ of that particular graph:
$\langle c_{a}(\theta^{*}_{1},\dots,\theta^{*}_{K})\rangle=c_{a}(G^{*})\qquad
a=1,\dots,K$ (6)
The above parameter choice automatically maximises the probability $P(G^{*})$
to obtain the real network $G^{*}$ under the model considered, and is
therefore in accordance with the _maximum likelihood principle_ mylikelihood .
The graph Hamiltonian $H(G)$, which represents a sort of energy or cost
associated to the graph $G$, is a linear combination of the constraints.
Clearly, $P(G_{1})=P(G_{2})$ if and only if $H(G_{1})=H(G_{2})$, which means
that graphs with the same energy are equiprobable (and vice versa). The
transformation mapping $G_{1}$ into a different graph $G_{2}$ with
$H(G_{2})=H(G_{1})$ is a symmetry of the Hamiltonian. Any such transformation
changes the topology of the graph but preserves the values of the constraints
appearing in the Hamiltonian (and is therefore more general than permutations
of vertices).
Graph ensembles provide an ideal framework to study stochastic graph
symmetries, that we defined in Reference symmetry1 . An exact symmetry of a
real network $G^{*}$ is a transformation mapping $G^{*}$ to itself (for
instance, an automorphism if the transformation considered is a vertex
permutation symmetry ; quotient ; redundancy ; symmetry_wtw ). By contrast, a
stochastic symmetry is associated with an ensemble of graphs, rather than with
a single one. In particular, we can define a graph ensemble as stochastically
symmetric under a transformation if the latter maps each graph $G_{1}$ into an
equiprobable subgraph $G_{2}$ with $P(G_{1})=P(G_{2})$. Maximally random
graphs with constraints are therefore stochastically symmetric under
transformations that are symmetries of the Hamiltonian. If a real network
$G^{*}$ is well reproduced by a stochastically symmetric ensemble, then we can
denote $G^{*}$ as stochastically symmetric (under the same transformations
involved in the symmetry of the ensemble, or _under the model considered_ for
brevity). That is, while $G^{*}$ is exactly symmetric under its automorphisms,
it is stochastically symmetric under the transformations defining an ensemble
of which $G^{*}$ is a typical member. By contrast, if the ensemble is not a
good model of the real network,then those transformations are not stochastic
symmetries of $G^{*}$. We will encounter both situations later on.
Two important examples of maximally random graphs that we will use as null
models are the Erdős-Rényi random graph model and the configuration model.
Here we consider the undirected versions of both models, and we will
generalize them to the directed case later on. To avoid confusion with their
directed counterparts, here we use a different notation with respect to our
presentation in Reference symmetry1 . The adjacency matrix of an undirected
graph will be denoted as $B$, with entries $b_{ij}=1$ if an undirected link
between vertices $i$ and $j$ is there, and $b_{ij}=0$ otherwise.
In the undirected Erdős-Rényi random graph model, the only constraint is the
total number of undirected links $L^{u}=\sum_{i<j}b_{ij}$. Thus the
Hamiltonian reads
$H(G)=\theta L^{u}(G)$ (7)
and its symmetries are the transformations mapping a graph $G$ into another
(equiprobable) graph with the same number of links. The probability $P(G)$
factorizes in terms of the probability
$q\equiv\frac{e^{-\theta}}{1+e^{-\theta}}$ (8)
that a link is there between any two vertices. If the model is interpreted as
a null model of the real network $G^{*}$, the parameter $q$ must be set to the
particular value $q^{*}$ such that
$\langle L^{u}\rangle=q^{*}\frac{N(N-1)}{2}=L^{u}(G^{*})$ (9)
ensuring that, in accordance with the maximum likelihood principle
mylikelihood , the expected number of links $\langle L^{u}\rangle$ coincides
with the number of links $L^{u}(G^{*})$ of $G^{*}$.
In the configuration model, the constraints are the degrees of all vertices,
i.e. the _degree sequence_ $\\{k_{i}\\}$, where $k_{i}\equiv\sum_{j\neq
i}b_{ij}$. Therefore the Hamiltonian takes the form
$H(G)=\sum_{i=1}^{N}\theta_{i}k_{i}(G)$ (10)
and its symmetries are the transformations that map a graph into a different
one with the same degree sequence, i.e. those explored by the _local rewiring
algorithm_ symmetry1 . The probability that vertices $i$ and $j$ are connected
is no longer uniform across all pairs of vertices, and reads
$q_{ij}=\frac{w_{i}w_{j}}{1+w_{i}w_{j}}$ (11)
where $w_{i}\equiv e^{-\theta_{i}}$. If the configuration model is used as a
null model of a real network $G^{*}$, then the parameters
$\\{w_{1},\dots,w_{N}\\}$ must be set to the values
$\\{w^{*}_{1},\dots,w^{*}_{N}\\}$ solving the following $N$ coupled equations
$\langle k_{i}\rangle=\sum_{j\neq
i}\frac{w^{*}_{i}w^{*}_{j}}{1+w^{*}_{i}w^{*}_{j}}=k_{i}(G^{*})\qquad\forall i$
(12)
ensuring that the expected degree sequence coincides with the observed one,
and maximising the likelihood to obtain $G^{*}$ mylikelihood ; myrandomization
.
### 2.2 Transpose Equivalence and Transpose Equiprobability
We now come to the description of _link reversal_ symmetry. There are two ways
in which one can formulate link reversal symmetry in directed networks. The
first, simpler definition is the exact invariance of a single graph under the
inversion of the direction defined on each of its edges. Under this
definition, the graph is perfectly symmetric if all of its edges are
bidirectional. If $A$ is the adjacency matrix of a directed graph ($a_{ij}=1$
if a directed edge from $i$ to $j$ is there, and $a_{ij}=0$ otherwise), then
the graph is exactly symmetric under link reversal if
$A=A^{T}$ (13)
where $A^{T}$ indicates the transpose of the matrix $A$. Clearly,
bidirectional graphs are equivalent to undirected graphs. In this sense, one
can say that real networks are found to be either symmetric (this is the case
of real-world undirected networks such as the Internet, protein interaction
graphs or friendship networks) or asymmetric (this is the case of
intrinsically directed networks such as food webs, the WWW, metabolic
networks, the World Trade Web, etc.). This first type of link reversal
symmetry will be denoted _transpose equivalence_ in what follows.
A second, novel definition of link reversal symmetry that we introduce here is
a stochastic one, in the sense discussed in Reference symmetry1 and briefly
recalled above. As any stochastic symmetry, it is associated to an ensemble of
equiprobable graphs. If each graph $G$ in the ensemble is identified with its
adjacency matrix $A$, we say that the ensemble is stochastically symmetric
under link reversal if
$P(A)=P(A^{T})$ (14)
This second definition is completely different from the first one. It does not
imply that any single graph $A$ in the ensemble is bidirectional, but that it
has the same probability of occurrence of its link-reversed $A^{T}$, i.e.
$P(A)=P(A^{T})$. The equiprobability of $A$ and $A^{T}$ has important effects
on the directionality of the expected topological properties across the
ensemble, but is perfectly consistent with the asymmetry of individual graphs
in the ensemble. If the ensemble considered is a maximally random graph model
defined by a Hamiltonian $H(G)$ (see Section 2.1), then Equation (14) is
equivalent to
$H(A)=H(A^{T})$ (15)
showing that link reversal is a symmetry of the Hamiltonian. In accordance
with our general definition of stochastic symmetry symmetry1 recalled in
Section 2.1, we can also define a single graph $G^{*}$ as stochastically
symmetric under link reversal if it is a typical member of (i.e. it is well
modelled by) an ensemble which is stochastically symmetric under link
reversal. In simpler words, the graph $A$ is stochastically symmetric under
link reversal if it is statistically equivalent to its link-reversed $A^{T}$.
This second, stochastic type of link reversal symmetry will be denoted
_transpose equiprobability_ in what follows.
The dichotomy existing between transpose equivalence and transpose
equiprobability, the different underlying mechanisms they might reveal, and
the relation they have to many of the symmetries we have discussed in
Reference symmetry1 (including ensemble equiprobability, statistical
equivalence and dependence on external or hidden vertex properties) make link
reversal symmetry an ideal candidate to discuss in more detail in what
follows. Moreover, link reversal symmetry is tightly related to the problem of
_reciprocity_. Therefore, before presenting a deeper study of this symmetry,
in the next section we study the problem of reciprocity in great detail.
## 3 Reciprocity of Directed Networks
Reciprocity is the tendency of pairs of vertices to be connected by two mutual
links pointing in opposite directions, a particular type of correlation found
in directed networks wasserman ; myreciprocity ; mymultispecies . Depending on
the nature of the network, reciprocity is related to various important
phenomena, such as ecological symbiosis in food webs, reversibility of
biochemical reactions in metabolic networks, bidirectionality of chemical
synapses in neural networks, synonymy in networks of dictionary terms,
mutuality of psychological associations in networks of freely linked words,
reciprocity of hyperlinks in the WWW, crossed financial ownership in
shareholding networks, economic interdependence of countries in the
international trade network, and so on myreciprocity . In this section, we
study link reversal symmetry in great detail. We first discuss the problem of
the definition of proper reciprocity measures, present the analysis of the
reciprocity structure of real networks, and define some theoretical concepts
useful to interpret the observed patterns. Then, in Section 4 we highlight the
relation existing between reciprocity, the two types of link reversal symmetry
defined in Section 2.2, and other symmetries we introduced. In Section 5 we
finally apply all these concepts to the empirical analysis of the World Trade
Web.
### 3.1 The Traditional Approach to Reciprocity
The study of reciprocity has a long tradition in social science wasserman as
a way to quantify how many ‘ties’ (directed links) are reciprocated in a
social network of ‘actors’ (vertices). The _reciprocal link_ of a directed
link pointing from $i$ to $j$ is a link pointing from $j$ to $i$. A link is
_reciprocated_ if its reciprocal one is present in the network. In terms of
the adjacency matrix of the graph, two reciprocated links are present between
$i$ and $j$ if and only if $a_{ij}=a_{ji}=1$. In the example shown in Figure
1a, the edges between vertices $A$ and $B$, as well as those between $A$ and
$D$, are reciprocated. All other edges are not reciprocated. Therefore, while
the total number of directed links is given by
$L=\sum_{i\neq j}a_{ij}$ (16)
the number of reciprocated links is
$L^{\leftrightarrow}=\sum_{i\neq j}a_{ij}a_{ji}$ (17)
Since $0\leq L^{\leftrightarrow}\leq L$, the traditional definition of the
reciprocity of a network is
$r\equiv\frac{L^{\leftrightarrow}}{L}$ (18)
so that $0\leq r\leq 1$. Although not usually remarked, it is important to
notice that whether the value of $r$ can actually span the entire range
between $0$ and $1$ depends on the link density (or _connectance_) of the
network, defined as
$\bar{a}\equiv\frac{\sum_{i\neq j}a_{ij}}{N(N-1)}=\frac{L}{N(N-1)}$ (19)
We shall comment more about the effects of $\bar{a}$ on the allowed values of
the reciprocity later on. Note that the requirement $i\neq j$ in Equations
(16), (17) and (19) arises from the assumption of no self-loops (links
starting and ending at the same vertex) in the network. If self-loops are
present, we assume that they are ignored and therefore not computed in $L$ and
$L^{\leftrightarrow}$. This is because self-loops would give a nonzero
contribution to both $L$ and $L^{\leftrightarrow}$, even if they are not a
true signature of reciprocity. Two networks with the same topology apart from
a different number of self-loops should not be considered as having different
degrees of reciprocity myreciprocity .
Figure 1: (a) Example of a directed network with $N=6$ vertices. Here $L=9$,
$L^{\leftrightarrow}=4$ and the maximum possible number of directed links is
$N(N-1)=30$. (b) The undirected version of the same network. Here $L^{u}=7$
and the maximum possible number of undirected links is $N(N-1)/2=15$.
As for any topological property, a given value of $r$ is only significant with
respect to some null model. This is because, even in a network where directed
links are drawn completely at random, a certain number of reciprocated
connections will be formed. As we shall discuss in more detail in Section 4,
in such an uncorrelated network $r$ is simply equal to the average probability
that _any_ two vertices are connected by a directed link, i.e. to the
connectance defined in Equation (19):
$r_{rand}=\bar{a}$ (20)
Comparing the value of $r$ with that of $r_{rand}$ allows to assess if mutual
links occur more ($r>r_{rand}$) or less ($r<r_{rand}$) often than expected by
chance. This is the traditional approach to the study of reciprocity in social
networks, which has been more recently extended to other networks such as the
WWW, e-mail networks and the World Trade Web myreciprocity .
### 3.2 An Improved Definition
Although the comparison of $r$ with $r_{rand}$ is a safe method to detect
nonrandom reciprocity in a particular network, it is completely unadapted to
compare the reciprocity of networks with different link density, or to assess
the evolution of reciprocity in a single network with time-varying density
myreciprocity . This is because $r$ is not an absolute quantity, and its value
has only a relative meaning with respect to $r_{rand}=\bar{a}$. The reference
value for $r$ unavoidably varies as the density $\bar{a}$ varies. Therefore it
is not possible to order various networks, or various snapshots of the same
network, according to their value of $r$. In order to overcome this problem, a
new definition of reciprocity was proposed myreciprocity as the Pearson
correlation coefficient between the symmetric entries of the adjacency matrix:
$\rho\equiv\frac{\sum_{i\neq j}(a_{ij}-\bar{a})(a_{ji}-\bar{a})}{\sum_{i\neq
j}(a_{ij}-\bar{a})^{2}}=\frac{r-\bar{a}}{1-\bar{a}}$ (21)
where the second equality comes from an explicit calculation making use of
Equations (16)– (18) and of the property ${(a_{ij})^{2}=a_{ij}}$. The range of
$\rho$, as for any correlation coefficient, is $-1\leq\rho\leq 1$ (see however
our discussion below for more details on the allowed values of $\rho$). It is
possible to write down an expression for the statistical error associated to a
single measurement of $\rho$ on a particular network myreciprocity .
Unlike $r$, $\rho$ is an absolute quantity, and the effects of link density
are already accounted for in it. In particular, its null value is
$\rho_{rand}=0$ (22)
irrespective of the value of $\bar{a}$. The sign of $\rho$ alone is enough to
distinguish between positively correlated (or _reciprocal_) networks where
there are more reciprocated links than expected by chance ($\rho>0$) and
negatively correlated (or _antireciprocal_) networks where there are fewer
reciprocated links than expected by chance ($\rho<0$). The null case $\rho=0$
(consistently with the statistical error) corresponds to uncorrelated or
_areciprocal_ networks. The existence of a unique reference scale allows to
order several networks according to their value of $\rho$, as shown in Table
1. Among the networks considered, one finds both positively and and negatively
correlated ones. Remarkably, such ordering reveals interesting empirical
patterns of reciprocity, since networks of the same kind are found to display
similar values of $\rho$. The positively correlated networks are, in
decreasing order of $\rho$ (see Table 1): all purely bidirectional
(undirected) networks such as the Internet ($\rho=1$), the 53 snapshots of the
World Trade Web from year 1948 to 2000 ($0.68\leq\rho\leq 0.95$), an instance
of the WWW ($\rho=0.5165$), two neural networks ($0.41\leq\rho\leq 0.44$), two
e-mail networks ($0.19\leq\rho\leq 0.23$), two word association networks
($0.12\leq\rho\leq 0.19$) and 43 metabolic networks ($0.006\leq\rho\leq
0.052$). In particular, for the 53 snapshots of the World Trade Web
considered, the use of $\rho$ allows to properly track the evolution of
reciprocity over time, as we shall discuss in Section 5. The negatively
correlated networks considered are two shareholding networks
($-0.0034\leq\rho\leq-0.0012$) and 28 food webs ($-0.13\leq\rho\leq-0.01$).
The case of minimum reciprocity will be discussed in Section 3.3.
The analysis reported above reveals that real networks display nontrivial
reciprocity patterns and are always either correlated or anticorrelated. This
result is very important, since theoretical studies have shown that a
nontrivial degree of reciprocity affects the properties of various dynamical
processes taking place on directed networks, such as epidemic spreading
recipr_newman , percolation recipr_boguna , and localization recipr_vinko .
The effects of reciprocity are even more interesting on scale-free networks,
where even an infinitely small fraction of bidirectional links was shown to
give rise to a phase transition characterized by the onset of a giant strongly
connected component recipr_boguna .
### 3.3 Minimum Reciprocity
As we mentioned, in principle the allowed range of $\rho$ is ${-1\leq\rho\leq
1}$. However, from Table 1 we note that while the most correlated directed
network in the set considered displays $\rho=0.95$, which is almost equal to
the largest possible value, the most anticorrelated one displays only
$\rho=-0.13$, which is quite far from the lower bound $\rho=-1$. Still, for
most of the 30 antireciprocal networks reported in the table the number of
reciprocated links is zero ($r=0$) and therefore the value of $\rho$ is the
minimum possible myreciprocity .
Network | Range of $\rho$
---|---
Perfectly reciprocal | $\rho=1$
World Trade Web (53 webs) | $0.68\leq\rho\leq 0.95$
World Wide Web (1 web) | $\rho=0.5165$
Neural Networks (2 webs) | $0.41\leq\rho\leq 0.44$
Email Networks (2 webs) | $0.19\leq\rho\leq 0.23$
Word Networks (2 webs) | $0.12\leq\rho\leq 0.19$
Metabolic Networks (43 webs) | $0.006\leq\rho\leq 0.052$
Areciprocal | $\rho=0$
Shareholding Networks (2 webs) | $-0.0034\leq\rho\leq-0.0012$
Food Webs (28 webs) | $-0.13\leq\rho\leq-0.01$
Perfectly antireciprocal | $\rho=-1$
Table 1: Empirical values of $\rho$ (in decreasing order), for the 133 real
networks analysed in Reference myreciprocity . The values reported show the
significant digits with respect to the statistical errors.
This seemingly puzzling outcome can be explained as follows. Note that
Equation (21) implies that even in a network with $r=0$ the value of $\rho$ is
always different from $-1$ unless $\bar{a}=1/2$. This occurs because
$\bar{a}=1/2$ is the only case allowing perfect anticorrelation: in order to
have $a_{ij}=1$ whenever $a_{ji}=0$, the adjacency matrix must be exactly
‘half-filled’ with unit entries, and the number of links must be half the
maximum possible one myreciprocity . Remarkably, for $\bar{a}\neq 1/2$ there
are two different cases. In the ‘sparse’ range $\bar{a}<1/2$, the minimum
value of $r$ is $r_{min}=0$ since it is always possible to place all the links
without having reciprocal pairs. Consequently, Equation (21) implies that
${\rho_{min}=\bar{a}/(\bar{a}-1)}$. By contrast, in the ‘dense’ range
$\bar{a}>1/2$ some links must be unavoidably placed between the same pairs of
vertices and therefore $r>0$. More precisely, since the number of vertex pairs
is ${N(N-1)/2}$, the minimum number of reciprocal links is given by twice the
number of links exceeding this number, or in other words
${L^{\leftrightarrow}_{min}=2[L-N(N-1)/2]}$. Consequently,
${r_{min}=2-1/\bar{a}}$ and ${\rho_{min}=(\bar{a}-1)/\bar{a}}$. Putting these
results together, we have
$r_{min}=\left\\{\begin{array}[]{ll}0&\quad\textrm{if $\bar{a}\leq 1/2$}\\\
&\\\ 2-\displaystyle\frac{1}{\bar{a}}&\quad\textrm{if
$\bar{a}>1/2$}\end{array}\right.$ (23)
and
$\rho_{min}=\left\\{\begin{array}[]{ll}\displaystyle\frac{\bar{a}}{\bar{a}-1}&\quad\textrm{if
$\bar{a}\leq 1/2$}\\\ &\\\
\displaystyle\frac{\bar{a}-1}{\bar{a}}&\quad\textrm{if
$\bar{a}>1/2$}\end{array}\right.$ (24)
Both trends, together with the simple behaviour of $r_{rand}=\bar{a}$ for an
antireciprocal network, are shown as functions of $\bar{a}$ in Figure 2.
Figure 2: Behaviour of $r_{rand}$, $r_{min}$ and $\rho_{min}$ as functions of
$\bar{a}$.
### 3.4 Related Topological Properties
In this section we introduce various topological properties related to the
reciprocity of a network. We will refer again to Figure 1 to illustrate many
of the properties discussed in this section. The local quantities that
characterize each vertex $i$ are the _in-degree_ $k^{in}_{i}$ and the _out-
degree_ $k^{out}_{i}$, defined as the number of in-coming and out-going links
respectively:
$\displaystyle k^{in}_{i}$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}a_{ji}$ (25) $\displaystyle k^{out}_{i}$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}a_{ij}$ (26)
In the example shown in Figure 1a, vertex $A$ has $k^{in}_{A}=4$ in-coming
links and $k^{out}_{A}=3$ out-going links. Unfortunately, these commonly used
quantities do not carry information about the reciprocity, since they do not
tell us if the in-coming and out-going links of a vertex $i$ ‘overlap’
completely, partly or not at all. As a way to measure the overlap between the
sets of in-coming and out-going links of a vertex $i$, the _reciprocated
degree_ $k^{\leftrightarrow}_{i}$ was defined myreciprocity ; mymultispecies ;
recipr_newman ; recipr_boguna as the number of ‘reciprocal neighbours’
(vertices joined by two reciprocal links) of $i$:
$k^{\leftrightarrow}_{i}\equiv\sum_{j\neq i}a_{ij}a_{ji}$ (27)
In the example shown in Figure 1a, vertex $A$ has $k^{\leftrightarrow}_{A}=2$
reciprocal neighbours. As extreme examples, in a purely bidirectional network
($\rho=1$) there is complete overlap and
${k^{\leftrightarrow}_{i}=k^{in}_{i}=k^{out}_{i}\ \forall i}$, while in a
purely unidirectional network ($\rho=\rho_{min}<0$) there is no overlap and
${k^{\leftrightarrow}_{i}=0\ \forall i}$. One could think of
$k^{\leftrightarrow}_{i}$ as the result of a kind of ‘attraction’ or
‘repulsion’ between the in-coming and out-going links of vertex $i$, and of
$\rho$ as an average strength of the corresponding (positive or negative)
interaction.
As we mentioned, the knowledge of $k^{in}_{i}$ and $k^{out}_{i}$ alone is not
enough to know $k^{\leftrightarrow}_{i}$. It only informs us about the maximum
possible overlap, which is
$(k^{\leftrightarrow}_{i})_{max}=\min\\{k^{in}_{i},k^{out}_{i}\\}$ (28)
In the case shown in Figure 1a, $(k^{\leftrightarrow}_{A})_{max}=3$. If the
total number $N$ of vertices is known, then $k^{in}_{i}$ and $k^{out}_{i}$ can
also tell us about the minimum overlap, which is
$(k^{\leftrightarrow}_{i})_{min}=\left\\{\begin{array}[]{ll}0&\quad\textrm{if}\
k^{in}_{i}+k^{out}_{i}\leq N-1\\\
k^{in}_{i}+k^{out}_{i}-(N-1)&\quad\textrm{if}\
k^{in}_{i}+k^{out}_{i}>N-1\end{array}\right.$ (29)
depending on the possibility to place in-coming and out-going links without
overlap. The above expression is the analogous of Equation (23) for individual
vertices. In the case shown in Figure 1a, $(k^{\leftrightarrow}_{A})_{min}=2$.
Indeed, in the example considered it would be impossible to achieve a value of
$k^{\leftrightarrow}_{A}$ smaller than that realised, given the values of
$k^{in}_{A}$ and $k^{out}_{A}$. It would also be impossible to achieve a value
of $k^{\leftrightarrow}_{A}$ larger than $3$. In general, even the joint
knowledge of the in- and out-degrees $\\{k^{in}_{i}\\}$ and
$\\{k^{out}_{i}\\}$ of all vertices, or similarly the joint degree
distribution $P(k^{in}=k,k^{out}=k^{\prime})$ that a randomly chosen vertex
has in-degree $k$ and out-degree $k^{\prime}$, cannot characterize the
reciprocity of the network. What can be extracted from these quantities is
only the maximum and minimum numbers of reciprocated links, an information
analogous to that leading to Equation (24).
By contrast, the _three_ degree sequences $\\{k^{in}_{i}\\}$,
$\\{k^{out}_{i}\\}$ and $\\{k^{\leftrightarrow}_{i}\\}$ specify the
connectivity properties including the reciprocity. Summing over all vertices
gives the same information as Equations (16) and (17), and $\rho$ can then be
easily computed. Alternatively, it is also possible to define the _non-
reciprocated in-degree_ $k^{\leftarrow}_{i}$ and the _non-reciprocated out-
degree_ $k^{\to}_{i}$ of a vertex $i$ as the number of in-coming and out-going
links that are not reciprocated respectively:
$\displaystyle k^{\leftarrow}_{i}$ $\displaystyle\equiv$
$\displaystyle\sum_{j\neq
i}a_{ji}(1-a_{ij})=k^{in}_{i}-k^{\leftrightarrow}_{i}$ (30) $\displaystyle
k^{\to}_{i}$ $\displaystyle\equiv$ $\displaystyle\sum_{j\neq
i}a_{ij}(1-a_{ji})=k^{out}_{i}-k^{\leftrightarrow}_{i}$ (31)
In the example shown in Figure 1a, vertex $A$ has $k^{\leftarrow}_{A}=2$ and
$k^{\to}_{A}=1$. The information specified by the three degree sequences
$\\{k^{\leftarrow}_{i}\\}$, $\\{k^{\to}_{i}\\}$ and
$\\{k^{\leftrightarrow}_{i}\\}$ is the same as that carried by
$\\{k^{in}_{i}\\}$, $\\{k^{out}_{i}\\}$ and $\\{k^{\leftrightarrow}_{i}\\}$.
Note that the _total degree_ $k^{tot}_{i}$ can be expressed in the equivalent
forms
$\displaystyle k^{tot}_{i}$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}(a_{ji}+a_{ij})$ $\displaystyle=$ $\displaystyle k^{in}_{i}+k^{out}_{i}$
$\displaystyle=$ $\displaystyle
k_{i}^{\leftarrow}+k_{i}^{\to}+2k_{i}^{\leftrightarrow}$
The above quantities also come into play whenever a directed graph is regarded
as an undirected one by simply ignoring the direction of the links. We will
consider this problem in a real-world case in Section 5. The undirected
projection of a directed graph is an undirected graph where each pair of
vertices is connected by an undirected edge if _at least one_ directed link
(irrespective of its direction) is present between them in the original
directed graph. Figure 1b reports the undirected version of the directed graph
of Figure 1a. If $A$ is the adjacency matrix of the original directed network,
then the adjacency matrix $B$ of the projected undirected network has entries
$b_{ij}=a_{ij}+a_{ji}-a_{ij}a_{ji}$ (33)
and is now symmetric, as for any undirected network. Each vertex $i$ in the
undirected graph is now simply characterized by its _undirected degree_
$k_{i}$:
$\displaystyle k_{i}$ $\displaystyle=$ $\displaystyle\sum_{j\neq i}b_{ji}$
$\displaystyle=$ $\displaystyle
k^{in}_{i}+k^{out}_{i}-k^{\leftrightarrow}_{i}$ $\displaystyle=$
$\displaystyle k^{\leftarrow}_{i}+k^{\to}_{i}+k^{\leftrightarrow}_{i}$
The number of links $L^{u}$ in the undirected network is
$L^{u}=\sum_{i<j}b_{ij}=\frac{1}{2}\sum_{i}k_{i}=L-\frac{1}{2}L^{\leftrightarrow}=\left(1-\frac{r}{2}\right)L$
(35)
and the link density, or connectance, of the undirected network is the ratio
between $L^{u}$ and the maximum number of undirected links, i.e.
$\bar{b}\equiv\frac{2\sum_{i<j}b_{ij}}{N(N-1)}=\frac{2L^{u}}{N(N-1)}=(2-r)\bar{a}$
(36)
which is in an interesting relation with Equation (19). From the above
equations, which can be checked explicitly in the example shown in Figure 1,
it is clear that the knowledge of $k^{in}_{i}$ and $k^{out}_{i}$ is not enough
to determine $k_{i}$. Again, a crucial role is played by
$k^{\leftrightarrow}_{i}$ and consequently by the reciprocity of the network.
For perfectly antireciprocal networks $k^{\leftrightarrow}_{i}=0$ and
$k_{i}=k^{tot}_{i}$, while for perfectly reciprocal ones
$k_{i}=k^{\leftrightarrow}_{i}=k^{tot}/2$. More in general, the knowledge of a
directed topological property is not enough to determine the corresponding
property in the projected undirected graph. The missing information is carried
by the reciprocity structure of the network.
In what follows, it will be useful to evaluate the expectation values of the
above quantities across various graph ensembles. Therefore, before discussing
specific cases, we briefly develop a formalism useful in an ensemble setting.
In a graph ensemble, each link has an associated probability of occurrence
symmetry1 . The information relevant to the reciprocity structure is captured
by two different probabilities. The first one is the _marginal_ probability
$p_{ij}\equiv p(i\to j)=\langle a_{ij}\rangle$ (37)
that a directed link from $i$ to $j$ is there, irrespective of the presence of
the reciprocal link. The second one is the _conditional_ probability $r_{ij}$
that a directed link from vertex $i$ to vertex $j$ is there _given that_ the
reciprocal link from $j$ to $i$ is there:
$r_{ij}\equiv p(i\to j|i\leftarrow j)$ (38)
The trivial case, where the occurrence of reciprocal links is only due to
chance, is when the event $i\leftarrow j$ has no influence on the event $i\to
j$, so that $r_{ij}$ is equal to the marginal probability $p_{ij}$. By
contrast, if $r_{ij}>p_{ij}$ ($r_{ij}<p_{ij}$), the presence of two mutual
links between $i$ and $j$ is more (less) likely than expected by chance.
From the two probabilities above, a range of properties related to the
reciprocity structure can be derived. For instance, the probability
$p^{\leftrightarrow}_{ij}$ that $i$ and $j$ are connected by two reciprocal
links is
$p^{\leftrightarrow}_{ij}\equiv p(i\to j\cap i\leftarrow j)=\langle
a_{ij}a_{ji}\rangle=r_{ij}p_{ji}=r_{ji}p_{ij}$ (39)
and the probability $p^{\to}_{ij}$ that a single link from $i$ to $j$ is
there, with no reciprocal one from $j$ to $i$, is
$p^{\to}_{ij}\equiv p(i\to j\cap i\nleftarrow j)=\langle
a_{ij}(1-a_{ji})\rangle=p_{ij}-p^{\leftrightarrow}_{ij}=p_{ij}(1-r_{ji})$ (40)
Consequently, the expected values of $k^{in}_{i}$, $k^{out}_{i}$ and
$k^{\leftrightarrow}_{i}$ are
$\displaystyle\langle k^{in}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}p_{ji}$ (41) $\displaystyle\langle
k^{out}_{i}\rangle$ $\displaystyle=$ $\displaystyle\sum_{j\neq i}p_{ij}$ (42)
$\displaystyle\langle k^{\leftrightarrow}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}r_{ij}p_{ji}$ (43)
Similarly, the expectation value of the total number of directed links is
$\langle L\rangle=\sum_{i\neq j}\langle a_{ij}\rangle=\sum_{i\neq j}p_{ij}$
(44)
and that of the number of reciprocated links is
$\langle L^{\leftrightarrow}\rangle=\sum_{i\neq j}\langle
a_{ij}a_{ji}\rangle=\sum_{i\neq j}p^{\leftrightarrow}_{ij}=\sum_{i\neq
j}r_{ij}p_{ji}$ (45)
Therefore we can write down an expression for the expected value of $r$ across
the ensemble:
$\langle r\rangle=\frac{\sum_{i\neq j}r_{ij}p_{ji}}{\sum_{i\neq j}p_{ij}}$
(46)
Similarly, the expected correlation coefficient $\rho$ can be expressed as
$\langle\rho\rangle=\frac{\sum_{i\neq j}p_{ij}r_{ji}-(\sum_{i\neq
j}p_{ij})^{2}/N(N-1)}{\sum_{i\neq j}p_{ij}-(\sum_{i\neq j}p_{ij})^{2}/N(N-1)}$
(47)
The above relations will be useful later on.
It is also possible to exploit $p_{ij}$, $r_{ij}$ and
$p_{ij}^{\leftrightarrow}$ to obtain the probability that an undirected link
$(i-j)$ exists between vertices $i$ and $j$ in the undirected projection of
the graph defined by Equation (33). If $q_{ij}\equiv p(i-j)$ denotes this
_undirected_ connection probability, then Equation (33) implies
$q_{ij}\equiv p(i-j)=\langle
b_{ij}\rangle=p_{ij}+p_{ji}-p^{\leftrightarrow}_{ij}=p_{ij}+p_{ji}-r_{ij}p_{ji}$
(48)
Therefore the expectation value of the undirected degree $k_{i}$ defined in
Equation (3.4) is
$\displaystyle\langle k_{i}\rangle$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}q_{ji}$ $\displaystyle=$ $\displaystyle\langle k^{in}_{i}\rangle+\langle
k^{out}_{i}\rangle-\langle k^{\leftrightarrow}_{i}\rangle$ $\displaystyle=$
$\displaystyle\langle k^{\leftarrow}_{i}\rangle+\langle
k^{\to}_{i}\rangle+\langle k^{\leftrightarrow}_{i}\rangle$
Similarly, the expected number of undirected links is
$\langle L^{u}\rangle=\sum_{i<j}q_{ij}=\frac{1}{2}\sum_{i}\langle
k_{i}\rangle=\langle L\rangle-\frac{1}{2}\langle L^{\leftrightarrow}\rangle$
(50)
and the expected undirected connectance is
$\langle\bar{b}\rangle=\frac{2\sum_{i<j}q_{ij}}{N(N-1)}=\frac{2\langle
L^{u}\rangle}{N(N-1)}$ (51)
## 4 Reciprocity, Link Reversal Symmetry, and Ensemble Equiprobability
We can now discuss the relation between the reciprocity of networks (3),
ensemble equiprobability (Section 2.1), and the two types of link reversal
symmetry defined in Section 2.2, i.e. _transpose equivalence_ and _transpose
equiprobability_. As we shall try to highlight, different invariances are
captured by different topological properties, including the two measures of
reciprocity we have introduced. This shows that an in-depth understanding of
graph symmetries can indicate more informative definitions of topological
properties. We start by stressing again that if $r=1$, or equivalently
$\rho=1$, then every edge is reciprocated. This means that the network has the
first type of link reversal invariance, i.e. transpose equivalence: the
adjacency matrix $A$ is symmetric ($A=A^{T}$). The quantities $r$ and $\rho$
are therefore both informative with respect to transpose equivalence. By
contrast, as we now show they carry different pieces of information about
ensemble equiprobability and, as a particular case of it, the second type of
link reversal invariance, i.e. transpose equiprobability. As we mentioned,
both symmetries are related to an ensemble of graphs rather than to a single
network. We can therefore exploit the expressions derived in Section 3.4 to
obtain the expected reciprocity structure in specific graph ensembles. The
natural class of graph ensembles relevant to this problem is the directed
version of the maximum-entropy models with specified constraints symmetry1
that we have briefly recalled in Section 2.1. These ensembles provide us with
a null model against which, as we anticipated in Section 3.1, it is important
to compare the empirically observed reciprocity. For directed networks,
(grand)canonical graph ensembles consist of $2^{N(N-1)}$ possible directed
graphs with no self-loops, each described by a Hamiltonian $H(G)$ and by the
corresponding maximum-entropy probability $P(G)=e^{-H(G)}/Z$.
As a first example, we consider the _directed random graph_ , which is the
directed analogue of the model defined by Equation (7) and corresponds to the
Hamiltonian
$H(G)=\theta L(G)=\theta\sum_{i\neq j}a_{ij}(G)$ (52)
where now $L(G)$ is the number of _directed_ links. In such a model, a
directed link from vertex $i$ to vertex $j$ is drawn with constant probability
$p\equiv e^{-\theta}/(1+e^{-\theta})$, independently of all other links. That
is, also the reciprocal link from $j$ to $i$ is drawn independently and with
the same probability $p$. Due to the statistical independence of reciprocal
edges, in this model the conditional probability $r_{ij}$ reduces to the
marginal one $p_{ij}$. Putting these results together, we have:
$r_{ij}=p_{ij}=p=\frac{e^{-\theta}}{1+e^{-\theta}}$ (53)
Inserting the above relation into Equation (46) one finds that the expected
value of $r$ is
$\langle r\rangle=p$ (54)
If, in analogy with the undirected random graph discussed in Section 2.1 and
according to the maximum likelihood principle mylikelihood , $p$ is set to the
value $p^{*}=L(G^{*})/N(N-1)$ producing a null model of a real network $G^{*}$
with $L(G^{*})$ directed links and connectance
$\bar{a}(G^{*})=L(G^{*})/N(N-1)$, then the expected value of $r$ in the
directed random graph model is
$r_{rand}=p^{*}=\bar{a}(G^{*})$ (55)
Similarly, the expected value of $\rho$ under the same null model is
$\rho_{rand}=\frac{r_{rand}-p^{*}}{1-p^{*}}=0$ (56)
The above results prove what we anticipated in Equations (20) and (22). Note
that the directed random graph model defined by Equation (52) is symmetric
under transpose equiprobability: since $\theta$ is a global parameter, one has
$H(A)=H(A^{T})$ (where $A$ denotes the adjacency matrix of graph $G$)
irrespective of the symmetry of the real network $G^{*}$. A consequence of
this invariance is that in the null model the expected in-degree and out-
degree of any vertex are equal:
$\langle k^{in}_{i}\rangle=\langle k^{out}_{i}\rangle=p^{*}(N-1)\qquad\forall
i$ (57)
irrespective of whether they are equal in the real network. Similarly, the
expectation values of all other directed properties are invariant under link
reversal, i.e. exchanging the inward and outward directions. We can also
rephrase the differences between $r$ and $\rho$ in terms of their performance
with respect to transpose equiprobability in the random graph model as
follows. The reciprocity measure $r$ is completely uninformative with respect
to transpose equiprobability, since its behaviour under even this simple null
model is not universal and depends on the link density of the network. By
contrast, $\rho$ is informative since the transpose equiprobability of the
directed random graph model translates into a universal value $\rho_{rand}=0$.
Another case of interest is the _directed configuration model_ , defined by a
generalisation of Equation (10) corresponding to the enforcement of both the
in-degree and the out-degree sequences $\\{k^{in}_{i}\\}$ and
$\\{k^{out}_{i}\\}$ as constraints:
$H(G)=\sum_{i}\left[\theta_{i}^{in}k^{in}_{i}(G)+\theta_{i}^{out}k^{out}_{i}(G)\right]=\sum_{i\neq
j}(\theta_{i}^{out}+\theta_{j}^{in})a_{ij}(G)$ (58)
In this model, two reciprocal edges are again statistically independent,
therefore the conditional probability $r_{ij}$ equals the marginal one
$p_{ij}$, which is
$r_{ij}=p_{ij}=\frac{x_{i}y_{j}}{1+x_{i}y_{j}}$ (59)
where $x_{i}\equiv e^{-\theta^{out}_{i}}$ and $y_{i}\equiv
e^{-\theta^{in}_{i}}$. The above expression generalises Equation (11) to
directed graphs. If the directed configuration model is used as a null model
of a real network $G^{*}$, a discussion similar to that leading to Equation
(12) in the undirected case shows that the parameter values $\\{x^{*}_{i}\\}$
and $\\{y^{*}_{i}\\}$ indicated by the maximum likelihood principle are those
satisfying the $2N$ coupled equations
$\displaystyle\langle k^{in}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\frac{x^{*}_{j}y^{*}_{i}}{1+x^{*}_{j}y^{*}_{i}}=k^{in}_{i}(G^{*})\qquad\forall
i$ (60) $\displaystyle\langle k^{out}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\frac{x^{*}_{i}y^{*}_{j}}{1+x^{*}_{i}y^{*}_{j}}=k^{out}_{i}(G^{*})\qquad\forall
i$ (61)
ensuring that both the expected in-degree and out-degree sequences equal the
empirical ones. Note that, unlike the directed random graph, in this model the
transpose equiprobability symmetry does not hold: Equation (58) implies that
in general $H(A)\neq H(A^{T})$. Only if $\theta^{in}_{i}=\theta^{out}_{i}$, or
equivalently $x_{i}=y_{i}$, then $H(A)=H(A^{T})$. From Equations (60) and (61)
we see that this only occurs if $k^{in}_{i}(G^{*})=k^{out}_{i}(G^{*})$
$\forall i$, i.e. if in the real network the in-degree and the out-degree of
all vertices are equal. In such a case, in analogy with Equation (57), one has
that the inward and outward expected topological properties in the null model
are equal, and the transpose equiprobability symmetry holds. However, if
$k^{in}_{i}(G^{*})\neq k^{out}_{i}(G^{*})$ for some $i$, the transpose
equiprobability symmetry does not hold.
In the directed configuration model, all the graphs with the same in- and out-
degree sequences are equiprobable, irrespective of the number of mutual links
arising in them. This produces a trivial reciprocity structure. As an example,
consider Figure 3, which portrays various directed generalisations of the
_local rewiring algorithm_ introduced for undirected networks symmetry1 . If
$H(G)$ is defined by Equation (58), then each graph on the left has the same
probability of occurrence as the corresponding graph on the right, since
$H(G_{1})=H(G_{2})$, $H(G_{3})=H(G_{4})$ and $H(G_{5})=H(G_{6})$. However,
while the two graphs $G_{1}$ and $G_{2}$, and similarly the two graphs $G_{3}$
and $G_{4}$, have the same reciprocity, the graphs $G_{5}$ and $G_{6}$ have
different reciprocity, even if they occur with the same probability in the
ensemble defined by the model. This means that the reciprocity structure of
the network is not preserved across the ensemble, just like any other property
except the in- and out-degree sequences, as required by the model. This result
confirms our discussion in Section 3.4, where we showed that the two degree
sequences $\\{k^{in}_{i}\\}$ and $\\{k^{out}_{i}\\}$ alone do not specify the
reciprocity of the network. In analogy with the discussion leading to
Equations (55) and (56) for the directed random graph model, it is possible to
study the reciprocity generated by chance in the configuration model as the
result of specifying given degree distributions vinko . Conversely, it is also
possible to study the different problem of the influence of reciprocal links
on the degree distribution and degree correlations vinko2 .
In order to generate an ensemble with nontrivial reciprocity, one needs to
enforce an additional constraint in the Hamiltonian. One quite general
possibility mymultispecies is, according to our discussion in Section 3.4, to
specify the _three_ degree sequences $\\{k^{in}_{i}\\}$, $\\{k^{out}_{i}\\}$
and $\\{k^{\leftrightarrow}_{i}\\}$:
$\displaystyle H(G)$ $\displaystyle=$
$\displaystyle\sum_{i}\left[\theta_{i}^{in}k^{in}_{i}(G)+\theta_{i}^{out}k^{out}_{i}(G)+\theta_{i}^{\leftrightarrow}k^{\leftrightarrow}_{i}(G)\right]$
(62) $\displaystyle=$ $\displaystyle\sum_{i\neq
j}(\theta_{i}^{out}+\theta_{j}^{in})a_{ij}(G)+\sum_{i<j}(\theta_{i}^{\leftrightarrow}+\theta_{j}^{\leftrightarrow})a_{ij}(G)a_{ji}(G)$
In the example shown in Figure 3, in this model we still have
$H(G_{1})=H(G_{2})$ and $H(G_{3})=H(G_{4})$, but now $H(G_{5})\neq H(G_{6})$
since the reciprocal degree sequence $\\{k^{\leftrightarrow}_{i}\\}$ of the
graphs $G_{5}$ and $G_{6}$ is different. So the addition of the extra term
breaks the previous ensemble equiprobability symmetry of the Hamiltonian and
restricts it to smaller equivalence classes. This implies that now the
conditional and marginal connection probabilities are different: if we define
$x_{i}\equiv e^{-\theta^{out}_{i}}$, $y_{i}\equiv e^{-\theta^{in}_{i}}$ and
$z_{i}\equiv e^{-\theta^{\leftrightarrow}_{i}}$ it can be shown mymultispecies
that
$\displaystyle p^{\to}_{ij}$ $\displaystyle=$
$\displaystyle\frac{x_{i}y_{j}}{1+x_{i}y_{j}+x_{j}y_{i}+x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}$
(63) $\displaystyle p^{\leftrightarrow}_{ij}$ $\displaystyle=$
$\displaystyle\frac{x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}{1+x_{i}y_{j}+x_{j}y_{i}+x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}$
(64)
so that
$\displaystyle p_{ij}$ $\displaystyle=$ $\displaystyle
p^{\to}_{ij}+p^{\leftrightarrow}_{ij}=\frac{x_{i}y_{j}+x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}{1+x_{i}y_{j}+x_{j}y_{i}+x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}$
(65) $\displaystyle r_{ij}$ $\displaystyle=$
$\displaystyle\frac{p^{\leftrightarrow}_{ij}}{p_{ji}}=\frac{x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}{x_{i}y_{j}+x_{i}x_{j}y_{i}y_{j}z_{i}z_{j}}=\frac{x_{j}y_{i}z_{i}z_{j}}{1+x_{j}y_{i}z_{i}z_{j}}$
(66)
Figure 3: In the directed version of the configuration model, the local
rewiring algorithm symmetry1 has various generalizations. If one requires
that only the two degree sequences $\\{k^{in}_{i}\\}$ and $\\{k^{out}_{i}\\}$
are preserved, with $H(G)$ defined by Equation (58), then each graph on the
left has the same probability of occurrence as the corresponding graph on the
right, since $H(G_{1})=H(G_{2})$, $H(G_{3})=H(G_{4})$ and $H(G_{5})=H(G_{6})$.
By additionally requiring that also $\\{k^{\leftrightarrow}_{i}\\}$ is
preserved, and redefining $H(G)$ as in Equation (62), the above symmetry of
the Hamiltonian is broken: $G_{5}$ and $G_{6}$ are no longer equiprobable
since now $H(G_{5})\neq H(G_{6})$.
In this case the maximum likelihood principle mylikelihood indicates that, in
order to provide a null model of a real network $G^{*}$, the parameters
$\\{x_{i}\\}$, $\\{y_{i}\\}$ and $\\{z_{i}\\}$ must be set to the particular
values $\\{x^{*}_{i}\\}$, $\\{y^{*}_{i}\\}$ and $\\{z^{*}_{i}\\}$ satisfying
the $3N$ coupled equations
$\displaystyle\langle k^{in}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\frac{x^{*}_{i}y^{*}_{j}+x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}{1+x^{*}_{i}y^{*}_{j}+x^{*}_{j}y^{*}_{i}+x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}=k^{in}_{i}(G^{*})\qquad\forall
i$ (67) $\displaystyle\langle k^{out}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\frac{x^{*}_{j}y^{*}_{i}+x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}{1+x^{*}_{i}y^{*}_{j}+x^{*}_{j}y^{*}_{i}+x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}=k^{out}_{i}(G^{*})\qquad\forall
i$ (68) $\displaystyle\langle k^{\leftrightarrow}_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\frac{x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}{1+x^{*}_{i}y^{*}_{j}+x^{*}_{j}y^{*}_{i}+x^{*}_{i}x^{*}_{j}y^{*}_{i}y^{*}_{j}z^{*}_{i}z^{*}_{j}}=k^{\leftrightarrow}_{i}(G^{*})\qquad\forall
i$ (69)
ensuring that the expectation values of the three degree sequences equal the
empirical ones. Again, we see that in this model the transpose equiprobability
symmetry only holds if the real network $G^{*}$ has
$k^{in}_{i}(G^{*})=k^{out}_{i}(G^{*})$ $\forall i$. In such a case, from the
above equations one finds $x^{*}_{i}=y^{*}_{i}$ $\forall i$ which also implies
$p_{ij}=p_{ji}$ and $H(A)=H(A^{T})$ so that all the expected topological
properties have inward/outward invariance. Otherwise, the symmetry does not
hold. A particular case of the above model turns out to empirically describe
the World Trade Web, as we discuss in the next section.
## 5 Symmetries, Symmetry Breaking and the Evolution of World Trade
We now present an important real-world application of the concepts introduced
so far, i.e. the evolution of the international trade network. The World Trade
Web (WTW in the following) is the global network of import/export trade
relationships among all world countries serrano ; mywtw ; myalessandria ;
myinterplay ; 5 ; giorgiowtw . We already encountered the WTW in Section 3.2
among the other networks reported in Table 1. In the WTW, a vertex represents
one country and a directed link represents the existence (during the period
considered, usually one year) of an export relationship from one country to
another country. The WTW is in principle a weighted network, since trade
intensities can be measured by their (highly heterogeneous) total monetary
values aggregated over the period. Therefore the properties of the network can
be measured on a weighted basis 5 ; giorgiowtw . However, here we will
consider the WTW as a binary network, and only refer to its purely topological
properties. As we will show, even this simple picture is extremely interesting
and allows an informative study of the international trade system. In
particular, we will study how the network has evolved in time starting from
the year 1950, and how a joint analysis of the trends displayed by different
topological properties inform us about the change in the underlying
symmetries. If the WTW is regarded as an undirected graph, its structural
properties are remarkably stable over time, and indicate that the network
displays a clear invariance under transformations that preserve its degree
sequence. On the other hand, when the directionality of trade is taken into
account, the above symmetry is broken and the intensity of this symmetry
breaking changes in time. A strong increase in reciprocity is observed,
clearly evidencing that a major structural change started taking place from
the late 1970’s onwards. The symmetry concepts developed in Reference
symmetry1 and in the previous sections will be employed to suggest, or rule
out, possible explanations for the observed evolution of the WTW. In
particular, we identify as candidate explanations a strong embedding in
economic space and a spatial symmetry breaking in the production system, which
is known to have occurred starting from the late 1970’s 7 ; 8 and could
therefore explain the simultaneous change in the reciprocity of the network.
Surprisingly, other mechanisms such as the increase in the number of trade
relationships, size effects and the formation of trade agreements are not
enough in order to explain the observed evolution of the symmetry properties
considered. This analysis highlights the importance of identifying the
behaviour of complex systems under different types of symmetries, and of
introducing suitable measures that succeed in distinguishing between the
latter.
### 5.1 Undirected Symmetries
Various empirical results describing the topology of the WTW can be combined
in order to have a detailed picture of the underlying symmetries. In this
section we consider the undirected projection of the network as defined in
Section 3.4, while in the next one we consider the WTW as a directed network.
A first interesting observation, that will be useful in the following, is that
the undirected connectance $\bar{b}$ defined by Equation (36) remains almost
constant during the time interval considered, as shown in Figure 4. This
happens despite the fact that the number $N(t)$ of world countries increases
significantly, due to a number of new independent states being formed between
1948 and 2000.
Figure 4: Evolution of the density $\bar{b}(t)$ of the undirected version of
the World Trade Web.
Importantly, the constancy of the connectance does not mean that the latter
characterises the WTW satisfactorily. If we use the random graph model as a
null model of the WTW, the undirected connection probability defined in
Equation (48) is uniform: $q_{ij}(t)=q(t)$. The maximum likelihood principle,
in accordance with Equation (9), indicates the following choice for this
probability:
$q^{*}(t)=\frac{2L^{u}(t)}{N(t)[N(t)-1]}$ (70)
However, the above choice generates trivial expectations which are not in
accordance with the empirical results, in particular a Binomial degree
distribution, a constant (uncorrelated with the degree) average nearest
neighbour degree and a constant clustering coefficient. This means that the
ensemble equiprobability invariance of the random graph model under
transformations preserving the total number of links is not a stochastic
symmetry of the WTW in the sense explained in Section 2.1.
By contrast, an important finding myrandomization ; mywtw is that, in every
snapshot of the network within the time window considered, the undirected
projection of the WTW is always remarkably well reproduced by the
configuration model. This means that, according to Equation (11), the
probability that a trade relationship exists (irrespective of its direction)
between two countries $i$ and $j$ in a given year $t$ is
$q_{ij}(t)=\frac{w^{*}_{i}(t)w^{*}_{j}(t)}{1+w^{*}_{i}(t)w^{*}_{j}(t)}$ (71)
where the parameters $\\{w^{*}_{i}(t)\\}$ are the solution of the $N$ coupled
equations
$\langle k_{i}(t)\rangle=\sum_{j\neq
i}\frac{w^{*}_{i}(t)w^{*}_{j}(t)}{1+w^{*}_{i}(t)w^{*}_{j}(t)}=k_{i}(t)\qquad\forall
i$ (72)
which are equivalent to Equation (12). The accordance between the
configuration model and the real undirected WTW has been checked by studying
various higher-order properties, including the average nearest neighbour
degree and the clustering coefficient of all vertices, and confirming that
they are excellently reproduced by the model myrandomization ; mywtw . The
undirected WTW is therefore a good example of a network whose higher-order
properties can be traced back to low-level constraints. According to our
discussion in Section 2.1, this implies that the ensemble equiprobability
invariance displayed by the configuration model under transformations
preserving the degree sequence symmetry1 is a stochastic symmetry of the real
WTW. In turn, this implies that in every snapshot of the WTW all vertices with
the same degree $k$ are statistically equivalent symmetry1 . That is, the
overall symmetry of the network under permutations of vertex labels is broken
down to distinct universality classes consisting of vertices with the same
degree. This is evident from the fact that, in passing from the random graph
model (where all vertices are statistically equivalent) to the configuration
model (where all vertices with the same degree are statistically equivalent),
the connection probability changes from Equations (70) to (71) and therefore
acquires a dependence on the variables $w_{i}^{*}$ and $w_{j}^{*}$, which in
turn depend on the degree sequence through Equation (72). Unlike $q(t)$, the
probability $q_{ij}(t)$ is not uniform across all pairs of vertices, but only
across pairs of vertices with the same pair of degrees $k_{i}$ and $k_{j}$. As
shown in Equation (51), the following relation holds between the expected
connectance $\langle\bar{b}\rangle$ and the probability $q_{ij}(t)$:
$\langle\bar{b}(t)\rangle=\frac{2\sum_{i<j}q_{ij}(t)}{N(N-1)}=\frac{2}{N(N-1)}\sum_{i<j}\frac{w^{*}_{i}(t)w^{*}_{j}(t)}{1+w^{*}_{i}(t)w^{*}_{j}(t)}$
(73)
Therefore the observed stationarity of $\bar{b}$ shown in Figure 4 indicates
that, despite $q_{ij}(t)$ varies greatly among pairs of world countries and
also over time, its average across all pairs of countries remains remarkably
stable.
The accordance between the undirected WTW and the configuration model means
that the degree sequence is extremely informative, since its knowledge allows
one to obtain correct expectations about all other topological properties.
This implies that, in order to explain the WTW topology at an undirected
level, it is enough to explain its degree sequence. Thus reproducing the
degree sequence should be the target of any model of the WTW topology, an
important point that we will discuss in Section 5.3. Whatever the cause of the
empirical degree sequence of WTW, this cause is the symmetry-breaking
phenomenon restricting the invariance of the network to degree-preserving
transformations.
### 5.2 Directed Symmetries
We now come to the description of the WTW as a directed network, which
involves additional information. Note that, since the configuration model
reproduces the real WTW topology, and since in this model different pairs of
vertices are statistically independent, then also the directed version of the
model must be reproduced by a model with independent pairs of vertices. What
remains to be clarified is whether the possible events that can occur within a
single pair of vertices are also statistically independent, i.e. whether the
conditional connection probability $r_{ij}$ and the marginal connection
probability $p_{ij}$ defined in Section 3.4 are equal. In other words, we need
to characterise the reciprocity structure of the network.
To this end, a first useful result is that, irrespective of the year $t$
considered, the in-degree and the out-degree of every vertex are empirically
found to be approximately equal myreciprocity ; myalessandria ; myinterplay ,
i.e.
$k^{in}_{i}(t)\approx k^{out}_{i}(t)\qquad\forall i$ (74)
A second empirical result is that the reciprocated degree
$k^{\leftrightarrow}_{i}(t)$ defined in Equation (27) is always proportional
to the total degree $k^{tot}_{i}(t)=k^{in}_{i}(t)+k^{out}_{i}(t)$, with a
time-dependent proportionality coefficient myreciprocity ; myalessandria :
$k^{\leftrightarrow}_{i}(t)\propto k^{tot}_{i}(t)\qquad\forall i$ (75)
This result is shown in Figure 5 for various years $t$.
Figure 5: Dependence of the reciprocated degree $k^{\leftrightarrow}_{i}$ on
the total degree $k^{tot}_{i}=k^{in}_{i}+k^{out}_{i}$ in various snapshots of
the WTW (from bottom to top: $t=1975,1980,1985,1990,1995,2000$). The different
curves have been shifted vertically for better visibility.
Taken together, these two results inform us about the structure of the
connection probabilities $p_{ij}$, $r_{ij}$ and $p_{ij}^{\leftrightarrow}$
introduced in Section 3.4. Indeed, since $\langle
k^{in}_{i}(t)\rangle=\sum_{j\neq i}p_{ji}(t)$ and $\langle
k^{out}_{i}(t)\rangle=\sum_{j\neq i}p_{ij}(t)$, the result in Equation (74)
can be rephrased as
$p_{ij}(t)\approx p_{ji}(t)\qquad r_{ij}(t)\approx r_{ji}(t)$ (76)
Similarly, since $\langle k^{\leftrightarrow}_{i}(t)\rangle=\sum_{j\neq
i}r_{ij}(t)p_{ji}(t)$, Equation (75) implies that $r_{ij}(t)$ is independent
of $i$ and $j$, i.e. the conditional connection probability is uniform:
$r_{ij}(t)\approx r_{0}(t)$ (77)
The latter determines the proportionality coefficient relating the
reciprocated degree to the total degree as in Equation (75):
$\langle k_{i}^{\leftrightarrow}(t)\rangle=r_{0}(t)\sum_{j\neq
i}p_{ij}(t)=r_{0}(t)\langle k^{out}_{i}(t)\rangle=\frac{r_{0}(t)}{2}\langle
k^{tot}_{i}(t)\rangle$ (78)
Moreover, Equation (46) implies that the expected reciprocity of the network
at time $t$ coincides with the conditional connection probability:
$\langle r(t)\rangle\approx r_{0}(t)$ (79)
This result can be confirmed independently,by measuring the observed
reciprocity $r(t)$ and checking that it is indeed approximately equal to the
proportionality coefficient $r_{0}(t)$ relating the reciprocated degree to the
total degree as in Equation (78), obtained from a linear fit of the trends
shown in Figure 5 myreciprocity . We will show more empirical results about
the reciprocity in Section 5.4.
The uniformity of $r_{ij}(t)$ implies that the marginal connection probability
must be different from the conditional one. Otherwise, the WTW would be well
reproduced by the directed random graph model introduced in Section 4, with
$p(t)\approx r_{0}(t)$. This possibility is ruled out by the fact that the two
approximate equalities $p_{ij}(t)\approx p_{ji}(t)$ and $r_{ij}(t)\approx
r_{0}(t)$, if inserted into Equation (48), imply
$q_{ij}(t)\approx[2-r_{0}(t)]p_{ij}(t)$ (80)
which, together with Equation (71), implies that the WTW is well described by
the marginal connection probability
$p_{ij}(t)\approx\frac{1}{2-r_{0}(t)}q_{ij}(t)=\frac{1}{2-r_{0}(t)}\frac{w^{*}_{i}(t)w^{*}_{j}(t)}{1+w^{*}_{i}(t)w^{*}_{j}(t)}$
(81)
The above marginal probability is not uniform as the directed random graph
model would predict, and is necessarily different from the uniform conditional
connection probability $r_{0}(t)$. This means that the reciprocity of the WTW,
whatever the snapshot considered, is nontrivial. As another consequence, this
result implies that a good model of the WTW is not even provided by the
directed configuration model defined by Equation (58), because the latter
predicts $r_{ij}=p_{ij}$ as shown in Equation (59). Therefore the directed
representation of the WTW does not display, as a stochastic symmetry, the
ensemble equiprobability invariance under transformations that preserve the
degree sequences $\\{k^{in}_{i}\\}$ and $\\{k^{out}_{i}\\}$.
As discussed in Section 4, a step forward the simple directed configuration
model is provided by the model defined by Equation (62), i.e. a maximum-
entropy ensemble of graphs with constraints given by the three degree
sequences $\\{k^{out}_{i}\\}$, $\\{k^{in}_{i}\\}$,
$\\{k^{\leftrightarrow}_{i}\\}$ controlled by the Lagrange multipliers
$\\{\theta^{out}_{i}\\}$, $\\{\theta^{in}_{i}\\}$,
$\\{\theta^{\leftrightarrow}_{i}\\}$ or equivalently $\\{x_{i}\\}$,
$\\{y_{i}\\}$, $\\{z_{i}\\}$. We now prove various theoretical relations
describing what is implied when a uniform conditional connection probability
$r_{ij}=r_{0}$ is assumed as a further ingredient of this model, and show that
these relations are in excellent agreement with all the empirical properties
of the WTW discussed above, and reconcile the undirected picture with the
directed one. For brevity, in our notation we drop the dependence of the
various quantities on the time $t$. First, note that, due to the equality
$p_{ij}r_{ji}=p_{ji}r_{ij}$ appearing for instance in Equation (39),
$r_{ij}=r_{0}$ implies
$p_{ij}=p_{ji}$ (82)
and automatically predicts both $\langle k^{in}_{i}\rangle=\langle
k^{out}_{i}\rangle$ and $\langle
k^{\leftrightarrow}_{i}\rangle=(r_{0}/2)\langle k^{tot}_{i}\rangle$. In other
words, the constancy of $r_{ij}$ implies the symmetry of $p_{ij}$ and is
enough to simultaneously explain the two empirical properties of the WTW
reported in Equations (74) and (75). As another consequence, one has
$x_{i}=y_{i}$ $\forall i$ in Equation (65) so that Equation (66) becomes
$r_{ij}=\frac{x_{i}x_{j}z_{i}z_{j}}{1+x_{i}x_{j}z_{i}z_{j}}$ (83)
But under our hypothesis the above expression must be a constant $r_{0}$,
which is only possible if $x_{i}z_{i}=y_{i}z_{i}=\alpha$ where $\alpha$ is a
constant. This implies
$x_{i}=y_{i}=\alpha
z_{i}^{-1}\qquad\Leftrightarrow\qquad\theta^{out}_{i}=\theta^{in}_{i}=-\ln\alpha-\theta^{\leftrightarrow}_{i}$
(84)
Therefore among the three parameters $x_{i}$, $y_{i}$, $z_{i}$ there is only
an independent one (say $x_{i}$). This allows us to rewrite Equations (63) and
(64) as
$\displaystyle p^{\to}_{ij}$ $\displaystyle=$
$\displaystyle\frac{x_{i}x_{j}}{1+(2+\alpha^{2})x_{i}x_{j}}$ (85)
$\displaystyle p^{\leftrightarrow}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}x_{i}x_{j}}{1+(2+\alpha^{2})x_{i}x_{j}}$ (86)
and Equations (65) and (66) as
$\displaystyle p_{ij}$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha^{2})x_{i}x_{j}}{1+(2+\alpha^{2})x_{i}x_{j}}$
(87) $\displaystyle r_{ij}$ $\displaystyle=$ $\displaystyle
r_{0}=\frac{\alpha^{2}}{1+\alpha^{2}}$ (88)
The last equation, if inserted into Equation (46), implies
$\langle r\rangle=r_{ij}=\frac{\alpha^{2}}{1+\alpha^{2}}$ (89)
which clearly shows that in this model the expected reciprocity $r$ coincides
with the conditional probability $r_{ij}$ and is uniquely determined by
$\alpha$. Thus all the above quantities could be expressed as functions of
$r_{0}$ (or $\langle r\rangle$) rather than $\alpha$, by exploiting the
inverse relation
$\alpha=\sqrt{\frac{r_{0}}{1-r_{0}}}=\sqrt{\frac{\langle r\rangle}{1-\langle
r\rangle}}$ (90)
Equation (48) implies that under the above model the connection probability in
the undirected projection of the network is
$q_{ij}=p_{ij}+p_{ji}-r_{ij}p_{ji}=\frac{(2+\alpha^{2})x_{i}x_{j}}{1+(2+\alpha^{2})x_{i}x_{j}}$
(91)
Together with Equations (87) and (90), the above relation implies
$q_{ij}=\frac{2+\alpha^{2}}{1+\alpha^{2}}p_{ij}=(2-r_{0})p_{ij}$ (92)
which exactly reproduces the empirical property of the WTW shown in Equation
(80). Note that, if the parameters $\alpha$ and $\\{x_{i}\\}$ are tuned to the
values $\alpha^{*}$ and $\\{x^{*}_{i}\\}$ fitting the model to the real
network, Equation (91) coincides with Equation (71) once $\alpha^{*}$ is
reabsorbed into $w^{*}_{i}$ as follows:
$w_{i}^{*}=x_{i}^{*}\sqrt{2+(\alpha^{*})^{2}}$ (93)
That is, once the value of $\alpha^{*}$ enforcing the observed value of the
reciprocity is fixed, the values of $\\{x_{i}^{*}\\}$ determine the undirected
degree sequence exactly as $\\{w_{i}^{*}\\}$ in the undirected configuration
model. This important result indicates that the undirected version of the
directed model considered here coincides with the undirected configuration
model, and thus reconciles the directed and undirected descriptions. Note that
this is not true in general: for instance, the undirected version of the
directed configuration model defined by Equation (58) does _not_ coincide with
the undirected configuration model. It is the nontrivial structure of the
reciprocity of the WTW, manifest in the uniformity of $r_{ij}$, that ensures
this property. This result can be confirmed by noticing that Equation (84)
implies that the Hamiltonian of the model, which in general has the form in
Equation (62), in this case becomes
$\displaystyle H(G)$ $\displaystyle=$
$\displaystyle\sum_{i}[\theta_{i}k_{i}^{out}+\theta_{i}k_{i}^{in}+(\theta_{0}-\theta_{i})k_{i}^{\leftrightarrow}]$
(94) $\displaystyle=$
$\displaystyle\sum_{i}\theta_{i}k_{i}+\theta_{0}L^{\leftrightarrow}$
where we have defined $\theta_{i}\equiv\theta_{i}^{in}=\theta_{i}^{out}=-\ln
x_{i}$ and $\theta_{0}\equiv-\ln\alpha$. The above expression highlights that
the constraints required in order to reproduce all the topological properties
of the WTW discussed so far are the undirected degree sequence $\\{k_{i}\\}$
and the number of reciprocated links $L^{\leftrightarrow}$, or equivalently
the reciprocity $r$. If the maximum likelihood principle mylikelihood is
applied to this model, it is straightforward to show that the parameters
reproducing a given snapshot of the WTW must be set to the particular values
$\alpha^{*}=e^{-\theta_{0}^{*}}$ and
$\\{x^{*}_{i}\\}=\\{e^{-\theta^{*}_{i}}\\}$ satisfying the following $N+1$
coupled equations
$\displaystyle\langle k_{i}\rangle$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}q_{ij}=\sum_{j\neq
i}\frac{[2+(\alpha^{*})^{2}]x^{*}_{i}x^{*}_{j}}{1+[2+(\alpha^{*})^{2}]x^{*}_{i}x^{*}_{j}}=k_{i}\qquad\forall
i$ (95) $\displaystyle\langle L^{\leftrightarrow}\rangle$ $\displaystyle=$
$\displaystyle\sum_{i\neq j}r_{ji}p_{ij}=\sum_{j\neq
i}\frac{(\alpha^{*})^{2}x^{*}_{i}x^{*}_{j}}{1+[2+(\alpha^{*})^{2}]x^{*}_{i}x^{*}_{j}}=L^{\leftrightarrow}$
(96)
The first of the two expressions above indeed coincides with the condition
fixing the values of $\\{w_{i}\\}$ in the undirected configuration model as in
Equation (72), under the identification given by Equation (93). The second
expression allows to enforce any value of the reciprocity $r$ as additional
constraint, thanks to the extra parameter $\alpha=e^{-\theta_{0}}$. Note that
this graph ensemble is stochastically symmetric under link reversal, since
$H(A)=H(A^{T})$. Furthermore, since the real WTW is well reproduced by the
aforementioned graph ensemble, it is also stochastically symmetric under link
reversal.
We have therefore shown that the topology of the WTW for any year $t$ since
1950 is completely reproduced by specifying the undirected degree sequence
$\\{k_{i}(t)\\}$ and the reciprocity $r(t)$. This implies that, in order to
explain why the WTW displays the structure we observe, it is enough to explain
these two topological properties. However, while assessing the relevance of
some structural features is a rigorous procedure as we have shown so far,
explaining them in terms of underlying mechanisms involves a higher degree of
uncertainty and subjective interpretation. Bearing this in mind, in what
follows we suggest possible explanations for the two structurally informative
ingredients of the WTW topology. These should be intended as candidate
hypotheses rather than certain mechanisms. Nonetheless, since the symmetry of
the effect must be at least that of the cause, the symmetry analysis carried
out in the preceding sections can be exploited to safely rule out explanations
that do not fulfill this principle.
### 5.3 Topological Space and Embedding Spaces
We start by considering here the undirected degree sequence, while in the next
section we focus on the reciprocity. The accordance between the configuration
model and the undirected projection of the WTW, i.e. the stochastic symmetry
of the WTW under transformations preserving the degree sequence, can be
rephrased as the finding that the network is the result of a random matching
process between the edges attached to every vertex. Vertices connect to each
other as the mere result of the constraint on their degrees. The larger their
degrees, the higher the probability that two vertices are connected, with no
higher-order effect on the topology. This implies that, whatever the factor
responsible for the observed degree of a country, it must similarly respect
the symmetry and be such that _the more a country is endowed with this factor,
the larger its degree and the higher its probability to connect to other
vertices, with no higher-order effect on the topology other than those implied
by the degree sequence_. If we denote the hidden factor as $h$, and its value
for vertex $i$ as $h_{i}$, the above statement can be rephrased as _the larger
$h_{i}$, the larger the expected value of $\langle k_{i}\rangle$_; similarly,
_the larger $h_{i}$ and $h_{j}$, the larger the undirected connection
probability $q_{ij}$_. Therefore the hidden values $\\{h_{i}\\}$ must play
exactly the same role as that of the Lagrange multipliers $\\{w_{i}\\}$
controlling the expected degrees of all vertices in the undirected
configuration model as in Equation (72). More in general, the Lagrange
multiplier $w_{i}$ could be a monotonic function $f(h_{i})$ of the hidden
factor $h_{i}$. The above consideration suggests at least two ways to test
whether any empirically observable quantity is indeed a good candidate as the
hidden factor determining the degree sequence in a given snapshot of the WTW.
First, one can solve the $N$ coupled equations in Equation (72) and obtain the
set of values $\\{w_{i}^{*}(t)\\}$ which _are_ the exact values of the
Lagrange multipliers enforcing the observed degree sequence in year $t$, and
then check whether a candidate quantity $h(t)$, with empirically observed
values $\\{h_{i}(t)\\}$, is indeed in some approximate functional dependence
with these multipliers, i.e.
$w^{*}_{i}(t)\approx f[h_{i}(t),h_{0}(t)]\quad\forall i$ (97)
where $f$ can in general depend on, besides $h_{i}(t)$, a global time-
dependent parameter $h_{0}(t)$ setting the scale of the dependence. As a
second alternative, one could _assume_ the functional dependence
$w_{i}(t)=f[h_{i}(t),h_{0}(t)]$, rewrite $q_{ij}(t)$ as
$q_{ij}(t)=\frac{w_{i}(t)w_{j}(t)}{1+w_{i}(t)w_{j}(t)}=\frac{f[h_{i}(t),h_{0}(t)]f[h_{j}(t),h_{0}(t)]}{1+f[h_{i}(t),h_{0}(t)]f[h_{j}(t),h_{0}(t)]}$
(98)
and apply the maximum likelihood principle to the resulting model, which now
has only $h_{0}(t)$ as a free parameter since the values $\\{h_{i}(t)\\}$ are
empirically accessible. This leads to the single equation
$\langle
L^{u}(t)\rangle=\sum_{i<j}\frac{f[h_{i}(t),h^{*}_{0}(t)]f[h_{j}(t),h^{*}_{0}(t)]}{1+f[h_{i}(t),h^{*}_{0}(t)]f[h_{j}(t),h^{*}_{0}(t)]}=L^{u}(t)\quad\forall
i$ (99)
fixing the value of $h_{0}(t)$ for each year $t$ and replacing Equation (72).
The goodness of the assumed dependence can be tested by checking whether
Equation (98), with the value $h^{*}_{0}(t)$ inserted in it, reproduces the
properties of the real network, in the same way as Equation (72) is used to
assess the goodness of the configuration model. Clearly, the first procedure
is preferable as it leaves the determination of the form of
$f[h_{i}(t),h_{0}(t)]$ at the end: once the values $\\{w_{i}^{*}(t)\\}$ are
found exactly, one can study the dependence of the latter on various candidate
quantities $h$, and with different functional forms. The second procedure
requires from the beginning the assumption one wants to test, and is therefore
less accurate; nonetheless, it could represent a further test of the
hypothesis if the output of the first method is used as the input in the
second one.
Both the approaches described above have been used to look for hidden factors
explaining the degree sequence, and consequently the entire topology, of the
undirected WTW mylikelihood ; mywtw . The result of this analysis is that the
Gross Domestic Product (GDP in what follows) is a very good candidate factor.
If $h_{i}(t)$ is identified with the empirical GDP value of country $i$ in
year $t$, then an approximate linear relationship between $h_{i}(t)$ and the
value $w_{i}^{*}(t)$ obtained from Equation (72) for the same year is observed
mylikelihood . This means that Equation (97) reduces to the simplest possible
functional form
$w^{*}_{i}(t)\approx h_{i}(t)\sqrt{h_{0}(t)}\quad\forall i$ (100)
where the proportionality factor has been denoted as $\sqrt{h_{0}(t)}$ for
convenience. This indicates that the probability that a trade relationship
(whatever its direction) exists between countries $i$ and $j$ in year $t$ is
$q_{ij}(t)\approx\frac{h_{0}(t)h_{i}(t)h_{j}(t)}{1+h_{0}(t)h_{i}(t)h_{j}(t)}$
(101)
This result is confirmed by assuming the above form of the connection
probability, using Equation (99) to find the value $h^{*}_{0}(t)$ generating
the observed number of links, and checking that indeed the empirical
properties of the undirected WTW are reproduced mywtw . This result highlights
that the larger (in economic terms) a country, the higher its probability to
connect to other countries. According to our discussion at the end of Section
5.1, since the GDP is responsible for the degree sequence of the WTW, it
represents the symmetry-breaking variable restricting the invariance of the
network to equal-degree (or similarly equal-GDP) equivalence classes. Contrary
to what one could expect on the basis of the spatial embedding of the WTW, no
significant dependence is found on other factors such as distance, membership
to common geographic areas or trade associations, etc.
The above result is very instructive in the light of the relation between
network structure and symmetry. What we should bear in mind, when we consider
symmetry breaking in the field of network theory, is that symmetry
(invariance) is hard to depict unless we use analytical tools. Our
imagination, intended as the faculty of forming images, has been educated to
depict shapes in Euclidean spaces. Whenever we must traduce shapes from
Euclidean to topological spaces, we are inevitably biased by the fact the we
tend to recall the Euclidean representation of forms in the new space. This
overlapping of spaces generates misrepresentation. To better stress out the
conundrum of spaces’ inequality representation, in Figure 6 we picture the
trade network of Europe (EU-15), as it would appear in topological space (left
panel) and in Euclidean space (middle panel), assuming that trades travel
mainly on the road network road . While the Euclidean representation of the
road network, except for the scale and a certain degree of abstraction, is
conformal to the real system’s shape, the corresponding representation of the
trade network in a metric space (the plane), is purely conventional. Indeed,
we could have represented the same network in several different ways, e.g.,
arraying in a circle or randomly scattering nodes. We could actually produce
_ad libitum_ different Euclidean embeddings of the same graph. The
topological-Euclidean dichotomy is further complicated by the fact the system
represented by the WTW network is immersed in an economic space, involving
variables and relations in large number, that are not detected in the network.
Consider for example the traveling time of goods, which is determined by
several exogenous factors. Traveling time, together with energy efficiency and
labor costs, is among the major factors affecting shipping costs. In the right
panel of Figure 6 we show a ‘metric representation’ of the space modification
due to traveling time times . It is noteworthy that in an economic space
distances are not merely Euclidean and the compound metric is made by length,
time, labor costs and energy units at the minimum.
Indeed, the result that the WTW is excellently reproduced by a connection
probability that uniquely depends on the GDP indicates that the space
modification is even more extreme, as at a global level geographic distances
appear to play almost no role. In regular lattices the overall permutation
symmetry of vertices is broken by positions in Euclidean space and is
restricted to invariances of lesser order such as translational symmetry
symmetry1 . In geographically embedded networks such as that shown in the
middle panel of Figure 6, the irregularity of the geography further restricts
the symmetry properties. When additional variables are also taken into
account, even stronger distortions take place as in the right panel of Figure
6, and in the case of the WTW we are in an extreme situation where the
symmetry-breaking variable is virtually only the GDP, and the distance
dependence practically disappears. The properties of the network must be
therefore interpreted in economic space rather than geographic space. Still,
in this space we find a remarkable symmetry: countries with the same GDP are
statistically equivalent symmetry1 , and pairs of countries with the same
couple of GDP values have the same probability to trade. In other words, we
can rephrase the symmetry properties of the WTW we discussed in Section 5.1 in
terms of the GDP values rather than the degree sequence. This invariance is
preserved despite the heterogeneity of the GDP across world countries
increases in time myinterplay , which means that the intensity of the GDP-
induced symmetry breaking also increases. And, despite the latter determines
ever-increasing divergences between the values of the connection probability
$q_{ij}$ across pairs of countries, the average probability
$2\sum_{i<j}q_{ij}/N(N-1)$ remains almost constant as indicated by the
stationarity of the undirected connectance shown in Figure 4.
Figure 6: European (EU-15) trade network as it would appear in topological
space (left panel), in Euclidean space assuming that trades travel mainly on
the road network (middle panel, after road ), and after also taking into
account the space modification due to traveling times (right panel, after
times ).
### 5.4 The Reciprocation Process of World Trade and Spatial Symmetry
Breaking
As we mentioned, the second ingredient required in order to explain the
topological properties of the WTW is the reciprocity $r$, which coincides with
the conditional connection probability $r_{0}$ as indicated by Equation (79).
While we have shown that the marginal connection probability varies greatly
among different pairs of vertices, a property that can be traced back to the
heterogeneous degrees and possibly explained by the GDP values, the
conditional connection probability is uniform and must therefore be related to
a completely different mechanism. The heterogeneity of vertex degrees, or of
GDP values, is completely reflected in the marginal connection probability
while it is not reflected at all in the conditional connection probability and
in the reciprocity. To better understand the problem, we now consider the
temporal evolution of the reciprocity and show how this may suggest possible
explanations.
As clear from Equations (75) and (78), the proportionality constant
$r_{0}(t)/2$ between $k^{\leftrightarrow}_{i}(t)$ and $k^{tot}_{i}(t)$ is
time-dependent. As Equation (79) indicates, this means that the reciprocity
$r(t)=r_{0}(t)$ of the network must also change in time. In Figure 7 we show
the empirical evolution of $r(t)$. Indeed, we find that the reciprocity of the
WTW has evolved dramatically during the period considered. In particular, we
see that $r(t)$ has been fluctuating about a constant value from 1950 to the
late 1970’s. Then, from the late 1970’s to the late 1990’s, a steady increase
of $r(t)$ took place. More importantly, this occurred despite the density of
undirected trade relationships (the undirected connectance $\bar{b}$ shown in
Figure 4) remained approximately constant during the same period. This
indicates that, from the late 1970’s on, there has been an establishment of
many new directed trade relationships mainly between countries that had
already been trading in the opposite direction, rather than between countries
that had not been trading at all. That is, the reciprocation process of
unidirectional trade channels has dominated the formation of new trade
relationships between non-interacting countries.
Figure 7: Evolution of the reciprocity measures $r(t)$ (blue points) and
$\rho(t)$ (red points) in the directed version of the World Trade Web.
The above results have implications for the evolution of the symmetry
properties of the network. As it was highlighted in the previous sections, the
equiprobability symmetry of the configuration model allows a statistical
interpretation of the real undirected WTW. Higher order topological variables
can be explained by the degree sequence. The invariance of the Hamiltonian is
preserved through time, as reflected by the stationarity of some topological
variables (Figure 4). The stationarity is however disrupted as we change lens
and move from the undirected to the directed graph. Reciprocity determines a
clear symmetry breaking in the directed analogue of the above invariance, as
the in- and out-degrees alone are no longer enough to characterise the
network. The intensity of this symmetry breaking evolves in time, as evident
in the trend of the reciprocity $r(t)$, and even more of $\rho(t)$. Also,
while the second type of link reversal symmetry (transpose equiprobability) is
approximately unchanged over time, since the approximate equality
$k^{in}_{i}(t)\approx k^{out}_{i}(t)$ holds throughout the interval
considered, the adjacency matrix suddenly starts becoming _more symmetric_ :
$A$ and $A^{T}$ become more and more similar in time, indicating that the WTW
has undergone a strong evolution towards higher levels of link reversal
symmetry of the first type (transpose equivalence). The reason of such a
sudden change is obscure. The evolution may have been either driven by other
topological variables, i.e. it was endogenous to the network, or determined by
some hidden variables, thus exogenous to the network. In other words: the
change either pertains thoroughly the topological space or comes from an
‘outer embedding space’, where the exogenous variables belong to, that shapes
the topological space symmetry1 . Besides, it may also be possible that the
symmetry breaking actually occurred in this latter space and consequently
affected the topological space. In what follows we explore this problem in
more detail.
A first natural explanation of the above empirical result could be looked for
in an overall increase in the number of directed trade relationships during
the period considered, a possibility consistent with the globalisation
process. Note that in our case Equation (36) implies that, while the
undirected connectance $\bar{b}(t)$ is approximately constant, the directed
connectance $\bar{a}(t)$ (hence the density of directed trade relationships)
of the WTW has indeed increased significantly due to the observed increase of
$r(t)$. In order to understand whether the observed increase in reciprocity is
merely due to the overall increase in link density, it is important to recall
our discussion in Section 3.2, where we stressed the importance of using
$\rho$ instead of $r$ since the former washes away density effects. Since in
this case the link density $\bar{a}(t)$ changes in time, using $\rho(t)$
instead of $r(t)$ is also important in order to correctly quantify the
temporal evolution of the reciprocity. In Figure 7, besides $r(t)$, we also
show the evolution of $\rho(t)$. Unlike $r(t)$, the behaviour of $\rho(t)$ is
informative and clearly shows that the increase in density cannot explain the
increase in reciprocity. Remarkably, the evolution of $\rho(t)$ is even more
pronounced than that of $r(t)$, indicating that the change in density
determines an underestimation of the steep increase in reciprocity, if the
latter is measured by $r$ rather than by $\rho$. The same consideration
applies even if one takes into account the fact that, according to our results
discussed above, the increase in the density of directed trade relationships
has occurred differentially across world countries, i.e. not uniformly as in a
directed random graph model with increasing connection probability $p(t)$ but
rather as in a directed configuration model with heterogeneous probabilities
$p_{ij}(t)$. If the observed increase in reciprocity were merely due to a
differential, rather than homogeneous, increase of link density, then we would
observe $r_{ij}\approx p_{ij}$ as discussed in Section 5.2. By contrast, the
uniformity of $r$ rules out this possibility. In other words, the inadequacy
of the random graph model and the configuration model in reproducing the
observed properties of the WTW rules out the possibility that the increase in
reciprocity is due to the globalisation process, at least the component of the
latter that is responsible for an (either homogeneous or differential)
increase in the density of directed trade relationships.
As a second hypothesis, one could consider the establishment of new trade
agreements (preferentially between countries that had only unidirectional
trade relationships, and determining the reciprocation of the latter) as a
possible explanation for the increase in the density of reciprocated links.
However, trade agreements do not explain the uniformity of the conditional
connection probability $r_{ij}(t)=r_{0}(t)$. For all years, the latter is
empirically found to be the same across all pairs of vertices, which is in
contrast with what expected from the formation of trade agreements: an
increased value of $r_{ij}(t)$ for pairs of countries signing the agreement,
determining an increased heterogeneity of $r_{ij}(t)$ across all pairs.
Therefore the evolution in $r$ cannot be explained by the formation of trade
agreements. The uniformity of the conditional connection probability also
indicates that other factors such as size, distance, etc. appear to be not
enough in order to explain how the reciprocity of world trade has evolved.
The above considerations show that the reciprocation of preexisting
unidirectional relationships appears to have occurred massively, however with
no preference for nearby or richer countries, and in a way which cannot be
traced back to an overall increase in the number of trade relationships and
trade agreements. We stress again that all these factors must have had an
impact on international trade patterns, especially on the intensity of trade
relationships, however at a purely topological level they appear to be
dominated by a different mechanism, which is uniform across all pairs of
countries. In simplified terms, the evolution of the reciprocity of the WTW
could be approximated by a process where, with time-varying but country-
independent probability, a unidirectional trade relationship existing at time
$t$ becomes reciprocated in the following year. Among the possible underlying
mechanisms that could generate this process, we must look for one displaying a
temporal trend which is synchronous with the one followed by the reciprocity
of the WTW and shown in Figure 7. To this end, it is useful to recall that in
the case of the WTW, vertices and links are samples of vertices and links of a
larger underlying network. Indeed, countries themselves do not trade; rather,
firms and consumers trade. Hence there are at least two submerged, and much
larger, networks: one of goods—final products—and one of production
factors—raw materials and semi-products (together with a third network related
to the service market). The WTW may be considered as an overlapping map of
these two networks. While the two hypotheses advanced above mainly concerned
the network of final products, one could look for an explanation relative to
the production network (a network composed by factories as vertices and
productive means as links). The hypothesis that symmetry breaking occurred in
the economic space of the industrial sector in a period starting between the
1970’s to the early 1980’s, with a significant worldwide impact on the
productive structure, has been recently advanced 7 ; 8 . This transition was
due on one hand to decreasing energy costs of transport means and on the other
hand to raising labor costs. Firms therefore were stimulated to provide
production factors outside the division and began dispersing the productive
chain outside the company boundaries, sometimes abroad. This process, named by
economists _outsourcing_ , transformed the space of firms from a Euclidean
space, where providers where separated from the production plant by physical
distances, to an economic space where physical distances were secondary to
other variables (i.e., changed the metric of economic space) 8 . This process
was further reinforced by specialization and technological enhancement, and
was one of the driving forces of globalisation. Note that, when a firm extends
its productive chain outside the national boundaries, new links may appear in
the trade network. This process can determine an increase in reciprocity, if
the new countries entering the production process already import from the
firm’s country. This mechanism can therefore provide a candidate explanation,
from the production side of network flows, for the observed increase in
reciprocity, which is also temporally consistent with the empirical trend. If
this hypothesis, which must be further investigated, is correct, we would have
faced a symmetry breaking in economic space affecting topological space,
partially determining the phase transition we observe, and at the correct
moment in time.
## 6 Conclusions
In this paper we took full advantage of our investigation of the symmetry
properties of real networks symmetry1 to perform a detailed study in a
specific case. We exploited our concept of stochastic graph symmetries to
introduce a new definition of link reversal symmetry, i.e. transpose
equiprobability. We showed that, when combined with other symmetry properties
of directed networks, stochastic link reversal symmetry allows an improved
understanding of the reciprocity of real networks. In particular, we have
studied various symmetry properties of the World Trade Web across its
evolution. Our analysis also sheds a light in a specific case on the symmetry
properties of networks and symmetry breaking in network topology. We found
that a space-symmetry approach to network theory may provide new insights into
the complex structure of the underlying system. We also made a conjecture
about the interplay between different spaces embedding the system, captured by
the topological space, that may lie behind some dramatic changes observed in
the detected topological variables. We believe that spatial symmetry breaking
deserves more attention as it may lead to new perspectives in understanding
complexity evolution and specifically, those kind of transformations
characterized by a sudden leap in the complexity of the structure. Network
theory represents a theoretical framework that enables holistic analyses and
is suited to detect ongoing dynamics between the system’s components and the
surrounding environment. In other words, network theory is a paradigm that
considers the system as a whole and distributes its functioning in space and
time. The analysis of the resulting symmetries carries information about the
system.
More then twenty years ago, Marshall McLuhan, in the field of communication
theory, advocated the need to overcome the constrains of conventional theories
about communication, according to him relegated to an Euclidean and ‘visual’
space, to achieve a new theory based on an ubiquitous and synchronous space,
in his words: an ‘acoustic space’. He advanced the point that space (and not
just time) is an agent of communication. According to him, printed texts have
educated us to a sequential type of communication, whereas the electronic age
developed spatial communication: actors communicate in the same time with the
environment and mutually between them 9 . Nevertheless, in his opinion,
communication theory did not follow changes in communication media. In his own
words: ‘The basis of all contemporary Western theories of communication—the
Shannon-Weaver model—is a characteristic example of left-hemisphere lineal
bias. It ignores the surrounding environment as a kind of pipeline model of a
hardware container for software content. It stresses the idea of inside and
outside and assumes that communication is a literal matching rather than
making’ 10 .
We believe network theory is a paradigm intrinsically ‘spatial’ and ‘global’,
that best suits the need for a holistic theory versus a sequential theory,
that it could further benefit from the interaction with other disciplines and
concepts. We considered here the case of symmetry and symmetry breaking, and
showed that a formalisation of the relation between these phenomena and
network properties is intriguing and informative, but at present still
incomplete. One of the present limitations is due to the fact that the studied
network is often a mere map of a larger, underlying network, embedded in
Euclidean or non-Euclidean spaces. Symmetry breaking may occur in a different
space, that is only indirectly represented in the topological space. This
indirect consequence complicates a clear understanding of the underlying
process, but stochastic symmetries appear to capture patterns that are
unaccessible to exact symmetries. Future research must explore this scenario
more thoroughly, and possibly shed light on the relation between network
dynamics, symmetry breaking, the causal chain and its premises.
## Acknowledgements
D.G. acknowledges financial support from the European Commission 6th FP
(Contract CIT3-CT-2005-513396), Project: DIME - Dynamics of Institutions and
Markets in Europe. We also thank Tiziano Squartini for his help.
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|
arxiv-papers
| 2010-09-22T22:04:02 |
2024-09-04T02:49:13.015297
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Franco Ruzzenenti, Diego Garlaschelli, Riccardo Basosi",
"submitter": "Diego Garlaschelli",
"url": "https://arxiv.org/abs/1009.4489"
}
|
1009.4500
|
∎
11institutetext: M. Saadatfar 22institutetext: T. J. Senden33institutetext: A.
P. Sheppard 44institutetext: Research School of Physics and Engineering, The
Australian National University, Canberra ACT 0200 Australia 55institutetext:
A. J. Kabla 66institutetext: Department of Engineering, University of
Cambridge, Trumpington street, CB2 1PZ, Cambridge, UK
# Mapping forces in 3D elastic assembly of grains.
Mohammad Saadatfar Adrian P. Sheppard Tim J. Senden
Alexandre J. Kabla
###### Abstract
Our understanding of the elasticity and rheology of disordered materials, such
as granular piles, foams, emulsions or dense suspensions relies on improving
experimental tools to characterize their behaviour at the particle scale.
While 2D observations are now routinely carried out in laboratories, 3D
measurements remain a challenge. In this paper, we use a simple model system,
a packing of soft elastic spheres, to illustrate the capability of X-ray
microtomography to characterise the internal structure and local behaviour of
granular systems. Image analysis techniques can resolve grain positions,
shapes and contact areas; this is used to investigate the material’s
microstructure and its evolution upon strain. In addition to morphological
measurements, we develop a technique to quantify contact forces and estimate
the internal stress tensor. As will be illustrated in this paper, this opens
the door to a broad array of static and dynamical measurements in 3D
disordered systems.
###### Keywords:
granular matter force measurement 3D image processing tomography segmentation
Despite the practical importance of macroscopic disordered materials such as
soil, foams and emulsions the rules that govern their mechanical behaviour
remain poorly understood. In the case of granular materials, these studies are
essential to the advancement of related industrial processes and also to the
prediction of often catastrophic geological phenomena (avalanches of slurries,
earthquakes). Although empirical constitutive equations are nowadays able to
partly model the response of these systems, a unified multiscale framework
does not exist yet. Over the past 20 years, foams and granular systems have
emerged as a model system for low temperature glasses Langer:00 . Many
theories and experiments have been designed to address the relationship
between the internal structure and macroscopic behaviour of such systems.
These theories and models often use approaches derived from statistical
physics Zhang:05a ; OHern:01 . Although the behaviour of foams seems today
rather well understood Weaire2010 , the complexity of granular matters
behaviour at the microscopic scale has prevented a proper physical model of
granular systems. A large part of the problem arises from the strongly
nonlinear contact law between rigid bodies coupled with the dynamical nature
of the contact network. In this context, the spatial distribution and temporal
evolution of the force network has become a highly sought after quantity that
expands from micro scale (grain-grain contacts) to macro scale (granular
assembly).
Early simulations have provided important insights on the internal force
distributions of a granular pile Radjai:96 ; Coppersmith:96 ; Makse:00a . Two-
dimensional (2D) experiments managed to keep up with the pace of advancement
in simulations Drescher:72 ; Howell:99 ; Kolb:04 , 3D experiments however,
have been limited to measurements of the distribution of forces only at the
boundaries of the containers Brockbank:97 ; Mueth:98 ; Blair:01 ; Corwin:05 .
These experimental methods could not have access to the spatial arrangement of
the contact force network in the bulk of granular assemply. Moreover they were
unable to determine structural features such as force chains and arching which
have been postulated as the signature of jamming Lois:07 .
In recent years, a range of tools have been developed to apprehend the 3D
nature of bulk properties. X-ray or MRI techniques have allowed to study
dynamic properties of granular systems such as flow profiles and shear
banding, at a mesoscopic scale Mantle:01 ; Alex:05 ; Alex:09 ; Dijksman:10 .
3D reconstructions of compressed emulsion systems using confocal microscopy
have provided the first measurements of the bulk force distributions Brujic:03
; Bruji:05 by estimating the contact force from contact geometry. A similar
technique has also been used to characterise spatial correlations of forces
inside 3D piles of frictionless liquid droplets Zhou06 . Although these
observations are very valuable, the systems used are significantly different
from real granular pile which has very different contact laws and frictional
properties.
X-ray computed tomography has been used for static characterisation of large
packings of spherical grains Silb:02 ; Aste:05 ; saadatfarthesis . In this
study, we apply this technique to a model system made of soft millimetric
elastic beads. In addition to the measurement of traditional structural
quantities (packing fraction, coordination number etc.), the contact force is
measured from the contact area between grains which can be integrated at a
mesoscopic scale to estimate the local stress tensor in the pile. We then
apply a series of external forces to our model system and investigate its
response to these controlled loadings. This approach delivers results
consistent with the existing literature and is promising as a generic tool to
study the local, non-linear mechanics of granular assemblies.
## 1 Experiment and methodology
### 1.1 Experimental setup
Figure 1: Sketch of the compression cell and consecutive stages of
compression.
The packing studied in this paper is made of relatively monodisperse spherical
rubber balls of diameter $3.10\pm 0.05~{}mm$ 111Styrene-Butadine-Rubber (SBR);
purchases from Mid-Atlntic Rubber Co., USA. The maximum and minimum ball
diameter measured in the packing are $3.19mm$ and $2.85mm$ respectively. The
grains are made of commercial rubber with a shear modulus estimated at
$850kPa$ .
A cylindrical PMMA container, with internal diameter of $44mm$ is used as a
compression cell (see figure 1). The inner wall of the cylinder is lubricated
with canola oil to reduce the friction. $2020$ rubber elastic balls are then
poured into this chamber. The latter is then closed at the top by a piston of
diameter slightly smaller than the container; it can therefore move freely
without touching the PMMA cylinder and without allowing beads to leave the
container. A horizontal platform is rigidly connected by a shaft to the piston
so that a load can be applied to the sample by placing weights on it. A pair
of springs connecting the piston to the container is used to make sure that,
in the absence of a load, the piston does not fall down due to its own weight.
The compression cell is attached to a motorised rotation stage located between
a high-resolution microfocus X-ray source ($80kV$ accelerating voltage and
$200\mu A$ beam current and a CCD detector of the size $67mm\times 67mm$,
$2048\times 2048$ pixels of size $33.6\mu m$ each Arthur:04a ; Arthur:04b .
The compression cell is rotated by about $0.125$ degree increments around its
vertical axis and a radiographic projection of the packing is taken after each
rotation with an exposure time of $18$ s. A total of $2880$ projections are
taken for a complete rotation of the specimen. It takes approximately $15$
hours to scan the volume for each case. After the completion of each scan,
tomographic cone beam reconstructions are performed on these $2880$
projections using the a Feldkamp algorithm Feldkamp84 . Tomograms of about
$2000^{3}$ voxels are obtained at $27$ microns resolutions.
We have recorded and analysed the internal geometry of the packing under three
different loading conditions.
Stage 0:
Pre-compression stage with no extra loading; the piston is held in balance
just above the top of the packing without making contact with it (see Fig. 1).
Rubber balls are at rest, only bearing their own weight ($41.1g$). The
combined weight of the piston and the platform ($600g$) is fully balanced by
the pull of the springs so that at this point there is no external force being
exerted onto the pile.
Stage 1:
The platform is loaded with $150g$($1.47N$). By taking into account the
opposite force applied by the springs, a net weight of $115g$ is applied to
the packing, resulting in a compressive strain of $\epsilon=3.78\%$. We define
the axial strains as the ratio of the displacement of piston to the initial
height of the packing (engineering strains).
Stage 2:
A total of $900g$($8.83N$) is placed on the platform resulting in a net $780g$
weight onto the packing. The total axial strain at that stage is
$\epsilon=7.83\%$.
### 1.2 3D Image analysis: segmentation and partitioning
The tomographic image consists of a cubic array of reconstructed linear X-ray
attenuation coefficient values, each corresponding to a finite volume cube
(voxel) of the sample. The first step in analysing this data is to
differentiate the attenuation map into distinct pore and grain phases.
Ideally, one would wish to have a bi-modal distribution giving unambiguous
phase separation of the pore and solid phase peaks. This simple phase
extraction is possible in our rubber ball pack. The intensity histogram of the
tomogram (Fig. 2(a)) shows two distinct peaks associated with the two phases
(beads and air). The peak centred around $9000$ is associated with the beads
and the lower peak around $4000$ is associated with the pore phase.
(a)
Figure 2: (a) Normalized X-ray density histogram of the full image volume of
the rubber ball sample. (b) Gray scale X-ray density map of a slice of rubber
ball sample. (c) The same slices after phase separation.
The segmentation algorithm Adrian:04 uses both the attenuation value and its
gradient to detect interfaces between the two phases (grain-void). In
particular, a sharp gradient of attenuation is required to detect an
interface. This method, after adjustment Saadatfar:06 , provides a very robust
detection of grain boundaries, but is of course limited to void spaces larger
than the voxel size. In particular, if the gap between two grains is very
small (order of a voxel), the density gradient across the gap between these
two grains will not appear as steep as a typical grain-void interface. This
can result in the detection of false contact regions in the segmented datasets
which consequently gives rise to the following undesired morphological
effects: i) detection of false contacts between grains, and ii) the surface
area of real contacts appear larger than their actual values. As detailled
later in this paper, this effect can be corrected in spherical grain piles by
offsetting the apparent contact area of the grains.
Figure 3: Different stages of the watershed algorithm. (a) Two overlapping
discs. (b) EDT of the two overlapping discs. (c) Identifying the local maxima
(seeds). (d) and (e) Regions grow from the initial seeds by having pixels on
their boundary added to them. (f) Discs separated.
The next step is to reconstruct and label individual grains from the geometry
of the solid phase. The basic assumption in the grain identification algorithm
is that the boundaries between grains which are not isolated coincide with the
watershed surfaces of the Euclidean distance map of grains (distance to the
nearest grain boundary) Vincent:91 . The entire grain space can be thought of
as the union of spheres centred on every grain voxel. Each sphere radius is
given by the Euclidean distance value of the voxel at its centre. The next
stage is identifying all the voxels that are not covered by any larger
Euclidean spheres.
Each one of these voxels, which are at the maxima of the distance function in
their local neighbourhood, then forms a seed that will grow into a single
grain in the following stage of the algorithm. Fig. 3 illustrates a simple 2D
case where 2 overlapping discs are separated. The seed regions essentially
grow by having voxels on their boundary added to them. Voxels that lie on the
boundaries of the regions are processed in reverse Euclidean distance order,
i.e. voxels with high distance values are processed first. When a voxel is
processed, it is assigned to the region on whose boundary it lies, or, if it
lies on more than one region boundary, the region whose boundary it first
became part of. At the end of the algorithm, the grain space will be
partitioned into grains whose boundaries lie on the watershed surfaces of the
Euclidean distance function saadatfarthesis . The watershed based grain
partitioning algorithm is computationally expensive. Hence it is parallelised
(using an implementation of the ”time warp” discrete event simulation protocol
Soille:03 ), so it can be used to analyse very large datasets ($\sim
2000^{3}$).
## 2 Measurements
Figure 4: A subvolume of the elastic ball compaction experiment. The contact
zone between touching grains (dark patches) provides a measurement of contact
force.
The 3D watershed algorithm provides us with large amounts of information about
the pile structure. Each grain has been labelled, its location and shape are
known, its direct neighbours have been identified, and the geometry of the
contact they share can been extracted as well. This data can now be processed
to extract physical and mechanical properties. In this section, we present a
few methods implemented in the context of granular systems. These can be
separated into structural characteristics and mechanical measurements. The
latter involves kinematic and force measurements that rely on prior knowledge
of a contact law for individual grains.
### 2.1 Structural quantities
#### 2.1.1 Packing fraction
One of the most readily measurable quantities from 3D images is the packing
fraction. Packing fraction or apparent density is defined as the ratio of the
volume of balls to the total volume. Table 1 summarises the packing fractions
measured directly from the digital images. The packing fractions we measure on
the initial pile is $57\%$, which is in the lower end of the spectrum for
monodisperse beads. Such a density is achievable in our system due to the
large frictional coefficient of rubber-rubber contacts. As the loading is
increased, the volume fraction increases up to $62\%$.
Compression stage | $0$ | $1$ | $2$
---|---|---|---
Loading [gr] | $0$ | $115$ | $780$
Strain [%] | $0$ | $3.78$ | $7.83$
Packing fraction | $0.57$ | $0.59$ | $0.62$
Avg. coord. no. | $6.28\pm 1.64$ | $6.63\pm 1.58$ | $7.13\pm 1.49$
raw
Avg. coord. no. | $4.99\pm 1.80$ | $5.32\pm 1.66$ | $5.92\pm 1.56$
filtered
Table 1: Details of the compression progression.
#### 2.1.2 Radial Distribution Function
Figure 5: Autocorrelation function of the packing at different stages of
compression normalized by the volume of a single grain and the packing
fraction. Note the shift in the peaks as the pressure increases.
From the shapes of the individual grains calculated by the Watershed
algorithm, the coordinates of each grain’s centre of mass can be accurately
measured. This data is in particular required to characterize the
microstructure of granular systems and to calculate the autocorrelation
function of the grain centres.
The radial distribution function (RDF) is a measure of the degree of
separation of grain centres and their density at a given distance. It is
calculated by counting the number of grains $N$ that are separated from a
given sphere by a distance in the interval $[r,r+dr]$, where $r$ is the
distance between grain centres. For large $r$ the number density of the grain
centres found in the interval $[r,r+dr]$ approaches the average density of
grains in the packing, $\frac{3}{4\pi{R}^{3}}\rho$, where $\rho$ is the
packing fraction and $R$ is average grain radius. In our calculations, we
normalize RDF by the average density. Figure 5 shows the RDF in our packings
throughout compression progression. The RDF of the system before insertion of
any external force (stage 0) shows a prominent peak at $r\approx D_{g}$ where
$D_{g}$ is the average grain diameter ($D_{g}=2\times R\approx 3.10mm$). The
second peak appears at $r\approx 1.95D_{g}$ and a sub-peak approximately at
$r=\sqrt{3}D_{g}$. As the vertical load increases (stage 1 and stage 2), the
peak of the RDF widens and shifts to the left due to the compression of
touching spheres, hence shortening the centre-centre distance between them
(see Fig. 5(left)).
(a)
(b)
Figure 6: Histogram of coordination number as a function of loading for
different threshold values: (a) No threshold is applied (using the raw data
after watershed algorithm). (b) After a threshold value of $0.35~{}mm^{2}$ is
subtracted from all contact surface areas. As a result the average
coordination number is reduced in all three stages.
The magnitude of the shift is about $1\%$ and $2\%$ for stages $1$ and $2$.
This is to be compared with the macroscopic axial strain which is of the other
of 4% and 8% without any deformation along the order directions. This suggests
that the internal compression is more isotropic than the loading.
#### 2.1.3 Mean coordination
The grain partitioning also provides us with the list of contact areas between
grains, from which it is possible to analyze the contact network of grains.
Using first the raw data from the watershed algorithm, we have measured the
coordination (number of contacts) of each grain and plotted its distribution
for each stage of compression (see Figure 6(a)). This plot shows a significant
increase of grain coordination as the loading increases. It is notable that as
pressure increases, the distributions move to higher values and get slightly
narrower (see Table 1). This increase in the coordination number is achieved
by re-organisation of grains in the packing when they are compressed and also
grain compression/deformation process which reduces the overall grain-grain
distance. A relatively large number of grains lose their contacts with some of
their immediate neighbours while almost the same amount gain new contacts as
the compression progresses. We have shown the histogram of lost/gained
contacts in the inset of figure 6. Negative values on the horizontal axis
represent lost contacts while positive values show the number of newly gained
contacts. Nearly $50\%$ of grains retain their coordination number during the
compression without any changes.
Figure 7: (a) Network representation the full packing. (b) An internal view of
the packing illustrating the connecting grains
As discussed in section 3, the inherent finite resolution of the CT leads to
systematic bias in fine details such as grain contacts; if the gap between two
grains is in the order of, or less than, the voxel size, grains will appear to
be in contact in the segmented data. To correct for this intrinsic resolution
limitation, we offset all the contact areas by a small amount, which
essentially corresponds to the apparent contact area of two touching perfectly
spherical grains (see section 2.2.3 for detailed discussion). All contacts
whose apparent surface area is below this threshold, are therefore discarded.
Figure 6(b) shows a re-plot of figure 6(a) after applying the offset. The
isostatic limit for mechanically stable packing of spheres in 3D suggests a
connectivity of $4$ for frictional and $6$ for frictionless systems. In our
measurements, the average coordination number is $\simeq 5$ for stage $0$ (see
Fig.6(b)) which is in agreement with previous measurements bernal:60 ; Aste:05
.
From the knowledge of the whole contact network, a large number of other
morphological and topological quantities can be calculated, such as spatial
correlations in contact orientations, or contact anisotropy. Figure 7 shows a
reconstruction of the contact network through which forces propagate. The
statistical properties of such networks will be developed in further studies.
In what follows, we will focus on the determination of mechanical forces
within the pile, at the micro scale, as well as mesoscopic scale.
### 2.2 Kinematics and dynamics
#### 2.2.1 Displacement fields
Figure 8: Displacement field presented as vector fields (velocity fields) for
stage 1 (a) and 2 (b) of the experiment.
We calculate next the coordinates of the centroid of the grains for all three
scans and then track each individual grain throughout the experiment. As an
illustration, we render the 3D vector field derived from tracking grains
during the compression for both stages of compression, see figures 8(a-b).
(a) (b) (c)
Figure 9: Displacement field along the loading axis. (a) The overall vertical
displacement field is represented very well by an exponential equation. For
visual convenience every 30 data points is shown. (b) Longitudinal
displacement for the 1st stage and the overall displacement. Here we choose a
bin size of $0.4mm$ for averaging. (c) Average radial displacement of grains
measured from the distance between the centre of packing its outer boundary.
To gain a qualitative insight into the displacement field, we measure the
average displacement of grain centres in both longitudinal and radial
directions. Figure 9(a) demonstrates such displacement along the loading axis.
As expected, gradient of displacement from top to bottom shows that the system
is under vertical compression. The decrease of the slope with depth also
indicates that part of this stress is screened by the time it reaches the
bottom. A typical explanation of this is the Janssen effect Janssen:1895 ,
suggesting that friction on the lateral walls might be significant. To confirm
this, we studied the vertical displacement profile in the radial direction
(figure 9(b)) which confirms that grain movements closer to the outer region
of cylinder are smaller than that of the central region. Therefore a shear
component in the $(z,r)$ direction is expected. Another feature of the
displacement field is the weak but noticeable radial displacement of the grain
towards the boundary (see figure 9(c)). This radial displacement can be
clearly seen in figure 8(a).
#### 2.2.2 Contact mechanics
In addition to the direct measurement of individual grain displacements, a
number of other mechanically relevant features can be extracted from the
tomogram. Upon compression, we observed that the distance between grains
varies, as well as shape and contact areas between grains. These quantities
can a priori be used to quantify the force transmitted between the grains. In
this section, we describe the implementation of a contact model that is
suitable for stress calculations from 3D images.
Figure 10: Response of a single rubber ball to loading.
Contact mechanics between solid objects have been studied for over a century.
The seminal works of Hertz Hertz:1882 and Mindlin Mindlin:49 about the
contact mechanics of solid bodies have provided analytical solutions for
idealised cases, summarised in Johnson:87 . The contact force between two
elastic spheres can be calculated from the nonlinear Hertz-Mindlin model. The
expression of the normal force between two contacting elastic spherical grains
$m$ and $n$ with uncompressed radii $R_{m}$ and $R_{n}$, made of the same
material, is given by:
$f_{mn}=\frac{2}{3}\frac{4G}{1-\nu}R^{1/2}\xi_{mn}^{3/2},$ (1)
where $R$ is the geometric mean of $R_{m}$ and $R_{n}$,
$R=2R_{m}R_{n}/(R_{m}+R_{n})$, $\xi_{mn}$ is the normal overlap, or
penetration length, as depicted in Fig. 10a, $G$ is the shear modulus and
$\nu$ the Poisson ratio of the grains, equal to $0.5$ for rubber Zhang:05a .
Figure 11: Solid lines: the mean vertical stress as a function of contact area
offset. Dotted lines: the applied load, 740 Pa for stage 1, and 5000 Pa for
stage 2.
We have tested the mechanical response of our grains by compressing a few
individual grains between two steel plates and measuring the force required as
a function of the gap between the plates (Fig. 10). The response of the grains
is consistent with the Hertz model, and we have extracted from these graphs
the value of the grains shear modulus, $850kPa$, which is a reasonable value
for a commercial rubber.
The Hertz-Mindlin theory therefore provides us with a suitable model to
calculate the contact force from the grain geometry, as long as we have a good
estimate of the normal overlap $\xi$ for each contact. Two options can be
considered at this stage. i) The normal overlap can be estimated from the
distance between grain centres. If the grains are located at positions
$\underline{r}_{m}$ and $\underline{r}_{n}$, assuming grains remain spherical,
$\xi_{mn}=\frac{1}{2}[R_{m}+R_{n}-|\mathbf{\underline{r}_{m}-\underline{r}_{n}}|]$.
The autocorrelation function shows that we expect this quantity to be of the
order of a few percent of the grain diameter. Although the grain location is
determined accurately, calculation of $\xi$ can be fairly inaccurate due to
the anisotropy of the imposed strain causing the grains to deviate
significantly from a spherical shape into more ellipsoidal geometries. The
centre-centre distance is therefore not a good approach to estimate the
contact geometry. ii) Our image processing protocols deliver a sensitive
measurement of the contact area (Fig. 4), which provides us with a more
reliable way to calculate the forces independently of the centre-centre
distances. The contact area $(s_{mn})$, between grains labelled by $m$ and
$n$, is related with the normal overlap $\xi_{mn}$ by:
$\xi_{mn}=\frac{s_{mn}}{2\pi R}$. This purely local measurement of the force
can be applied to any grain geometry, as long as the local curvature of the
grain near the contact point can be estimated as well.
#### 2.2.3 Force distribution and stress field
As discussed above, the finite resolution of the CT causes the contact areas
to be systematically larger than their real values so this measurement needs
to be carefully calibrated. There is de facto a small distance $\delta$, of
the order of the voxel size, such that if two surfaces are separated by less
than $\delta$, they will appear in contact after segmentation. This results in
a systematic enlargement of the contact area, and even detection of false
contacts if the separation between grains is less than $\delta$. In the case
of spherical elastic grains, the contact area after segmentation would
correspond to the geometrical overlap of two grains with each radius being
increased by $\delta/2$. Since the contact area is linear with the contact
overlap $\xi$, the effect that $\delta$ has on the surface area is equivalent
to a systematic offset $s_{off}=\pi R\delta$. However, $\delta$ is a priori
unknown and has to be determined by calibrating our measurements.
Figure 12: Histograms of normal forces in the sample for all three loading
stages.
The mean stress in the vertical direction at the vicinity of the upper
interface can be estimated in two independent ways for each loading step;
either from the knowledge of the loading mass and geometry of the setup, or by
using the contact forces to calculate the stress tensor in the bulk of the
pile. We consider a subvolume $\delta v$. Each contact contained in this
volume bears a force denoted $\mathbf{\underline{f}_{mn}}$, measured from the
contact area $s_{mn}$ between grains indexed by $m$ and $n$. The centre to
centre vector is noted $\mathbf{\underline{l}_{m,n}}$. The components of the
stress tensor, indexed by $i$ and $j$, are then obtained from the following
expression:
$\sigma_{ij}=\frac{1}{\delta v}\sum_{m,n}\;f^{i}_{mn}l^{j}_{mn}$ (2)
where the sum is over all contacting pairs of grains in the volume $\delta v$.
In order to measure the offset $\delta$ required to compensate for the
segmentation error, we have calculated, using the upper third of the sample,
the mean vertical stress $\sigma_{zz}$ we would obtain for a range of values
of the surface offset $s_{off}=\pi R\delta$ (Fig. 11). A suitable choice
should provide the value of the normal stress consistent with the loading
applied on the sample (i.e. where the solid line intersects the dotted line on
Fig. 11). Based on this graph, we select $s_{off}\approx$ 0.25 mm2, that
nearly sets the normal stress to its expected value for the high load value,
where the calibration is the most reliable due to the large values of the
force and increased number of contacts. This value corresponds, as expected,
to the size of a single voxel, confirming the consistency of the method. For
stage 0 and 1, however, based on this single threshold calibration, the
overall mean stress appears to have been overestimated.
Figure 13: Stress fields in a plane section of the sample for various loads.
From top to bottom, i) cartoons showing the location of the section in the
sample, along a diameter, and various stress components reported, ii) the
stress map in the azimuthal direction $\sigma_{\theta\theta}$, in the radial
direction $\sigma_{rr}$, in the vertical direction $\sigma_{zz}$, and shear
stress $\sigma_{rz}$.
Once this tool is calibrated, it can be used to probe a number of statistical
quantities in the pile. Figure 12 shows the distribution of normal forces in
the pile for the three different loads studied here. The 3D bulk measurement
of the contact normal forces exhibit a number of features that are
characteristic of granular systems. In particular, a salient feature of force
measurement presented in this study is that the distributions are primarily
exponential for large forces. This is in agreement with earlier measurements
in 2D piles Coppersmith:97 ; Liu:95 or at a 3D interface Bruji:05 and also
the numerical studies of dynamics of granulated systems Radjai:96 . Another
feature often reported in granular dynamics studies OHern:01 ; Majmudar:05 is
the presence of a peak or a plateau at low forces. However, this region of the
distribution is also where the forces are the most sensitive to image
resolution, segmentation and thresholding. We believe a much higher spatial
resolution is required before conclusions can be made about the precise shape
of the force distribution in the region of low forces, in particular since the
mean stress could not be adjusted.
The stress field in the pile is represented by a series of color maps in Fig.
13. These stress maps confirm a number of expected results. In the absence of
any loading (stage 0), we observe a slight increase of the stress from top to
bottom, in agreement with the fact that only the weight of the grains
themselves acts at this stage. The same pattern is observed in all directions.
The vertical component of the stress tensor shows an increase with the
loading, as well as the radial and azimuthal components. The magnitude of the
vertical component is about twice as large as the other two. Another finding
of interest concerns the existence of the shear component $(r,z)$. It is also
notable that both vertical and radial components of the stress tensor have
larger values near the boundary of the confining cylindrical cell at the 2nd
stage of compression. Grains at the centre experience a reduced compression,
due to their ability to move slightly towards the sides. Lateral grains are
however under stronger radial and vertical displacement gradients. These maps
are therefore in full agreement with the displacement field measured from the
tracking of grains for high loadings (Fig 9).
## Conclusion
We have presented in this paper the first measurements of internal stresses in
a dry granular system resolved at the single grain scale. We have used here a
simple geometry that allows us to calibrate and validate our measurements. In
particular, we are now able to quantify, for the same pile, a number of
characteristic features. They include mainly the evolution of the coordination
number, the 2-point correlation function, the internal displacement fields,
force distributions and stress fields, all as a function of loading. Taken
together, all these measurements will enable us to better unravel the
micromechanical behaviour of granular systems, in particular in quasistatic
regimes. It is worth pointing out as well that by measuring the force based on
the contact geometry, we are able to deal with non symmetric and polydisperse
grains.
Although we have proven that these measurements are realistic and achievable,
a number of improvements are still required in order to study more complex
cases. We need, in particular, to increase both the spatial statistics (number
of grains) and the resolution at the contact scale which will provide a better
calibration of the force and therefore a more reliable force measurement in
the pile. This will also improve our measurements of coordination number and
other geometric quantities. Increased image resolution will also allow the use
of stiffer grains which in turn will widen the range of suitable materials to
use for the grains. These improvements are achievable in the near future using
high resolution nano-focus CT combined with large panel detectors, providing
at least a five fold improvement in resolution and speed. The contact model
also needs further refinements, so that tangential forces can be estimated
from the orientation of the contact zone with respect to the centre-to-centre
line. These technical advances, combined with the analysis tools presented
here, will contribute to the understanding of a number of open problems.
Internal force distribution can now be studied in the bulk, reaching the low
force region of the distribution. The mechanics of the material and in
particular its mechanical stability rely on the internal organisation of the
grain contacts and also the force network. These are quantities that can be
directly analysed in 3D, not only statically, but also dynamically by
monitoring how an applied load affects each individual grain contact.
The kinematic study of the material can also be qualitatively and
quantitatively extended. Not only the local strain can be measured, but the
non-affine part of the displacement field is also directly accessible. The
latter is increasingly thought to be related with the bulk material
properties, in particular when the system is at the onset of rigidity
Wyart2008 . How such ideas can apply to 3D frictional granular systems remains
to be investigated. In particular, answering such questions will require an
extension to our current method so that we can study grain rotation. This
would allow us to test the importance of micropolar elasticity in the
mechanics of granular systems.
###### Acknowledgements.
The authors wish to thank for the techincal support of ANU’s X-ray tomography
team in particular Christoph Arns and Mark Knackstedt. We thant ANU
Supercomputer Facility and NCI for their generous allocation of computing
time. We also thank Ajay Limay and Jose Maoricio for assisting with some of
the visualisations in this article. MS acknowledges useful discussions with
Nicolas Francois and Tomaso Aste. Financial support for this work through a
grant from Australian Research Council, Project DP0881458, is greatefully
acknowledged.
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|
arxiv-papers
| 2010-09-22T23:36:47 |
2024-09-04T02:49:13.030521
|
{
"license": "Public Domain",
"authors": "Mohammad Saadatfar and Adrian P. Sheppard and Tim J. Senden and\n Alexandre J. Kabla",
"submitter": "Mohammad Saadatfar",
"url": "https://arxiv.org/abs/1009.4500"
}
|
1009.4663
|
∎
11institutetext: T. Kallman 22institutetext: NASA Goddard Space Flight Center,
Greenbelt, MD 20771
Tel.: 301-286-3680 22email: Timothy.R.Kallman@nasa.gov
# Modeling of Photoionized Plasmas
T. R. Kallman
(Received: date / Accepted: date)
###### Abstract
In this paper I review the motivation and current status of modeling of
plasmas exposed to strong radiation fields, as it applies to the study of
cosmic X-ray sources. This includes some of the astrophysical issues which can
be addressed, the ingredients for the models, the current computational tools,
the limitations imposed by currently available atomic data, and the validity
of some of the standard assumptions. I will also discuss ideas for the future:
challenges associated with future missions, opportunities presented by
improved computers, and goals for atomic data collection.
## 1 Introduction
The general problem of calculating the reprocessing of ionizing continuum
radiation from a star or compact object into longer wavelength lines and
diffuse continuum has broad importance in astrophysics. Photoionization
modeling is generally thought to include any situation in which the dominant
ionization and excitation mechanism is photons from an external source. In
addition, key ingredients of such models are applicable more broadly, to
calculating the photoelectric opacity and the flux transmitted through any
astrophysical gas. In the context of X-ray astronomy, photoionization is
important in sources which contain a compact object, notably active galactic
nuclei (AGN), X-ray binaries, and cataclysmic variables.
Many of the same principles apply to modeling the properties of diffuse
nebulae illuminated by UV radiation from stars, i.e. H II regions and
planetary nebulae. Discussions of model line emission from nebulae date from
Zanstra (1927), but Seaton and collaborators laid the groundwork for modern
numerical modeling by calculating many atomic quantities of importance to
nebular modeling (Seaton, 1958, 1959; Burgess and Seaton, 1960). The subject
has been reviewed extensively by Osterbrock & Ferland (2006), and others (see
references in Kallman (2001)) In this volume, many of the relevant ideas have
been discussed by Foster et al. (2011). This paper will not repeat this
material, but rather will describe the specific developments modeling of
photoionized plasmas which are relevant to X-ray astronomy.
## 2 What is Photoionization
For the purposes of subsequent discussion, typical photoionization models
calculate the ionization, excitation, and heating of cosmic gas by an external
source of photons. The gas is generally assumed to be in a time-steady balance
between ionization and recombination, and between heating and cooling. If the
gas is optically thin, then a useful scaling parameter is the ionization
parameter, defined in terms of the ratio of the incident ionizing flux to the
gas density or pressure. We adopt the definition $\xi=4\pi F/n$ where $F$ is
the incident energy flux integrated between 1 and 1000 Ry, and $n$ is the gas
number density Tarter et al. (1969); various other definitions are also in
use. When the gas is isobaric, the appropriate ionization parameter is
proportional to the ratio ionizing flux/gas pressure. We adopt the combination
$\xi/T$ or $\Xi=\xi/(kTc)$ Krolik et al. (1981) for this purpose. This much of
photoionization modeling is common to today’s models for X-ray sources and to
more traditional models used for nebulae.
Prior to the launch of $Chandra$ and $XMM-Newton$, it was recongnized that
objects such as AGN and X-ray binaries would have spectral features with
diagnostic application when observed in the X-ray band. These included
$K\alpha$ fluorescence lines from iron seen from many objects (Gottwald et
al., 1995; Asai et al., 2000), and the warm absobers in the soft X-ray band
from AGN (Reynolds, 1997). The quantum leap in sensitivity and spectral
resolution represented by $Chandra$ and $XMM-Newton$ revealed that other new
ingredients are needed in order to make the models useful for quantitative
study of X-ray sources. This motivated a great deal of work in the treatment
of physical processes previously neglected, and in the accuracy and
comprehensiveness of atomic data. A summary of some of these areas will occupy
the remainder of this paper.
## 3 Radiation Transport
Although numerical methods for radiation transfer are well establised (Hubeny,
2001), their implementation has the potential to be very computationally
expensive. Also, radiation transfer depends sensitively on the geometrical
arrangement of the gas and the sources of illuminating radiation, so it is
difficult to make a calculation which has wide applicability. A calculation
must treat the transfer of ionizing continuum into the photoionized gas and
transfer of cooling or reprocessed radiation out of the gas. For the purposes
of X-ray astronomy, it is also useful to calculate the entire synthetic
spectrum produced by the photoionized plasma. This can then be used for direct
fitting to data using tools such as xspec
111http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec, isis
222http://space.mit.edu/CXC/isis or sherpa 333http://cxc.harvard.edu/sherpa/.
Traditional treatments of radiation transfer make several simplifying
assumptions which allow for efficient calculation and wider applicability of
model results. These include: (i) Simplified geometry, such as a plane
parallel slab or a spherical shell; (ii) Use of single-stream transport of the
illuminating radiation; and (iii) Use of escape probability transport of
resonance lines. We note that X-ray resonance line optical depths are
typically much smaller than optical or UV line depths, so the effects of line
transfer approximations are reduced compared with traditional nebular
photoionization calculations.
Recent progress in radiation transfer for X-ray photoionization has centered
around the application of sophisticated accurate transfer treatments for
certain limited specialized problems. These include the adoption of the
accelerated lambda iteration method in the Titan code (Chevallier et al.,
2006; Różańska et al., 2006; Gonçalves et al., 2007). Recently, the Monte
Carlo method has been used by Sim et al. (2008, 2010) to treat the transfer in
AGN broad absorption line (BAL) and warm absorber flows. This latter work
combines radiation transfer with the detailed geometry derived from a
numerical hydrodynamic model. It has led to important insights into the origin
of spectral features such as high velocity lines, and X-ray emission lines.
This work illustrates the importance of consideration of the detailed
geometrical distribution of the gas, and the inherent limitations of
traditional simplified geometrical assumptions.
## 4 Atomic Data and Comprehensiveness
The other half of the challenge pf current photoionization modeling is almost
entirely associated with atomic data. It has long been realized that
comprehensive atomic data is crucial to accurately calculating the thermal
properties of gases hotter than $\sim 10^{4}$K Kwan & Krolik (1981), and that
this outweighs considerations such as the treatment of radiation transfer for
traditional nebular problems. This implies consistent treatment of various
ionization/excitation processes, both radiative and collisional, and their
inverses, including inner shell processes, for all of the ions of the $\geq$
10 most abundant elements. Similar arguments apply to the calculation of the
X-ray opacity of partially ionized gases. Furthermore, the need to calculate
synthetic spectra introduces a further requirement of accuracy on atomic data:
observed X-ray spectra can have spectral resolution of
$\varepsilon/\Delta\varepsilon\simeq 1000$, and in order to fit to such data,
synthetic spectra must employ wavelengths and ionization potentials which are
accurate at this level. Such precision cannot be achieved by current atomic
physics calculations, and requires dedicated experiments.
A great deal of work has been done to calculate and measure atomic energy
levels, cross sections and transition probabilities for the purposes of
astrophysical X-ray photoionization modeling. A detailed summary would require
a dedicated paper; many areas of progress have been described by Beiersdorfer
(2003); Kallman and Palmeri (2007); Foster et al. (2011). Notable are the
measurements carried out by the electron beam ion trap (EBIT) and storage
rings in Germany and Sweden, calculations using the FAC, HULLAC, R-Matrix and
Autostructure codes. Some of this work will be highlighted in the following
sections.
## 5 Recent Topics
### 5.1 Iron M-shell Unresolved Transition Array (UTA)
The $Chandra$ spectrum of the Seyfert 1 galaxy NGC 3783 (Kaspi et al., 2002),
and spectra of similar objects, has led to significant revisions of our
understanding of these sources. It has also pointed out several physical
effects which were not included in previous photoionization models. One
example is the importance of the many line features produced by transitions in
iron between the levels with principle quantum numbers 2 and 3 (the L and M
shells), notably from ions with 10 or more electrons, where the L shell is
full. Previously these lines had only been considered in ions with partially
filled L shells. Typical ions have many such lines ($\sim$ 10 – 100), grouped
together in wavelength, and the ensemble is called the iron M shell unresolved
transition array (UTA). The importance of these lines was first pointed out by
Behar et al. (2001), who also showed that they provide an ionization
diagnostic for relatively low ionization material. That is, the centroid
shifts from 16 – 17$\AA$ to 15 – 16$\AA$ as ionization increases owing to the
change from 2p-3d transitions to 2s-3p transitions as the 3p shell opens. This
is illustrated in figure 1, taken from Behar et al. (2001).
Figure 1: Iron M shell UTA absorption for 3 different ionization parameters.
From Behar et al. (2001).
These features are detected in most Seyfert galaxy warm absorbers, and are
important diagnostics of ionization. In particular, in the case of NGC 3783,
they clearly indicate the presence of low ionization iron Fe2+ – Fe8+, and the
relative absence of iron Fe9+ – Fe16+.
### 5.2 Fluorescence Lines
Fluorescence lines can be emitted by gas over a wide range of ionization
states. Very useful tables of the properties of these lines were provided by
Kaastra & Mewe (1993). Also, it was pointed out by Palmeri et al. (2003) that
the $K\beta/K\alpha$ ratio is a sensitive diagnostic of ionization, and this
has been applied to observed spectra by Yaqoob et al. (2007) and others.
Figure 2 illustrates the variety of iron K emission feature profiles and
strengths that can arise for various ionization states Kallman et al. (2004).
Figure 2: Model K shell line spectra of iron for various ionization
parameters (see below for definition). Emissivity units are arbitrary, but are
the same for all panels. Kallman et al. (2004).
### 5.3 Resonance Scattering
In traditional nebular modeling it was typically assumed that the gas was
spherically symmetric around the continuum source and stationary. If so,
resonance scattering of the incident continuum is expected to have negligible
effect on the spectrum seen by a distant observer. There is a cancellation
between continuum photons removed from and those added to the beam of
continuum we observed from the central source. In this case, the region of the
spectrum emergent from the cloud which includes a resonance line would show
neither a deficit nor an excess at the line energy. Of course, photoionization
followed by recombination leads to a redistribution of the energy of the
photons absorbed by ionization, and so would produce an excess at the line
energy.
It is now recognized that many photoionization-dominated sources observed in
X-rays have reprocessing gas which is not spherically distributed around the
continuum source. In the case of a non-spherical scattering region, the
cancellation between photons scattered out of and into the observed beam is
not exact, and thus scattering can have an effect on the observed spectrum.
The effect depends, crudely, on the relation between the column density
between the continuum source and infinity averaged over 4$\pi$ steradians, as
compared with the column density along the observer’s line of sight to the
continuum source. If the observer’s line of sight traverses more gas than the
spherical-average, then resonance scattering will result in a net removal of
photons from the observed radiation field and the spectrum will have a deficit
at the line energy. This could be termed the ’net absorption’ case. If the
observer’s line of sight traverses less gas than the spherical-average, then
resonance scattering will result in a net addition of photons to the observed
radiation field and the spectrum will have an excess at the line energy. Such
apparent ’net emission’ can be confused with emission due to recombination or
electron impact excitation, though these can be distinguished by differing
dependence on atomic quantities such as oscillator strengths or collision
cross sections. Furthermore, the resonance scattering cross section is much
greater than the background cross sections for photoionization, so line are
expected to saturate at column densities which are small compared to the
column densities where the continuum can penetrate. Recombination emission
will dominate at higher column densities and scattered emission will dominate
at lower column densities. Kinkhabwala et al. (2002) have shown that that
resonance scattering has a distinct signature in the line ratios and
line/continuum ratio in emitting plasmas. This is illustrated in figure 3,
which shows the transition from a scattering-dominated to a recombination-
dominated spectrum as a function of the column density of the plasma. The left
column corresponds to what would be observed in the ’net absorption’ case, and
the right column corresponds to ’net emission’. At low column densities
radiative pumping of resonance lines dominates because the cross sections are
larger than for photoionization, and the apparent line emission in the ’net
emission’ case is due to resonance scattering. At high column densities the
emission is dominated by recombination.
Figure 3: Scattering-dominated spectrum from Kinkhabwala et al. (2002). Left
panel shows spectrum seen in transmission, right panel shows spectrum seen in
reflection.
### 5.4 Dielectronic Recombination
It was pointed out by Netzer (2004) and Kraemer et al. (2004) that the
ionization balance needed to fit the K lines of Si in the spectrum of NGC3783
was discrepant from that needed to fit the iron UTA lines. It was suggested
that this was due to the use of inaccurate dielectronic recombination rates
for iron, and it was postulated that the rates where were larger by $\sim$10\.
This suggestion was confirmed, qualitatively, by Badnell (2006), who performed
distorted wave calculations of the rates and obtained rate coefficients which
were even greater than those suggested by Netzer (2004). Experimental
confirmation was demonstrated for Fe13+ Schmidt et al. (2006), and laboratory
measurements have been made for ions down to Fe7+ Schippers et al. (2010).
The implications of these new rates are illustrated in figures 4 and 5. These
show the iron ionization balance and the fit of the resulting synthetic
spectrum to the 800 ksec $Chandra$ observation of the Seyfert 1 galaxy NGC
3783 (Kaspi et al., 2002). These are shown for two choices of dielectronic
recombination rates: the new rates, adopting the fits from Badnell (2006) in
the upper panels, and the rates which were previously used (these are
described in Kallman and Bautista (2001); Bautista and Kallman (2001)). The
fits to the spectrum utilize two components of gas, each with a single
ionization parameter, and both using the same velocity and line broadening.
These are: v$\simeq$ 800 km/s relative to the galaxy, and v${}_{turb}\simeq$
300 km/s. Figure 5 shows this fit using the new rates in the regions
containing the iron M shell UTA and the Si K lines, showing general agreement.
The ionization parameters are log$\xi$=2 and 1 for the new rates. The
ionization balance produced by the older rates require log$\xi$=2 and -0.5 in
order to fit the iron UTA. The Si K lines are missing from the model since the
ionization state of Si is too low to allow K$\alpha$ absorption at
log$\xi$=-0.5. The fit to the model using the older rates is not shown. The
older rates cannot fit the the iron UTA, near 17 $\AA$ and the Si K likes,
near 7$\AA$ simultaneously, while the newer ones can.
Figure 4: Ionization balance for Fe, ion fractions on the vertical axis vs.
ionization parameter . New Badnell (2006) rates for dielectronic recombination
were used in the calculations shown in upper panel, older rates were used in
the calculations shown in the lower panel. Verical axis is log(ion fraction)
relative to the total for iron. Horizontal axis is log$\xi$. Highest ion
fraction is H-like (Fe${25+}$), corresponding to right-most curve, and lowest
is neutral, corresponding to left-most curve, in both panels.
Figure 5: Fits to the NGC3783 spectrum using Badnell (2006) rates for iron
dielectronic recombination. Left panel: region including the iron UTA near 17
$\AA$. Right panel: region showing the Si K lines near 6.8-7.2 $\AA$.
### 5.5 Thermal Instability
A possible explanation for the two phase behavior found in the fits to the NGC
3783 spectrum is thermal instability. This is due to the properties of the
cooling function in the gas which allows for two stable and one unstable
solutions to the thermal equilibrium equation. This has been discussed in
detail by many authors, notably in the context of quasar broad line clouds
Krolik et al. (1981); Mathews & Ferland (1987). The physical origin is the the
fact that the net cooling function has strong temperature dependence in some
regions of temperature, and weaker dependence in other regions. This behavior
is illustrated in figure 6, which shows surfaces of constant net cooling
(defined below) in the temperature-ionization parameter plane for a
photoionized gas. Here and in all the examples in this section we adopt a
$\Gamma$=2 power law for the ionizing spectrum illuminating the gas. That is,
the illuminating flux is $F_{\varepsilon}\propto\varepsilon^{1-\Gamma}$ erg
cm-2 s-1 erg-1. This figure shows the equilibrium surface as a solid curve,
and illustrates the difference between constant density and constant pressure
gases. In the constant density case there is only a single value of
equilibrium temperature for a given ionization parameter, while in the
constant pressure case there is a narrow region of ionization parameter where
three equilibria are possible.
Thermal instablility is associated with regions where the net cooling function
$\Lambda(T)$ is a decreasing function of temperature. A stable temperature
solution is characterized by a cooling function which is an increasing
function of tempeature; a positive temperature perturbation about a stable
equilibrium temperature will lead to increased cooling, which will restore the
gas to equilibrium, with the corresponding behavior for a negative temperature
perturbation. Perturbations about an unstable temperature will tend to run
away toward higher or lower temperature until a stable region of the cooling
curve is reached. The net cooling function of a photoionized gas is globally
increasing over the temperature range from $\sim$ 1000 K – 108K, owing to
strong hydrogen cooling at low temperature, and strong inverse Compton cooling
at high temperature. Instability is associated with a local maximum to the
function, and so there must always be an odd number of thermal equilibrium
solutions.
The instability is more likely to occur when the gas is isobaric, rather than
isochoric. Isochoric gas heated by a radiation field such as that shown here,
i.e. the spectral energy distribution characteristic of AGN, is predicted to
be thermally stable; a radiation field which is flat, or deficient in soft
photons, is more likely to produce thermal instability. The presence of the
thermal instability depends on interesting things: the shape of the ionizing
spectrum (SED) from IR through the X-rays, the atomic rates, elemental
abundances, and (weakly) in the gas density. This suggests possible diagnostic
use.
The same result is displayed in a different way in figure 7, which plots the
heating and cooling functions per hydrogen nucleus vs. temperature for various
values of the ionization parameter. The left panel shows this for constant
density gas and the right panel is for constant pressure gas. In the constant
density case the curves correspond to various values of the ionization
parameter $\xi$, while in the constant pressure case the curves correspond to
values of the ionization parameter $\Xi$. The figure shows the equilibrium
solutions as colored dots: green dots correspond to solutions which are
unique, i.e. values of ionization parameter for which there is only one
solution; and blue dots where solutions are not unique.
The existence of thermal instability depends on the validity of the assumption
of thermal (and ionization) equilibrium, and it is important to consider this
when using it for quantitative work. Thermal equilibrium requires that the
timescale for heating and cooling, the thermal timescale, be less than other
relevant timescales. In the case of warm absorbers, which are flowing out from
the AGN center, this includes the gas flow timescale.
The net cooling rate per nucleus in a photoionized gas can be written
$L_{net}=n\Lambda-H$ (1)
where $\Lambda$ is the cooling rate coefficient, and the heating rate can be
written:
$H=H_{X}+H_{C}$ (2)
$H_{X}$ and $H_{C}$ are the photoelectric and Compton heating rates,
respectively, and can be written:
$H_{C}\simeq n\xi\sigma_{C}\frac{<\varepsilon>-4kT}{m_{e}c^{2}}$ (3)
where $T$ is the electron kinetic temperature, $<\varepsilon>$ is the mean
photon energy, $\sigma_{C}$ is the Compton cross section, $n$ is the gas
number density. An approximation to the photoelectric heating rate is Blondin
(1994):
$H_{X}\simeq 1.5\times
10^{-21}n_{cm3}^{2}\xi^{1/4}T_{K}^{1/2}\left(1-\frac{T}{T_{x}}\right){\rm~{}erg~{}s^{-1}}$
(4)
where $T_{x}\simeq 10^{6}$K is a typical value for power law ionizing spectra
and $T_{K}$ is temperature in units of K and $n_{cm3}$ is density in units of
cm-3 . An approximation to the cooling rate is Blondin (1994):
$\Lambda\simeq 3.3\times 10^{-27}T_{K}^{1/2}+1.7\times
10^{-18}\xi^{-1}T_{K}^{-1/2}e^{-\frac{1.3\times
10^{5}K}{T_{K}}}{\rm~{}erg~{}cm^{3}~{}s^{-1}}$ (5)
The first term is due to bremsstrahlung and dominates at temperatures
$T_{x}\leq T\leq T_{C}$. At constant pressure in this range the cooling rate
per particle has tempeature dependence $n\Lambda\propto T^{-1/2}$, and such
gas will be thermally unstable according to the criterion of Field (1965):
$dL/dT<0$.
The cooling time is
$t_{cool}=\frac{3kT}{L}$ (6)
and using the result from equation 5
$t_{cool}\simeq 10^{16}n^{-1}T_{5}^{3/2}\xi_{2}\rm{s}$ (7)
where $T_{5}$ is the temperature in units of $10^{5}$K and $\xi_{2}$ is the
ionization parameter in units of $10^{2}$ erg cm s-1. The density $n$ is
poorly constrained, but likely values are in the range $10^{4}$ – $10^{8}$.
Figure 7 shows more accurate numerical calculation of these rates as a
function of T and $\xi$. This shows the difference between constant density
and constant pressure calculations, and shows the regions of thermal
instability. Constant pressure is more likely to lead to thermal instability
owing to the added inverse dependence of density on temperature, thereby
leading to regions where the net cooling decreases with increasing
temperature.
Figure 6: Contours of constant net cooling-heating in the T-$\xi$ plane. Upper
panel: constant density; lower panel: constant pressure. Equilibrium is shown
as solid curve.
Figure 7: Heating (read) and cooling (black) rates vs. temperature. The upper
panel corresponds to constant density gas, and the curves are for values of
log$\xi$ 0 – 5. The lower panel corresponds to contant pressure gas and the
curves are for values of log($\Xi$) -1 – 2. Vertical axis is log(Rate per
hydrogen nuclues) in erg s-1. Equilibrium values are shown as dots, green dots
for thermally stable solutions. Blue dots correspond to equilibrium values
which are either unstable, or which are stable but occur for values of $\Xi$
which also have unstable solutions.
The timescale for gas to flow outward in the warm absorber is
$t_{flow}=\frac{R}{v}\simeq 10^{11}R_{18}v_{7}^{-1}\rm{s}$ (8)
where R is the typical flow lengthscale, $R_{18}$ is $R/10^{18}$ cm, and $v$
is the flow velocity. It is clear from these estimates that the timescales for
cooling and flow can be comparable. Thus, the assumption of thermal
equilibrium must be carefully evaluated before the thermal instability is used
as a diagnistic.
Now we can test this for a more realistic model of the warm absorber. This is
a 2.5 dimensional (3 dimensional axisymmetric) hydrodynamic calculation of the
evaporation from the torus responsible for the obscuration in Seyfert 2
galaxies. In this model the torus is heated by a non-thermal $\Gamma$=2 power
law from the black hole. The warm absorber is formed as gas is evaporated and
flows out (radiative driving is included) The thermodynamics of X-ray heating
and radiative cooling is included, and the dynamics are calculated as pure
hydrodynamics, no magnetohydrodynamic effects are included. The synthetic
spectrum is also calculated from these simulations. These have been published
in Dorodnitsyn et al. (2008); Dorodnitsyn & Kallman (2009). It is interesting
to consider what happens in the T vs $\xi/T$ plane in such a model. This is
illustrated in figure 8, which shows the loci of $T$ vs $\xi/T$ for such a
simulation. The effect of thermal instability is seen for temperatures near
$10^{5.5}$K, where there are fewer points.
Figure 8: Locus of points in evaporative torus hydrodynamic model. This
snapshot illustrates that many zones are out of thermal equilibrium; time
depdendent effects are important. The thermally unstable regions are generally
avoided, but not completely. Equilibrium curve is shown in red.
This model shows that the thermal properties of warm absorber gas are
considerably more complicated than most current models. Departures from
thermal equilibrium are important. At the highest temperatures adiabatic
cooling is important. This is a snapshot at one particular time in the
simulation; the loci of points are not static, but are constantly moving as
material is evaporated from the torus, moves outward and cools. It illustrates
that many zones are out of thermal equilibrium; time dependent effects are
important. The thermally unstable regions are generally avoided, but not
completely. Thus, simple stability models provide a very approximate guide for
where the gas ends up. They may overestime the ionization parameter, since
non-equilibrium gas may be at a lower ionization parameter than would be
inferred from fitting to equilibrium models. We should not be surprised to see
gas in unstable regions. Furthermore, the appearance varies on flow timescale;
the same model may look different when viewed in many different objects.
## 6 The Future
Fitting models of photoionized plasmas to $Chandra$ and $XMM-Newton$ spectra
provides insights to the nature of warm absorbers and related structures:
their degree of ionization, density, location, composition and kinematics.
However, the models are likely still incomplete in important ways. For
example, X-ray grating spectra with good statistics seldom give truly
statistically acceptable fits to standard models. Typically, $\chi^{2}$ per
degree of freedom is $\sim$2 or greater. Possible reasons include: missing
lines in the atomic database, incorrect treatment of line broadening,
incorrect ionization balance, overly idealized assumptions (such as ionization
equilibrium), inaccurate treatments of radiative transfer or geometrical
effects. In addition to thinking about these things, modelers need to prepare
for the next generation of X-ray instruments, which will likely have improved
sensitivity in the iron K energy band. This will allow quantitative study of
lines from trace elements such as Cr and Mn. Another frontier is low
ionization material; these instruments may detect inner shell fluorescence
from many elements with $Z>10$. Time dependent effects deserve more
exploration, as do more user friendly general purpose radiative transfer
models. Photoionization modelers will be helped by laboratory measurements and
atomic theory if this results in more accurate and comprehensive line
wavelengths. The quest for good fits also depends on accurate instrumental
calibration, including the response to narrow features, and accurate
calibration of continuum for accurate subtraction. Users of photoionization
models can provide feedback to modelers and others in order to detect errors
and improve the user interface.
###### Acknowledgements.
I thank Dr. Ehud Behar and Dr. Daniel Savin for constructive comments, and my
collaborators: M. Bautista, A. Dorodnitsyn, J. Garcia, C. Mendoza, P. Palmeri,
P. Quinet, and M. Witthoeft.
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|
arxiv-papers
| 2010-09-23T17:29:43 |
2024-09-04T02:49:13.041949
|
{
"license": "Public Domain",
"authors": "T. R. Kallman",
"submitter": "T. Kallman",
"url": "https://arxiv.org/abs/1009.4663"
}
|
1009.4673
|
# Noise Performance of Lumped Element Direct Current Superconducting Quantum
Interference Device Amplifiers in the 4 GHz-8 GHz Range111This paper is a
contribution of the U.S. government and is not subject to U.S. copyright.
Lafe Spietz lafe.spietz@boulder.nist.gov Kent Irwin Minhyea Lee José
Aumentado National Institute of Standards and Technology, Boulder, Colorado
80305, USA
###### Abstract
We report on the noise of a lumped element Direct Current Superconducting
Quantum Interference Device amplifier. We show that the noise temperature in
the 4 GHz-8 GHz range over ranges of 10’s of MHz is below 1 kelvin (three
photons of added noise), characterize the overall behavior of the noise as a
function of bias parameters, and discuss potential mechanisms which determine
the noise performance in this amplifier. We show that this device can provide
more than a factor of 10 improvement in practical system noise over existing
phase-preserving microwave measurement systems in this frequency band.
The measurement needs of the quantum information community have created a
rapidly growing demand for ultra-low-noise microwave amplifiers. The typical
frequencies of interest for quantum information are in the 4 GHz–8 GHz range.
In general, the first stage cryogenic amplifier sets the overall speed of the
measurement, which can be critical for quantum information experiments. The
commercial state of the art in cryogenic microwave amplifiers uses High
Electron Mobility Transistor (HEMT) technology, and typically have a noise
temperature of 2-4 K, but yield practical system noise temperatures in the
10-20 K range, as discussed below. A rapidly growing subfield has emerged
using superconducting electronics to build pre-amplifiers for these
semiconductor-based cryogenic amplifiers.
In the 4 GHz–8 GHz range, this has primarily consisted of parametric
amplifiers. Parametric amplifiers have demonstrated noise below the standard
quantum limit and to have sufficient gain to overcome the following stages of
amplifier noise manuel_2008 . However, they have practical bandwidth typically
well below 1 MHz and do not amplify both phase quadratures of a signal.
Furthermore, parametric amplifiers require a microwave bias, increasing
significantly the complexity of operation. Amplifiers using Superconducting
Quantum Interference Devices (SQUIDs) have been shown to have near-quantum-
limited noise performance in a variety of circumstances mueck_quantum ,
primarily at frequencies below 2 GHz, or with limited gain prokopenko1 . We
have previously shown over 27 GHz of gain-bandwidth product, and gains up to
30 dB in microwave amplifiers using SQUIDs spietz_squidamp2 , and have
characterized the input impedance of these devices spietz_squidamp1 . Here we
show noise performance in a SQUID amplifier with less than three photons per
mode of added noise over 10’s of MHz of instantaneous bandwidth in the 4 GHz–8
GHz range.
Before discussing the SQUID amplifier in detail, we address the distinction
between system noise and amplifier noise in practical microwave measurements.
When loss of a linear factor A at temperature T is placed between a signal and
an amplifier of noise temperature $T_{n}^{amp}$, the combination of the
amplifier and the loss acts like an amplifier with lower gain and higher
noise. The system noise temperature is
$T_{n}^{sys}=AT_{n}^{amp}+(A-1)T,$ (1)
and the gain is suppressed by the factor A pozar . Because HEMT amplifiers
dissipate 10’s to 100’s of milliwatts of power, they must be at the 4 K stage
in a dilution refrigerator instead of at the base temperature due to cooling
power considerations. This forces the amplifier to be physically separated
from the experiment by 10’s of cm of cable, as well as by components such as
isolators and couplers. This in turn forces the noise temperature observed in
typical experiments to be considerably higher than the intrinsic noise of the
amplifiers. Since the integration time required for a given signal to noise
ratio scales with the square of the noise temperature dicke , the factor of 10
practical system noise improvements shown with this amplifier can lead to as
much as a factor of 100 shorter measurement time, making dramatic improvements
in the types of experiments which are accessible with microwaves at very low
temperatures.
Figure 1: Experimental layout. This shows the apparatus with the SNTJ. For the
y-factor measurements, the apparatus before the directional coupler is
replaced by the heated microwave termination. The transfer switch connected to
amplifier by superconducting cables allows for calibrated gain measurements.
Use of double isolators at both the input and output of the SQUID amplifier
realistically simulates a typical quantum measurement experiment.
The devices measured here were lumped element DC-SQUID amplifiers, impedance
matched at the input with quarter wave resonators of varying length to set the
frequency, and impedance matched to 50 $\Omega$ at the output using series
inductors and capacitors to ground, all on-chip, as described in
spietz_squidamp1 ; spietz_squidamp2 .
We have measured the system noise of our SQUID amplifiers in two different
ways: the y-factor method and using a Shot Noise Tunnel Junction (SNTJ)inprep
. Characterization of input impedance, bandwidth, and gain of these amplifiers
is reported in previous work spietz_squidamp1 ; spietz_squidamp2 . Figure 1
shows the apparatus used in this experiment, which was intended to
realistically replicate the circumstances of a typical cryogenic microwave
measurement, including the double isolator between the input of the SQUID
amplifier and the noise source. The y-factor method is the more conceptually
straightforward and widely accepted, but the SNTJ method is much faster, and
is thus much more useful for acquiring data into the rich structure of the
noise as a function of the two bias parameters: the flux bias and the current
bias. We used the y-factor measurements as well as shot noise to characterize
the 7.06 GHz amplifier shown in Figure 2 as well as the 1.7 GHz amplifier, and
just shot noise to characterize amplifiers at 6.3, 6.8, and 7.2 GHz.
The SNTJ consists of a normal metal tunnel junction matched to 50 $\Omega$
over a broad band, and biased through a bias tee so that a known direct
current can be applied to the junction while the microwave noise is measured
by the amplifier shotnoisescience . Because the physics of this device is well
understood, this measurement allows for the rapid determination of both the
physical temperature of the sample and the noise temperature of the amplifier.
A detailed description of the use of the SNTJ for amplifier calibration will
be given elsewhere. The most important aspect of it for the purpose of this
paper is its speed and ease of use relative to the y-factor method. Our
y-factor measurements were carried out by placing a heater, a ruthenium oxide
thermometer and a microwave terminator in thermal contact with each other at
the end of approximately 10 cm of niobium semi-rigid coaxial cable with a
copper wire providing a known thermal conductance to the base temperature of
the dilution refrigerator. Changing this temperature by enough to get useful
data on noise temperature took well over an hour for each thermal cycle. The
SNTJ, in contrast, can be switched in well under a millisecond, allowing for
noise temperature measurements limited primarily by the speed of the microwave
noise measurement, which is generally on the order of a second for a single
trace from the spectrum analyzer.
Figure 2: System noise temperature as a function of frequency at a good bias
point for three amplifiers with different lengths of input resonator. The blue
line is a 2 mm long input resonator, the green line is 1.6 mm, and the red
line is 1.4 mm. Inset shows zoomed in noise temperature as a function of
frequency around the optimal noise point for the 1.6 mm amplifier. These data
were taken using the y-factor method for the 1.6 mm device and the SNTJ method
for the other devices, with the configuration of double circulators,
directional coupler, and transfer switch shown in Figure 1.
Figure 2 shows noise temperature data for three of our amplifiers at an
optimal bias point, as measured using the y-factor method and the SNTJ method.
Typical system noise temperatures without the SQUID amplifier are
approximately 12 K, which is the upper limit of the temperature axis in the
figure. As shown in Fig. 2, there is more than a factor of 10 improvement at
the best frequency, and an improvement of more than a factor of 3 over 400 MHz
of bandwidth. This broadband improvement of the order of $\sqrt{10}$
corresponds to approximately a factor of 10 increase in speed in most
microwave measurements making this a very useful amplifier for a large class
of cryogenic microwave measurements. Amplifiers with optimal operation
frequencies at 1.7 GHz, 6.3 GHz, 6.8 GHz, 7 GHz, and 7.2 GHz were all
measured, and all showed their best system noise temperature to be close to 1
K, or three added noise photons at 7 GHz. Given that the double isolators at
the input of the amplifiers can have as much as 1 dB of loss, this implies
that the intrinsic amplifier noise at the optimal operating point is at least
as low as 0.76 K, or 4.5 times the standard quantum limit (where we define the
standard quantum limit to be one half the photon energy divided by the
Boltzmann constant). caves1982 .
We now discuss qualitatively the physical mechanism of the noise in these
amplifiers and how we believe it could be improved in the future. The noise of
DC-SQUIDs has been studied in detail both theoretically and experimentally for
lower frequency operation clarkebook ; martinisandclarke . In the literature,
it is found that the noise depends linearly on the temperature of the shunt
resistors, which is in general higher than the physical temperature of the
amplifier packaging due to Joule heating wellstood . It is also found that the
mixed-down noise from the Josephson frequency is a very significant factor in
the overall noise level of the SQUID. We observe in all our SQUID amplifiers
that the noise and gain have peaks and dips that correspond to certain values
of the voltage on the device, hence to certain specific values of the
Josephson frequency. We speculate that these peaks and dips correspond to
resonances from higher harmonics of the input resonator. This theory is
consistent with the fact that we see a larger number of harmonics in both the
voltage and the in-band microwave characteristics in amplifiers with lower
fundamental frequency input resonators than we do in the higher frequency
amplifiers.
Figure 3: System noise temperature in kelvin as a function of current through
the flux bias coil and signal frequency for an amplifier with a 1.4 mm
resonator. Again, we see useful signal-to-noise improvement over several
hundred MHz frequency range. Also, note the strong dependence of optimal
frequency, bandwidth, and minimum noise temperature on flux bias. The current
bias is fixed at 200 $\mu$A, which puts the SQUID in finite-voltage state.
Figure 3 shows how noise temperature depends on both frequency and flux bias
for a fixed current bias. First, note that it is possible to change the
optimal operating frequency of the amplifier by changing the bias. We believe
that this is due to the complex feedback dynamics of the SQUID, which change
the effective reactance at the end of the input resonator, shifting the
optimal signal frequency as well as the bandwidth mueck_feedback ;
spietz_squidamp2 . Second, note that the bias point of the SQUID is absolutely
critical in achieving optimal noise performance, and that care must be taken
to use the amplifier at the correct bias point for a given application. The
narrow optimal point for the noise performance corresponds to a resonance in
the voltage-current characteristics as well, both in the SQUID shown and in
all of the other measured SQUIDs. The exact nature of these resonances at
higher harmonics and how they can be designed to improve noise performance
will be explored in future work.
In conclusion, we have both shown useful noise characteristics in a practical
microwave SQUID amplifier in the 4 GHz-8 GHz frequency range and have begun to
learn what we need to in order to improve the performance in the future. The
bandwidth, operating frequency control, flatness of gain and noise, overall
noise performance, and sensitivity to bias are all strong functions of how
well controlled the microwave impedance environment is from the signal
frequency up to at least the Josephson frequency of 20-30 GHz. We plan to
improve all of these figures of merit in future designs by measuring the
scattering parameters of the lumped element SQUID in a calibrated way, and
then using standard microwave engineering techniques to control the impedance
environment as necessary to obtain the desired characteristics, in much the
same way that an amplifier designer would with a transistor-based amplifier.
While the amplifier presented in this paper and previous work spietz_squidamp1
; spietz_squidamp2 is already proving to be very useful for microwave quantum
measurement experiments, these improvements will lead to an amplifier with
even broader applicability by increasing the ease-of-use and ultimate noise
performance considerably.
## References
* (1) M. Castellanos-Beltran, K. Irwin, G. Hilton, L. Vale, and K. Lehnert, Nature Physics 4, 929 (2008).
* (2) M. Mück, J. B. Kycia, and J. Clarke, Applied Physics Letters 78, 967 (2001).
* (3) M. Tarasov, G. Prokopenko, V. Koshelets, I. Lapitskaya, and L. Filippenko, Applied Superconductivity, IEEE Transactions on 5, 3226 (1995).
* (4) L. Spietz, K. Irwin, and J. Aumentado, Applied Physics Letters 95, 092505 (2009), eprint 0908.4250.
* (5) L. Spietz, K. Irwin, and J. Aumentado, Applied Physics Letters 93, 082506 (2008).
* (6) D. M. Pozar, _Microwave Engineering_ (John Wiley and Sons, New York, 2005).
* (7) R. H. Dicke, Review of Scientific Instruments 17, 268 (1946).
* (8) Unpublished.
* (9) L. Spietz, K. W. Lehnert, I. Siddiqi, and R. J. Schoelkopf, Science 300, 1929 (2003).
* (10) C. M. Caves, Phys. Rev. D 26, 1817 (1982).
* (11) J. Clarke and A. I. Braginski, _The SQUID Handbook_ (John Wiley and Sons, Weinheim, Germany, 2004).
* (12) J. M. Martinis and J. Clarke, Journal of Low Temperature Physics 61, 227 (1985).
* (13) F. C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B 49, 5942 (1994).
* (14) M. Mück, D. Hover, S. Sendelbach, and R. McDermott, Applied Physics Letters 94, 132509 (2009).
|
arxiv-papers
| 2010-09-23T18:17:48 |
2024-09-04T02:49:13.050010
|
{
"license": "Public Domain",
"authors": "Lafe Spietz, Kent Irwin, Minhyea Lee, Jose Aumentado",
"submitter": "Lafe Spietz",
"url": "https://arxiv.org/abs/1009.4673"
}
|
1009.4766
|
# Efficient $\ell_{1}/\ell_{q}$ Norm Regularization
Jun Liu Jieping Ye
(September 23, 2010)
###### Abstract
Sparse learning has recently received increasing attention in many areas
including machine learning, statistics, and applied mathematics. The mixed-
norm regularization based on the $\ell_{1}/\ell_{q}$ norm with $q>1$ is
attractive in many applications of regression and classification in that it
facilitates group sparsity in the model. The resulting optimization problem
is, however, challenging to solve due to the structure of the
$\ell_{1}/\ell_{q}$-regularization. Existing work deals with special cases
including $q=2,\infty$, and they can not be easily extended to the general
case. In this paper, we propose an efficient algorithm based on the
accelerated gradient method for solving the $\ell_{1}/\ell_{q}$-regularized
problem, which is applicable for all values of $q$ larger than $1$, thus
significantly extending existing work. One key building block of the proposed
algorithm is the $\ell_{1}/\ell_{q}$-regularized Euclidean projection (EP1q).
Our theoretical analysis reveals the key properties of EP1q and illustrates
why EP1q for the general $q$ is significantly more challenging to solve than
the special cases. Based on our theoretical analysis, we develop an efficient
algorithm for EP1q by solving two zero finding problems. Experimental results
demonstrate the efficiency of the proposed algorithm.
## 1 Introduction
Regularization has played a central role in many machine learning algorithms.
The $\ell_{1}$-regularization has recently received increasing attention, due
to its sparsity-inducing property, convenient convexity, strong theoretical
guarantees, and great empirical success in various applications. A well-known
application of the $\ell_{1}$-regularization is the Lasso [32]. Recent studies
in areas such as machine learning, statistics, and applied mathematics have
witnessed growing interests in extending the $\ell_{1}$-regularization to the
$\ell_{1}/\ell_{q}$-regularization [2, 7, 14, 23, 29, 37, 38]. This leads to
the following $\ell_{1}/\ell_{q}$-regularized minimization problem:
$\min_{\mathbf{W}\in\mathbb{R}^{p}}f(\mathbf{W})\equiv
l(\mathbf{W})+\lambda\varpi(\mathbf{W}),$ (1)
where $\mathbf{W}\in\mathbb{R}^{p}$ denotes the model parameters, $l(\cdot)$
is a convex loss dependent on the training samples and their corresponding
responses, $\mathbf{W}=[\mathbf{w}_{1}^{\rm T},\mathbf{w}_{2}^{\rm
T},\ldots,\mathbf{w}_{s}^{\rm T}]^{\rm T}$ is divided into $s$ non-overlapping
groups, $\mathbf{w}_{i}\in\mathbb{R}^{p_{i}},i=1,2.\ldots,s$, $\lambda>0$ is
the regularization parameter, and
$\varpi(\mathbf{W})=\sum_{i=1}^{s}\|\mathbf{w}_{i}\|_{q}$ (2)
is the $\ell_{1}/\ell_{q}$ norm with $\|\cdot\|_{q}$ denoting the vector
$\ell_{q}$ norm ($q\geq 1$). The $\ell_{1}/\ell_{q}$-regularization belongs to
the composite absolute penalties (CAP) [38] family. When $q=1$, the problem
(1) reduces to the $\ell_{1}$-regularized problem. When $q>1$, the
$\ell_{1}/\ell_{q}$-regularization facilitates group sparsity in the resulting
model, which is desirable in many applications of regression and
classification.
The practical challenge in the use of the $\ell_{1}/\ell_{q}$-regularization
lies in the development of efficient algorithms for solving (1), due to the
non-smoothness of the $\ell_{1}/\ell_{q}$-regularization. According to the
black-box Complexity Theory [25, 26], the optimal first-order black-box method
[25, 26] for solving the class of nonsmooth convex problems converges as
$O(\frac{1}{\sqrt{k}})$ ($k$ denotes the number of iterations), which is slow.
Existing algorithms focus on solving the problem (1) or its equivalent
constrained version for $q=2,\infty$, and they can not be easily extended to
the general case. In order to systematically study the practical performance
of the $\ell_{1}/\ell_{q}$-regularization family, it is of great importance to
develop efficient algorithms for solving (1) for any $q$ larger than $1$.
### 1.1 First-Order Methods Applicable for (1)
When treating $f(\cdot)$ as the general non-smooth convex function, we can
apply the subgradient descent [5, 25, 26]:
$\mathbf{X}_{i+1}=\mathbf{X}_{i}-\gamma_{i}\mathbf{G}_{i},$ (3)
where $\mathbf{G}_{i}\in\partial f(\mathbf{X}_{i})$ is a subgradient of
$f(\cdot)$ at $\mathbf{X}_{i}$, and $\gamma_{i}$ a step size. There are
several different types of step size rules, and more details can be found in
[5, 25]. Subgradient descent is proven to converge, and it can yield a
convergence rate of $O(1/\sqrt{k})$ for $k$ iterations. However, SD has the
following two disadvantages: 1) SD converges slowly; and 2) the iterates of SD
are very rarely at the points of non-differentiability [7], thus it might not
achieve the desirable sparse solution (which is usually at the point of non-
differentiability) within a limited number of iterations.
Coordinate Descent [33] and its recent extension—Coordinate Gradient Descent
(CGD) can be applied for optimizing the non-differentiable composite function
[34]. Coordinate descent has been applied for the $\ell_{1}$-norm regularized
least squares [9], $\ell_{1}/\ell_{\infty}$-norm regularized least squares
[16], and the sparse group Lasso [10]. Coordinate gradient descent has been
applied for the group Lasso logistic regression [21]. Convergence results for
CD and CGD have been established, when the non-differentiable part is
separable [33, 34]. However, there is no global convergence rate for CD and
CGD (Note, CGD is reported to have a _local_ linear convergence rate under
certain conditions [34, Theorem 4]). In addition, it is not clear whether CD
and CGD are applicable for solving the problem (1) with an arbitrary $q\geq
1$.
Fixed Point Continuation [12, 31] was recently proposed for solving the
$\ell_{1}$-norm regularized optimization (i.e.,
$\varpi(\mathbf{W})=\|\mathbf{W}\|_{1}$). It is based on the following fixed
point iteration:
$\mathbf{X}_{i+1}=\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{X}_{i}-\tau
l^{\prime}(\mathbf{X}_{i})),$ (4)
where
$\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{W})=\mbox{sgn}(\mathbf{W})\odot\max(\mathbf{W}-\lambda\tau,0)$
is an operator and $\tau>0$ is the step size. The fixed point iteration (4)
can be applied to solve (1) for any convex penalty $\varpi(\mathbf{W})$, with
the operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$ being defined as:
$\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{W})=\arg\min_{\mathbf{X}}\frac{1}{2}\|\mathbf{X}-\mathbf{W}\|_{2}^{2}+\lambda\tau\varphi(\mathbf{X}).$
(5)
The operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$ is called the
proximal operator [13, 22, 36], and is guaranteed to be non-expansive. With a
properly chosen $\tau$, the fixed point iteration (4) can converge to the
fixed point $\mathbf{X}^{*}$ satisfying
$\mathbf{X}^{*}=\mathcal{P}^{\varpi}_{\lambda\tau}(\mathbf{X}^{*}-\tau
l^{\prime}(\mathbf{X}^{*})).$ (6)
It follows from (5) and (6) that,
$\mathbf{0}\in\mathbf{X}^{*}-(\mathbf{X}^{*}-\tau
l^{\prime}(\mathbf{X}^{*}))+\lambda\tau\partial\varpi(\mathbf{X}^{*}),$ (7)
which together with $\tau>0$ indicates that $\mathbf{X}^{*}$ is the optimal
solution to (1). In [3, 27], the gradient descent method is extended to
optimize the composite function in the form of (1), and the iteration step is
similar to (4). The extended gradient descent method is proven to yield the
convergence rate of $O(1/k)$ for $k$ iterations. However, as pointed out in
[3, 27], the scheme in (4) can be further accelerated for solving (1).
Finally, there are various online learning algorithms that have been developed
for dealing with large-scale data, e.g., the truncated gradient method [15],
the forward-looking subgradient [7], and the regularized dual averaging [35]
(which is based on the dual averaging method proposed in [28]). When applying
the aforementioned online learning methods for solving (1), a key building
block is the operator $\mathcal{P}^{\varpi}_{\lambda\tau}(\cdot)$.
### 1.2 Main Contributions
In this paper, we develop an efficient algorithm for solving the
$\ell_{1}/\ell_{q}$-regularized problem (1), for any $q\geq 1$. More
specifically, we develop the GLEP1q algorithm111GLEP1q stands for Group
Sparsity Learning via the $\ell_{1}/\ell_{q}$-regularized Euclidean
Projection., which makes use of the accelerated gradient method [3, 27] for
minimizing the composite objective functions. GLEP1q has the following two
favorable properties: (1) It is applicable to any smooth convex loss
$l(\cdot)$ (e.g., the least squares loss and the logistic loss) and any $q\geq
1$. Existing algorithms are mainly focused on
$\ell_{1}/\ell_{2}$-regularization and/or
$\ell_{1}/\ell_{\infty}$-regularization. To the best of our knowledge, this is
the first work that provides an efficient algorithm for solving (1) with any
$q\geq 1$; and (2) It achieves a global convergence rate of
$O(\frac{1}{k^{2}})$ ($k$ denotes the number of iterations) for the smooth
convex loss $l(\cdot)$. In comparison, although the methods proposed in [1, 6,
16, 29] converge, there is no known convergence rate; and the method proposed
in [21] has a _local_ linear convergence rate under certain conditions [34,
Theorem 4]. In addition, these methods are not applicable for an arbitrary
$q\geq 1$.
The main technical contribution of this paper is the development of an
efficient algorithm for computing the $\ell_{1}/\ell_{q}$-regularized
Euclidean projection (EP1q), which is a key building block in the proposed
GLEP1q algorithm. More specifically, we analyze the key theoretical properties
of the solution of EP1q, based on which we develop an efficient algorithm for
EP1q by solving two zero finding problems. In addition, our theoretical
analysis reveals why EP1q for the general $q$ is significantly more
challenging than the special cases such as $q=2$. We have conducted
experimental studies to demonstrate the efficiency of the proposed algorithm.
### 1.3 Related Work
We briefly review recent studies on $\ell_{1}/\ell_{q}$-regularization, most
of which focus on $\ell_{1}/\ell_{2}$-regularization and/or
$\ell_{1}/\ell_{\infty}$-regularization.
$\ell_{1}/\ell_{2}$-Regularization: The group Lasso was proposed in [37] to
select the groups of variables for prediction in the least squares regression.
In [21], the idea of group lasso was extended for classification by the
logistic regression model, and an algorithm via the coordinate gradient
descent [34] was developed. In [29], the authors considered joint covariate
selection for grouped classification by the logistic loss, and developed a
blockwise boosting Lasso algorithm with the boosted Lasso [39]. In [1], the
authors proposed to learn the sparse representations shared across multiple
tasks, and designed an alternating algorithm. The Spectral projected-gradient
(Spg) algorithm was proposed for solving the $\ell_{1}/\ell_{2}$-ball
constrained smooth optimization problem [4], equipped with an efficient
Euclidean projection that has expected linear runtime. The
$\ell_{1}/\ell_{2}$-regularized multi-task learning was proposed in [18], and
the equivalent smooth reformulations were solved by the Nesterov’s method
[26].
$\ell_{1}/\ell_{\infty}$-Regularization: A blockwise coordinate descent
algorithm [33] was developed for the mutli-task Lasso [16]. It was applied to
the neural semantic basis discovery problem. In [30], the authors considered
the multi-task learning via the $\ell_{1}/\ell_{\infty}$-regularization, and
proposed to solve the equivalent $\ell_{1}/\ell_{\infty}$-ball constrained
problem by the projected gradient descent. In [24], the authors considered the
multivariate regression via the $\ell_{1}/\ell_{\infty}$-regularization,
showed that the high-dimensional scaling of
$\ell_{1}/\ell_{\infty}$-regularization is qualitatively similar to that of
ordinary $\ell_{1}$-regularization, and revealed that, when the overlap
parameter is large enough ($>2/3$), $\ell_{1}/\ell_{\infty}$-regularization
yields the improved statistical efficiency over $\ell_{1}$-regularization.
$\ell_{1}/\ell_{q}$-Regularization: In [6], the authors studied the problem of
boosting with structural sparsity, and developed several boosting algorithms
for regularization penalties including $\ell_{1}$, $\ell_{\infty}$,
$\ell_{1}/\ell_{2}$, and $\ell_{1}/\ell_{\infty}$. In [38], the composite
absolute penalties (CAP) family was introduced, and an algorithm called iCAP
was developed. iCAP employed the least squares loss and the
$\ell_{1}/\ell_{\infty}$ regularization, and was implemented by the boosted
Lasso [39]. The multivariate regression with the
$\ell_{1}/\ell_{q}$-regularization was studied in [17]. In [23], a unified
framework was provided for establishing consistency and convergence rates for
the regularized $M$-estimators, and the results for $\ell_{1}/\ell_{q}$
regularization was established.
### 1.4 Notation
Throughout this paper, scalars are denoted by italic letters, and vectors by
bold face letters. Let $\mathbf{X},\mathbf{Y},\ldots$ denote the
$p$-dimensional parameters, $\mathbf{x}_{i},\mathbf{y}_{i},\ldots$ the
$p_{i}$-dimensional parameters of the $i$-th group, and $x_{i}$ the $i$-th
component of $\mathbf{x}$. We denote $\bar{q}=\frac{q}{q-1}$, and thus $q$ and
$\bar{q}$ satisfy the following relationship:
$\frac{1}{\bar{q}}+\frac{1}{q}=1$. We use the following componentwise
operators: $\odot$, $|\cdot|$ and ${\rm sgn}(\cdot)$. Specifically,
$\mathbf{z}=\mathbf{x}\odot\mathbf{y}$ denotes $z_{i}=x_{i}y_{i}$;
$\mathbf{y}=|\mathbf{x}|$ denotes $y_{i}=|x_{i}|$; and $\mathbf{y}={\rm
sgn}(\mathbf{x})$ denotes $y_{i}={\rm sgn}(x_{i})$, where ${\rm sgn}(\cdot)$
is the signum function: ${\rm sgn}(t)=1$ if $t>0$; ${\rm sgn}(t)=0$ if $t=0$;
and ${\rm sgn}(t)=-1$ if $t<0$.
## 2 The Proposed GLEP1q Algorithm
In this section, we present the proposed GLEP1q algorithm for solving (1) in
the batch learning setting. The main technical contribution lies in the
development of an efficient algorithm for the $\ell_{1}/\ell_{q}$-regularized
Euclidean projection. Specifically, we analyze the key theoretical properties
of the projection in Section 2.1, and show that the projection can be computed
by solving two zero finding problems in Section 2.2. Note that, one can
develop the online learning algorithm for (1) using the online learning
algorithms discussed in the last section, where the
$\ell_{1}/\ell_{q}$-regularized Euclidean projection is also a key building
block.
We first construct the following model for approximating the composite
function $\mathcal{M}(\cdot)$ at the point $\mathbf{X}$ [3, 27]:
$\displaystyle\mathcal{M}_{L,\mathbf{X}}(\mathbf{Y})=[\mbox{loss}(\mathbf{X})+\langle\mbox{loss}^{\prime}(\mathbf{X}),\mathbf{Y}-\mathbf{X}\rangle]+\lambda\varpi(\mathbf{Y})+\frac{L}{2}\|\mathbf{Y}-\mathbf{X}\|_{2}^{2},$
(8)
where $L>0$. In the model $\mathcal{M}_{L,\mathbf{X}}(\mathbf{Y})$, we apply
the first-order Taylor expansion at the point $\mathbf{X}$ (including all
terms in the square bracket) for the smooth loss function $l(\cdot)$, and
directly put the non-smooth penalty $\varpi(\cdot)$ into the model. The
regularization term $\frac{L}{2}\|\mathbf{Y}-\mathbf{X}\|_{2}^{2}$ prevents
$\mathbf{Y}$ from walking far away from $\mathbf{X}$, thus the model can be a
good approximation to $f(\mathbf{Y})$ in the neighborhood of $\mathbf{X}$.
The accelerated gradient method is based on two sequences
$\\{\mathbf{X}_{i}\\}$ and $\\{\mathbf{S}_{i}\\}$ in which
$\\{\mathbf{X}_{i}\\}$ is the sequence of approximate solutions, and
$\\{\mathbf{S}_{i}\\}$ is the sequence of search points. The search point
$\mathbf{S}_{i}$ is the affine combination of $\mathbf{X}_{i-1}$ and
$\mathbf{X}_{i}$ as
$\mathbf{S}_{i}=\mathbf{X}_{i}+\beta_{i}(\mathbf{X}_{i}-\mathbf{X}_{i-1}),$
(9)
where $\beta_{i}$ is a properly chosen coefficient. The approximate solution
$\mathbf{X}_{i+1}$ is computed as the minimizer of
$\mathcal{M}_{L_{i},\mathbf{S}_{i}}(\mathbf{Y})$:
$\mathbf{X}_{i+1}=\arg\min_{\mathbf{Y}}\mathcal{M}_{L_{i},\mathbf{S}_{i}}(\mathbf{Y}),$
(10)
where $L_{i}$ is determined by line search, e.g., the Armijo-Goldstein rule so
that $L_{i}$ should be appropriate for $\mathbf{S}_{i}$.
Algorithm 1 GLEP1q: Group Sparsity Learning via the
$\ell_{1}/\ell_{q}$-regularized Euclidean Projection
0: $\lambda_{1}\geq 0,\lambda_{2}\geq 0,L_{0}>0,\mathbf{X}_{0},k$
0: $\mathbf{X}_{k+1}$
1: Initialize $\mathbf{X}_{1}=\mathbf{X}_{0}$, $\alpha_{-1}=0$,
$\alpha_{0}=1$, and $L=L_{0}$.
2: for $i=1$ to $k$ do
3: Set $\beta_{i}=\frac{\alpha_{i-2}-1}{\alpha_{i-1}}$,
$\mathbf{S}_{i}=\mathbf{X}_{i}+\beta_{i}(\mathbf{X}_{i}-\mathbf{X}_{i-1})$
4: Find the smallest $L=L_{i-1},2L_{i-1},\ldots$ such that
$f(\mathbf{X}_{i+1})\leq\mathcal{M}_{L,\mathbf{S}_{i}}(\mathbf{X}_{i+1}),$
where
$\mathbf{X}_{i+1}=\arg\min_{\mathbf{Y}}\mathcal{M}_{L,\mathbf{S}_{i}}(\mathbf{Y})$
5: Set $L_{i}=L$ and $\alpha_{i+1}=\frac{1+\sqrt{1+4\alpha_{i}^{2}}}{2}$
6: end for
The algorithm for solving (1) is presented in Algorithm 1. GLEP1q inherits the
optimal convergence rate of $O(1/k^{2})$ from the accelerated gradient method.
In Algorithm 1, a key subroutine is (10), which can be computed as
$\mathbf{X}_{i+1}=\pi_{1q}(\mathbf{S}_{i}-l^{\prime}(\mathbf{S}_{i})/L_{i},\lambda/L_{i})$,
where $\pi_{1q}(\cdot)$ is the $\ell_{1}/\ell_{q}$-regularized Euclidean
projection (EP1q) problem:
$\pi_{1q}(\mathbf{V},\lambda)=\arg\min_{\mathbf{X}\in\mathbb{R}^{p}}\frac{1}{2}\|\mathbf{X}-\mathbf{V}\|_{2}^{2}+\lambda\sum_{i=1}^{s}\|\mathbf{x}_{i}\|_{q}.$
(11)
The efficient computation of (11) for any $q>1$ is the main technical
contribution of this paper. Note that the $s$ groups in (11) are independent.
Thus the optimization in (11) decouples into a set of $s$ independent
$\ell_{q}$-regularized Euclidean projection problems:
$\pi_{q}(\mathbf{v})=\arg\min_{\mathbf{x}\in\mathbb{R}^{n}}\left(g(\mathbf{x})=\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{q}\right),$
(12)
where $n=p_{i}$ for the $i$-th group. Next, we study the key properties of
(12).
### 2.1 Properties of the Optimal Solution to (12)
The function $g(\cdot)$ is strictly convex, and thus it has a unique
minimizer, as summarized below:
###### Lemma 1
The problem (12) has a unique minimizer.
Next, we show that the optimal solution to (12) is given by zero under a
certain condition, as summarized in the following theorem:
###### Theorem 1
$\pi_{q}(\mathbf{v})=\mathbf{0}$ if and only if
$\lambda\geq\|\mathbf{v}\|_{\bar{q}}$.
Proof: Let us first compute the directional derivative of $g(\mathbf{x})$ at
the point $\mathbf{0}$:
$Dg(\mathbf{0})[\mathbf{u}]=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}[g(\alpha\mathbf{u})-g(\mathbf{0})]=-\langle\mathbf{v},\mathbf{u}\rangle+\lambda\|\mathbf{u}\|_{q},$
where $\mathbf{u}$ is a given direction. According to the Hölder’s inequality,
we have
$|\langle\mathbf{u},\mathbf{v}\rangle|\leq\|\mathbf{u}\|_{q}\|\mathbf{v}\|_{\bar{q}},\forall\mathbf{u}.$
Therefore, we have
$Dg(\mathbf{0})[\mathbf{u}]\geq 0,\forall\mathbf{u},$ (13)
if and only if $\lambda\geq\|\mathbf{v}\|_{\bar{q}}$. The result follows,
since (13) is the necessary and sufficient condition for $\mathbf{0}$ to be
the optimal solution of (12). $\Box$
Next, we focus on solving (12) for $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. We
first consider solving (12) in the case of $1<q<\infty$, which is the main
technical contribution of this paper. We begin with a lemma that summarizes
the key properties of the optimal solution to the problem (12):
###### Lemma 2
Let $1<q<\infty$ and $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. Then,
$\mathbf{x}^{*}$ is the optimal solution to the problem (12) if and if only it
satisfies:
$\mathbf{x}^{*}+\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}{\mathbf{x}^{*}}^{(q-1)}=\mathbf{v},$
(14)
where $\mathbf{y}\equiv\mathbf{x}^{(q-1)}$ is defined component-wisely as:
$y_{i}={\rm sgn}(x_{i})|x_{i}|^{q-1}$. Moreover, we have
$\pi_{q}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\pi_{q}(|\mathbf{v}|),$ (15)
${\rm sgn}(\mathbf{x}^{*})={\rm sgn}(\mathbf{v}),$ (16)
$0<|x^{*}_{i}|<|v_{i}|,\forall i\in\\{i|v_{i}\neq 0\\}.$ (17)
Proof: Since $\lambda<\|\mathbf{v}\|_{\bar{q}}$, it follows from Theorem 1
that the optimal solution $\mathbf{x}^{*}\neq\mathbf{0}$. $\|\mathbf{x}\|_{q}$
is differentiable when $\mathbf{x}\neq\mathbf{0}$, so is $g(\mathbf{x})$.
Therefore, the sufficient and necessary condition for $\mathbf{x}^{*}$ to be
the solution of (12) is $g^{\prime}(\mathbf{x}^{*})=0$, i.e., (14). Denote
$c^{*}\equiv\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}>0$. It follows from (14) that
(15) holds, and
$\mbox{sgn}(x^{*}_{i})\left(|x^{*}_{i}|+c^{*}|x^{*}_{i}|^{q-1}\right)=v_{i},$
(18)
from which we can verify (16) and (17). $\Box$
Figure 1: Illustration of the failure of the fixed point iteration
$\mathbf{x}=\mathbf{v}-\lambda\|\mathbf{x}\|_{q}^{1-q}{\mathbf{x}}^{(q-1)}$
for solving (12). We set $\mathbf{v}=[1,3]^{\rm T}$ and the starting point
$\mathbf{x}=[1,3]^{\rm T}$. The vertical axis denotes the values of $x_{1}$
during the iterations.
It follows from Lemma 2 that i) if $v_{i}=0$ then $x_{i}^{*}=0$; and ii)
$\pi_{q}(\mathbf{v})$ can be easily obtained from $\pi_{q}(|\mathbf{v}|)$.
Thus, we can restrict our following discussion to $\mathbf{v}>\mathbf{0}$,
i.e., $v_{i}>0,\forall i$. It is clear that, the analysis can be easily
extended to the general $\mathbf{v}$. The optimality condition in (14)
indicates that $\mathbf{x}^{*}$ might be solved via the fixed point iteration
$\mathbf{x}=\eta(\mathbf{x})\equiv\mathbf{v}-\lambda\|\mathbf{x}\|_{q}^{1-q}{\mathbf{x}}^{(q-1)},$
which is, however, not guaranteed to converge (see Figure 1 for examples), as
$\eta(\cdot)$ is not necessarily a contraction mapping [14, Proposition 3]. In
addition, $\mathbf{x}^{*}$ cannot be trivially solved by firstly guessing
$c=\|\mathbf{x}\|_{q}^{1-q}$ and then finding the root of $\mathbf{x}+\lambda
c{\mathbf{x}}^{(q-1)}=\mathbf{v}$, as when $c$ increases, the values of
$\mathbf{x}$ obtained from $\mathbf{x}+\lambda
c{\mathbf{x}}^{(q-1)}=\mathbf{v}$ decrease, so that
$c=\|\mathbf{x}\|_{q}^{1-q}$ increases as well (note that $1-q<0$).
### 2.2 Computing the Optimal Solution $\mathbf{x}^{*}$ by Zero Finding
In the following, we show that $\mathbf{x}^{*}$ can be obtained by solving two
zero finding problems. Below, we construct our first auxiliary function
$h_{c}^{v}(\cdot)$ and reveal its properties:
###### Definition 1 (Auxiliary Function $h_{c}^{v}(\cdot)$ )
Let $c>0$, $1<q<\infty$, and $v>0$. We define the auxiliary function
$h_{c}^{v}(\cdot)$ as follows:
$h_{c}^{v}(x)=x+cx^{q-1}-v,0\leq x\leq v.$ (19)
###### Lemma 3
Let $c>0$, $1<q<\infty$, and $v>0$. Then, $h_{c}^{v}(\cdot)$ has a unique root
in the interval $(0,v)$.
Proof: It is clear that $h_{c}^{v}(\cdot)$ is continuous and strictly
increasing in the interval $[0,v]$, $h_{c}^{v}(0)=-v<0$, and
$h_{c}^{v}(v)=cv^{q-1}>0$. According to the Intermediate Value Theorem,
$h_{c}^{v}(\cdot)$ has a unique root lying in the interval $(0,v)$. This
concludes the proof. $\Box$
###### Corolary 1
Let $\mathbf{x},\mathbf{v}\in\mathbb{R}^{n}$, $c>0$, $1<p<\infty$, and
$\mathbf{v}>\mathbf{0}$. Then, the function
$\varphi_{c}^{\mathbf{v}}(\mathbf{x})=\mathbf{x}+c\mathbf{x}^{(q-1)}-\mathbf{v},\mathbf{0}<\mathbf{x}<\mathbf{v}$
(20)
has a unique root.
Let $\mathbf{x}^{*}$ be the optimal solution satisfying (14). Denote
$c^{*}=\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}$. It follows from Lemma 2 and
Corollary 1 that $\mathbf{x}^{*}$ is the unique root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ defined in (20), provided that the
optimal $c^{*}$ is known. Our methodology for computing $\mathbf{x}^{*}$ is to
first compute the optimal $c^{*}$ and then compute $\mathbf{x}^{*}$ by
computing the root of $\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$. Next, we show how
to compute the optimal $c^{*}$ by solving a single variable zero finding
problem. We need our second auxiliary function $\omega(\cdot)$ defined as
follows:
###### Definition 2 (Auxiliary Function $\omega(\cdot)$)
Let $1<q<\infty$ and $v>0$. We define the auxiliary function $\omega(\cdot)$
as follows:
$c=\omega(x)=(v-x)/x^{q-1},0<x\leq v.$ (21)
###### Lemma 4
In the interval $(0,v]$, $c=\omega(x)$ is i) continuously differentiable, ii)
strictly decreasing, and iii) invertible. Moreover, in the domain
$[0,\infty)$, the inverse function $x=\omega^{-1}(c)$ is continuously
differentiable and strictly decreasing.
Proof: It is easy to verify that, in the interval $(0,v]$, $c=\omega(x)$ is
continuously differentiable with a non-positive gradient, i.e.,
$\omega^{\prime}(x)<0$. Therefore, the results follow from the Inverse
Function Theorem. $\Box$
It follows from Lemma 4 that given the optimal $c^{*}$ and $\mathbf{v}$, the
optimal $\mathbf{x}^{*}$ can be computed via the inverse function
$\omega^{-1}(\cdot)$, i.e., we can represent $\mathbf{x}^{*}$ as a function of
$c^{*}$. Since $\lambda\|\mathbf{x}^{*}\|_{q}^{1-q}-c^{*}=0$ by the definition
of $c^{*}$, the optimal $c^{*}$ is a root of our third auxiliary function
$\phi(\cdot)$ defined as follows:
###### Definition 3 (Auxiliary Function $\phi(\cdot)$)
Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, and
$\mathbf{v}>\mathbf{0}$. We define the auxiliary function $\phi(\cdot)$ as
follows:
$\phi(c)=\lambda\psi(c)-c,c\geq 0,$ (22)
where
$\psi(c)=\left(\sum_{i=1}^{n}(\omega_{i}^{-1}(c))^{q}\right)^{\frac{1-q}{q}},$
(23)
and $\omega_{i}^{-1}(c)$ is the inverse function of
$\omega_{i}(x)=(v_{i}-x)/x^{q-1},0<x\leq v_{i}.$ (24)
Recall that we assume $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$ (otherwise the
optimal solution is given by zero from Theorem 1). The following lemma
summarizes the key properties of the auxiliary function $\phi(\cdot)$:
###### Lemma 5
Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, and
$\epsilon=(\|\mathbf{v}\|_{\bar{q}}-\lambda)/\|\mathbf{v}\|_{\bar{q}}.$ (25)
Then, $\phi(\cdot)$ is continuously differentiable in the interval
$[0,\infty)$. Moreover, we have
$\phi(0)=\lambda\|\mathbf{v}\|_{q}^{1-q}>0,\phi(\overline{c})\leq 0,$
where
$\overline{c}=\max_{i}c_{i},$ (26)
$c_{i}=\omega_{i}(v_{i}\epsilon),i=1,2,\ldots,n.$ (27)
Proof: From Lemma 4, the function $\omega_{i}^{-1}(c)$ is continuously
differentiable in $[0,\infty)$. It is easy to verify that
$\omega_{i}^{-1}(c)>0,\forall c\in[0,\infty)$. Thus, $\phi(\cdot)$ in (22) is
continuously differentiable in $[0,\infty)$.
It is clear that $\phi(0)=\lambda\|\mathbf{v}\|_{q}^{1-q}>0$. Next, we show
$\phi(\overline{c})\leq 0$. Since $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, we
have
$0<\epsilon<1.$ (28)
It follows from (24), (26), (27) and (28) that
$0<c_{i}\leq\overline{c},\forall i$. Let
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\rm T}$ be the root of
$\varphi_{\overline{c}}^{\mathbf{v}}(\cdot)$ (see Corollary 1). Then,
$x_{i}=\omega^{-1}_{i}(\overline{c})$. Since $\omega^{-1}_{i}(\cdot)$ is
strictly decreasing (see Lemma 4), $c_{i}\leq\overline{c}$,
$v_{i}\epsilon=\omega_{i}^{-1}(c_{i})$, and
$x_{i}=\omega_{i}^{-1}(\overline{c})$, we have
$x_{i}\leq v_{i}\epsilon.$ (29)
Combining (24), (29), and $\overline{c}=\omega_{i}(x_{i})$, we have
$\overline{c}\geq v_{i}(1-\epsilon)/x_{i}^{q-1}$, since $\omega_{i}(\cdot)$ is
strictly decreasing. It follows that
$x_{i}\geq\left(\frac{v_{i}(1-\epsilon)}{\overline{c}}\right)^{\frac{1}{q-1}}$.
Thus, the following holds:
$\psi(\overline{c})=\left(\sum_{i=1}^{n}(\omega_{i}^{-1}(\overline{c}))^{q}\right)^{\frac{1-q}{q}}=\left(\sum_{i=1}^{n}x_{i}^{q}\right)^{\frac{1-q}{q}}\leq\frac{\overline{c}}{\|\mathbf{v}\|_{\bar{q}}(1-\epsilon)},$
which leads to
$\phi(\overline{c})=\lambda\psi(\overline{c})-\overline{c}\leq\overline{c}\left(\frac{\lambda}{\|\mathbf{v}\|_{\bar{q}}(1-\epsilon)}-1\right)=0,$
where the last equality follows from (25). $\Box$
###### Corolary 2
Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, and $\underline{c}=\min_{i}c_{i}$, where $c_{i}$’s
are defined in (27). We have $0<\underline{c}\leq\overline{c}$ and
$\phi(\underline{c})\geq 0$.
Following Lemma 5 and Corollary 2, we can find at least one root of
$\phi(\cdot)$ in the interval $[\underline{c},\overline{c}]$. In the following
theorem, we show that $\phi(\cdot)$ has a unique root:
###### Theorem 2
Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$, and
$\mathbf{v}>\mathbf{0}$. Then, in $[\underline{c},\overline{c}]$,
$\phi(\cdot)$ has a unique root, denoted by $c^{*}$, and the root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ is the optimal solution to (12).
Proof: From Lemma 5 and Corollary 2, we have $\phi(\overline{c})\leq 0$ and
$\phi(\underline{c})\geq 0$. If either $\phi(\overline{c})=0$ or
$\phi(\underline{c})=0$, $\overline{c}$ or $\underline{c}$ is a root of
$\phi(\cdot)$. Otherwise, we have $\phi(\underline{c})\phi(\overline{c})<0$.
As $\phi(\cdot)$ is continuous in $[0,\infty)$, we conclude that $\phi(\cdot)$
has a root in $(\underline{c},\overline{c})$ according to the Intermediate
Value Theorem.
Next, we show that $\phi(\cdot)$ has a unique root in the interval
$[0,\infty)$. We prove this by contradiction. Assume that $\phi(\cdot)$ has
two roots: $0<c_{1}<c_{2}$. From Corollary 1,
$\varphi_{c_{1}}^{\mathbf{v}}(\cdot)$ and
$\varphi_{c_{2}}^{\mathbf{v}}(\cdot)$ have unique roots. Denote
$\mathbf{x}^{1}=[x_{1}^{1},x_{2}^{1},\ldots,x_{n}^{1}]^{\rm T}$ and
$\mathbf{x}^{2}=[x_{1}^{2},x_{2}^{2},\ldots,x_{n}^{2}]^{\rm T}$ as the roots
of $\varphi_{c_{1}}^{\mathbf{v}}(\cdot)$ and
$\varphi_{c_{2}}^{\mathbf{v}}(\cdot)$, respectively. We have
$0<x_{i}^{1},x_{i}^{2}<v_{i},\forall i$. It follows from (22-24) that
$\displaystyle\mathbf{x}^{1}+\lambda\|\mathbf{x}^{1}\|_{q}^{1-q}{\mathbf{x}^{1}}^{(q-1)}-\mathbf{v}=\mathbf{0},$
$\displaystyle\mathbf{x}^{2}+\lambda\|\mathbf{x}^{2}\|_{q}^{1-q}{\mathbf{x}^{2}}^{(q-1)}-\mathbf{v}=\mathbf{0}.$
According to Lemma 2, $\mathbf{x}^{1}$ and $\mathbf{x}^{2}$ are the optimal
solution of (12). From Lemma 1, we have $\mathbf{x}^{1}=\mathbf{x}^{2}$.
However, since $x_{i}^{1}=\omega_{i}^{-1}(c_{1})$,
$x_{i}^{2}=\omega_{i}^{-1}(c_{2})$, $\omega_{i}^{-1}(\cdot)$ is a strictly
decreasing function in $[0,\infty)$ by Lemma 4, and $c_{1}<c_{2}$, we have
$x_{i}^{1}>x_{i}^{2},\forall i$. This leads to a contradiction. Therefore, we
conclude that $\phi(\cdot)$ has a unique root in
$[\underline{c},\overline{c}]$.
From the above arguments, it is clear that, the root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$ is the optimal solution to (12). $\Box$
###### Remark 1
When $q=2$, we have
$\underline{c}=\overline{c}=\frac{\lambda}{\|\mathbf{v}\|_{2}-\lambda}$. It is
easy to verify that $\phi(\underline{c})=\phi(\overline{c})=0$ and
$\pi_{2}(\mathbf{v})=\frac{\|\mathbf{v}\|_{2}-\lambda}{\|\mathbf{v}\|_{2}}\mathbf{v}.$
(30)
Therefore, when $q=2$, we obtain a closed-form solution.
### 2.3 Solving the Zero Finding Problem by Bisection
Let $1<q<\infty$, $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$,
$\mathbf{v}>\mathbf{0}$, $\overline{v}=\max_{i}v_{i}$,
$\underline{v}=\min_{i}v_{i}$, and $\delta>0$ be a small constant (e.g.,
$\delta=10^{-8}$ in our experiments). When $q>2$, we have
$\underline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\overline{v}^{q-2}}\quad\mbox{
and }\quad\overline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\underline{v}^{q-2}}.$
When $1<q<2$, we have
$\underline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\underline{v}^{q-2}}\quad\mbox{
and }\quad\overline{c}=\frac{1-\epsilon}{\epsilon^{q-1}\overline{v}^{q-2}}.$
If either $\phi(\overline{c})=0$ or $\phi(\underline{c})=0$, $\overline{c}$ or
$\underline{c}$ is the unique root of $\phi(\cdot)$. Otherwise, we can find
the unique root of $\phi(\cdot)$ by bisection in the interval
$(\underline{c},\overline{c})$, which costs at most
$N=\log_{2}\frac{(1-\epsilon)|\overline{v}^{q-2}-\underline{v}^{q-2}|}{\epsilon^{q-1}\overline{v}^{q-2}\underline{v}^{q-2}\delta}$
iterations for achieving an accuracy of $\delta$. Let $[c_{1},c_{2}]$ be the
current interval of uncertainty, and we have computed $\omega_{i}^{-1}(c_{1})$
and $\omega_{i}^{-1}(c_{2})$ in the previous bisection iterations. Setting
$c=\frac{c_{1}+c_{2}}{2}$, we need to evaluate $\phi(c)$ by computing
$\omega_{i}^{-1}(c),i=1,2,\ldots,n$. It is easy to verify that
$\omega_{i}^{-1}(c)$ is the root of $h_{c}^{v_{i}}(\cdot)$ in the interval
$(0,v_{i})$. Since $\omega_{i}^{-1}(\cdot)$ is a strictly decreasing function
(see Lemma 4), the following holds:
$\omega_{i}^{-1}(c_{2})<\omega_{i}^{-1}(c)<\omega_{i}^{-1}(c_{1}),$
and thus $\omega_{i}^{-1}(c)$ can be solved by bisection using at most
$\log_{2}\frac{\omega_{i}^{-1}(c_{2})-\omega_{i}^{-1}(c_{1})}{\delta}<\log_{2}\frac{v_{i}}{\delta}\leq\log_{2}\frac{\overline{v}}{\delta}$
iterations for achieving an accuracy of $\delta$. For given
$\mathbf{v},\lambda$, and $\delta$, $N$ and $\overline{v}$ are constant, and
thus it costs $O(n)$ for finding the root of $\phi(\cdot)$. Once $c^{*}$, the
root of $\phi(\cdot)$ is found, it costs $O(n)$ flops to compute
$\mathbf{x}^{*}$ as the unique root of $\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$.
Therefore, the overall time complexity for solving (12) is $O(n)$.
We have shown how to solve (12) for $1<q<\infty$. For $q=1$, the problem (12)
is reduced to the one used in the standard Lasso, and it has the following
closed-form solution [3]:
$\pi_{1}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-\lambda,0).$
(31)
For $q=\infty$, the problem (12) can computed via (31), as summarized in the
following theorem:
###### Theorem 3
Let $q=\infty$, $\bar{q}=1$, and $0<\lambda<\|\mathbf{v}\|_{\bar{q}}$. Then we
have
$\pi_{\infty}(\mathbf{v})={\rm sgn}(\mathbf{v})\odot\min(|\mathbf{v}|,t^{*}),$
(32)
where $t^{*}$ is the unique root of
$h(t)=\sum_{i=1}^{n}\max(|v_{i}|-t,0)-\lambda.$ (33)
Proof: Making use of the property that
$\|\mathbf{x}\|_{\infty}=\max_{\|\mathbf{y}\|_{1}\leq
1}\langle\mathbf{y},\mathbf{x}\rangle$, we can rewrite (12) in the case of
$q=\infty$ as
$\min_{\mathbf{x}}\max_{\mathbf{y}:\|\mathbf{y}\|_{1}\leq\lambda}s(\mathbf{x},\mathbf{y})\equiv\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\langle\mathbf{y},\mathbf{x}\rangle.$
(34)
The function $s(\mathbf{x},\mathbf{y})$ is continuously differentiable in both
$\mathbf{x}$ and $\mathbf{y}$, convex in $\mathbf{x}$ and concave in
$\mathbf{y}$, and the feasible domains are solids. According to the well-known
von Neumann Lemma [25], the min-max problem (34) has a saddle point, and thus
the minimization and maximization can be exchanged. Setting the derivative of
$s(\mathbf{x},\mathbf{y})$ with respect to $\mathbf{x}$ to zero, we have
$\mathbf{x}=\mathbf{v}-\mathbf{y}.$ (35)
Thus we obtain the following problem:
$\min_{\mathbf{y}:\|\mathbf{y}\|_{1}\leq\lambda}\frac{1}{2}\|\mathbf{y}-\mathbf{v}\|_{2}^{2},$
(36)
which is the problem of the Euclidean projection onto the $\ell_{1}$ ball [4,
6, 20]. It has been shown that the optimal solution $\mathbf{y}^{*}$to (36)
for $\lambda<\|\mathbf{v}\|_{1}$ can be obtained by first computing $t^{*}$ as
the unique root of (33) in linear time, and then computing $\mathbf{y}^{*}$ as
$\mathbf{y}^{*}={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-t^{*},0).$ (37)
It follows from (35) and (37) that (32) holds. $\Box$
We conclude this section by summarizing the main steps for solving the
$\ell_{q}$-regularized Euclidean projection in Algorithm 2.
Algorithm 2 Epq: $\ell_{q}$-regularized Euclidean projection
0: $\lambda>0,q\geq 1,\mathbf{v}\in\mathbb{R}^{n}$
0:
$\mathbf{x}^{*}=\pi_{q}(\mathbf{v})=\arg\min_{\mathbf{x}\in\mathbb{R}^{n}}\frac{1}{2}\|\mathbf{x}-\mathbf{v}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{q}$
1: Compute $\bar{q}=\frac{q}{q-1}$
2: if $\|\mathbf{v}\|_{\bar{q}}\leq\lambda$ then
3: Set $\mathbf{x}^{*}=\mathbf{0}$, return
4: end if
5: if $q=1$ then
6: Set $\mathbf{x}^{*}={\rm sgn}(\mathbf{v})\odot\max(|\mathbf{v}|-\lambda,0)$
7: else if $q=2$ then
8: Set
$\mathbf{x}^{*}=\frac{\|\mathbf{v}\|_{2}-\lambda}{\|\mathbf{v}\|_{2}}\mathbf{v}$
9: else if $q=\infty$ then
10: Obtain $t^{*}$, the unique root of $h(t)$, via the improved bisection
method [20]
11: Set $\mathbf{x}^{*}={\rm sgn}(\mathbf{v})\odot\min(|\mathbf{v}|,t^{*})$
12: else
13: Compute $c^{*}$, the unique root of $\phi(c)$, via bisection in the
interval $[\underline{c},\overline{c}]$ (Theorem 2)
14: Obtain $\mathbf{x}^{*}$ as the unique root of
$\varphi_{c^{*}}^{\mathbf{v}}(\cdot)$
15: end if
## 3 Experiments
We have conducted experiments to evaluate the efficiency of the proposed
algorithm using both synthetic and real-world data. We set the regularization
parameter as $\lambda=r\times\lambda_{\max}^{q}$, where $0<r\leq 1$ is the
ratio, and $\lambda_{\max}^{q}$ is the maximal value above which the
$\ell_{1}/\ell_{q}$-norm regularized problem (1) obtains a zero solution (see
Theorem 1). We try the following values for $q$: $1.25,1.5,1.75,2,2.33,3,5$,
and $\infty$. The source codes, included in the SLEP package [19], are
available online222http://www.public.asu.edu/~jye02/Software/SLEP/.
### 3.1 Simulation Studies
We use the synthetic data to study the effectiveness of the
$\ell_{1}/\ell_{q}$-norm regularization for reconstructing the jointly sparse
matrix under different values of $q>1$. Let $A\in\mathbb{R}^{m\times d}$ be a
measurement matrix with entries being generated randomly from the standard
normal distribution, $X^{*}\in\mathbb{R}^{d\times k}$ be the jointly sparse
matrix with the first $\tilde{d}<d$ rows being nonzero and the remaining rows
exactly zero, $Y=AX^{*}+Z$ be the response matrix, and
$Z\in\mathbb{R}^{m\times k}$ is the noise matrix whose entries are drawn
randomly from the normal distribution with mean zero and standard deviation
$\sigma=0.1$. We treat each row of $X^{*}$ as a group, and estimate $X^{*}$
from $A$ and $Y$ by solving the following $\ell_{1}/\ell_{q}$-norm regularized
problem:
$X=\arg\min_{W}\frac{1}{2}\|AW-Y\|_{F}^{2}+\lambda\sum_{i=1}^{d}\|W^{i}\|_{q},$
where $W^{i}$ denotes the $i$-th row of $W$. We set $m=100$, $d=200$, and
$\tilde{d}=k=50$. We try two different settings for $X^{*}$, by drawing its
nonzero entries randomly from 1) the uniform distribution in the interval
$[0,1]$ and 2) the standard normal distribution.
Figure 2: Performance of the $\ell_{1}/\ell_{q}$-norm regularization for
reconstructing the jointly sparse $X^{*}$. The nonzero entries of $X^{*}$ are
drawn randomly from the uniform distribution for the plots in the first row,
and from the normal distribution for the plots in the second row. Plots in the
first two rows show $\|X-X^{*}\|_{F}$, the Frobenius norm difference between
the solution and the truth; and plots in the third row show the
$\ell_{2}$-norm of each row of the solution $X$.
We compute the solutions corresponding to a sequence of decreasing values of
$\lambda=r\times\lambda_{\max}^{q}$, where $r=0.9^{i-1}$, for
$i=1,2,\ldots,100$. In addition, we use the solution corresponding to the
$0.9^{i}\times\lambda_{\max}^{q}$ as the “warm” start for
$0.9^{i+1}\times\lambda_{\max}^{q}$. We report the results in Figure 2, from
which we can observe: 1) the distance between the solution $X$ and the truth
$X^{*}$ usually decreases with decreasing values of $\lambda$; 2) for the
uniform distribution (see the plots in the first row), $q=1.5$ performs the
best; 3) for the normal distribution (see the plots in the second row),
$q=1.5,1.75,2$ and 3 achieve comparable performance and perform better than
$q=1.25$, 5 and $\infty$; 4) with a properly chosen threshold, the support of
$X^{*}$ can be exactly recovered by the $\ell_{1}/\ell_{q}$-norm
regularization with an appropriate value of $q$, e.g., $q=1.5$ for the uniform
distribution, and $q=2$ for the normal distribution; and 5) the recovery of
$X^{*}$ with nonzero entries drawn from the normal distribution is easier than
that with entries generated from the uniform distribution.
The existing theoretical results [17, 23] can not tell which $q$ is the best;
and we believe that the optimal $q$ depends on the distribution of $X^{*}$, as
indicated from the above results. Therefore, it is necessary to conduct the
distribution-specific theoretical studies (note that the previous studies
usually make no assumption on $X^{*}$). The proposed GLEP1q algorithm shall
help verify the theoretical results to be established.
### 3.2 Performance on the Letter Data Set
We apply the proposed GLEP1q algorithm for multi-task learning on the Letter
data set [29], which consists of 45,679 samples from 8 default tasks of two-
class classification problems for the handwritten letters: c/e, g/y, m/n, a/g,
i/j, a/o, f/t, h/n. The writings were collected from over 180 different
writers, with the letters being represented by $8\times 16$ binary pixel
images. We use the least squares loss for $l(\cdot)$.
Figure 3: Computational time (seconds) comparison between GLEP1q ($q=2$) and
Spg under different values of $\lambda=r\times\lambda_{\max}^{q}$ and $m$.
#### 3.2.1 Efficiency Comparison with Spg
We compare GLEP1q with the Spg algorithm proposed in [4]. Spg is a specialized
solver for the $\ell_{1}/\ell_{2}$-ball constrained optimization problem, and
has been shown to outperform existing algorithms based on blockwise coordinate
descent and projected gradient. In Figure 3, we report the computational time
under different values of $m$ (the number of samples) and
$\lambda=r\times\lambda_{\max}^{q}$ ($q=2$). It is clear from the plots that
GLEP1q is much more efficient than Spg, which may attribute to: 1) GLEP1q has
a better convergence rate than Spg; and 2) when $q=2$, the EP1q in GLEP1q can
be computed analytically (see Remark 1), while this is not the case in Spg.
#### 3.2.2 Efficiency under Different Values of $q$
We report the computational time (seconds) of GLEP1q under different values of
$q$, $\lambda=r\times\lambda_{\max}^{q}$ and $m$ (the number of samples) in
Figure 4. We can observe from this figure that the computational time of
GLEP1q under different values of $q$ (for fixed $r$ and $m$) is comparable.
Together with the result on the comparison with Spg for $q=2$, this experiment
shows the promise of GLEP1q for solving large-scale problems for any $q\geq
1$.
Figure 4: Computation time (seconds) of GLEP1q under different values of $m$,
$q$ and $r$.
Figure 5: The balanced error rate achieved by the $\ell_{1}/\ell_{q}$
regularization under different values of $q$. The title of each plot indicates
the percentages of samples used for training, validation, and testing.
#### 3.2.3 Performance under Different Values of $q$
We randomly divide the Letter data into three non-overlapping sets: training,
validation, and testing. We train the model using the training set, and tune
the regularization parameter $\lambda=r\times\lambda_{\max}^{q}$ on the
validation set, where $r$ is chosen from $\\{10^{-1},5\times 10^{-2},2\times
10^{-2},1\times 10^{-2},5\times 10^{-3},2\times 10^{-3},1\times 10^{-3}\\}$.
On the testing set, we compute the balanced error rate [11]. We report the
results averaged over 10 runs in Figure 5. The title of each plot indicates
the percentages of samples used for training, validation, and testing. The
results show that, on this data set, a smaller value of $q$ achieves better
performance.
## 4 Conclusion
In this paper, we propose the GLEP1q algorithm for solving the
$\ell_{1}/\ell_{q}$-norm regularized problem, for any $q\geq 1$. The main
technical contribution of this paper is the efficient algorithm for the
$\ell_{1}/\ell_{q}$-norm regularized Euclidean projection (EP1q), which is a
key building block of GLEP1q. Specifically, we analyze the key theoretical
properties of the solution of EP1q, based on which we develop an efficient
algorithm for EP1q by solving two zero finding problems. Our analysis also
reveals why EP1q for the general $q$ is significantly more challenging than
the special cases such as $q=2$.
In this paper, we focus on the efficient implementation of the
$\ell_{1}/\ell_{q}$-regularized problem. We plan to study the effectiveness of
the $\ell_{1}/\ell_{q}$ regularization under different values of $q$ for real-
world applications in computer vision and bioinformatics. We also plan to
conduct the distribution-specific [8] theoretical studies for different values
of $q$.
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|
arxiv-papers
| 2010-09-24T05:53:28 |
2024-09-04T02:49:13.059287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Liu, Jieping Ye",
"submitter": "Jun Liu",
"url": "https://arxiv.org/abs/1009.4766"
}
|
1009.4856
|
Astronomy Letters, 2010, Vol. 36, No. 11, pp. 816822.
Stars Outside the Hipparcos List Closely
Encountering the Solar System
V.V. Bobylev
Pulkovo Astronomical Observatory, Russian Academy of Sciences, St-Petersburg
Abstract–Based on currently available kinematic data, we have searched for
stars outside the Hipparcos list that either closely encountered in the past
or will encounter in the future the Solar system within several parsecs. For
the first time, we have identified two single stars, GJ 3379 (G 099–049) and
GJ 3323 (LHS 1723), as candidate for a close encounter with the solar orbit.
The star GJ 3379 could encounter the Sun more closely to a minimum distance
$d_{min}=1.32\pm 0.03$ pc at time $t_{min}=-163\pm 3$ thousand years. We have
found two potential candidates for a close encounter that have only
photometrical distances: the white dwarf SSSPM J1549–3544 without any data on
its radial velocity and the L-dwarf SDSS J1416+1348. The probabilities of
their penetration into the Oort cloud region are 0.09 (at a model radial
velocity $|V_{r}|=50$ km s-1) and 0.05, respectively.
## 1 INTRODUCTION
The Oort comet cloud (Oort 1950) is presumed to be located at the outer
boundaries of the Solar system. It is highly likely that the cloud has a
spherical shape and a radius of about $1\times 10^{5}$ AU (0.485 pc). Close
encounters of Galactic field stars with the Solar system play an important
role in the dynamical evolution of the Oort cloud. In particular, such
passages of stars can provoke the formation of comet showers that reach the
region of the giant planets (Emelyanenko et al. 2007; Leto et al. 2008;
Rickman et al. 2008). Several researchers associate the traces of comet
bombardments of the Earth with the impact of such showers (Wickramasinghe and
Napier 2008).
The question about the encounters of stars with the Sun within distances
$r<2-5$ pc was considered by Matthews (1997) and Mülläri and Orlov (1996)
using various ground-based observations and by Garcia-Sanchez et al. (1999;
2001), Dybczyński (2006) and Bobylev (2010) based on Hipparcos (1997) data in
combination with stellar radial velocities. About 160 Hipparcos stars are
known from the solar neighborhood 50 pc in radius that either encountered or
will encounter the Solar system within $r<5$ pc in a time interval of $\pm$10
Myr.
The goal of this study is to search for candidate stars closely encountering
the Sun based on currently available kinematic data on stars that do not
belong to the Hipparcos catalogue. Indeed, since the paper by Mülläri and
Orlov (1996), who analyzed all stars from the catalog by Gliese and Jahreiß
(1991), new observational data have appeared. In this paper, we solve the
problem of statistical simulations by taking into account the random errors in
the input data in order to estimate the probability of a star penetrating into
the Oort cloud region.
## 2 THE METHODS
### 2.1 Orbit construction
We use a rectangular Galactic coordinate system with the axes directed away
from the observer toward the Galaxy center $(l$=$0^{\circ}$, $b$=$0^{\circ},$
the $X$ axis), in the direction of Galactic rotation
$(l=90^{\circ},b=^{\circ},$ the $Y$ axis), and toward the North Pole
$(b=90^{\circ}$, the $Z$ axis). The corresponding space velocity components of
an object $U,V,W$ are also directed along the $X,Y,Z$ axes. The epicyclic
approximation (Lindblad, 1927) allows the stellar orbits to be constructed in
a coordinate system rotating around the Galactic center. The equations are
(Fuchs et al., 2006)
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill X(t)=X(0)+{U(0)\over\kappa}\sin(\kappa
t)+{V(0)\over 2B}(1-\cos(\kappa t)),\hfill\hbox to0.0pt{\hss(}1)\cr
0.0pt{\hfil$\displaystyle\hfill Y(t)=Y(0)+2A\biggl{(}X(0)+{V(0)\over
2B}\biggr{)}t-{\Omega_{0}\over B\kappa}V(0)\sin(\kappa
t)+{2\Omega_{0}\over\kappa^{2}}U(0)(1-\cos(\kappa t)),\hfill\cr
0.0pt{\hfil$\displaystyle\hfill Z(t)={W(0)\over\nu}\sin(\nu t)+Z(0)\cos(\nu
t),\hfill\crcr}}}}$
where $t$ is the time in Myr (we proceed from the ratio pc/Myr = 0.978 km
s-1); $A$ and $B$ are the Oort constants; $\kappa=\sqrt{-4\Omega_{0}B}$ is the
epicyclic frequency; $\Omega_{0}$ is the angular velocity of Galactic rotation
for the local standard of rest, $\Omega_{0}=A-B$; $\nu=\sqrt{4\pi G\rho_{0}}$
is the vertical oscillation frequency, where $G$ is the gravitational constant
and $\rho_{0}$ is the star density in the solar neighborhood.
The parameters $X(0),Y(0),Z(0)$ and $U(0),V(0),W(0)$ in the system of
equations (1) denote the current stellar positions and velocities. The
displacement of the Sun from the Galactic plane is taken to be $Z(0)=17$ pc
(Joshi, 2007). We calculate the velocities $U,V,W$ relative to the local
standard of rest (LSR) with $(U,V,W)_{LSR}=(10.00,5.25,7.17)$ km s-1 (Dehnen
and Binney, 1998).
At present, the question about the specific values of the Suns peculiar
velocity relative to the local standard of rest $(U,V,W)_{LSR}$ is being
actively debated (Francis and Anderson, 2009; Binney, 2010). Arguments for
increasing the velocity $V_{LSR}$ from 5 km s-1 to $\approx 11$ km s-1 are
adduced. Since, in our case, both the solar orbit and the stellar orbit are
constructed with these values, the influence of the adopted $V_{LSR}$ is
virtually imperceptible. This is confirmed by good agreement between the
encounter parameters of, for example, the star GJ 710 and other stars obtained
with various values of $(U,V,W)_{LSR}$ by Bobylev (2010) and Garcia-Sanchez et
al. (2001).
We took $\rho_{0}=0.1~{}M_{\odot}/$pc3 (Holmberg and Flinn, 2004), which gives
$\nu=74$ km s-1 kpc-1. We used the following Oort constants found by Bobylev
et al. (2008): $A=15.5\pm 0.3$ km s-1 kpc-1 and $B=-12.2\pm 0.7$ km s-1 kpc-1;
$\kappa=37$ km s-1 kpc-1 corresponds to these values.
Note that we neglect the gravitational interaction between the star and the
Sun.
### 2.2 Statistical Simulations
In accordance with the method of Monte Carlo statistical simulations, we
compute a set of orbits for each object by taking into account the random
errors in the input data. For each star, we compute the encounter parameter
between the stellar and solar orbits, $d=\sqrt{\Delta X^{2}(t)+\Delta
Y^{2}(t)+\Delta Z^{2}(t)}$). We characterize the time of the closest encounter
by two numbers, $d_{min}$ and $t_{min}$. The errors in the stellar parameters
are assumed to be distributed normally with a dispersion $\sigma$. The errors
are added to the equatorial coordinates, proper motion components, parallax,
and radial velocity of the star. We separately consider stars with
spectroscopic distance estimates obtained with typical errors of 2030%. In
this case, the random errors are added to the distance during the simulations.
## 3 RESULTS AND DISCUSSION
### 3.1 Stars with Trigonometric Parallaxes
First, let us consider the solar neighborhood about 10 pc in radius using the
list of 100 nearest stars from the Chilean RECONS site
(http://www.chara.gsu.edu/RECONS/). This list reflects the results published
before January 1, 2009. The trigonometric parallax of each star in the list
was calculated as a weighted mean of the results of several (from one to four)
observations. We are interested in ten RECONS stars that are not Hipparcos
ones (see Table 1). Other RECONS stars are ten M dwarfs without any radial
velocity estimates and the remaining stars are Hipparcos ones.
The equatorial coordinates and proper motions of the stars in Table 1 were
taken from Salim and Gould (2003) and Lépine and Shara (2005), where the
proper motions were calculated by comparing the 2MASS stellar positions with
those from the Palomar Observatory Sky Survey.
We took the radial velocities from the works of various authors. Note that,
according to the original first-class measurements by Nidever et al. (2002),
two stars, GL 406 and GL 905, exhibit a high stability in an observing time
interval of 1–2 yr. The internal measurement error is $\pm 0.1$ km s-1;
therefore, they were suggested as candidates for standards to determine the
radial velocities. Since these stars are M type dwarfs, the external error in
$V_{r}$ is $\pm 0.4$ km s-1. We took the radial velocity of the center of mass
for the triple system GL 866 ABC from the plot in Delfosse et al. (1999). Note
that, on the whole, the radial velocities for the stars listed in the table
differ insignificantly from those in the catalog by Gliese and Jahreiß (1991).
Only the star GJ 473, for which the discrepancy in $V_{r}$ is two orders of
magnitude, constitutes an exception. We will discuss this situation below.
For each star, we constructed its orbit relative to the Sun in the time
interval from –2 Myr to +2 Myr. Data on six stars with an encounter parameter
$d<3$ pc are presented in Table 2 and the trajectories of these stars are
shown in Fig.1.
(1) The parameters of the GL 905 encounter with the solar orbit found here are
in good agreement with the estimates by Mülläri and Orlov (1996),
$d_{min}=0.95\pm 0.11$ pc and $t_{min}=36.3\pm 1.4$ thousand years; our values
have considerably smaller random errors due to the currently available data
being very accurate.
Note that the encounter parameter for Proxima Cen is $d_{min}=0.89\pm 0.02$ pc
(Bobylev, 2010) or $d_{min}=0.95\pm 0.04$ pc (Garcia-Sanchez et al., 2001),
which was obtained with a slightly different radial velocity. As we see from
Fig.1, GL 905 has a chance to be even slightly closer to the Sun than Proxima
Cen in $\approx$37 thousand years. At present, we know only two stars with a
smaller (than that of GL 905) encounter parameter: GJ 710=HIP 89825 with
$d_{min}=0.31\pm 0.17$ pc (Bobylev, 2010) and HIP 85661 with $d_{min}=0.94\pm
0.71$ pc (Garcia-Sanchez et al., 2001).
(2) We have identified the stars GJ 3379 (G 099–049) and GJ 3323 (LHS 1723) as
candidates for a close encounter with the solar orbit for the first time.
Their trigonometric parallaxes with a relative error $e_{\pi}/\pi<1\%$ were
first determined only recently by Henry et al. (2006). Their radial velocities
obtained by analyzing the published data from Montes et al. (2001) are also
accurate.
(3) As our statistical simulations show, for all six stars from Table 2, the
probability of their penetration into the Oort cloud region is essentially
zero.
### 3.2 Stars with Indirect Distance Estimates
We do not set the objective to survey all stars with spectrophotometric
distance estimates. Note only two interesting stars, SSSPM J1549–3544 and SDSS
J1416+1348, which we have been able to reveal using data from Scholz et al.
(2004) and Schmidt et al. (2010a; 2010b).
(1) According to Scholz et al. (2004), the kinematic data for SSSPM J1549–3544
are: $\alpha=15^{h}48^{m}40^{s}.23$,
$\delta=-35^{\circ}44^{\prime}25^{\prime\prime}.4$,
$\mu_{\alpha}\cos\delta=-591\pm 8$ mas yr-1, $\mu_{\delta}=-538\pm 5$ mas
yr-1, and $d_{spec}=34$ pc with an error of 1 pc. This star may be the single
cool white dwarf closest to the Sun (closer than the well-known Van Maanen
star, $d=4.3$ pc). Scholz et al. (2004) obtained the distance $d_{spec}=4\pm
1$ pc using 2MASS photometry and $d_{spec}=3\pm 1$ pc using less accurate SSS
(SuperCOSMOS Sky Survey) photometry. Since there are no radial velocity data,
the space velocities $(U,V,W)$ of this star were estimated by Scholz et al.
(2004) for three model radial velocities, –50, 0, and 50 km s-1.
Following Scholz et al. (2004), we will use $d=4\pm 1$ pc ($e_{d}/d=25\%$) and
various model radial velocities for this star. Several trajectories are shown
in Fig. 2. As we see from the figure, the sign of the radial velocity
determines whether the star could encounter the Sun in the past (positive
sign) or in the future (negative sign). We clearly see that the encounter with
the Sun becomes increasingly close with increasing magnitude of the radial
velocity. Because of the distance estimation error $\pm$1 pc, formally there
is a chance of very close encounters. For example, when the plot is shifted
vertically downward by $\approx$1 pc, the stellar trajectories highlighted by
the heavy lines fall at the boundary of the Oort cloud.
Table 3 gives two results obtained at fixed radial velocities of the star with
the addition of random errors to the proper motion components and the distance
$d$. We constructed 10 000 model orbits for each case and in $\approx$900
cases the star falls into the Oort cloud region, $d_{min}\leq 0.485$ pc, then
$P_{1}=900/10000$.
(2) According to Schmidt et al. (2010a, 2010b), the kinematic data for SDSS
J1416+1348 are: $\alpha=14^{h}16^{m}24^{s}.08$,
$\delta=13^{\circ}48^{\prime}26^{\prime\prime}.7$,
$\mu_{\alpha}\cos\delta=88.0\pm 2.8$ mas yr-1, $\mu_{\delta}=139.9\pm 1.3$ mas
yr-1, and $V_{r}=-42.2\pm 5.1$ km s-1. The most reliable distance estimate,
$d=8.0\pm 1.6$ pc ($e_{d}/d=20\%$), was derived by Schmidt et al. (2010a) as a
mean of five photometric and spectroscopic determinations from 2MASS infrared
data and SDSS optical data.
(3) Our statistical simulations show that both SSSPM J1549–3544 and SDSS
J1416+1348 have a nonzero probability of penetrating into the Oort cloud
region: $P_{1}=0.09$ and $P_{1}=0.05,$ respectively (the last column of Table
3). Both stars are of great interest in the problem being solved.
We may conclude that determining the trigonometric parallaxes and radial
velocities of these stars is topical.
(4) Let us now turn to the list of 25 stars from the catalog by Gliese and
Jahreiß (1991) that were revealed by Mülläri and Orlov (1996) as candidates
for a close encounter with the Solar system. Since 16 of them are Hipparcos
stars, the results of the analysis of their trajectories are presented in
Garcia-Sanchez et al. (2001) and Bobylev (2010).
Three stars are located within $d<10$ pc; these are GJ 905, GJ 473, and GJ
3166. Since the remaining six stars are farther than 10 pc, they were not
included in the RECONS list. We have already discussed the encounter
parameters of GJ 905 in the previous section.
The star GJ 3166 (designated as No 456 NN in Mülläri and Orlov, 1996) is of
great interest, because, according to the estimate by Mülläri and Orlov
(1996), it can encounter the Sun to a record distance $d_{min}=0.16$ pc at
time $t_{min}=1600$ thousand years. How ever, as it turned, there is no
information about its proper motion in the catalog by Gliese and
Jahreiß(1991). Therefore, its space velocities $U,V,W$ were calculated by
assuming the tangential velocity to be zero, $V_{t}=0.$
Taking $d=20.8$ pc (photometric distance), $\mu_{t}=0$ mas yr-1, and
$V_{r}=-12$ km s-1 for GJ 3166, as in the catalog by Gliese and Jahreiß(1991),
we find the encounter parameters $d_{min}=0.18$ pc and $t_{min}=1708$ thousand
years, which confirm the result by Mülläri and Orlov. For the same distance
and radial velocity but taking the proper motion components
$\mu_{\alpha}\cos\delta=-103.3\pm 8.1$ mas yr-1 and $\mu_{\delta}=-67.7\pm
9.2$ mas yr-1 from the UCAC3 catalog (Zacharias et al., 2009), we find
completely different encounter parameters, $d_{min}=15$ pc and $t_{min}=590$
thousand years, which are already less interesting in our problem. Note that
the absolute proper motions of this star are also available in the XPM catalog
(Fedorov et al., 2009): $\mu_{\alpha}\cos\delta=-87.2$ mas yr-1 and
$\mu_{\delta}=-68.1$ mas yr-1. These were determined by comparing the 2MASS
and Palomar Observatory Sky Survey positions referenced to galaxies with a
mean error of about 6 mas yr-1 in each coordinate (Bobylev et al., 2010).
Finally, according to the estimate by Mülläri and Orlov (1996), GJ 473 can
encounter the Sun very closely, $d_{min}=0.29$ pc, at time $t_{min}=7.5$
thousand years. These parameters were obtained using the radial velocity
$V_{r}=-553.7$ km s-1 (Gliese and Jahreiß, 1991). GJ 473 (LHS 333=FL Vir=Wolf
424 AB) is a close binary system with a known orbit (Torres et al., 1999). The
measurements by Tinney and Reid (1998) performed with a high-resolution
spectrometer yield the systems heliocentric radial velocity $V_{r}=0.9\pm 1.7$
km s-1. Note that previous measurements also gave a low radial velocity for
this system: $V_{r}=-5\pm 5$ km s-1 (GCRV, Wilson 1953). The radial velocity
in the compilation by Gliese and Jahreiß(1991) is probably erroneous. Since
the encounter parameters of GJ 473 calculated with its new radial velocity
$V_{r}=0.9\pm 1.7$ km s-1, $d_{min}=6\pm 5$ pc and $t_{min}=-3\pm 6$ thousand
years, are no longer the close encounter parameters, this star was not
included in Table 2.
## 4 CONCLUSIONS
Based on currently available kinematic data, we searched for stars that either
encountered or will encounter the solar neighborhood within less than 3 pc. We
considered stars outside the Hipparcos list. For each of them, there is an
estimate of its trigonometric parallax with a relative error
$e_{\pi}/\pi<2\%,$ radial velocity, and proper motion components. We found six
such stars that are an important supplement to the list of Hipparcos stars
closely encountering the Solar system (Garcia-Sanchez et al., 2001; Bobylev,
2001).
For the first time, two single stars, GJ 3379 (G 099–049) and GJ 3323 (LHS
1723), have been identified as candidates for a close encounter with the solar
orbit.
We confirmed the remarkable result by Mülläri and Orlov (1996) that the star
GL 905 could encounter the Sun fairly closely: $d_{min}=0.93\pm 0.01$ pc and
$t_{min}=37.1\pm 0.2$ thousand years.
Two unique stars are located in the immediate solar neighborhood ($d<10$
pc)—the white dwarf SSSPM J1549–3544 and the L dwarf SDSS J1416+1348. Our
statistical simulations showed that both of them have a nonzero probability of
penetrating into the Oort cloud region: $P_{1}=0.09$ and $P_{1}=0.05$,
respectively. Determining the trigonometric parallaxes and radial velocities
for these stars is topical. This task can be accomplished using both ground-
based and space (e.g., GAIA) observations.
Based on new data, we showed that the stars GJ 473 and GJ 3166 are not
suitable candidates for a very close encounter with the Solar system as was
presumed previously.
ACKNOWLEDGMENTS
I am grateful to A.T. Bajkova for the software package for statistical
simulations. The SIMBAD database and the RECONS site were very helpful in the
work. This work was supported by the Russian Foundation for Basic Research
(project nos. 08–02–00400 and 09–02–90443–Ukr_f) and in part by the “Origin
and Evolution of Stars and Galaxies” Program of the Presidium of the Russian
Academy of Sciences.
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Translated by Shtaerman
Table 1: Stars and multiple systems with trigonometric parallaxes
Star | $\alpha_{J2000},$ | $\mu_{\alpha},$ mas yr-1 | $\pi,$ mas | $V_{r},$ km s-1 | Ref*
---|---|---|---|---|---
| $\delta_{J2000}$ | $\mu_{\delta},$ mas yr-1 | | |
GJ 65 AB | $1^{h}39^{m}01^{s}.54$ | $3296\pm 5$ | $373.7\pm 2.7$ | $29.0\pm 4.6$ | e
| $-17^{o}57^{\prime}01^{\prime\prime}.8$ | $563\pm 5$ | | |
GJ 299 | $8^{h}11^{m}57^{s}.58$ | $1099\pm 8$ | $146.3\pm 3.1$ | $-35\pm 5$ | e
| $8^{o}46^{\prime}22^{\prime\prime}.1$ | $-5123\pm 8$ | | |
GJ 388 AB | $10^{h}19^{m}36^{s}.28$ | $-502\pm 8$ | $204.6\pm 2.8$ | $11.6\pm 0.3$ | e
| $19^{o}52^{\prime}12^{\prime\prime}.1$ | $-43\pm 8$ | | |
GJ 406 | $10^{h}56^{m}28^{s}.86$ | $-3842\pm 8$ | $419.1\pm 2.1$ | $19.5\pm 0.4$ | a
| $7^{o}00^{\prime}52^{\prime\prime}.8$ | $-2727\pm 8$ | | |
GJ 473 AB | $12^{h}33^{m}17^{s}.38$ | $-1730\pm 8$ | $227.9\pm 4.6$ | $0.9\pm 1.7$ | d
| $8^{o}46^{\prime}22^{\prime\prime}.1$ | $203\pm 8$ | | |
GJ 866 ABC | $22^{h}38^{m}33^{s}.73$ | $2314\pm 5$ | $289.5\pm 4.4$ | $-50\pm 1$ | c
| $-15^{o}17^{\prime}57^{\prime\prime}.3$ | $2295\pm 5$ | | |
GJ 905 | $23^{h}41^{m}54^{s}.99$ | $111\pm 8$ | $316.0\pm 1.1$ | $-78.0\pm 0.4$ | a
| $44^{o}10^{\prime}40^{\prime\prime}.8$ | $-1584\pm 8$ | | |
GJ 1111 | $8^{h}29^{m}49^{s}.35$ | $-1112\pm 5$ | $275.8\pm 3.0$ | $9.0\pm 0.5$ | b
| $26^{o}46^{\prime}33^{\prime\prime}.7$ | $-611\pm 4$ | | |
GJ 3323 | $5^{h}01^{m}57^{s}.47$ | $-550\pm 5$ | $187.9\pm 1.3$ | $42.0\pm 0.1$ | b
| $-6^{o}56^{\prime}45^{\prime\prime}.9$ | $-533\pm 5$ | | |
GJ 3379 | $6^{h}00^{m}03^{s}.50$ | $311\pm 3$ | $190.9\pm 1.9$ | $30.0\pm 0.1$ | b
| $2^{o}42^{\prime}23^{\prime\prime}.67$ | $-42\pm 3$ | | |
Note. The radial velocities were taken from the following papers: (a) Nidever
(2002), (b) Montes et al.( 2001), (c) Delfosse et al. (1999), (d) Tinney and
Reid (1998), (e) Barbier-Brossat and Figon (2000); the parallaxes are given
according to the RECONS list of 100 nearest stars.
Table 2: Candidates for a close encounter with the Solar system Star | SP | $M/M_{\odot}$ | $d_{min},$ pc | $t_{min},$ thousand yr
---|---|---|---|---
GJ 905 | M5.5V | 0.12 | $0.93\pm 0.01$ | $37.1\pm 0.2$
GJ 3379 | M3.5V | 0.19 | $1.32\pm 0.03$ | $-163\pm 3$
GJ 65 AB | M5.5V/M6.0V | 0.11/0.10 | $2.21\pm 0.12$ | $-29\pm 2$
GJ 406 | M6.0V | 0.09 | $2.24\pm 0.02$ | $-15\pm 1$
GJ 3323 | M4.5V | 0.15 | $2.25\pm 0.05$ | $-104\pm 3$
GJ 866 ABC | M5.0V/–/– | 0.11/0.11/0.10 | $2.52\pm 0.07$ | $32.3\pm 0.3$
Table 3: Stars with spectroscopic or photometric distance estimates Star | SP | $M/M_{\odot}$ | $d_{min},$ | $t_{min},$ | $V_{r},$ | $P_{1}$
---|---|---|---|---|---|---
| | | pc | thousand yr | km s-1 |
SSSPM J1549$-$3544 | $>$DC11 | $\approx$ 0.5 | $1.21\pm 0.58$ | $-72\pm 6$ | $+50$ | 0.09
SSSPM J1549$-$3544 | | | $1.21\pm 0.58$ | $+72\pm 6$ | $-50$ | 0.09
SDSS J1416+1348 | L5V | $\approx$ 0.08 | $1.24\pm 0.65$ | $186\pm 44$ | $-42.2\pm 5.1$ | 0.05
Fig. 1. Trajectories of the stars relative to the Sun: (1) GJ 866, (2) GJ 406,
(3) GJ 65, (4) GJ 3323; for GJ 905 and GJ 3379, we give the model trajectories
computed by taking into account the random errors in the observational data
(1000 realizations). The trajectories hatch the 3$\sigma$ confidence regions,
the Oort cloud region is shaded, the dotted line indicates the trajectory of
Proxima Cen.
Fig. 2. Model trajectories of the star SSSPM J15493544 relative to the Sun:
$d=4$ pc at $V_{r}=+50$ km s-1 (1), $V_{r}=+25$ km s-1 (2), $V_{r}=-50$ km s-1
(3), and $V_{r}=-25$ km s-1 (4).
|
arxiv-papers
| 2010-09-24T14:45:38 |
2024-09-04T02:49:13.069173
|
{
"license": "Public Domain",
"authors": "Vadim V. Bobylev",
"submitter": "Anisa Bajkova",
"url": "https://arxiv.org/abs/1009.4856"
}
|
1009.4970
|
# Doubly Exponential Solutions for Randomized Load Balancing Models with
Markovian Arrival Processes and Phase-Type Service Times
Quan-Lin Li1 John C.S. Lui2
1 School of Economics and Management Sciences
Yanshan University, Qinhuangdao 066004, China
2 Department of Computer Science & Engineering
The Chinese University of Hong Kong, Shatin, N.T, Hong Kong
###### Abstract
In this paper, we provide a novel matrix-analytic approach for studying doubly
exponential solutions of randomized load balancing models (also known as
_supermarket models_) with Markovian arrival processes (MAPs) and phase-type
(PH) service times. We describe the supermarket model as a system of
differential vector equations by means of density dependent jump Markov
processes, and obtain a closed-form solution with a doubly exponential
structure to the fixed point of the system of differential vector equations.
Based on this, we show that the fixed point can be decomposed into the product
of two factors inflecting arrival information and service information, and
further find that the doubly exponential solution to the fixed point is not
always unique for more general supermarket models. Furthermore, we analyze the
exponential convergence of the current location of the supermarket model to
its fixed point, and apply the Kurtz Theorem to study density dependent jump
Markov process given in the supermarket model with MAPs and PH service times,
which leads to the Lipschitz condition under which the fraction measure of the
supermarket model weakly converges the system of differential vector
equations. This paper gains a new understanding of how workload probing can
help in load balancing jobs with non-Poisson arrivals and non-exponential
service times.
Keywords: Randomized load balancing, supermarket model, matrix-analytic
approach, doubly exponential solution, density dependent jump Markov process,
Markovian Arrival Process (MAP), phase-type (PH) distribution, fixed point,
exponential convergence, Lipschitz condition.
## 1 Introduction
Randomized load balancing, where a job is assigned to a server from a small
subset of randomly chosen servers, is very simple to implement, and can
surprisingly deliver better performance (for example reducing collisions,
waiting times, backlogs) in a number of applications, such as data centers,
hash tables, distributed memory machines, path selection in networks, and task
assignment at web servers. One useful model extensively used to study
randomized load balancing schemes is the supermarket model. In the supermarket
model, a key result by Vvedenskaya, Dobrushin and Karpelevich [42] indicated
that when each Poisson arriving job is assigned to the shortest one of $d\geq
2$ randomly chosen queues with exponential service times, the equilibrium
queue length can decay doubly exponentially in the limit as the population
size $n\rightarrow\infty$, and the stationary fraction of queues with at least
$k$ customers is $\rho^{\frac{d^{k}-1}{d-1}}$, which indicates a substantially
exponential improvement over the case for $d=1$, where the tail of stationary
queue length in the corresponding M/M/1 queue is $\rho^{k}$. At the same time,
the exponential improvement is also illustrated by another key work in which
Luczak and McDiarmid [21] studied the maximum queue length in the supermarket
model with Poisson arrivals and exponential service times.
The distributed load balancing strategies in which individual job decisions
are based on information on a limited number of other processors, have been
studied by analytical methods in Eager, Lazokwska and Zahorjan [9, 10, 11] and
by trace-driven simulations in Zhou [47]. Based on this, the supermarket
models can be developed by using either queueing theory or Markov processes.
Most of recent research deals with a simple supermarket model with Poisson
arrivals and exponential service times by means of density dependent jump
Markov processes. The methods used in the recent literature are based on
determining the behavior of the supermarket model as its population size grows
to infinity, and its behavior is naturally described as a system of
differential equations whose fixed point leads to a closed-form solution with
a doubly exponential structure. Readers may refer to, such as, Azar, Broder,
Karlin and Upfal [3], Vvedenskaya, Dobrushin and Karpelevich [42] and
Mitzenmacher [26, 27].
Certain generalizations of the supermarket models have been explored, for
example, in studying simple variations by Mitzenmacher and Vöcking [34],
Mitzenmacher [28, 29, 32], Vöcking [41], Mitzenmacher, Richa, and Sitaraman
[33] and Vvedenskaya and Suhov [43]; in discussing load information by
Mirchandaney, Towsley, and Stankovic [35], Dahlin [7] and Mitzenmacher [31,
33]; and in mathematical analysis by Graham [12, 13, 14], Luczak and Norris
[23] and Luczak and McDiarmid [21, 22]. Using fast Jackson networks, Martin
and Suhov [25], Martin [24], Suhov and Vvedenskaya [40] studied supermarket
mall models, where each node in a Jackson network is replaced by $N$ parallel
servers, and a job joins the shortest of $d$ randomly chosen queues at the
node to which it is directed. For non-Poisson arrivals or for non-exponential
service times, Li, Lui and Wang [19] discussed the supermarket model with
Poisson arrivals and PH service times, and indicated that the fixed point
decreases doubly exponentially, where the stationary phase-type environment is
shown to be a crucial factor. Bramson, Lu and Prabhakar [4] provided a
modularized program based on ansatz for treating the supermarket model with
Poisson arrivals and general service times, and Li [18] further discussed this
supermarket model by means of a system of integral-differential equations, and
illustrated that the fixed point decreases doubly exponentially and that the
heavy-tailed service times do not change the doubly exponential solution to
the fixed point.
For the PH distribution, readers may refer to Neuts [36, 37] and Li [17]. The
MAP is a useful mathematical model, for example, for describing bursty
traffic, self similarity and long-range dependence in modern computer
networks, e.g., see Adler, Feldman and Taqqu [1]. For detail information of
the MAP, readers may refer to Chapter 5 in Neuts [37], Lucantoni [20], Chapter
1 in Li [17], and three excellent overviews by Neuts [39], Chakravarthy [5]
and Cordeiro and Kharoufeh [6]. In computer networks, Andersen and Nielsen [2]
applied the MAP to describe long-range dependence, and Yoshihara, Kasahara and
Takahashi [46] analyzed self-similar traffic by means of a Markov-modulated
Poisson process.
It is interesting to answer whether or how non-Poisson arrivals or non-
exponential service times can disrupt doubly exponential solutions to the
fixed points in supermarket models. To that end, this paper studies a
supermarket model with MAPs and PH service times, and shows that there still
exists a doubly exponential solution to the fixed point. The main
contributions of the paper are threefold. The first one is to provide a novel
matrix-analytic approach to study the supermarket model with MAPs and PH
service times. Based on density dependent jump Markov processes, the
supermarket model is described as a system of differential vector equations
whose fixed point has a closed-form solution with a doubly exponential
structure. The second one is to obtain a crucial result that the fixed point
can be decomposed into the product of two factors inflecting arrival
information and service information, which indicates that the doubly
exponential solution to the fixed point can exist extensively, but it is not
always unique for more general supermarket models. The third one is to analyze
exponential convergence of the current location of the supermarket model to
its fixed point. Not only does the exponential convergence indicate the
existence of the fixed point, but it also shows that such a convergent process
is very fast. To study the limit behavior of the supermarket model as its
population size goes to infinity, this paper applies the Kurtz Theorem to
study density dependent jump Markov process given in the supermarket model
with MAPs and PH service times, which leads to the Lipschitz condition under
which the fraction measure of the supermarket model weakly converges the
system of differential vector equations.
The remainder of this paper is organized as follows. In Section 2, we first
describe a supermarket model with MAPs and PH service times. Then the
supermarket model is described as a systems of differential vector equations
in terms of density dependent jump Markov processes. In Section 3, we first
introduce a fixed point of the system of differential vector equations, and
set up a system of nonlinear equations satisfied by the fixed point. Then we
provide a closed-form solution with a doubly exponential structure to the
fixed point, and show that the fixed point can be decomposed into the product
of two factors inflecting arrival information and service information. In
Section 4, we provide an important observation in which the doubly exponential
solution to the fixed point is not always unique for more general supermarket
models. In Section 5, we study exponential convergence of the current location
of the supermarket model to its fixed point. In Section 6, we apply the Kurtz
Theorem to study density dependent jump Markov process given in the
supermarket model with MAPs and PH service times, which leads to the Lipschitz
condition under which the fraction measure of the supermarket model weakly
converges the system of differential vector equations. Some concluding remarks
are given in Section 7.
## 2 Supermarket Model Description
In this section, we first provide a supermarket model with MAPs and PH service
times. Then the supermarket model is described as a system of differential
vector equations based on density dependent jump Markov processes.
We first introduce some notation as follows. Let $A\otimes B$ be the
_Kronecker product_ of two matrices $A=(a_{i,j})$ and $B=(b_{i,j})$, that is,
$A\otimes B=(a_{i,j}B)$; $A\oplus B$ the _Kronecker sum_ of $A$ and $B$, that
is, $A\oplus B=A\otimes I+I\otimes B$. We denote by $A\odot B$ the _Hadamard
Product_ of $A$ and $B$ as follows:
$A\odot B=\left(a_{i,j}b_{i,j}\right).$
Specifically, for $k\geq 2$, we have
$A^{\odot k}=\underset{k\text{ matrix }A}{\underbrace{A\odot A\odot\cdots\odot
A}}.$
For a vector $a=\left(a_{1},a_{2},\ldots,a_{m}\right)$, we write
$a^{\odot\frac{1}{d}}=\left(a_{1}^{\frac{1}{d}},a_{2}^{\frac{1}{d}},\ldots,a_{m}^{\frac{1}{d}}\right).$
Now, we describe the supermarket model, which is abstracted as a multi-server
multi-queue queueing system. Customers arrive at a queueing system of $n>1$
servers as a MAP with an irreducible matrix descriptor $\left(nC,nD\right)$ of
size $m_{A}$. Let $\gamma$ be the stationary probability vector of the
irreducible Markov chain $C+D$. Then the stationary arrival rate of the MAP is
given by $n\lambda=n\gamma De$, where $e$ is a column vector of ones with a
suitable size. The service time of each customer is of phase type with an
irreducible representation $\left(\alpha,T\right)$ of order $m_{B}$, where the
row vector $\alpha$ is a probability vector whose $j$th entry is the
probability that a service begins in phase $j$ for $1\leq j\leq m_{B}$; $T$ is
an $m_{B}\times m_{B}$ matrix whose $\left(i,j\right)^{th}$ entry is denoted
by $t_{i,j}$ with $t_{i,i}<0$ for $1\leq i\leq m_{B}$, and $t_{i,j}\geq 0$ for
$1\leq i,j\leq m_{B}$ and $i\neq j$. Let $T^{0}=-Te\gvertneqq 0$. When a PH
service time is in phase $i$, the transition rate from phase $i$ to phase $j$
is $t_{i,j}$, the service completion rate is $t_{i}^{0}$, and the output rate
from phase $i$ is $\mu_{i}=-t_{i,i}$. At the same time, the expected service
time is given by $1/\mu=-\alpha T^{-1}e$. Each arriving customer chooses
$d\geq 1$ servers independently and uniformly at random from the $n$ servers,
and waits for service at the server which currently contains the fewest number
of customers. If there is a tie, servers with the fewest number of customers
will be chosen randomly. All customers in every server will be served in the
first-come-first service (FCFS) manner. We assume that all the random
variables defined above are independent, and that this system is operating in
the region $\rho=\lambda/\mu<1$. Please see Figure 1 for an illustration of
such a supermarket model.
Figure 1: The supermarket model wherein each customer can probe $d$ servers
The following lemma, which is stated without proof, provides an intuitively
sufficient condition under which the supermarket model is stable. Note that
this proof can be given by a simple comparison argument with the queueing
system in which each customer queues at a random server (i.e., where $d=1$).
When $d=1$, each server acts like a MAP/PH/1 queue which is stable if
$\rho=\lambda/\mu<1$, see chapter 5 in Neuts [37]. The comparison argument is
similar to those in Winston [45] and Weber [44], thus we can obtain two useful
results: (1) the shortest queue is optimal due to the assumptions on MAPs and
PH service times; and (2) the size of the longest queue in the supermarket
model is stochastically dominated by the size of the longest queue in a set of
$n$ independent MAP/PH/1 queues.
###### Lemma 1
The supermarket model with MAPs and PH service times is stable if
$\rho=\lambda/\mu<1.$
We define $n_{k}^{\left(i,j\right)}\left(t\right)$ as the number of queues
with at least $k$ customers who include the customer in service, the MAP in
phase $i$ and the PH service time in phase $j$ at time $t\geq 0$. Clearly,
$0\leq n_{k}^{\left(i,j\right)}\left(t\right)\leq n$ for $1\leq i\leq m_{A}$,
$1\leq j\leq m_{B}$ and $k\geq 0$. Let
$x_{n}^{\left(i\right)}\left(0,t\right)=\frac{n_{0}^{\left(i\right)}\left(t\right)}{n}$
and for $k\geq 1$
$x_{n}^{\left(i,j\right)}\left(k,t\right)=\frac{n_{k}^{\left(i,j\right)}\left(t\right)}{n},$
which is the fraction of queues with at least $k$ customers, the MAP in phase
$i$ and the PH service time in phase $j$ at time $t\geq 0$. Using the
lexicographic order we write
$X_{n}\left(0,t\right)=\left(x_{n}^{\left(1\right)}\left(0,t\right),x_{n}^{\left(2\right)}\left(0,t\right),\ldots,x_{n}^{\left(m_{A}\right)}\left(0,t\right)\right)$
and for $k\geq 1$
$\displaystyle X_{n}\left(k,t\right)=$
$\displaystyle(x_{n}^{\left(1,1\right)}\left(k,t\right),x_{n}^{\left(1,2\right)}\left(k,t\right),\ldots,x_{n}^{\left(1,m_{B}\right)}\left(k,t\right);\ldots;$
$\displaystyle
x_{n}^{\left(m_{A},1\right)}\left(k,t\right),x_{n}^{\left(m_{A},2\right)}\left(k,t\right),\ldots,x_{n}^{\left(m_{A},m_{B}\right)}\left(k,t\right)),$
$X_{n}\left(t\right)=\left(X_{n}\left(0,t\right),X_{n}\left(1,t\right),X_{n}\left(2,t\right),\ldots\right).$
The state of the supermarket model may be described by the vector
$X_{n}\left(t\right)$ for $t\geq 0$. Since the arrival process to the queueing
system is a MAP and the service time of each customer is of phase type, the
stochastic process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ describing
the state of the supermarket model is a Markov process whose state space is
given by
$\displaystyle\Omega_{n}=$
$\displaystyle\\{\left(g_{n}^{\left(0\right)},g_{n}^{\left(1\right)},g_{n}^{\left(2\right)}\ldots\right):g_{n}^{\left(0\right)}\text{
is a probability vector, }g_{n}^{\left(k-1\right)}\geq
g_{n}^{\left(k\right)}\geq 0$ $\displaystyle\text{ for }k\geq 2\text{,}\text{
and }ng_{n}^{\left(l\right)}\text{ is a vector of nonnegative integers for
}l\geq 0\\}.$
Let
$s_{0}^{\left(i\right)}\left(n,t\right)=E\left[x_{0}^{\left(i\right)}\left(n,t\right)\right]$
and $k\geq 1$
$s_{k}^{\left(i,j\right)}\left(n,t\right)=E\left[x_{k}^{\left(i,j\right)}\left(n,t\right)\right].$
Using the lexicographic order we write
$S_{0}\left(n,t\right)=\left(s_{0}^{\left(1\right)}\left(n,t\right),s_{0}^{\left(2\right)}\left(n,t\right),\ldots,s_{0}^{\left(m_{A}\right)}\left(n,t\right)\right)$
and for $k\geq 1$
$\displaystyle S_{k}\left(n,t\right)=$
$\displaystyle(s_{k}^{\left(1,1\right)}\left(n,t\right),s_{k}^{\left(1,2\right)}\left(n,t\right),\ldots,s_{k}^{\left(1,m_{B}\right)}\left(n,t\right);\ldots;$
$\displaystyle
s_{k}^{\left(m_{A},1\right)}\left(n,t\right),s_{k}^{\left(m_{A},2\right)}\left(n,t\right),\ldots,s_{k}^{\left(m_{A},m_{B}\right)}\left(n,t\right)),$
$S\left(n,t\right)=\left(S_{0}\left(n,t\right),S_{1}\left(n,t\right),S_{2}\left(n,t\right),\ldots\right).$
As shown in Martin and Suhov [25] and Luczak and McDiarmid [21], the Markov
process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ is asymptotically
deterministic as $n\rightarrow\infty$. Thus the limits
$\lim_{n\rightarrow\infty}E\left[x_{0}^{\left(i\right)}\left(n,t\right)\right]$
and
$\lim_{n\rightarrow\infty}E\left[x_{k}^{\left(i,j\right)}\left(n,t\right)\right]$
always exist by means of the law of large numbers. Based on this, we write
$S_{0}\left(t\right)=\lim_{n\rightarrow\infty}S_{0}\left(n,t\right),$
for $k\geq 1$
$S_{k}\left(t\right)=\lim_{n\rightarrow\infty}S_{k}\left(n,t\right),$
and
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right).$
Note that $S_{0}\left(t\right)$ and $S_{k}\left(t\right)$ are two row vectors
of order $m_{A}$ and $m_{A}m_{B}$, respectively. Let
$X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$. Then it is
easy to see from the MAPs and the PH service times that
$\left\\{X\left(t\right),t\geq 0\right\\}$ is also a Markov process whose
state space is given by
$\Omega=\left\\{\left(g^{\left(0\right)},g^{\left(1\right)},g^{\left(2\right)},\ldots\right):g^{\left(0\right)}\text{
is a probability vector},g^{\left(k-1\right)}\geq g^{\left(k\right)}\geq
0\right\\}.$
If the initial distribution of the Markov process
$\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ approaches the Dirac delta-
measure concentrated at a point $g\in$ $\Omega$, then the limit
$X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$ is concentrated
on the trajectory $S_{g}=\left\\{S\left(t\right):t\geq 0\right\\}$. This
indicates a law of large numbers for the time evolution of the fraction of
queues of different lengths. Furthermore, the Markov process
$\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ converges weakly to the
fraction vector
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$
as $n\rightarrow\infty$, or for a sufficiently small $\varepsilon>0$,
$\lim_{n\rightarrow\infty}P\left\\{||X_{n}\left(t\right)-S\left(t\right)||\geq\varepsilon\right\\}=0,$
where $||a||$ is the $L_{\infty}$-norm of vector $a$.
The following proposition shows that the sequence
$\left\\{S_{k}\left(t\right),k\geq 0\right\\}$ is monotone decreasing, while
its proof is easy by means of the definition of $S\left(t\right)$.
###### Proposition 1
For $1\leq k<l$
$S_{l}\left(t\right)<S_{k}\left(t\right)$
and
$S_{l}\left(t\right)e<S_{k}\left(t\right)e<S_{0}\left(t\right)e=1.$
In what follows we set up a system of differential vector equations satisfied
by the fraction vector $S\left(t\right)$ by means of density dependent jump
Markov processes.
We first provide an example to indicate how to derive the differential vector
equations. Consider the supermarket model with $n$ servers, and determine the
expected change in the number of queues with at least $k$ customers over a
small time period of length d$t$. The probability vector that an arriving
customer joins a queue of size $k-1$ during this time period is given by
$\left[S_{k-1}^{\odot d}\left(n,t\right)\left(D\otimes I\right)+S_{k}^{\odot
d}\left(n,t\right)\left(C\otimes I\right)\right]\cdot n\text{d}t,$
since each arriving customer chooses $d\geq 1$ servers independently and
uniformly at random from the $n$ servers, and waits for service at the server
which currently contains the fewest number of customers. Similarly, the
probability vector that a customer leaves a server queued by $k$ customers
during this time period is given by
$\left[S_{k}\left(n,t\right)\left(I\otimes
T\right)+S_{k+1}\left(n,t\right)\left(I\otimes T^{0}\alpha\right)\right]\cdot
n\text{d}t.$
Therefore, we can obtain
$\displaystyle\text{d}E\left[n_{k}\left(n,t\right)\right]=$
$\displaystyle\left[S_{k-1}^{\odot d}\left(n,t\right)\left(D\otimes
I\right)+S_{k}^{\odot d}\left(n,t\right)\left(C\otimes I\right)\right]\cdot
n\text{d}t$ $\displaystyle+\left[S_{k}\left(n,t\right)\left(I\otimes
T\right)+nS_{k+1}\left(n,t\right)\left(I\otimes T^{0}\alpha\right)\right]\cdot
n\text{d}t,$
which leads to
$\displaystyle\frac{\text{d}S_{k}\left(n,t\right)}{\text{d}t}=$ $\displaystyle
S_{k-1}^{\odot d}\left(n,t\right)\left(D\otimes I\right)+S_{k}^{\odot
d}\left(n,t\right)\left(C\otimes I\right)$
$\displaystyle+S_{k}\left(n,t\right)\left(I\otimes
T\right)+S_{k+1}\left(n,t\right)\left(I\otimes T^{0}\alpha\right).$ (1)
Using a similar analysis for Equation (1), we can obtain a system of
differential vector equations for the fraction vector
$S\left(n,t\right)=\left(S_{0}\left(n,t\right),S_{1}\left(n,t\right),S_{2}\left(n,t\right),\ldots\right)$
as follows:
$S_{0}\left(n,t\right)e=1,$ (2)
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(n,t\right)=S_{0}^{\odot
d}\left(n,t\right)C+S_{1}\left(n,t\right)\left(I\otimes T^{0}\right),$ (3)
$\displaystyle\frac{\mathtt{d}}{\text{d}t}S_{1}\left(n,t\right)=$
$\displaystyle S_{0}^{\odot
d}\left(n,t\right)\left(D\otimes\alpha\right)+S_{1}^{\odot
d}\left(n,t\right)\left(C\otimes I\right)$
$\displaystyle+S_{1}\left(n,t\right)\left(I\otimes
T\right)+S_{2}\left(n,t\right)\left(I\otimes T^{0}\alpha\right),$ (4)
and for $k\geq 2$
$\displaystyle\frac{\text{d}S_{k}\left(n,t\right)}{\text{d}t}=$ $\displaystyle
S_{k-1}^{\odot d}\left(n,t\right)\left(D\otimes I\right)+S_{k}^{\odot
d}\left(n,t\right)\left(C\otimes I\right)$
$\displaystyle+S_{k}\left(n,t\right)\left(I\otimes
T\right)+S_{k+1}\left(n,t\right)\left(I\otimes T^{0}\alpha\right).$ (5)
Noting that the limit $\lim_{n\rightarrow\infty}S_{k}\left(n,t\right)$ exists
for $k\geq 0$ and taking $n\rightarrow\infty$ in the both sides of the system
of differential vector equations (2) to (5), we can easily obtain a system of
differential vector equations for the fraction vector
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$
as follows:
$S_{0}\left(t\right)e=1,$ (6)
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)=S_{0}^{\odot
d}\left(t\right)C+S_{1}\left(t\right)\left(I\otimes T^{0}\right),$ (7)
$\displaystyle\frac{\mathtt{d}}{\text{d}t}S_{1}\left(t\right)=$ $\displaystyle
S_{0}^{\odot d}\left(t\right)\left(D\otimes\alpha\right)+S_{1}^{\odot
d}\left(t\right)\left(C\otimes I\right)$
$\displaystyle+S_{1}\left(t\right)\left(I\otimes
T\right)+S_{2}\left(t\right)\left(I\otimes T^{0}\alpha\right),$ (8)
and for $k\geq 2$,
$\displaystyle\frac{\text{d}S_{k}\left(t\right)}{\text{d}t}=$ $\displaystyle
S_{k-1}^{\odot d}\left(t\right)\left(D\otimes I\right)+S_{k}^{\odot
d}\left(t\right)\left(C\otimes I\right)$
$\displaystyle+S_{k}\left(t\right)\left(I\otimes
T\right)+S_{k+1}\left(t\right)\left(I\otimes T^{0}\alpha\right).$ (9)
###### Remark 1
(a) For the supermarket model, many papers, such as Mitzenmacher [26] and
Luczak and McDiarmid [21], assumed that the arrival process is Poisson with
rate $n\lambda$. As a direct generalization of the Poisson arrivals with rate
$n\lambda$, this paper uses a MAP with an irreducible matrix descriptor
$\left(nC,nD\right)$ of size $m_{A}$ whose stationary arrival rate is given by
$n\lambda=n\gamma De$.
(b) When there are $n$ servers in the supermarket model, we may use a more
general MAP with an irreducible matrix descriptor $\left(C_{n},D_{n}\right)$
of size $m_{A}$, where
$\lim_{n\rightarrow\infty}\frac{C_{n}}{n}=C,\text{ \
}\lim_{n\rightarrow\infty}\frac{D_{n}}{n}=D,$
and $\left(C,D\right)$ is also the irreducible matrix descriptor of a MAP. It
is easy to see from the above analysis that we can also obtain the system of
differential vector equations (6) to (9) with respect to the more general MAP.
## 3 Doubly Exponential Solution
In this section, we provide a novel matrix-analytic approach for computing the
fixed point of the system of differential vector equations (6) to (9), and
give a closed-form solution with a doubly exponential structure to the fixed
point.
A row vector $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is called a
fixed point of the fraction vector $S\left(t\right)$ if
$\lim_{t\rightarrow+\infty}S\left(t\right)=\pi$. In this case, it is easy to
see that
$\lim_{t\rightarrow+\infty}\left[\frac{\mathtt{d}}{\text{d}t}S\left(t\right)\right]=0.$
Therefore, as $t\rightarrow+\infty$ the system of differential vector
equations (6) to (9) can be simplified as a system of nonlinear equations as
follows:
$\pi_{0}e=1,$ (10) $\pi_{0}^{\odot d}C++\pi_{1}\left(I\otimes T^{0}\right)=0,$
(11) $\pi_{0}^{\odot d}\left(D\otimes\alpha\right)+\pi_{1}^{\odot
d}\left(C\otimes I\right)+\pi_{1}\left(I\otimes T\right)+\pi_{2}\left(I\otimes
T^{0}\alpha\right)=0,$ (12)
and for $k\geq 2$,
$\pi_{k-1}^{\odot d}\left(D\otimes I\right)+\pi_{k}^{\odot d}\left(C\otimes
I\right)+\pi_{k}\left(I\otimes T\right)+\pi_{k+1}\left(I\otimes
T^{0}\alpha\right)=0.$ (13)
It is very challenging to solve the system of nonlinear equations (10) to
(13). Here, our goal is to derive a closed-form solution with a doubly
exponential structure to the fixed point $\pi=(\pi_{0},\pi_{1},\pi_{2},...)$
through a novel matrix-analytic approach.
It follows from Equations (12) and (13) that
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}C\otimes I&D\otimes I&&&\\\
&C\otimes I&D\otimes I&&\\\ &&C\otimes I&D\otimes I&\\\
&&&\ddots&\ddots\end{array}\right)$ (18) $\displaystyle+$
$\displaystyle\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)\left(\begin{array}[c]{cccc}I\otimes
T&&&\\\ I\otimes T^{0}\alpha&I\otimes T&&\\\ &I\otimes T^{0}\alpha&I\otimes
T&\\\ &&\ddots&\ddots\end{array}\right)$ (23)
$\displaystyle=-\left(\pi_{0}^{\odot
d}\left(D\otimes\alpha\right),0,0,\ldots\right).$ (24)
Let
$A=\left(\begin{array}[c]{cccc}I\otimes T&&&\\\ I\otimes T^{0}\alpha&I\otimes
T&&\\\ &I\otimes T^{0}\alpha&I\otimes T&\\\
&&\ddots&\ddots\end{array}\right).$
Then it is easy to check that the matrix $A$ is invertible, and
$-A^{-1}=\left(\begin{array}[c]{ccccc}I\otimes\left(-T\right)^{-1}&&&&\\\
I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left(-T\right)^{-1}&&&\\\
I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left(-T\right)^{-1}&&\\\
I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left[e\alpha\left(-T\right)^{-1}\right]&I\otimes\left(-T\right)^{-1}&\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right),$
$\left(\begin{array}[c]{ccccc}C\otimes I&D\otimes I&&&\\\ &C\otimes I&D\otimes
I&&\\\ &&C\otimes I&D\otimes I&\\\
&&&\ddots&\ddots\end{array}\right)\left(-A^{-1}\right)=\left(\begin{array}[c]{ccccc}R&V&&&\\\
W&R&V&&\\\ W&W&R&V&\\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right),$
and
$\left(\pi_{0}^{\odot
d}\left(D\otimes\alpha\right),0,0,\ldots\right)\left(-A^{-1}\right)=\left(\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right],0,0,\ldots\right),$
where
$V=D\otimes\left(-T\right)^{-1},$
$W=\left(C+D\right)\otimes\left[e\alpha\left(-T\right)^{-1}\right]$
and
$R=C\otimes\left(-T\right)^{-1}+D\otimes\left[e\alpha\left(-T\right)^{-1}\right].$
Thus it follows from (24) that
$\displaystyle\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)=$
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}R&V&&&\\\ W&R&V&&\\\ W&W&R&V&\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$ (29)
$\displaystyle+\left(\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right],0,0,\ldots\right),$ (30)
which leads to a new system of nonlinear equations as follows:
$\pi_{1}=\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right]+\pi_{1}^{\odot
d}R+\sum_{j=2}^{\infty}\pi_{j}^{\odot d}W$ (31)
and for $k\geq 2$,
$\pi_{k}=\pi_{k-1}^{\odot d}V+\pi_{k}^{\odot
d}R+\sum_{j=k+1}^{\infty}\pi_{j}^{\odot d}W.$ (32)
Now, we need to omit the two terms $\pi_{l}^{\odot d}R$ for $l\geq 1$ and
$\sum_{j=k}^{\infty}\pi_{j}^{\odot d}W$ for $k\geq 2$ in Equations (31) and
(32). Note that the Markov chain $C+D$ is positive recurrent, we assume that
the system of nonlinear equations (31) and (32) has a closed-form solution
$\pi_{0}=\theta\gamma^{\odot\frac{1}{d}}$ (33)
and for $k\geq 1$
$\pi_{k}=r\left(k\right)\left(\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}\right),$
(34)
where $\theta=1/\gamma^{\odot\frac{1}{d}}e$, and $r\left(k\right)$ is a
positive constant for $k\geq 1$. Then it follows from (31), (32) and (34) that
$\displaystyle
r\left(1\right)\left(\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}\right)=$
$\displaystyle\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right]+r^{d}\left(1\right)\left(\gamma\otimes\alpha\right)R$
$\displaystyle+\sum_{j=2}^{\infty}r^{d}\left(j\right)\left(\gamma\otimes\alpha\right)W;$
(35)
and for $k\geq 2$,
$\displaystyle
r\left(k\right)\left(\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}\right)=$
$\displaystyle
r^{d}\left(k-1\right)\left(\gamma\otimes\alpha\right)V+r^{d}\left(k\right)\left(\gamma\otimes\alpha\right)R$
$\displaystyle+\sum_{j=k+1}^{\infty}r^{d}\left(j\right)\left(\gamma\otimes\alpha\right)W.$
(36)
Note that
$\displaystyle\left(\gamma\otimes\alpha\right)W$
$\displaystyle=\left(\gamma\otimes\alpha\right)\left\\{\left(C+D\right)\otimes\left[e\alpha\left(-T\right)^{-1}\right]\right\\}$
$\displaystyle=\gamma\left(C+D\right)\otimes\alpha\left[e\alpha\left(-T\right)^{-1}\right]$
and
$\gamma\left(C+D\right)=0,$
it is clear that
$\left(\gamma\otimes\alpha\right)W=0.$
Similarly, we can compute
$\displaystyle\left(\gamma\otimes\alpha\right)R$
$\displaystyle=\left(\gamma\otimes\alpha\right)\left\\{C\otimes\left(-T\right)^{-1}+D\otimes\left[e\alpha\left(-T\right)^{-1}\right]\right\\}$
$\displaystyle=\gamma\left(C+D\right)\otimes\alpha\left(-T\right)^{-1}=0.$
It follows from (35) and (36) that
$\pi_{1}=\pi_{0}^{\odot d}D\otimes\left[\alpha\left(-T\right)^{-1}\right]$
(37)
or
$r\left(1\right)\left(\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}\right)=\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right];$ (38)
and for $k\geq 2$
$\pi_{k}=\pi_{k-1}^{\odot d}\left(D\otimes\left(-T\right)^{-1}\right)$ (39)
or
$r\left(k\right)\left(\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}\right)=r^{d}\left(k-1\right)\left(\gamma\otimes\alpha\right)\left(D\otimes\left(-T\right)^{-1}\right).$
(40)
Let $\omega=1/\alpha^{\odot\frac{1}{d}}e$. Then $0<\theta,\omega<1$ due to
$\gamma^{\odot\frac{1}{d}}e>1$ and $\alpha^{\odot\frac{1}{d}}e>1$. Note that
$\pi_{0}=\theta\gamma^{\odot\frac{1}{d}}$, $\lambda=\gamma De$,
$1/\mu=\alpha\left(-T\right)^{-1}e$ and $\rho=\lambda/\mu$, it follows from
(38) and (40) that
$r\left(1\right)=\theta^{d}\left(\theta\omega\rho\right)$ (41)
and for $k\geq 2$
$\displaystyle r\left(k\right)$
$\displaystyle=r^{d}\left(k-1\right)\theta\omega\rho$
$\displaystyle=\left[r\left(1\right)\right]^{d^{k-1}}\left(\theta\omega\rho\right)^{d^{k-2}+d^{k-3}+\cdots+1}$
$\displaystyle=\theta^{d^{k}}\left(\theta\omega\rho\right)^{d^{k-1}+d^{k-2}+\cdots+1}$
$\displaystyle=\theta^{d^{k}}\left(\theta\omega\rho\right)^{\frac{d^{k}-1}{d-1}}.$
(42)
It is easy to see from (33), (34) and (42) that
$\pi_{0}=\theta\cdot\gamma^{\odot\frac{1}{d}}$ (43)
and for $k\geq 1$
$\pi_{k}=\theta^{d^{k}}\left(\theta\omega\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}.$
(44)
Now, we use (43) and (44) to check Equations (11) and (37) that
$\left\\{\begin{array}[c]{l}\pi_{0}^{\odot d}C+\pi_{1}\left(I\otimes
T^{0}\right)=0,\\\ \pi_{1}=\pi_{0}^{\odot
d}D\otimes\left[\alpha\left(-T\right)^{-1}\right],\end{array}\right.$
which leads to
$\pi_{0}^{\odot d}\left(C+D\right)=0.$ (45)
Obviously, $\pi_{0}=\theta\cdot\gamma^{\odot\frac{1}{d}}$ is a non-zero
nonnegative solution to Equation (45), and $\pi_{0}e=1$.
Summarizing the above analysis, the following theorem describes a closed-form
solution with a doubly exponential structure to the fixed point.
###### Theorem 1
The fixed point $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is given by
$\pi_{0}=\theta\cdot\gamma^{\odot\frac{1}{d}},$
and for $k\geq 1,$
$\pi_{k}=\theta^{d^{k}}\left(\theta\omega\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}.$
The following corollary indicates that the fixed point can be decomposed into
the product of two factors inflecting arrival information and service
information. Based on this, it is easy to see the role played by the arrival
and service processes in the fixed point.
###### Corollary 2
The fixed point $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ can be
decomposed into the product of two factors inflecting arrival information and
service information
$\pi_{k}=\left\\{\theta^{\frac{d^{k+1}-1}{d-1}}\lambda^{\frac{d^{k}-1}{d-1}}\cdot\gamma^{\odot\frac{1}{d}}\right\\}\otimes\left\\{\left(\frac{\omega}{\mu}\right)^{\frac{d^{k}-1}{d-1}}\cdot\alpha^{\odot\frac{1}{d}}\right\\},\text{
\ }k\geq 0.$
###### Remark 2
We consider a supermarket model with Poisson arrivals with rate $\lambda$ and
exponential service times with rate $\mu$, which has been extensively analyzed
in the literature. Obviously, $\pi_{0}=1$. It follows from (24) that
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}-\lambda&\lambda&&&\\\
&-\lambda&\lambda&&\\\ &&-\lambda&\lambda&\\\
&&&\ddots&\ddots\end{array}\right)\newline $ $\displaystyle+$
$\displaystyle\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)\left(\begin{array}[c]{cccc}-\mu&&&\\\
\mu&-\mu&&\\\ &\mu&-\mu&\\\
&&\ddots&\ddots\end{array}\right)=-\left(\lambda,0,0,\ldots\right),$
which leads to
$\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)=\left(\pi_{1}^{\odot
d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}0&\rho&&&\\\ &0&\rho&&\\\
&&0&\rho&\\\ &&&\ddots&\ddots\end{array}\right)+\left(\rho,0,0,\ldots\right).$
Thus we obtain
$\pi_{1}=\rho$
and for $k\geq 2$,
$\pi_{k}=\rho\pi_{k-1}^{\odot
d}=\rho^{d^{k-1}+d^{k-2}+\cdots+1}=\rho^{\frac{d^{k}-1}{d-1}},$
which is the same as Lemma 3.2 in Mitzenmacher [27].
Based on Theorem 1, we now compute the expected sojourn time $T_{d}$ that a
tagged arriving customer spends in the supermarket model. For the PH service
time $X$ with an irreducible representation $\left(\alpha,T\right)$, the
residual time $X_{R}$ of $X$ is also of phase type with an irreducible
representation $\left(\tau,T\right)$, where $\tau$ is the stationary
probability vector of the Markov chain $T+T^{0}\alpha$. Thus, we have
$E\left[X\right]=\alpha\left(-T\right)^{-1}e=\frac{1}{\mu},\text{ \
}E\left[X_{R}\right]=\tau\left(-T\right)^{-1}e.$
For the PH service times, a tagged arriving customer is the $k$th customer in
the corresponding queue with probability $\pi_{k-1}^{\odot d}e-\pi_{k}^{\odot
d}e$. Thus it is easy to see that the expected sojourn time of the tagged
arriving customer is given by
$\displaystyle E\left[T_{d}\right]$ $\displaystyle=\left(\pi_{0}^{\odot
d}e-\pi_{1}^{\odot
d}e\right)E\left[X\right]+\sum_{k=1}^{\infty}\left(\pi_{k}^{\odot
d}e-\pi_{k+1}^{\odot
d}e\right)\left[E\left[X_{R}\right]+kE\left[X\right]\right]$
$\displaystyle=\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}\pi_{1}^{\odot
d}e+E\left[X\right]\sum_{k=0}^{\infty}\pi_{k}^{\odot d}e$
$\displaystyle=\theta^{d^{2}}\left(\theta\omega\rho\right)^{d}\left(\tau-\alpha\right)\left(-T\right)^{-1}e+\frac{1}{\mu}\sum_{k=0}^{\infty}\theta^{d^{k+1}}\left(\theta\omega\rho\right)^{\frac{d^{k+1}-d}{d-1}}.$
When the arrival process and the service time distribution are Poisson and
exponential, respectively, it is clear that $\alpha=\tau=1$ and
$\theta=\omega=1$, thus we have
$E\left[T_{d}\right]=\frac{1}{\mu}\sum_{k=0}^{\infty}\rho^{\frac{d^{k+1}-d}{d-1}},$
which is the same as Corollary 3.8 in Mitzenmacher [27].
In what follows we provide an example to indicate how the expected sojourn
time $E[T_{d}]$ depends on the choice number $d$. We assume that $m=2$ and
$C=\left(\begin{array}[c]{cc}-10&7\\\ 4&-9\end{array}\right),\ \
D=\left(\begin{array}[c]{cc}1&2\\\ 3&2\end{array}\right),$
and the service times are exponential with service rate $\mu=5,10,20$,
respectively. It is seen from Figure 2 that the expected sojourn time
$E[T_{d}]$ decreases very fast as the choice number $d$ increases.
Figure 2: $E[T_{d}]$ vs $d$ for the MAP
## 4 An Important Observation
In this section, we analyze a special supermarket model with Poisson arrivals
an PH service times, and obtain two different doubly exponential solutions to
the fixed point. Based on this, we give an important observation, namely that
the doubly exponential solution to the fixed point is not always unique for
more general supermarket models.
When the arrival process is Poisson, it follows from (10) to (13) that
$\pi_{0}=1,$ (46) $-\lambda+\pi_{1}T^{0}=0,$ (47)
$\lambda\alpha-\lambda\pi_{1}^{\odot d}+\pi_{1}T+\pi_{2}T^{0}\alpha=0,$ (48)
and for $k\geq 2$,
$\lambda\pi_{k-1}^{\odot d}\left(t\right)-\lambda\pi_{k}^{\odot
d}\left(t\right)+\pi_{k}\left(t\right)T+\pi_{k+1}\left(t\right)T^{0}\alpha=0.$
(49)
For the system of nonlinear equations (46) to (49), we can provide two
different doubly exponential solutions to the fixed point
$\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$.
### 4.1 The first doubly exponential solution
The first doubly exponential solution has been given in Section 3. Here, we
simply list the crucial derivations for the special supermarket model.
It follows from (24) that
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}-\lambda&\lambda&&&\\\
&-\lambda&\lambda&&\\\ &&-\lambda&\lambda&\\\
&&&\ddots&\ddots\end{array}\right)+\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)\left(\begin{array}[c]{cccc}T&&&\\\
T^{0}\alpha&T&&\\\ &T^{0}\alpha&T&\\\ &&\ddots&\ddots\end{array}\right)$
$\displaystyle=-\left(\lambda\alpha,0,0,\ldots\right),$
which leads to
$\displaystyle\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)=$
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}R&V&&&\\\ &R&V&&\\\ &&R&V&\\\
&&&\ddots&\ddots\end{array}\right)$
$\displaystyle+\left(\lambda\alpha\left(-T\right)^{-1},0,0,\ldots\right),$
where
$V=\lambda\left(-T\right)^{-1}$
and
$R=\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}.$
Thus we obtain
$\pi_{1}=\lambda\alpha\left(-T\right)^{-1}+\pi_{1}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]$ (50)
and for $k\geq 2$
$\pi_{k}=\pi_{k-1}^{\odot
d}\left[\lambda\left(-T\right)^{-1}\right]+\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right].$ (51)
To omit the term $\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]$ for $k\geq
1$, we assume that $\left\\{\pi_{k},k\geq 1\right\\}$ has the following
expression
$\pi_{k}=r\left(k\right)\alpha^{\odot\frac{1}{d}}.$
In this case, we have
$\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]=r^{d}\left(k\right)\alpha\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]=0,$
thus it follows from (50) and (51) that
$\pi_{1}=\lambda\alpha\left(-T\right)^{-1}$ (52)
and for $k\geq 2$
$\pi_{k}=\pi_{k-1}^{\odot d}\left[\lambda\left(-T\right)^{-1}\right].$ (53)
It follows from (52) that
$r\left(1\right)\alpha^{\odot\frac{1}{d}}=\lambda\alpha\left(-T\right)^{-1},$
which follows that
$r\left(1\right)=\omega\rho.$
It follows from (53) that
$r\left(k\right)\alpha^{\odot\frac{1}{d}}=r^{d}\left(k-1\right)\alpha\left[\lambda\left(-T\right)^{-1}\right]$
which follows that
$r\left(k\right)=r^{d}\left(k-1\right)\omega\rho=\left(\omega\rho\right)^{\frac{d^{k}-1}{d-1}}.$
Therefore, we can obtain
$\pi_{0}=1$
and for $k\geq 1$
$\pi_{k}=\left(\omega\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\alpha^{\odot\frac{1}{d}}.$
(54)
### 4.2 The second doubly exponential solution
The second doubly exponential solution was given in Li, Lui and Wang [19],
thus we provide some crucial computational steps.
It follows from (47) that
$\pi_{1}T^{0}=\lambda.$ (55)
To solve Equation (55), we denote by $\tau$ the stationary probability vector
of the irreducible Markov chain $T+T^{0}\alpha$. Obviously, we have
$\tau T^{0}=\mu,$ $\frac{\lambda}{\mu}\tau T^{0}=\lambda.$ (56)
Thus, we obtain
$\pi_{1}=\frac{\lambda}{\mu}\tau=\rho\cdot\tau.$
Using $\pi_{0}=1$ and $\pi_{1}=\rho\cdot\tau$, it follows from Equation (48)
that
$\lambda\alpha-\lambda\rho^{d}\cdot\tau^{\odot d}+\rho\cdot\tau
T+\pi_{2}T^{0}\alpha=0,$
which leads to
$\lambda-\lambda\rho^{d}\cdot\tau^{\odot d}e+\rho\cdot\tau Te+\pi_{2}T^{0}=0.$
Note that $\tau Te=-\mu$ and $\rho=\lambda/\mu$, we obtain
$\pi_{2}T^{0}=\lambda\rho^{d}\tau^{\odot d}e.$
Let $\psi=\tau^{\odot d}e$. Then it is easy to see that
$\psi\in\left(0,1\right)$, and
$\pi_{2}T^{0}=\lambda\psi\rho^{d}.$
Using similar analysis on Equation (55), we have
$\pi_{2}=\frac{\lambda\psi\rho^{d}}{\mu}\tau=\psi\rho^{d+1}\cdot\tau.$
Based on $\pi_{1}=\rho\cdot\tau$ and $\pi_{2}=\psi\rho^{d+1}\cdot\tau$, it
follows from Equation (49) that for $k=2$,
$\lambda\rho^{d}\cdot\tau^{\odot
d}-\lambda\psi^{d}\rho^{d^{2}+d}\cdot\tau^{\odot d}+\psi\rho^{d+1}\cdot\tau
T+\pi_{3}T^{0}\alpha=0,$
which leads to
$\lambda\psi\rho^{d}-\lambda\psi^{d+1}\rho^{d^{2}+d}+\psi\rho^{d+1}\cdot\tau
Te+\pi_{3}T^{0}=0,$
thus we obtain
$\pi_{3}T^{0}=\lambda\psi^{d+1}\rho^{d^{2}+d}.$
Using a similar analysis on Equation (55), we have
$\pi_{3}=\frac{\lambda\psi^{d+1}\rho^{d^{2}+d}}{\mu}\tau=\psi^{d+1}\rho^{d^{2}+d+1}\cdot\tau.$
Now, we assume that
$\pi_{k}=\psi^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\tau$ is
correct for the cases with $l=k$. Then for $l=k+1$ we have
$\displaystyle\lambda$
$\displaystyle\psi^{d^{k-2}+d^{k-3}+\cdots+d}\rho^{d^{k-1}+d^{k-2}+\cdots+d}\cdot\tau^{\odot
d}-\lambda\psi^{d^{k-1}+d^{k-2}+\cdots+d}\rho^{d^{k}+d^{k-1}+\cdots+d}\cdot\tau^{\odot
d}$
$\displaystyle+\psi^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\tau
T+\pi_{k+1}T^{0}\alpha=0,$
which leads to
$\displaystyle\lambda$
$\displaystyle\psi^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d}-\lambda\psi^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}$
$\displaystyle+\psi^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\tau
Te+\pi_{k+1}T^{0}=0,$
thus we obtain
$\pi_{k+1}T^{0}=\lambda\psi^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}.$
By a similar analysis to (55) and (56), we have
$\displaystyle\pi_{k+1}$
$\displaystyle=\frac{\lambda\psi^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}}{\mu}\tau$
$\displaystyle=\psi^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d+1}\cdot\tau.$
Therefore, by induction we can obtain
$\pi_{0}=1,$
and for $k\geq 1$
$\pi_{k}=\psi^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\tau.$
(57)
### 4.3 An important observation
Now, we have given two expressions (54) and (57) for the fixed point. In what
follows we provide some examples to indicate that the two expressions may be
different from each other.
Example one: When the PH service time is exponential, it is easy to see that
$\alpha=\tau=1$, which leads to that $\omega=\psi=1$. Thus the fixed point is
given by
$\pi_{k}=\rho^{\frac{d^{k}-1}{d-1}},k\geq 1.$
In this case, (54) is the same as (57).
Example two: When the service time is an $m$-order Erlang distribution with an
irreducible representation $(\alpha,T)$, where
$\alpha=\left(1,0,\ldots,0\right)$
and
$T=\left(\begin{array}[c]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ &&&&-\eta\end{array}\right),\text{ \ \
}T^{0}=\left(\begin{array}[c]{c}0\\\ 0\\\ \vdots\\\ 0\\\
\eta\end{array}\right).$
We have
$\alpha^{\odot\frac{1}{d}}=\left(1,0,\ldots,0\right)$
and
$\omega=\frac{1}{\alpha^{\odot\frac{1}{d}}e}=1.$
Thus the first doubly exponential solution is given by
$\pi_{k}^{\left(\text{F}\right)}=\rho^{\frac{d^{k}-1}{d-1}}\cdot\left(1,0,\ldots,0\right),\text{
\ }k\geq 1.$ (58)
It is clear that
$T+T^{0}\alpha=\left(\begin{array}[c]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ \eta&&&&-\eta\end{array}\right),$
which leads to the stationary probability vector of the Markov chain
$T+T^{0}\alpha$ as follows:
$\tau=\left(\frac{1}{m},\frac{1}{m},\ldots,\frac{1}{m}\right),$ $\mu=\tau
T^{0}=\frac{\eta}{m},$ $\rho=\frac{\lambda}{\mu}=\frac{m\lambda}{\eta}$
and
$\psi=m\left(\frac{1}{m}\right)^{d}=m^{1-d}.$
Thus the second doubly exponential solution is given by
$\displaystyle\pi_{k}^{\left(\text{S}\right)}$
$\displaystyle=\psi^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots,\frac{1}{m}\right)$
$\displaystyle=\rho^{\frac{d^{k}-1}{d-1}}\cdot\left(m^{-d^{k-1}},m^{-d^{k-1}},\ldots,m^{-d^{k-1}}\right),\text{
\ }k\geq 1.$ (59)
It is clear that (58) and (59) are different from each other for $k,m,d\geq
2$, and
$\frac{\pi_{k}^{\left(\text{F}\right)}e}{\pi_{k}^{\left(\text{S}\right)}e}=m^{d^{k-1}-1}.$
It is clear that
$\pi_{k}^{\left(\text{F}\right)}e\neq\pi_{k}^{\left(\text{S}\right)}e$ for
$k,m,d\geq 2$.
###### Remark 3
For the supermarket model with Poisson arrivals and PH service times, we have
obtained two different doubly exponential solutions to the fixed point. It is
interesting but difficult to be able to find another new doubly exponential
solution to the fixed point. Furthermore, we believe that it is an open
problem how to give all doubly exponential solutions to the fixed point for
more general supermarket models.
## 5 Exponential Convergence
In this section, we provide an upper bound for the current location
$S\left(t\right)$ of the supermarket model, and study exponential convergence
of the current location $S\left(t\right)$ to its fixed point $\pi$.
For the supermarket model, the initial point $S\left(0\right)$ can affect the
current location $S\left(t\right)$ for each $t>0$, since the arrival and
service processes are under a unified structure through a sample path
comparison. To explain this, it is necessary to provide some notation for
comparison of two vectors. Let $a=\left(a_{1},a_{2},a_{3},\ldots\right)$ and
$b=\left(b_{1},b_{2},b_{3},\ldots\right)$. We write $a\prec b$ if
$a_{k}<b_{k}$ for some $k\geq 1$ and $a_{l}\leq b_{l}$ for $l\neq k,l\geq 1$;
and $a\preceq b$ if $a_{k}\leq b_{k}$ for all $k\geq 1$.
Now, we can easily obtain the following useful proposition, while the proof is
clear by means of a sample path analysis, and thus is omitted here.
###### Proposition 2
If $S\left(0\right)\preceq\widetilde{S}\left(0\right)$, then
$S\left(t\right)\preceq\widetilde{S}\left(t\right)$ for $t>0$.
Based on Proposition 2, the following theorem shows that the fixed point $\pi$
is an upper bound of the current location $S\left(t\right)$ for all $t\geq 0$.
###### Theorem 3
For the supermarket model, if there exists some $k$ such that
$S_{k}\left(0\right)=0$, then the sequence
$\left\\{S_{k}\left(t\right)\right\\}$ for all $t\geq 0$ has an upper bound
sequence $\left\\{\pi_{k}\right\\}$ which decreases doubly exponentially, that
is, $S\left(t\right)\preceq\pi$ for all $t\geq 0$.
Proof: Let
$\widetilde{S}_{k}\left(0\right)=\pi_{k},\text{ \ }k\geq 0.$
Then for each $k\geq 0$,
$\widetilde{S}_{k}\left(t\right)=\widetilde{S}_{k}\left(0\right)=\pi_{k}$ for
all $t\geq 0$, since
$\widetilde{S}\left(0\right)=\left(\widetilde{S}_{0}\left(0\right),\widetilde{S}_{1}\left(0\right),\widetilde{S}_{2}\left(0\right),\ldots\right)=\pi$
is a fixed point for the supermarket model. If $S_{k}\left(0\right)=0$ for
some $k$, then $S_{k}\left(0\right)\prec\widetilde{S}_{k}\left(0\right)$.
Again, if $S_{j}\left(0\right)\preceq\widetilde{S}_{j}\left(0\right)$ for all
$j\neq k$, then $S\left(0\right)\preceq\widetilde{S}\left(0\right)$. It is
easy to see from Proposition 2 that
$S_{k}\left(t\right)\preceq\widetilde{S}_{k}\left(t\right)=\pi_{k}$ for all
$k\geq 0$ and $t\geq 0$. Thus we obtain that for all $k\geq 0$ and $t\geq 0$
$S_{k}\left(t\right)\leq\pi_{k}=\theta^{d^{k}}\left(\theta\omega\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\gamma^{\odot\frac{1}{d}}\otimes\alpha^{\odot\frac{1}{d}}.$
Since $0<\theta,\omega,\rho<1$, $\left\\{\pi_{k}\right\\}$ decreases doubly
exponentially. This completes the proof.
To show exponential convergence, we define a Lyapunov function
$\Phi\left(t\right)$ as
$\Phi\left(t\right)=\sum_{k=0}^{\infty}w_{k}\left(\pi_{k}-S_{k}\left(t\right)\right)e,$
(60)
where $\left\\{w_{k}\right\\}$ is a positive scalar sequence with $w_{k}\geq
w_{k-1}\geq w_{0}=1$ for $k\geq 2$.
The following theorem measures the distance of the current location
$S\left(t\right)$ to the fixed point $\pi$ for $t\geq 0$, and illustrates that
this distance will quickly come close to zero with exponential convergence.
Hence, it shows that from any suitable starting point, the supermarket model
can be quickly close to the fixed point, that is, there always exists a fixed
point in the supermarket model.
###### Theorem 4
For $t\geq 0$,
$\Phi\left(t\right)\leq c_{0}e^{-\delta t},$
where $c_{0}$ and $\delta$ are two positive constants. In this case, the
Lyapunov function $\Phi\left(t\right)$ is exponentially convergent.
Proof: It is seen from (60) that
$\frac{d}{dt}\Phi\left(t\right)=-\sum_{k=0}^{\infty}w_{k}\frac{d}{dt}S_{k}\left(t\right)e.$
It follows from Equations (6) to (9) that
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle-
w_{0}\left[S_{0}^{\odot d}\left(t\right)C+S_{1}\left(t\right)\left(I\otimes
T^{0}\right)\right]e$ $\displaystyle-w_{1}[S_{0}^{\odot
d}\left(t\right)\left(D\otimes\alpha\right)+S_{1}^{\odot
d}\left(t\right)\left(C\otimes I\right)$
$\displaystyle+S_{1}\left(t\right)\left(I\otimes
T\right)+S_{2}\left(t\right)\left(I\otimes T^{0}\alpha\right)]e$
$\displaystyle-\sum_{k=2}^{\infty}w_{k}[S_{k-1}^{\odot
d}\left(t\right)\left(D\otimes I\right)+S_{k}^{\odot
d}\left(t\right)\left(C\otimes I\right)$
$\displaystyle+S_{k}\left(t\right)\left(I\otimes
T\right)+S_{k+1}\left(t\right)\left(I\otimes T^{0}\alpha\right)]e.$
By means of $Ce=-De$ and $Te=-T^{0}$, we can obtain
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle-
w_{0}\left[S_{0}^{\odot
d}\left(t\right)\left(-De\right)+S_{1}\left(t\right)\left(e\otimes
T^{0}\right)\right]$ $\displaystyle-w_{1}[S_{0}^{\odot
d}\left(t\right)\left(De\right)+S_{1}^{\odot
d}\left(t\right)\left(\left(-De\right)\otimes e\right)$
$\displaystyle+S_{1}\left(t\right)\left(e\otimes\left(-T^{0}\right)\right)+S_{2}\left(t\right)\left(e\otimes
T^{0}\right)]$ $\displaystyle-\sum_{k=2}^{\infty}w_{k}[S_{k-1}^{\odot
d}\left(t\right)\left(\left(De\right)\otimes e\right)+S_{k}^{\odot
d}\left(t\right)\left(\left(-De\right)\otimes e\right)$
$\displaystyle+S_{k}\left(t\right)\left(e\otimes\left(-T^{0}\right)\right)+S_{k+1}\left(t\right)\left(e\otimes
T^{0}\right)].$ (61)
Let
$S_{0}^{\odot
d}\left(t\right)\left(De\right)=c_{0}\left(t\right)\cdot\left[\pi_{0}-S_{0}\left(t\right)\right]e,$
for $k\geq 1$
$S_{k}^{\odot d}\left(t\right)\left(\left(De\right)\otimes
e\right)=c_{k}\left(t\right)\cdot\left[\pi_{k}-S_{k}\left(t\right)\right]e$
and
$S_{k}\left(t\right)\left(e\otimes
T^{0}\right)=d_{k}\left(t\right)\cdot\left[\pi_{k}-S_{k}\left(t\right)\right]e.$
Then it follows from (61) that
$\displaystyle\frac{d}{dt}\Phi\left(t\right)$
$\displaystyle=-\left(w_{1}-w_{0}\right)c_{0}\left(t\right)\cdot\left[\pi_{0}-S_{0}\left(t\right)\right]e$
$\displaystyle-\sum_{k=1}^{\infty}\left[w_{k+1}c_{k}\left(t\right)-w_{k}\left(c_{k}\left(t\right)+d_{k}\left(t\right)\right)+w_{k-1}d_{k}\left(t\right)\right]\cdot\left[\pi_{k}-S_{k}\left(t\right)\right]e.$
Let
$w_{0}=1,$ $\left(w_{1}-w_{0}\right)c_{0}\left(t\right)\geq\delta w_{0}$
and
$w_{k+1}c_{k}\left(t\right)-w_{k}\left(c_{k}\left(t\right)+d_{k}\left(t\right)\right)+w_{k-1}d_{k}\left(t\right)\geq\delta
w_{k}.$
Then
$w_{1}\geq 1+\frac{\delta}{c_{0}\left(t\right)},$ $w_{2}\geq
w_{1}+\frac{\delta
w_{1}}{c_{1}\left(t\right)}+\frac{d_{1}\left(t\right)}{c_{1}\left(t\right)}\left(w_{1}-1\right)$
and for $k\geq 2$
$w_{k+1}\geq w_{k}+\frac{\delta
w_{k}}{c_{k}\left(t\right)}+\frac{d_{k}\left(t\right)}{c_{k}\left(t\right)}\left(w_{k}-w_{k-1}\right).$
Thus we have
$\frac{d}{dt}\Phi\left(t\right)\leq-\delta\sum_{k=0}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e.$
It follows from (60) that
$\frac{d}{dt}\Phi\left(t\right)\leq-\delta\Phi\left(t\right),$
which can leads to
$\Phi\left(t\right)\leq c_{0}e^{-\delta t}.$
This completes the proof.
###### Remark 4
We have provided an algorithm for computing the positive scalar sequence
$\left\\{w_{k}\right\\}$ with $1=w_{0}\leq w_{k-1}<w_{k}$ for $k\geq 2$ as
follows:
Step one:
$w_{0}=1.$
Step two:
$w_{1}=1+\frac{\delta}{c_{0}\left(t\right)}$
and
$w_{2}=w_{1}+\frac{\delta
w_{1}}{c_{1}\left(t\right)}+\frac{d_{1}\left(t\right)}{c_{1}\left(t\right)}\left(w_{1}-1\right)$
Step three: for $k\geq 2$
$w_{k+1}=w_{k}+\frac{\delta
w_{k}}{c_{k}\left(t\right)}+\frac{d_{k}\left(t\right)}{c_{k}\left(t\right)}\left(w_{k}-w_{k-1}\right).$
This illustrates that $w_{k}$ is a function of time $t$ for $k\geq 1$. Note
that $\lambda,\delta,c_{k}\left(t\right),d_{l}\left(t\right)>0$, it is clear
that for $k\geq 2$
$1=w_{0}\leq w_{k-1}<w_{k}.$
## 6 A Lipschitz Condition
In this section, we apply the Kurtz Theorem to study density dependent jump
Markov process given in the supermarket model with MAPs and PH service times,
which leads to the Lipschitz condition under which the fraction measure of the
supermarket model weakly converges the system of differential vector
equations.
The supermarket model can be analyzed by a density dependent jump Markov
process, where the density dependent jump Markov process is a Markov process
with a single parameter $n$ which corresponds to the population size. Kurtz’s
work provides a basis for density dependent jump Markov processes in order to
relate infinite-size systems of differential equations to corresponding
finite-size systems of differential equations. Readers may refer to Kurtz [16]
for more details.
In the supermarket model, when the population size is $n$, we write
$\text{Level }0:\text{ \ }E_{0}^{n}=\left\\{\left(0,i\right):1\leq i\leq
m_{A}\right\\}$
and for $k\geq 1$
$\text{Level }k:\text{ \ }E_{k}^{n}=\left\\{\left(k,i,j\right):1\leq i\leq
m_{A},1\leq j\leq m_{B}\right\\},$
$E_{n}=\bigcup\limits_{k=0}^{n}\left\\{\text{Level
}k\right\\}=\bigcup\limits_{k=0}^{n}\left\\{E_{k}^{n}\right\\}.$
In the state space $E_{n}$, the density dependent jump Markov process for the
supermarket model with MAPs and PH service times contains four classes of
state transitions as follows:
Class one $E_{k}^{n}$ $\underrightarrow{a}$ $E_{k}^{n}$:
$\left(0,i\right)\rightarrow\left(0,i^{\ast}\right)$ or
$\left(k,i,j\right)\rightarrow\left(k,i^{\ast},j\right)$, where $1\leq
i,i^{\ast}\leq m_{A}$;
Class two $E_{k}^{n}$ $\underrightarrow{s}$ $E_{k}^{n}$:
$\left(k,i,j\right)\rightarrow\left(k,i,j^{\ast}\right)$, where $1\leq
j,j^{\ast}\leq m_{B}$;
Class three $E_{k}^{n}$ $\underrightarrow{a}$ $E_{k+1}^{n}$:
$\left(0,i\right)\rightarrow\left(1,i,j\right)$ or
$\left(k,i,j\right)\rightarrow\left(k+1,i,j\right)$; and
Class four $E_{k}^{n}$ $\underrightarrow{s}$ $E_{k-1}^{n}$:
$\left(1,i,j\right)\rightarrow\left(0,i\right)$ or
$\left(k,i,j\right)\rightarrow\left(k-1,i,j\right)$.
Note that the transitions $\underrightarrow{a}$ and $\underrightarrow{s}$
express arrival transition and service transition, respectively.
We write
$s_{0}^{\left(i\right)}\left(n\right)=\left(\frac{0}{n},i\right),$
$S_{0}\left(n\right)=\left(s_{0}^{\left(1\right)}\left(n\right),s_{0}^{\left(2\right)}\left(n\right),\ldots,s_{0}^{\left(m_{A}\right)}\left(n\right)\right);$
and for $k\geq 1$
$s_{k}^{\left(i,j\right)}\left(n\right)=\left(\frac{k}{n},i,j\right)$
and
$S_{k}\left(n\right)=\left(s_{k}^{\left(1,1\right)}\left(n\right),s_{k}^{\left(1,2\right)}\left(n\right),\ldots,s_{k}^{\left(1,m_{B}\right)}\left(n\right);\ldots;s_{k}^{\left(m_{A},1\right)}\left(n\right),s_{k}^{\left(m_{A},2\right)}\left(n\right),\ldots,s_{k}^{\left(m_{A},m_{B}\right)}\left(n\right)\right).$
Note that the states of the density dependent jump Markov process can be
normalized and interpreted as measuring population densities
$S\left(n\right)=\left\\{S_{0}\left(n\right),S_{1}\left(n\right),S_{2}\left(n\right),\ldots\right\\},$
the transition rates of the Markov process depend only on these densities.
Let $\left\\{\widehat{X}_{n}\left(t\right):t\geq 0\right\\}$ be a density
dependent jump Markov process on the state space $E_{n}$ whose transition
rates corresponding to the above four cases are given by
$q_{\left(0,i\right)\rightarrow\left(0,i^{\ast}\right)}^{\left(n\right)}=n\beta_{i\rightarrow
i^{\ast}}\left(\frac{0}{n},i\right)=n\beta_{i\rightarrow
i^{\ast}}\left(s_{0}^{\left(i\right)}\left(n\right)\right),$
$q_{\left(k,i,j\right)\rightarrow\left(k,i^{\ast},j\right)}^{\left(n\right)}=n\beta_{i\rightarrow
i^{\ast}}\left(\frac{k}{n},i,j\right)=n\beta_{i\rightarrow
i^{\ast}}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right);$
$q_{\left(k,i,j\right)\rightarrow\left(k,i,j^{\ast}\right)}^{\left(n\right)}=n\beta_{j\rightarrow
j^{\ast}}\left(\frac{k}{n},i,j\right)=n\beta_{j\rightarrow
j^{\ast}}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right);$
$q_{\left(0,i\right)\rightarrow\left(1,i,j\right)}^{\left(n\right)}=n\beta_{0\rightarrow
1}\left(\frac{0}{n},i;j\right)=n\beta_{0\rightarrow
1}\left(s_{0}^{\left(i\right)}\left(n\right),j\right),$
$q_{\left(k,i,j\right)\rightarrow\left(k+1,i,j\right)}^{\left(n\right)}=n\beta_{k\rightarrow
k+1}\left(\frac{k}{n},i,j\right)=n\beta_{k\rightarrow
k+1}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right);$
$q_{\left(1,i,j\right)\rightarrow\left(0,i\right)}^{\left(n\right)}=n\beta_{1\rightarrow
0}\left(\frac{1}{n},i,j\right)=n\beta_{1\rightarrow
0}\left(s_{1}^{\left(i,j\right)}\left(n\right)\right),$
$q_{\left(k,i,j\right)\rightarrow\left(k-1,i,j\right)}^{\left(n\right)}=n\beta_{k\rightarrow
k-1}\left(\frac{k}{n},i,j\right)=n\beta_{k\rightarrow
k-1}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right).$
Let
$Q_{0}^{\left(n\right)}=\left(\beta_{i\rightarrow
i^{\ast}}\left(s_{0}^{\left(i\right)}\left(n\right)\right)\right)_{1\leq
i,i^{\ast}\leq m_{A}},$ $Q_{a,k}^{\left(n\right)}=\left(\beta_{i\rightarrow
i^{\ast}}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right)\right)_{1\leq
i,i^{\ast}\leq m_{A},1\leq j\leq m_{B}};$
$Q_{s,k}^{\left(n\right)}=\left(\beta_{j\rightarrow
j^{\ast}}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right)\right)_{1\leq
i\leq m_{A},1\leq j,j^{\ast}\leq m_{B}};$
$Q_{0}^{\left(n\right)}\left(A\right)=\left(\beta_{0\rightarrow
1}\left(s_{0}^{\left(i\right)}\left(n\right),j\right)\right)_{1\leq i\leq
m_{A},1\leq j\leq m_{B}},$
$Q_{k}^{\left(n\right)}\left(A\right)=\left(\beta_{k\rightarrow
k+1}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right)\right)_{1\leq i\leq
m_{A},1\leq j\leq m_{B}};$
$Q_{1}^{\left(n\right)}\left(S\right)=\left(\beta_{1\rightarrow
0}\left(s_{1}^{\left(i,j\right)}\left(n\right)\right)\right)_{1\leq i\leq
m_{A},1\leq j\leq m_{B}},$
$Q_{k}^{\left(n\right)}\left(S\right)=\left(\beta_{k\rightarrow
k-1}\left(s_{k}^{\left(i,j\right)}\left(n\right)\right)\right)_{1\leq i\leq
m_{A},1\leq j\leq m_{B}}.$
In the supermarket model, $\widehat{X}_{n}\left(t\right)$ is an unscaled
process which records the number of servers with at least $k$ customers for
$0\leq k\leq n$. We write
$Q^{\left(n\right)}=\left(\begin{array}[c]{ccccc}Q_{0}^{\left(n\right)}&Q_{0}^{\left(n\right)}\left(A\right)&&&\\\
Q_{1}^{\left(n\right)}\left(S\right)&Q_{a,1}^{\left(n\right)}+Q_{s,1}^{\left(n\right)}&Q_{1}^{\left(n\right)}\left(A\right)&&\\\
&Q_{2}^{\left(n\right)}\left(S\right)&Q_{a,2}^{\left(n\right)}+Q_{s,2}^{\left(n\right)}&Q_{2}^{\left(n\right)}\left(A\right)&\\\
&&\ddots&\ddots&\ddots\end{array}\right),$
where
$Q_{0}^{\left(n\right)}=\gamma^{T}\left[S_{0}\left(n\right)\right]^{\odot
d}C,$
$Q_{a,k}^{\left(n\right)}=\left(\gamma\otimes\tau\right)^{T}\left[S_{k}\left(n\right)\right]^{\odot
d}\left(C\otimes I\right);$
$Q_{s,k}^{\left(n\right)}=\left(\gamma\otimes\tau\right)^{T}S_{k}\left(n\right)\left(I\otimes
T\right);$
$Q_{0}^{\left(n\right)}\left(A\right)=\gamma^{T}\left[S_{0}\left(n\right)\right]^{\odot
d}\left(D\otimes\alpha\right),$
$Q_{k}^{\left(n\right)}\left(A\right)=\left(\gamma\otimes\tau\right)^{T}\left[S_{k}\left(n\right)\right]^{\odot
d}\left(D\otimes I\right);$
$Q_{1}^{\left(n\right)}\left(S\right)=\left(\gamma\otimes\tau\right)^{T}S_{1}\left(n\right)\left(I\otimes
T^{0}\right),$
$Q_{k}^{\left(n\right)}\left(S\right)=\left(\gamma\otimes\tau\right)^{T}S_{k}\left(n\right)\left(I\otimes
T^{0}\alpha\right).$
Using Chapter 7 in Kurtz [16] or Subsection 3.4.1 in Mitzenmacher [27], the
Markov process $\left\\{\widehat{X}_{n}\left(t\right):t\geq 0\right\\}$ with
transition rate matrix $\mathcal{Q}^{\left(n\right)}=$ $nQ^{\left(n\right)}$
is given by
$\widehat{X}_{n}\left(t\right)=\widehat{X}_{n}\left(0\right)+\sum_{b\in\mathbb{E}}l_{b}Y_{b}\left(n\int_{0}^{t}\beta_{l}\left(\frac{\widehat{X}_{n}\left(u\right)}{n}\right)\text{d}u\right),$
(62)
where $Y_{b}\left(x\right)$ for $b\in\mathbb{E}$ are independent standard
Poisson processes, $l_{b}$ is a positive integer with $l_{b}\leq\Re<+\infty$,
and
$\displaystyle\mathbb{E}=$
$\displaystyle\\{\left(0,i\right)\rightarrow\left(0,i^{\ast}\right),\left(k,i,j\right)\rightarrow\left(k,i^{\ast},j\right),\left(k,i,j\right)\rightarrow\left(k,i,j^{\ast}\right);\left(0,i\right)\rightarrow\left(1,i,j\right),$
$\displaystyle\left(k,i,j\right)\rightarrow\left(k+1,i,j\right)\text{ for
}k\geq
1;\left(1,i,j\right)\rightarrow\left(0,i\right),\left(l,i,j\right)\rightarrow\left(l-1,i,j\right)\text{
for }l\geq 2\\}$
Clearly, the jump Markov process in Equation (62) at time $t$ is determined by
the starting point and the transition rates which are integrated over its
history.
Let
$F\left(y\right)=\sum_{b\in\mathbb{E}\left(y\right)}l_{b}\beta_{b}\left(y\right),$
(63)
where
$\mathbb{E}\left(y\right)=\left\\{b\in\mathbb{E}:\text{ the transition
}b\text{ begins from state }y\right\\}.$
Taking $X_{n}\left(t\right)=n^{-1}\widehat{X}_{n}\left(t\right)$ which is an
appropriate scaled process, we have
$X_{n}\left(t\right)=X_{n}\left(0\right)+\sum_{b\in\mathbb{E}}l_{b}n^{-1}\widehat{Y}_{b}\left(n\int_{0}^{t}\beta_{b}\left(X_{n}\left(u\right)\right)\text{d}u\right)+\int_{0}^{t}F\left(X_{n}\left(u\right)\right)\text{d}u,$
(64)
where $\widehat{Y}_{b}\left(y\right)=Y_{b}\left(y\right)-y$ is a Poisson
process centered at its expectation.
Let $X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$ and
$x_{0}=\lim_{n\rightarrow\infty}X_{n}\left(0\right)$, we obtain
$X\left(t\right)=x_{0}+\int_{0}^{t}F\left(X\left(u\right)\right)\text{d}u,\text{
\ }t\geq 0,$ (65)
due to the fact that
$\lim_{n\rightarrow\infty}\frac{1}{n}\widehat{Y}_{b}\left(n\int_{0}^{t}\beta_{b}\left(X_{n}\left(u\right)\right)\text{d}u\right)=0$
by means of the law of large numbers. In the supermarket model, the
deterministic and continuous process $\left\\{X\left(t\right),t\geq
0\right\\}$ is described by the infinite-size system of differential vector
equations (6) to (9), or simply,
$\frac{d}{dt}X\left(t\right)=F\left(X\left(t\right)\right)$ (66)
with the initial condition
$X\left(0\right)=x_{0}.$ (67)
Now, we consider the uniqueness of the limiting deterministic process
$\left\\{X\left(t\right),t\geq 0\right\\}$ with (66) and (67), or the
uniqueness of the solution to the infinite-size system of differential vector
equations (6) to (9). To that end, a sufficient condition is Lipschitz, that
is, for some constant $M>0,$
$|F\left(y\right)-F\left(z\right)|\leq M||y-z||.$
In general, the Lipschitz condition is standard and sufficient for the
uniqueness of the solution to the finite-size system of differential vector
equations; while for the countable infinite-size case, readers may refer to
Theorem 3.2 in Deimling [8] and Subsection 3.4.1 in Mitzenmacher [27] for some
useful generalization.
To check the Lipschitz condition, by means of the law of large numbers we have
$\pi_{k}=\lim_{n\rightarrow\infty}S_{k}\left(n\right),\text{ \ \ }k\geq 0,$
which leads to
$Q=\left(\begin{array}[c]{ccccc}Q_{0}&Q_{0}\left(A\right)&&&\\\
Q_{1}\left(S\right)&Q_{a,1}+Q_{s,1}&Q_{1}\left(A\right)&&\\\
&Q_{2}\left(S\right)&Q_{a,2}+Q_{s,2}&Q_{2}\left(A\right)&\\\
&&\ddots&\ddots&\ddots\end{array}\right),$ (68)
where
$Q_{0}=\gamma^{T}\pi_{0}^{\odot d}C,$
$Q_{a,k}=\left(\gamma\otimes\tau\right)^{T}\pi_{k}^{\odot d}\left(C\otimes
I\right);$ $Q_{s,k}=\left(\gamma\otimes\tau\right)^{T}\pi_{k}\left(I\otimes
T\right);$ $Q_{0}\left(A\right)=\gamma^{T}\pi_{0}^{\odot
d}\left(D\otimes\alpha\right),$
$Q_{k}\left(A\right)=\left(\gamma\otimes\tau\right)^{T}\pi_{k}^{\odot
d}\left(D\otimes I\right);$
$Q_{1}\left(S\right)=\left(\gamma\otimes\tau\right)^{T}\pi_{1}\left(I\otimes
T^{0}\right),$
$Q_{k}\left(S\right)=\left(\gamma\otimes\tau\right)^{T}\pi_{k}\left(I\otimes
T^{0}\alpha\right).$
Let
$\zeta_{0}=\frac{\pi_{0}^{\odot d}De}{\pi_{0}e}$
and for $k\geq 1$
$\zeta_{k}=\frac{\pi_{k}^{\odot d}\left(De\otimes e\right)}{\pi_{k}e},$
$\eta_{k}=\frac{\pi_{k}\left(I\otimes T^{0}\alpha\right)}{\pi_{k}e}.$
Then $\zeta_{k},\eta_{k}>0$ for $k\geq 1$.
The following theorem shows that the supermarket model with MAPs and PH
service times satisfies the Lipschitz condition for analyzing the uniqueness
of the solution to the infinite-size system of differential vector equations
(6) to (9).
###### Theorem 5
The supermarket model with MAPs and PH service times satisfies the Lipschitz
condition.
Proof Let the state space of the Markov process $\left\\{X\left(t\right),t\geq
0\right\\}$ be
$\Omega=\left\\{\pi_{k}:k\geq 0\right\\}.$
For two arbitrary entries $y,z\in\Omega$, we have
$|F\left(y\right)-F\left(z\right)|\leq\sum_{a\in\mathbb{E}\left(y\right)\cap\mathbb{E}\left(z\right)}l_{a}|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq\Re\sum_{a\in\mathbb{E}\left(y\right)\cap\mathbb{E}\left(z\right)}|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|.$
Note that $a$ expresses either an arrival transition or a service transition
in the above four cases. When $a$ expresses an arrival transition, we can
analyze the function $\beta_{a}\left(y\right)$ from the two cases of arrival
transitions; while when $b$ expresses a service transition, the function
$\beta_{b}\left(y\right)$ can similarly be dealt with from the two cases of
service transitions.
When $a$ expresses an arrival transition, we analyze the function
$\beta_{a}\left(y\right)$ based on
$a\in\mathbb{E}\left(y\right)\cap\mathbb{E}\left(z\right)$ from the following
two cases.
Case one: $y=\pi_{0},z=\pi_{1}$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle=|\pi_{0}^{\odot d}Ce-\pi_{0}^{\odot
d}\left(D\otimes\alpha\right)e-\pi_{1}^{\odot d}\left(C\otimes I\right)e|$
$\displaystyle=|-\pi_{0}^{\odot d}De-\pi_{0}^{\odot d}De+\pi_{1}^{\odot
d}\left(De\otimes e\right)|$
$\displaystyle=|2\zeta_{0}\pi_{0}e-\zeta_{1}\pi_{1}e|$
$\displaystyle=|2\zeta_{0}-\zeta_{1}\pi_{1}e|.$
Taking
$M_{a}\left(0,1\right)\geq\frac{|2\zeta_{0}-\zeta_{1}\pi_{1}e|}{1-\pi_{1}e},$
it is clear that
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq
M_{a}\left(0,1\right)\left(1-\pi_{1}e\right)=M_{a}\left(0,1\right)\left(\pi_{0}e-\pi_{1}e\right).$
Note that $\pi_{0}$ and $\pi_{1}$ are two row vectors of sizes $m_{A}$ and
$m_{A}m_{B}$, respectively, in this case we write
$||y-z||=\pi_{0}e-\pi_{1}e.$
Thus we have
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq
M_{a}\left(0,1\right)||y-z||.$
Case two: $y=\pi_{k-1},z=\pi_{k}$ for $k\geq 2$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|=$
$\displaystyle|\pi_{k-2}^{\odot d}\left(D\otimes I\right)e+\pi_{k-1}^{\odot
d}\left(C\otimes I\right)e$ $\displaystyle-\pi_{k-1}^{\odot
d}\left(t\right)\left(D\otimes I\right)e-\pi_{k}^{\odot d}\left(C\otimes
I\right)e|$ $\displaystyle=$ $\displaystyle|\pi_{k-2}^{\odot
d}\left(t\right)\left(De\otimes e\right)-\pi_{k-1}^{\odot d}\left(De\otimes
e\right)$ $\displaystyle-\pi_{k-1}^{\odot d}\left(De\otimes
e\right)+\pi_{k}^{\odot d}\left(De\otimes e\right)|$ $\displaystyle=$
$\displaystyle|\zeta_{k-2}\pi_{k-2}e-2\zeta_{k-1}\pi_{k-1}e+\zeta_{k}\pi_{k}e|.$
Let
$M_{a}\left(k-1,k\right)\geq\frac{|\zeta_{k-2}\pi_{k-2}e-2\zeta_{k-1}\pi_{k-1}e+\zeta_{k}\pi_{k}e|}{||\pi_{k-1}-\pi_{k}||}.$
Then
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq
M_{a}\left(k-1,k\right)||\pi_{k-1}-\pi_{k}||.$
Based on the above two cases, taking
$M_{a}=\sup\left\\{M_{a}\left(0,1\right),M_{a}\left(k-1,k\right):k\geq
2\right\\},$
we obtain that for two arbitrary entries $y,z\in\Omega,$
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq M_{a}||y-z||.$ (69)
Similarly, when $b$ expresses a service transition, we can choose a positive
number $M_{b}$ such that for two arbitrary entries $y,z\in\Omega,$
$|\beta_{b}\left(y\right)-\beta_{b}\left(z\right)|\leq M_{b}||y-z||.$ (70)
Let $M=\Re\max\left\\{M_{a},M_{b}\right\\}$. Then it follows from (69) and
(70) that for two arbitrary entries $y,z\in\Omega,$
$|F\left(y\right)-F\left(z\right)|\leq M||y-z||.$
This completes the proof.
Based on Theorem 5, the following theorem easily follows from Theorem 3.13 in
Mitzenmacher [27].
###### Theorem 6
In the supermarket model with MAPs and PH service times,
$\left\\{X_{n}\left(t\right)\right\\}$ and $\left\\{X\left(t\right)\right\\}$
are respectively given by (64) and (65), we have
$\lim_{n\rightarrow\infty}\sup_{u\leq
t}|X_{n}\left(u\right)-X\left(u\right)|=0,\text{ \ }a.s.$
Proof It has been shown that in the supermarket model with MAPs and PH service
times, the function $F\left(y\right)$ for $y\in\Omega$ satisfies the Lipschitz
condition. At the same time, it is easy to take a subset
$\Omega^{\ast}\subset\Omega$ such that
$\left\\{X\left(u\right):u\leq t\right\\}\subset\Omega^{\ast}$
and
$\sup_{\begin{subarray}{c}y\in\Omega^{\ast}\\\
a\in\mathbb{E}\left(y\right)\end{subarray}}\beta_{a}\left(y\right)+\sup_{\begin{subarray}{c}y\in\Omega^{\ast}\\\
b\in\mathbb{E}\left(y\right)\end{subarray}}\beta_{b}\left(y\right)<+\infty,$
where $a$ and $b$ express an arrival transition and a service transition,
respectively. Thus, this proof can easily be completed by means of Theorem
3.13 in Mitzenmacher [27]. This completes the proof.
Using Theorem 3.11 in Mitzenmacher [27] and Theorem 6, the following theorem
for the expected sojourn time that an arriving tagged customer spends in an
initially empty supermarket model with MAPs and PH service times over the time
interval $\left[0,t\right]$.
###### Theorem 7
In the supermarket model with MAPs and PH service times, the expected sojourn
time that an arriving tagged customer spends in an initially empty system over
the time interval $\left[0,t\right]$ is bounded above by
$\theta^{d^{2}}\left(\theta\omega\rho\right)^{d}\left(\tau-\alpha\right)\left(-T\right)^{-1}e+\frac{1}{\mu}\sum_{k=0}^{\infty}\theta^{d^{k+1}}\left(\theta\omega\rho\right)^{\frac{d^{k+1}-d}{d-1}}+o\left(1\right),$
where $o\left(1\right)$ is understood as $n\rightarrow\infty$.
## 7 Concluding remarks
In this paper, we provide a novel matrix-analytic approach for studying doubly
exponential solutions of the supermarket models with MAPs and PH service
times. We describe the supermarket model as a system of differential vector
equations, and obtain a closed-form solution with a doubly exponential
structure to the fixed point of the system of differential vector equations.
Based on this, we shows that the fixed point can be decomposed into the
product of two factors inflecting arrival information and service information,
and indicate that the doubly exponential solution to the fixed point is not
always unique for more general supermarket models. Furthermore, we analyze the
exponential convergence of the current location of the supermarket model to
its fixed point, and apply the Kurtz Theorem to study density dependent jump
Markov process given in the supermarket model with MAPs and PH service times,
which leads to the Lipschitz condition under which the fraction measure of the
supermarket model weakly converges the system of differential vector
equations. Therefore, we gain a new and crucial understanding of how the
workload probing can help in load balancing jobs with either non-Poisson
arrivals or non-exponential service times.
Our approach given in this paper is useful in the study of load balancing in
data centers and multi-core servers systems. We expect that this approach will
be applicable to the study of other randomized load balancing schemes, for
example, analyzing a renewal arrival process or a general service time
distribution, discussing retrial service discipline and processor-sharing
discipline, and studying supermarket networks.
## Acknowledgements
The author are very grateful to Professors Åke Blomqvist and Juan Eloy Ruiz-
Castro whose comments have greatly improved the presentation of this paper.
John C.S. Lui was supported by the RGC grant. The work of Q.L. Li was
supported by the National Science Foundation of China under grant No. 10871114
and the National Grand Fundamental Research 973 Program of China under grant
No. 2006CB805901.
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|
arxiv-papers
| 2010-09-25T04:23:50 |
2024-09-04T02:49:13.079125
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan-Lin Li and John C.S. Lui",
"submitter": "Quan-Lin Li",
"url": "https://arxiv.org/abs/1009.4970"
}
|
1009.5021
|
# The Stationary Behaviour of Fluid Limits of Reversible Processes is
Concentrated on Stationary Points
Jean-Yves Le Boudec111EPFL IC-LCA2 - Lausanne, Switzerland,
###### Abstract
Assume that a stochastic process can be approximated, when some scale
parameter gets large, by a fluid limit (also called “mean field limit”, or
“hydrodynamic limit”). A common practice, often called the “fixed point
approximation” consists in approximating the stationary behaviour of the
stochastic process by the stationary points of the fluid limit. It is known
that this may be incorrect in general, as the stationary behaviour of the
fluid limit may not be described by its stationary points. We show however
that, if the stochastic process is reversible, the fixed point approximation
is indeed valid. More precisely, we assume that the stochastic process
converges to the fluid limit in distribution (hence in probability) at every
fixed point in time. This assumption is very weak and holds for a large family
of processes, among which many mean field and other interaction models. We
show that the reversibility of the stochastic process implies that any limit
point of its stationary distribution is concentrated on stationary points of
the fluid limit. If the fluid limit has a unique stationary point, it is an
approximation of the stationary distribution of the stochastic process.
## 1 Introduction
This paper is motivated by the use of fluid limits in models of interacting
objects or particles, in contexts such as communication and computer system
modelling prout07 , biology borghans1998tcellvaccination or game theory
benaim03 . Typically, one has a stochastic process $Y^{N}$, indexed by a size
parameter $N$; under fairly general assumptions, one can show that the
stochastic process $Y^{N}$ converges to a deterministic fluid limit $\varphi$
kurtz-81 . We are interested in the stationary distribution of $Y^{N}$,
assumed to exist and be unique, but which may be too complicated to be
computed explicitly. The “fixed point assumption” is then sometimes invoked
altman07 ; bianchi ; LCA-CONF-2006-015 ; kelly1991loss : it consists in
approximating the stationary distribution of $Y^{N}$ by a stationary point of
the deterministic fluid limit $\varphi$. In the frequent case where the fluid
limit $\varphi$ is described by an Ordinary Differential Equation (ODE), say
of the form $\dot{y}=F(y)$, the stationary points are obtained by solving
$F(y)=0$. If $Y^{N}$ is an empirical measure, convergence to a deterministic
limit implies propagation of chaos, i.e. the states of different objects are
asymptotically independent, and the distribution of any particular object at
any time is obtained from the fluid limit. Under the fixed point assumption,
the stationary distribution of one object is approximated by a stationary
point of the fluid limit.
A critique of the fixed point approximation method is formulated in
benaim2008class , which observes that one may only say, in general, that the
stationary distribution of $Y^{N}$ converges to a stationary distribution of
the fluid limit. For a deterministic fluid limit, a stationary distribution is
supported by the Birkhoff center of the fluid limit, which may be larger than
the set of stationary points. An example is given in benaim2008class where
the fluid limit has a unique stationary point, but the stationary distribution
of $Y^{N}$ does not converge to the Dirac mass at this stationary point; in
contrast, it converges to a distribution supported by a limit cycle of the
ODE. If the fluid limit has a unique limit point, say $y^{*}$, to which all
trajectories converge, then this unique limit point is also the unique
stationary point and the stationary distribution of $Y^{N}$ does converge to
the Dirac mass at $y^{*}$ (i.e. the fixed point approximation is then valid).
However, as illustrated in benaim2008class , this assumption may be difficult
to verify, as it often does not hold, and when it does, it may be difficult to
establish. For example, in EPFL-CONF-149779 it is shown that the fixed point
assumption does not hold for some parameter settings of a wireless system
analyzed in bianchi , due to limit cycles in the fluid limit.
In this paper we show that there is a class of systems for which such
complications may not arise, namely the class of reversible stochastic
processes. Reversibility is classically defined as a property of time
reversibility in stationary regime kelly1979reversibility . There is a large
class of processes that are known to be reversible, for example product-form
queuing networks with reversible routing, or stochastic processes in
kelly1991loss , which describes the occupancy of inter-city telecommunication
links; in Section 5 we give an example motivated by crowd dynamics. In such
cases, we show that the fluid limit must have stationary points, and any limit
point of the stationary distribution of $Y^{N}$ must be supported by the set
of stationary points. Thus, for reversible processes that have a fluid limit,
the fixed point approximation is justified.
## 2 Assumptions and Notation
### 2.1 A Collection of Reversible Random Processes
Let $E$ be a Polish space and let $d$ be a measure that metrizes $E$. Let
$\mathcal{P}(E)$ be the set of probability measures on $E$, endowed with the
topology of weak convergence. Let $\mathcal{C}_{b}(E)$ be the set of bounded
continuous functions from $E$ to $\mathbb{R}$, and similarly
$\mathcal{C}_{b}(E\times E)$ is the set of bounded continuous functions from
$E\times E$ to $\mathbb{R}$.
We are given a collection of probability spaces
$(\Omega^{N},\mathcal{F}^{N},\mathbb{P}^{N})$ indexed by $N=1,2,3,...$ and for
every $N$ we have a process $Y^{N}$ defined on
$(\Omega^{N},\mathcal{F}^{N},\mathbb{P}^{N})$. Time is continuous. Let
$D_{E}[0,\infty)$ be the set of cádlág functions $[0,\infty)\to E$; $Y^{N}$ is
then a stochastic process with sample paths in $D_{E}[0,\infty)$.
We denote by $Y^{N}(t)$ the random value of $Y^{N}$ at time $t\geq 0$. Let
$E^{N}\subset E$ be the support of $Y^{N}(0)$, so that
$\mathbb{P}^{N}(Y^{N}(0)\in E^{N})=1$.
We assume that, for every $N$, the process $Y^{N}$ is Feller, in the sense
that for every $t\geq 0$ and $h\in\mathcal{C}_{b}(E)$,
$\mathbb{E}^{N}\left.\left[h(Y^{N}(t))\right|Y^{N}(0)=y_{0}\right]$ is a
continuous function of $y_{0}\in E$. Examples of such processes are continuous
time Markov chains as in kurtz1970solutions , or linear interpolations of
discrete time Markov chains as in benaim-07 , or the projections of a Markov
process as in graham1993propagation .
###### Definition 1.
A probability $\Pi^{N}\in\mathcal{P}(E)$ is _invariant_ for $Y^{N}$ if
$\Pi^{N}(E^{N})=1$ and for every $h\in\mathcal{C}_{b}(E)$ and every $t\geq 0$:
$\int_{E}\mathbb{E}^{N}\left.\left[h\left(Y^{N}(t)\right)\right|Y^{N}(0)=y\right]\Pi^{N}(dy)=\int_{E}h(y)\Pi^{N}(dy)$
We are interested in reversible processes, i.e. processes that keep the same
stationary law under time reversal. A weak form of such a property is defined
as follows
###### Definition 2.
Assume $\Pi^{N}$ is a probability on $E$ such that $\Pi^{N}(E^{N})=1$, for
some $N$. We say that $Y^{N}$ is _reversible_ under $\Pi^{N}$ if for every
time $t\geq 0$ and any $h\in\mathcal{C}_{b}(E\times E)$:
$\int_{E}\mathbb{E}^{N}\left.\left[h\left(y,Y^{N}(t)\right)\right|Y^{N}(0)=y\right]\Pi^{N}(dy)=\int_{E}\mathbb{E}^{N}\left.\left[h\left(Y^{N}(t),y\right)\right|Y^{N}(0)=y\right]\Pi^{N}(dy)$
Note that, necessarily, $\Pi^{N}$ is an invariant probability for $Y^{N}$. If
$Y^{N}$ is an ergodic Markov process with enumerable state space, then
Definition 2 coincides with the classical definition of reversibility by Kelly
in kelly1979reversibility . Similarly, if $Y^{N}$ is a projection of a
reversible Markov process $X^{N}$, as in crametz1991limit , then $Y^{N}$ is
reversible under the projection of the stationary probability of $X^{N}$; note
that in such a case, $Y^{N}$ is not Markov.
### 2.2 A Limiting, Continuous Semi-Flow
Further, let $\varphi$ be a deterministic process, i.e. a measurable mapping
$\begin{array}[]{rccl}\varphi:&[0,\infty)\times E&\to&E\\\
&t,y_{0}&\mapsto&\varphi_{t}(y_{0})\end{array}$
We assume that $\varphi_{t}$ is a semi-flow, i.e.
1. 1.
$\varphi_{0}(y)=y$,
2. 2.
$\varphi_{s+t}=\varphi_{s}\circ\varphi_{t}$ for all $s\geq 0$ and $t\geq 0$,
and we say that $\varphi$ is “space continuous” if for every $t\geq 0$,
$\varphi_{t}(y)$ is continuous in $y$.
###### Definition 3.
We say that $y\in E$ is a _stationary point_ of $\varphi$ if
$\varphi_{t}(y)=y$ for all $t\geq 0$
In cases where $E$ is a subset of $\mathbb{R}^{d}$ for some integer $d$, the
semi-flow $\varphi$ may be an autonomous ODE, of the form $\dot{y}=F(y)$; here
the stationary points are the solutions of $F(y)=0$.
###### Definition 4.
We say that the semi-flow $\varphi$ is _reversible_ under the probability
$\Pi\in\mathcal{P}(E)$ if for every time $t\geq 0$ and any
$h\in\mathcal{C}_{b}(E\times E)$:
$\int_{E}h(y,\varphi_{t}(y))\Pi(dy)=\int_{E}h(\varphi_{t}(y),y)\Pi(dy)$ (1)
As we show in the next section, reversible semi-flows must concentrate on
stationary points.
### 2.3 Convergence Hypothesis
We assume that, for every fixed $t$ the processes $Y^{N}$ converge in
distribution to some space continuous deterministic process $\varphi$ as
$N\to\infty$ for every collection of converging initial conditions. More
precisely:
###### Hypothesis 1.
For every $y_{0}$ in $E$, every sequence $(y^{N}_{0})_{N=1,2,...}$ such that
$y^{N}_{0}\in E^{N}$ and $\lim_{N\rightarrow\infty}y^{N}_{0}=y_{0}$, and every
$t\geq 0$, the conditional law of $Y^{N}(t)$ given $Y^{N}(0)=y^{N}_{0}$
converges in distribution to the Dirac mass at $\varphi_{t}(y_{0})$. That is
$\lim_{N\rightarrow\infty}\mathbb{E}^{N}\left.\left[h(Y^{N}(t))\right|Y^{N}(0)=y_{0}^{N}\right]=h\circ\varphi_{t}(y_{0})$
(2)
for all $h\in\mathcal{C}_{b}(E)$ and any fixed $t\geq 0$. In the above,
$\varphi$ is a space continuous semi-flow.
Hypothesis 1 is commonly true in the context of fluid or mean field limits.
The stronger convergence results results in le2007generic ; kurtz1970solutions
; sandholm2006population ; prout07 imply that Hypothesis 1 is satisfied ; we
give a detailed example illustrating this in Section 5. Similarly,
benaim2008class gives very general conditions (called H1 to H5) that ensure
convergence of a stochastic process to its mean field limit; under these
conditions, Hypothesis 1 is automatically satisfied (the deterministic process
$\varphi$ is then an ODE). Note that the results in these references are
stronger than what we require in Hypothesis 1; for example in
kurtz1970solutions there is almost sure, uniform convergence for all
$t\in[0,T]$, for any $T\geq 0$; in graham1993propagation the convergence is
on the set of trajectories.
Under Hypothesis 1, $\varphi$ is called the _hydrodynamic limit_ or simply
_fluid limit_ of $Y^{N}$.
## 3 Reversible Semi-Flows Concentrate on Stationary Points
###### Theorem 1.
Let $\varphi$ be a space continuous semi-flow, reversible under $\Pi$. Let $S$
be the set of stationary points of $\varphi$. Then $\Pi$ is concentrated on
$S$, i.e. $\Pi(S)=1$.
Proof.
Step 1. Denote with $S^{c}$ the complement of the set of stationary points.
Take some fixed but arbitrary $y_{0}\in S^{c}$. By definition of $S$, there
exists some $\tau>0$ such that
$\varphi_{2\tau}(y_{0})\neq y_{0}$ (3)
Define $\varphi_{\tau}(y_{0})=y_{1}$, $\varphi_{\tau}(y_{1})=y_{2}$, so that
$y_{2}\neq y_{0}$.
For $y\in E$ and $\epsilon>0$ we denote with $B(y,\epsilon)$ the open ball
$=\left\\{x\in E,d(x,y)<\epsilon\right\\}$. Let $\epsilon=d(y_{0},y_{2})>0$
and let $B_{2}=B(y_{2},\epsilon/2)$. Since the semi-flow is space continuous,
there is some $\alpha_{1}>0$ such that $B_{1}=B(y_{1},\alpha_{1})$ and
$\varphi_{\tau}\left(B_{1}\right)\subset B_{2}$. Also let
$B_{1}^{\prime}=B(y_{1},\alpha_{1}/2)$. By the same argument, there exists
some $\alpha_{0}>0$ such that $\alpha_{0}<\epsilon/2$,
$B_{0}=B(y_{0},\alpha_{0})$ and $\varphi_{\tau}\left(B_{0}\right)\subset
B_{1}^{\prime}$. We have thus:
$\displaystyle\varphi_{\tau}\left(B_{0}\right)$ $\displaystyle\subset$
$\displaystyle B^{\prime}_{1}\subset B_{1}$
$\displaystyle\varphi_{\tau}\left(B_{1}\right)$ $\displaystyle\subset$
$\displaystyle B_{2}$ $\displaystyle B_{0}\cap B_{2}$ $\displaystyle=$
$\displaystyle\O$
Let $\xi$ be some continuous function $[0,+\infty)\to[0,1]$ such that
$\xi(u)=1$ whenever $0\leq u\leq 1/2$ and $\xi(u)=0$ whenever $u\geq 1$ (for
example take a linear interpolation). Now take
$h(y,z)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\xi\left(\frac{d(y_{0},y)}{\alpha_{0}}\right)\xi\left(\frac{d(y_{1},y)}{\alpha_{1}}\right)$
(4)
so that $h\in\mathcal{C}_{b}(E\times E)$ and
$\displaystyle h(y,z)=0$ $\displaystyle\mbox{ whenever }y\not\in
B_{0}\mathrm{\;or\;}z\not\in B_{1}$ $\displaystyle h(y,z)=1$
$\displaystyle\mbox{ whenever }d(y_{0},y)<\alpha_{0}/2\mathrm{\;and\;}z\in
B^{\prime}_{1}$
It follows that $h(\varphi_{\tau}(z),z)=0$ for every $z\in E$ and
$\int_{E}h(y,\varphi_{\tau}(y))\Pi(dy)\geq\Pi(B(y_{0},\alpha_{0}/2))$ (5)
Apply Definition 1, it comes $\Pi\left(B(y_{0},\alpha_{0}/2)\right)=0$; thus,
for any non stationary point $y_{0}$ there is some $\alpha>0$ such that
$\Pi\left(B(y_{0},\alpha)\right)=0$ (6)
Step 2. The space is polish thus also separable, i.e. has a dense enumerable
set, say $Q$.
For every $y\in S^{c}$ let $\alpha$ be as in Eq.(6) and pick some $q(y)\in Q$
and $n(y)\in\mathbb{N}$ s.t. $d(y,q(y))<\frac{1}{n(y)}<\alpha$. Thus $y\in
B(q(y)$, $\frac{1}{n(y)})$ and $\Pi\left(B(q(y),\frac{1}{n(y)})\right)=0$.
Let $F=\bigcup_{y\in S^{c}}(q(y),n(y))$. $F\subset Q\times\mathbb{N}$ thus $F$
is enumerable and
$\displaystyle S^{c}$ $\displaystyle\subset$ $\displaystyle\bigcup_{(q,n)\in
F}B\left(q,\frac{1}{n}\right)$
Thus
$0\leq\Pi(S^{c})\leq\sum_{(q,n)\in
F}\Pi\left(B\left(q,\frac{1}{n}\right)\right)=0$ (7)
$\Box$
Note that it follows that a semi-flow that does not have any stationary point
cannot be reversible under any probability.
## 4 Stationary Behaviour of Fluid Limits of Reversible Processes
###### Theorem 2.
Assume for every $N$ the process $Y^{N}$ is reversible under some probability
$\Pi^{N}$. Assume the convergence Hypothesis 1 holds and that
$\Pi\in\mathcal{P}(E)$ is a limit point (for weak convergence) of the sequence
$\Pi^{N}$. Then the fluid limit is reversible under $\Pi$. In particular, it
follows from Theorem 1 that $\Pi$ is concentrated on the set of stationary
points $S$ of the fluid limit $\varphi$.
Proof. All we need to show is that $\Pi$ verifies Definition 1. Let $N_{k}$ be
a subsequence such that $\lim_{k\rightarrow\infty}\Pi^{N_{k}}=\Pi$ in the weak
topology on $\mathcal{P}(E)$. By Skorohod’s representation theorem for Polish
spaces (ethier-kurtz-05, , Thm 1.8), there exists a common probability space
$(\Omega,\mathcal{F},\mathbb{P})$ on which some random variables $X^{k}$ for
$k\in\mathbb{N}$ and $X$ are defined such that
$\left\\{\begin{array}[]{l}\mbox{ law of }X^{k}=\Pi^{N_{k}}\\\ \mbox{ law of
}X=\Pi\\\ X^{k}\to X\;\mathbb{P}-\mbox{a.s.}\end{array}\right.$
Fix some $t\geq 0$ and $h\in\mathcal{C}_{b}(E\times E)$, and define, for
$k\in\mathbb{N}$ and $y\in E$
$\displaystyle a^{k}(y)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}$
$\displaystyle\mathbb{E}\left(\left.h\left(y,Y^{N_{k}}(t)\right)\right|Y^{N_{k}}(0)=y\right)$
$\displaystyle b^{k}(y)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}$
$\displaystyle\mathbb{E}\left(\left.h\left(Y^{N_{k}}(t),y\right)\right|Y^{N_{k}}(0)=y\right)$
Since $Y^{N}$ is reversible under $\Pi^{N_{k}}$:
$\int_{E}a^{k}(y)\Pi^{N_{k}}(dy)=\int_{E}b^{k}(y)\Pi^{N_{k}}(dy)$ (8)
Hypothesis 1 implies that
$\lim_{k\rightarrow\infty}a^{k}(x^{k})=h(x,\varphi_{t}(x))$ for every sequence
$x^{k}$ such that $x^{k}\in E^{N_{k}}$ and
$\lim_{k\rightarrow\infty}x^{k}=x\in E$. Now $X^{k}\in E^{N_{k}}$
$\mathbb{P}-$ almost surely, since the law of $X^{k}$ is $\Pi^{N_{k}}$ and
$Y^{N_{k}}$ is reversible under $\Pi^{N_{k}}$ . Further, $X^{k}\to X$
$\mathbb{P}-$ almost surely; thus
$\lim_{k\rightarrow\infty}a^{k}(X^{k})=h(X,\varphi_{t}(X))\;\;\;\mathbb{P}-\mbox{
almost surely}$ (9)
Now $a^{k}(X^{k})\leq\left\|h\right\|_{\infty}$ and, thus, by dominated
convergence:
$\lim_{k\rightarrow\infty}\mathbb{E}\left(a^{k}(X^{k})\right)=\mathbb{E}\left(h(X,\varphi_{t}(X))\right)$
(10)
and similarly for $b^{k}$. Thus
$\int_{E}h(y,\varphi_{t}(y))\Pi(dy)=\int_{E}h(\varphi_{t}(y),y)\Pi(dy)$ (11)
$\Box$
In particular, if the semi-flow has a unique stationary point, we have:
###### Corollary 1.
Assume the processes $Y^{N}$ are reversible under some probabilities
$\Pi^{N}$. Assume Hypothesis 1 holds and:
1. 1.
the sequence $(\Pi^{N})_{N=1,2,...}$ is tight;
2. 2.
the semi-flow $\varphi$ has a unique stationary point $y^{*}$.
It follows that the sequence $\Pi^{N}$ converges weakly to the Dirac mass at
$y^{*}$.
We leave the proof of the corollary to the reader (it follows in a classical
way from compactness arguments; the tightness condition implies that the set
$\left\\{\Pi^{N},N=1,2,3...\right\\}_{N}$ is relatively compact in
$\mathcal{P}(E)$).
Recall that tightness means that for every $\epsilon>0$ there is some compact
set $K\subset E$ such that $\Pi^{N}(K)\geq 1-\epsilon$ for all $N$. If $E$ is
compact then $(\Pi^{N})_{N=1,2,...}$ is necessarily tight, therefore condition
1 in the corollary is automatically satisfied. For mean field limits where $E$
is the simplex in finite dimension, the corollary says that, if the pre-limit
process is reversible, then the existence of a unique stationary point implies
that the Dirac mass at this stationary point is the limit of the stationary
probability of the pre-limit process.
Compare Corollary 1 to known results for the non reversible case benaim98 :
there we need that the fluid limit $\varphi$ has a unique limit point to which
all trajectories converge. In contrast, here, we need a much weaker
assumption, namely, the existence and uniqueness of a stationary point. It is
possible for a semi-flow to have a unique stationary point, without this
stationary point being a limit of all trajectories (for example because it is
unstable, or because there are stable limit cycles as in benaim2008class ). In
the reversible case, we do not need to show stability of the unique stationary
point $y^{*}$.
## 5 Example: Crowd Dynamics
In this section we give an example to illustrate the application of Theorem 2
– a detailed study of this example beyond the application of Theorem 2 is
outside the scope of this paper. We consider the crowd dynamics model of RoG03
. The model captures the emergence of crowds in a city. A city is modelled as
a fully connected bidirectional graph with $I$ vertices, every vertex
representing a square, where bars are located. There is a fixed total
population $N$. People spend some time in a square and once in a while decide
to leave a square and move to some other square. The original model is in
discrete time, and at every time slot, the probability that a tagged person
present in square $i$ leaves square $i$ is assume to be equal to
$(1-c)^{N_{i}(t)-1}$ where $N_{i}(t)$ is the population of square $i$ at time
$t$. In this equation, $c$ is the _chat_ probability, and this model thus
assumes that a person leaves a square when it has no one to chat with. The
model also assumes that departure events are independent. When a person leaves
a square $i$, it move to some other square $j$ according to Markov routing,
with probability $Q_{i,j}$ given by
$Q_{i,j}=\frac{1}{d(i)}$ (12)
where $d(i)$ is the degree of node $i$, i.e. the person picks a neighboring
square $j$ uniformly at random among all neighboring squares.
In RoG03 , the authors study by simulation the emergence of concentration in
one square. They also show that for regular graphs (i.e. when all vertices
have same degree) there is a critical value $c^{*}$ such that for $c>c^{*}$
concentrations occur, whereas for $c<c^{*}$ the stationary distribution of
people is uniform. The analysis is based on the study of stationary points for
the empirical distribution. Note that, as mentioned in the introduction, the
analysis with stationary points may, in general, miss the main part of the
stationary distribution, and it is quite possible that the stationary
distribution is not concentrated on stationary points (for example if there is
a limit cycle benaim2008class ). A fluid flow approximation is proposed in
Ma+11 , and similar results are found.
To understand whether the stationary point analysis is justified, we study the
large $N$ asymptotics for an appropriately rescaled version. To avoid
unnecessary complications, we replace the original model of RoG03 , which is
in discrete time, by its continuous time counterpart. The probability that a
tagged person present in square $i$ leaves square $i$ in a time slot is
replaced by the rate of service given by
$\mu^{N}(N_{i})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(1-c)^{N_{i}-1}$ (13)
i.e. the probability that a tagged person present in square $i$ leaves the
square in the next $dt$ seconds is $\mu^{N}(N_{i})dt+o(dt)$.
The corresponding continuous process $X^{N}(t)=(N_{1}(t),...,N_{I}(t))$ is a
Markov process on an enumerable state space. More precisely, it is a queuing
network of infinite server stations, with state dependent service rate and
with Markov routing. It follows from classical results on quasi-reversibility
that it has product-form (see for example (leboudec2010performance, , Chapter
8)), i.e. it is ergodic (since the graph of squares is fully connected and the
population is finite) and its stationary probability is given, for every
$(n_{1},...,n_{I})\in\mathbb{N}^{I}$ such that $n_{1}+...+n_{I}=N$ by
$\displaystyle P^{N}(n_{1},...,n_{I})$ $\displaystyle=$
$\displaystyle\eta^{N}\prod_{i=1}^{I}f^{N}_{i}(n_{i})$ (14)
In this formula, $\eta^{N}$ is a normalizing constant,
$f_{i}^{N}(n)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\theta_{i}^{n}}{n!}\prod_{m=1}^{n}\mu^{N}(m)$,
and the vector $\theta=(\theta_{1},...,\theta_{I})$ is the stationary
distribution of the Markov routing matrix $Q$, i.e. a normalized solution of
the equation $\theta Q=\theta$. Note that it follows from Eq.(12) that
$\theta_{i}=\frac{d(i)}{\sum_{j=1}^{I}d(j)}$ (15)
To apply Theorem 2, we need to show that $X^{N}$ is reversible. A product-form
queuing network is, in general, not reversible. However, it is so if the
Markov routing chain is reversible le1987interinput , which is the case here.
###### Theorem 3.
For every $N$, the process $X^{N}$ is reversible.
Proof. Take $\theta$ given by Eq.(15). Then
$\theta_{i}Q_{i,j}=\theta_{j}Q_{j,i}$ for any pair $(i,j)$, thus the Markov
chain with transition matrix $Q$ given by Eq.(12) is reversible. By
le1987interinput , it follows that the product-form queuing network $X^{N}$ is
reversible. $\Box$
In bortolussi2012revisiting , it is suggested to scale the chat probability as
$c=\frac{s}{N}$ (16)
in order to account for the fact that, for large populations, meetings tend to
be limited by space or size of the friend’s group. We use this scaling law and
consider the re-scaled process $Y^{N}$ of occupancy measures, i.e. given by
$Y^{N}(t)=\left(\frac{N_{1}(t)}{N},...\frac{N_{I}(t)}{N}\right)=\frac{1}{N}X^{N}(t)$
(17)
Obviously, for every $N$ the process $Y^{N}$ is Markov and is reversible.
Further, it converges to an ODE, as we see next.
To establish the convergence of $Y^{N}$, we compute its drift
$A^{N}(y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\lim_{dt\rightarrow
0}\frac{\mathbb{E}\left(\left.Y^{N}(t+dt)-y\right|Y^{N}(t)=y\right)}{dt}$ (18)
defined for every possible value $y$ of $Y^{N}(t)$. When the occupancy measure
is $y$, there are $N_{i}=Ny_{i}$ people in square $i$, and the rate of
departure from square $i$ is $N_{i}\mu(N_{i})=Ny_{i}\mu(Ny_{i})$; the delta to
the occupancy measure due to one person moving from square $i$ to square $j$
is $\frac{-e_{i}+e_{j}}{N}$, where $e_{i}$ is the row vector with a $1$ in
position $i$ and $0$ elsewhere. Therefore
$V^{N}(y)=\sum_{i,j}Ny_{i}\mu(Ny_{i})Q_{i,j}\frac{-e_{i}+e_{j}}{N}$
Taking into account Eq.(16), it comes
$V^{N}(y)=\sum_{i,j}y_{i}\left(1-\frac{s}{N}\right)^{Ny_{i}-1}Q_{i,j}\left(-e_{i}+e_{j}\right)$
(19)
Let $\Delta_{I}$ denote the simplex, i.e.
$\Delta_{I}=\left\\{y\in\mathbb{R}^{I},y_{i}\geq
0\mathrm{\;for\;all\;}i\mathrm{\;and\;}\sum_{i=1}^{I}y_{i}=1\right\\}$
When $N\to\infty$, $V^{N}(y)$ converges for every $y\in\Delta_{I}$ to
$V(y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i,j}y_{i}e^{-sy_{i}}Q_{i,j}\left(-e_{i}+e_{j}\right)$
(20)
This suggests that the fluid limit of $Y^{N}$, if it exists, would be the
deterministic process $y(t)$, with sample paths in $\mathbb{R}^{d}$, obtained
as solution of the ODE $\frac{dy}{dt}=V(y)$. As we show next, this is indeed
the case and follows from “Kurtz’s theorem” (sandholm2006population, , Theorem
9.2.1). Before that, we rewrite the ODE more explicitly as
$\frac{dy_{i}}{dt}=-y_{i}e^{-sy_{i}}+\sum_{j}y_{j}e^{-sy_{j}}Q_{j,i}$ (21)
###### Theorem 4.
Assume that the initial condition $y_{0}^{N}$ of the process $Y^{N}$ is
deterministic and converges to some $y_{0}\in\Delta_{I}$. Let $\varphi$ be the
semi-flow defined by the ODE (21), i.e. $\varphi_{t}(y_{0})$ is the solution
of the ODE (21) with initial condition $y(0)=y_{0}$ (this solution exists and
is unique by the Cauchy Lipschitz theorem). Then for each $T>0$ and
$\epsilon>0$:
$\lim_{N\rightarrow\infty}\mathbb{P}\left(\sup_{0\leq t\leq
T}\left\|Y^{N}(t)-\varphi_{t}(y_{0})\right\|>\epsilon\right)=0$ (22)
(the notation $\left\|\right\|$ stands for any norm on $\mathbb{R}^{I}$).
It follows that Hypothesis 1 is verified.
Proof. We apply Theorem 9.2.1 in sandholm2006population . We need to find a
sequence of numbers $\delta_{N}\to 0$ such that the following three conditions
hold:
$\displaystyle\lim_{N\rightarrow\infty}\sup_{y\in\Delta_{I}^{N}}\left\|V^{N}(y)-V(y)\right\|=0$
(23) $\displaystyle\sup_{N}\sup_{y\in\Delta^{N}_{I}}A^{N}(y)<\infty$ (24)
$\displaystyle\lim_{N\rightarrow\infty}\sup_{y\in\Delta^{N}_{I}}A^{N}_{\delta_{N}}(y)=0$
(25)
In the above, $\Delta^{N}_{I}$ is the set of feasible states of $Y^{N}$, i.e.
the set of $y\in\Delta_{I}$ such that $Ny$ is integer, $A^{N}(y)$ is the
expected norm of jump per time unit, and $A^{N}_{\delta_{N}}(y)$ is the
absolute expected norm of jump per time unit due to jumps travelling further
than $\delta_{N}$.
We now show Eq.(23). First consider the case $y\in\Delta^{N}_{I}$ such that
$y_{i}>0$ (thus we have $1/N\leq y_{i}\leq 1$). We apply the inequality
$\left|e^{-x}-e^{-x^{\prime}}\right|\leq\left|x-x^{\prime}\right|$, valid for
$x\geq 0$ and $x^{\prime}\geq 0$, to
$x=-(Ny_{i}-1)\log\left(1-\frac{s}{N}\right)$, $x^{\prime}=sy_{i}$ and obtain:
$\left|\left(1-\frac{s}{N}\right)^{Ny_{i}-1}-e^{-sy_{i}}\right|\leq\left|(Ny_{i}-1)\log\left(1-\frac{s}{N}\right)+sy_{i}\right|$
The right handside is convex in $y_{i}$ thus its maximum for $y_{i}\in[1/N,1$]
is obtained at one end of the interval. Thus
$\left|\left(1-\frac{s}{N}\right)^{Ny_{i}-1}-e^{-sy_{i}}\right|\leq
a_{N}(s)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\left[{s\over
N},\left|(N-1)\log\left(1-\frac{s}{N}\right)+s\right|\right]$
Second, multiply by $y_{i}$ and note that $y_{i}\leq 1$, it follows that,
whenever $y\in\Delta^{N}_{I}$ and $y_{i}>0$:
$\left|y_{i}\left(1-\frac{s}{N}\right)^{Ny_{i}-1}-y_{i}e^{-sy_{i}}\right|\leq
a_{N}(s)$
and this is also obviously true if $y_{i}=0$. It follows that
$\sup_{y\in\Delta^{N}_{I}}\left\|V^{N}(y)-V(y)\right\|\leq
a_{N}(s)\sum_{i,j}\left\|-e_{i}+e_{j}\right\|Q_{i,j}$
from where Eq.(23) follows since $\lim_{N\rightarrow\infty}a_{N}(s)=0$.
We now show Eq.(24). We take the sup norm on $\mathbb{R}^{d}$ so that
$\left\|-e_{i}+e_{j}\right\|=1$ for $i\neq j$; thus
$\displaystyle A^{N}(y)$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}$
$\displaystyle\sum_{i,j}y_{i}\left(1-\frac{s}{N}\right)^{Ny_{i}-1}Q_{i,j}\left\|-e_{i}+e_{j}\right\|$
$\displaystyle=$
$\displaystyle\sum_{i,j}y_{i}\left(1-\frac{s}{N}\right)^{Ny_{i}-1}Q_{i,j}\leq\sum_{i,j}Q_{i,j}=I$
which shows Eq.(24).
We now show Eq.(25). We take $\delta_{N}=\frac{1}{N}$. The jumps of $Y^{N}$
are of the form $\frac{-e_{i}+e_{j}}{N}$ and thus
$\left\|\frac{-e_{i}+e_{j}}{N}\right\|\leq\delta_{N}$. Thus
$A^{N}(y)_{\delta_{N}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i,j}y_{i}\left(1-\frac{s}{N}\right)^{Ny_{i}-1}Q_{i,j}\left\|-e_{i}+e_{j}\right\|\mathbf{1}_{\\{\left\|\frac{-e_{i}+e_{j}}{N}\right\|>\delta_{N}\\}}=0$
which trivially shows Eq.(25). By Theorem 9.2.1 in sandholm2006population ,
this establishes Eq.(22).
It follows (as a much weaker convergence) that for any fixed $T$, $Y^{N}(T)$
converges in probability to the deterministic $\varphi_{T}(y_{0})$. Thus there
is also convergence in law, since convergence in probability to a
deterministic variable implies convergence in distribution, i.e. Eq.(2) in
Hypothesis 1 is verified. It remains to see that $\varphi_{t}$ is space
continuous: this follows from the fact that the right-handside of the ODE is
Lipschitz continuous and from the Cauchy Lipschitz theorem. $\Box$
It follows from Theorem 2 that any limit point of the stationary probability
Eq.(14) is concentrated on the stationary points of the ODE (21). This
justifies a posteriori the method in RoG03 , which looked only at stationary
points.
For the case of a regular graph (this is the case studied analytically in
RoG03 ), the stationary points can be obtained explicitly (Theorem 6.1 in
bortolussi2012revisiting ). In particular, there is a critical value $s^{*}$
below which there is only one stationary point, equal to the uniform
distribution $y^{*}=(\frac{1}{I},...,\frac{1}{I})$ and above which there are
other stationary points. The critical value is given in
bortolussi2012revisiting ) and is equal to
$s^{*}=\min_{K=1,...,I-1}\min_{\alpha>1}\left((I-K)\alpha+K\phi(\alpha)\right)$
(26)
with $\phi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}-W_{0}(-xe^{-x})$,
$W_{0}$ being the Lambert-W function of index $0$. For example for $I=3$, the
critical value is $s^{*}\approx 2.7456$.
We can apply Corollary 1: since the state space $E=\Delta_{I}$ is compact, it
follows that for $s<s^{*}$, the stationary distribution given by Eq.(14), re-
scaled by $1/N$, converges as $N\to\infty$ to the uniform distribution. This
illustrates the interest of the reversibility results in this paper; we do not
need to show that all trajectories converge to the single stationary point –
its uniqueness and the reversibility argument are sufficient.
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|
arxiv-papers
| 2010-09-25T15:48:37 |
2024-09-04T02:49:13.091394
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jean-Yves Le Boudec",
"submitter": "Jean-Yves Le Boudec",
"url": "https://arxiv.org/abs/1009.5021"
}
|
1009.5074
|
# Backward Stochastic Differential Equations with Markov Chains and The
Application: Homogenization of PDEs System111This work was supported by
Natural Science Foundations of China (No. 10671112) and Shandong Province (No.
JQ200801 and 2008BS01024), the National Basic Research Program of China (973
Program, No. 2007CB814904) and the Science Fund for Distinguished Young
Scholars of Shandong University (No. 2009JQ004).
Huaibin Tang222Current address: INRIA-IRISA, Campus de Beaulieu, 35042 Rennes
Cedex, France. tanghuaibin@gmail.com School of Mathematics, Shandong
University, Jinan, Shandong 250100, P. R. China. Zhen Wu wuzhen@sdu.edu.cn
###### Abstract
Stemmed from the derivation of the optimal control to a stochastic linear-
quadratic control problem with Markov jumps, we study one kind of backward
stochastic differential equations (BSDEs) that the generator $f$ is affected
by a Markovian switching. Then, the case that the Markov chain is involved in
a large state space is considered. Following the classical approach, a
hierarchical approach is adopted to reduce the complexity and a singularly
perturbed Markov chain is involved. We will study the asymptotic property of
BSDE with the singularly perturbed Markov chain. At last, as an application of
our theoretical result, we show the homogenization of one system of partial
differential equations (PDEs) with a singularly perturbed Markov chain.
###### keywords:
BSDE , Markov chain , weak convergence , homogenization.
## 1 Introduction
The study of backward stochastic differential equations (BSDEs in short)
stemmed from stochastic control problem ([4]) in which a non-linear Ricatti
BSDE was introduced. Then, after the pioneering work of Pardoux and Peng
([17]) about the general BSDE which was driven by a Brownian motion, BSDEs
have been extensively studied in the last twenty years because of their
connections with mathematical finance, stochastic control, and partial
differential equations (PDEs in short), please refer to [10, 8, 7]. Since
then, many researchers devoted their work to more general BSDE, such as BSDE
driven by a Lévy process ([1, 15]) and BSDE with respect to both a Brownian
motion and a Poisson random measure ([3, 21, 11]). Recently, Cohen and Elliott
[5, 6] studied BSDE driven by the martingale part of a Markov chain and its
application in finance.
As studied in Zhang and Yin ([25]), we consider the stocks investment models
by virtue of hybrid geometric Brownian motion in which both the expected
return and volatility depend on a finite state Markov chain. To capture the
market trends as well as the various economic factors, a finite state Markov
chain $\alpha_{t}$, $t\geq 0$, is introduced to represent the general market
directions. If our object is to allocate assets into a number of stocks so as
to maximize an expected utility within a finite time horizon, it leads to an
optimal control problem. By virtue of maximum principle method, we need to
introduce an adjoint equation to deal with this optimization problem. The
adjoint equation and the state equation form forward backward stochastic
differential equations (FBSDEs in short) system. Here the adjoint equation
will be a kind of BSDE with a Markov chain.
Motivated by such adjoint equation, in our paper, we consider the following
BSDE with a Markov chain:
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s},\alpha_{s})ds-\int_{t}^{T}Z_{s}dB_{s},$
where $\alpha=\\{\alpha_{t};0\leq t\leq T\\}$ is a continuous-time Markov
chain independent of the Brownian motion $B$. It is noted that this BSDE is
different from the one studied in Cohen and Elliott ([5]), and it can be
considered as that its generator is disturbed by random environment and takes
a set of discrete values which can be described by a Markov chain.
When study the solvability for BSDE with a Markov chain, the classic method
with the contraction mapping cannot be directly used for the lack of suitable
filtration and corresponding Itô’s representation theorem. In this paper,
inspired by the method dealing with the BSDE with doubly Brownian motion
([19]), we construct a new filtration and give the corresponding extended
Itô’s representation theorem.
When various factors are considered, the underlying Markov chain inevitably
has a large state space, and the corresponding BSDE becomes increasingly
complicated. It is rationale that the change rates of states display a two-
time-scale behavior, a fast-time scale and a slow varying one. Under this
case, a small parameter $\varepsilon>0$ can be introduced and the singularly
perturbed Markov chain is involved.
In this paper, we consider the case that the states of the underlying Markov
chain are divided into a number of weakly irreducible classes such that the
Markov chain fluctuates rapidly among different states, and jumps less
frequently among those classes. To reduce the complexity, a small parameter
($\varepsilon>0$) is introduced to reflect the different rate of changes among
different states. As shown in Zhang and Yin ([23]), it leads to a singularly
perturbed Markovian models with two-time scale, the actual time $t$ and the
stretched time $\frac{t}{\varepsilon}$. By aggregating the states in each
irreducible class into a single, a limit aggregated Markov chain with a
considerably smaller space can be obtained and its asymptotic probability
distribution is studied. Such asymptotic theory has many applications in
optimal control problem and mathematical finance. We refer interested readers
to [25, 24].
In this paper, we will focus on the asymptotic property of BSDE with a
singularly perturbed Markov chain. Following the averaged approach that
aggregating the states according to their jump rates, we will show that the
distribution of $(Y_{t},\int_{t}^{T}Z_{s}d\bar{B}_{s})$ can be seen as an
asymptotic distribution to $(Y^{\varepsilon}_{t},$
$\int_{t}^{T}Z^{\varepsilon}_{s}dB_{s})$, where
$(Y^{\varepsilon},Z^{\varepsilon})$ and $(Y,Z)$ satisfy:
$Y^{\varepsilon}_{t}=\xi+\int_{t}^{T}f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})ds-\int_{t}^{T}Z^{\varepsilon}_{s}dB_{s}$
and
$Y_{t}=\xi+\int_{t}^{T}\bar{f}(s,Y_{s},\bar{\alpha}_{s})ds-\int_{t}^{T}Z_{s}d\bar{B}_{s}.$
Here $\bar{\alpha}$ and $\bar{f}(s,Y_{s},\bar{\alpha}_{s})$ are respectively
the limit aggregated Markov chain and the averaged generator with respect to
the quasi stationary distributions of the singularly perturbed Markov chain.
Compared to the original BSDE with the singularly perturbed Markov chain, the
limit BSDE depends on a Markov chain with a much smaller state space. Thus the
complexity is reduced.
It is well known that BSDEs provide a probabilistic representation for the
solution of a large class of quasi-linear second order partial differential
equations (PDEs in short) ([3, 18, 19, 20, 13]). Thus BSDEs provide a
probabilistic tool to study the homogenization of PDEs, which is the process
of replacing rapidly varying coefficients by new ones thus the solutions are
close. In this paper, as an application of our theoretical result, after
showing the relation between our BSDE and one system of semi-linear PDE, we
will show the homogenization result of one system of semi-linear PDE with a
singularly perturbed Markov chain.
This paper is organized as following. In section 2, we study the solvability
of BSDE with a Markov chain. Section 3 is devoted to the case that Markov
chain has a large space. Under Jakubowski S-topology ([9] ) which is weaker
than Skorohod’s topology, we present the asymptotic property of BSDE with a
singularly perturbed Markov chain. In section 4, we show the application of
our theoretical results in the homogenization of one system of semi-linear PDE
with a singularly perturbed Markov chain. For the terseness of the main text
of our paper, we put part of technical proofs for some results in Appendix.
## 2 BSDEs with Markov Chains
Let $(\Omega,{\cal{F}},P)$ be a probability space and $T>0$ be fixed.
$\\{{\cal{H}}_{t},0\leq t\leq T\\}$ is a filtration on the space satisfying
the usual condition. $B=\\{B_{t};0\leq t\leq T\\}$ with $B_{0}=0$ is a
$d$-dimensional ${\cal{H}}_{t}$-Brownian motion, and
$\alpha=\\{\alpha_{t};0\leq t\leq T\\}$ is a continuous-time Markov chain
independent of $B$ with the state space ${\cal{M}}=\\{1,2,\ldots,m\\}$.
Suppose the generator of the Markov chain $Q=(q_{ij})_{m\times m}$ is given by
$\begin{split}P\\{\alpha(t+\triangle)=j|\alpha(t)=i\\}=\left\\{\begin{array}[]{l
l}q_{ij}\triangle+o(\triangle),\ \textrm{ \quad\ \ if }i\neq j\\\
1+q_{ij}\triangle+o(\triangle),\ \textrm{ if
}i=j\end{array}\right.\end{split}$
where $\triangle>0$. Here $q_{ij}\geq 0$ is the transition rate from $i$ to
$j$ if $i\neq j$, while $q_{ii}=-\sum_{j=1,i\neq j}^{m}q_{ij}$.
Throughout this paper, we introduce the following notations: $|\cdot|$ is the
norm in the corresponding space; $A^{\prime}$ is the transpose of matrix $A$;
$L^{p}({\cal{H}}_{t};R^{n})$ is the space of $R^{n}$-valued
${\cal{H}}_{t}$-adapted random variable $\xi$ satisfying
$E(|\xi|^{p})<\infty$; $M_{\mathcal{H}_{t}}^{2}(0,T;$ $R^{n})$ denotes the
space of $R^{n}$-valued $\mathcal{H}_{t}$-adapted stochastic processes
$\varphi=\\{\varphi_{t};t\in[0,T]\\}$ satisfying
$E\int_{0}^{T}|\varphi_{t}|^{2}dt<\infty$;
$S_{\mathcal{H}_{t}}^{2}(0,T;R^{n})$ is the space of $R^{n}$-valued
${\cal{H}}_{t}$-adapted continuous stochastic processes
$\varphi=\\{\varphi_{t};t\in[0,T]\\}$ satisfying $E(\sup_{0\leq t\leq
T}|\varphi_{t}|^{2})$ $<\infty$.
### 2.1 Motivation
To study the stochastic optimal control problem with a Markov chain, we
introduce an adjoint equation, then the state equation and adjoint equation
form a kind of FBSDEs with a Markov chain. Here the adjoint equation will be a
BSDE with Markov chain. We give the following linear quadratic (LQ in short)
optimal control problem as an example.
Consider the following stochastic LQ control problem with Markov jumps
min. $\displaystyle
J(v)=\frac{1}{2}E\left(\int_{0}^{T}\left((x^{v}_{t})^{\prime}R(t,\alpha_{t})x^{v}_{t}+v^{\prime}_{t}N(t,\alpha_{t})v_{t}\right)dt+(x^{v}_{T})^{\prime}Q(\alpha_{T})x^{v}_{T}\right)$
(1a) s. t. $\displaystyle\left\\{\begin{array}[]{l
l}dx^{v}_{t}=\left(A(t,\alpha_{t})x^{v}_{t}+B(t,\alpha_{t})v_{t}\right)dt+\left(C(t,\alpha_{t})x^{v}_{t}+D(t,\alpha_{t})v_{t}\right)dB_{t}\\\
x^{v}_{0}=a\in R^{n}\end{array}\right.$ (1d)
where
$A(t,\alpha_{t})=A_{i}(t),B(t,\alpha_{t})=B_{i}(t),C(t,\alpha_{t})=C_{i}(t),D(t,\alpha_{t})=D_{i}(t),$
$R(t,\alpha_{t})=R_{i}(t),N(t,\alpha_{t})=N_{i}(t)$ when $\alpha_{t}=i\
(i=1,\cdots,m)$, and they are uniformly bounded $\mathcal{F}_{t}^{B}$-adapted
processes with appropriate dimensions. $Q(\alpha_{T})=Q_{i}$ when
$\alpha_{T}=i$ $(i=1,\cdots,m)$, and it is nonnegative symmetric matrices-
valued $\mathcal{F}_{T}^{B}$-measurable random variable. Besides, $R_{i}(t)$
is nonnegative symmetric matrices-valued, $N_{i}(t)$ is positive symmetric
matrices-valued and the inverse $N_{i}(t)^{-1}$ is bounded. The set of all
$\mathcal{H}_{t}$-adapted admissible controls is $\mathcal{U}_{ad}\equiv
M^{2}_{\mathcal{H}_{t}}(0,T;R^{n_{u}\times d})$, and our aim is to find an
admissible control $u$ such that $\displaystyle
J(u)=\inf_{v\in\mathcal{U}_{ad}}J(v)$.
There are many literatures on this kind of LQ optimal control problem with
Markov jumps (1) and its application, such as [26, 12, 24] and their
references. Different to their methods that constructing the optimal control
via the solution of Riccati equation, we will use the FBSDEs approach.
###### Theorem 2.1.
If the following FBSDE admits a unique solution $(x_{t},y_{t},z_{t})$
$\begin{split}\left\\{\begin{array}[]{l l
l}~{}~{}dx_{t}=\left(A(t,\alpha_{t})x_{t}+B(t,\alpha_{t})\left(-N^{-1}(t,\alpha_{t})\left(B^{\prime}(t,\alpha_{t})y_{t}+D^{\prime}(t,\alpha_{t})z_{t}\right)\right)\right)dt\\\
~{}~{}~{}~{}+\left(C(t,\alpha_{t})x_{t}+D(t,\alpha_{t})\left(-N^{-1}(t,\alpha_{t})\left(B^{\prime}(t,\alpha_{t})y_{t}+D^{\prime}(t,\alpha_{t})z_{t}\right)\right)\right)dB_{t},\\\
-dy_{t}=\left(A^{\prime}(t,\alpha_{t})y_{t}+C^{\prime}(t,\alpha_{t})z_{t}+R(t,\alpha_{t})x_{t}\right)dt-
z_{t}dB_{t},\\\ ~{}~{}~{}x_{0}=a,\ \
y_{T}=Q(\alpha_{T})x_{T}.\end{array}\right.\end{split}$ (2)
Then
$u_{t}=-N^{-1}(t,\alpha_{t})\left(B^{\prime}(t,\alpha_{t})y_{t}+D^{\prime}(t,\alpha_{t})z_{t}\right),0\leq
t\leq T$
is the unique optimal control for the LQ problem (1)
###### Proof.
Firstly, we will prove that $\\{u=u_{t};0\leq t\leq T\\}$ is an optimal
control for the LQ problem (1).
From the forward equation of (2), we can see that $x$ is the corresponding
system state trajectory of $u$. For an arbitrary admissible control $v$,
denote $x^{v}$ as the corresponding system state trajectory, then
$\displaystyle J(v)-J(u)$ $\displaystyle=$ $\displaystyle\
\frac{1}{2}E\Big{(}\int_{0}^{T}\big{(}(x_{t}^{v}-x_{t})^{\prime}R(t,\alpha_{t})(x_{t}^{v}-x_{t})+(v_{t}-u_{t})^{\prime}N(t,\alpha_{t})(v_{t}-u_{t})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+2x_{t}^{\prime}R(t,\alpha_{t})(x_{t}^{v}-x_{t})+2u_{t}^{\prime}N(t,\alpha_{t})(v_{t}-u_{t})\big{)}dt$
$\displaystyle\ \ \ \ \ \ \
+(x_{T}^{v}-x_{T})^{\prime}Q(\alpha_{T})(x_{T}^{v}-x_{T})+2x^{\prime}_{T}Q(\alpha_{T})(x_{T}^{v}-x_{T})\Big{)}.$
Applying Itô’s formula to $y^{\prime}_{t}(x_{t}^{v}-x_{t})$, we have
$\displaystyle Ex^{\prime}_{T}Q(\alpha_{T})(x_{T}^{v}-x_{T})$ $\displaystyle=$
$\displaystyle
E\int_{0}^{T}\big{(}\left(y^{\prime}_{t}B(t,\alpha_{t})+z^{\prime}_{t}D(t,\alpha_{t})\right)(v_{t}+N^{-1}(t,\alpha_{t})\left(B^{\prime}(t,\alpha_{t})y_{t}+D^{\prime}(t,\alpha_{t})z_{t}\right))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-x_{t}^{\prime}R(t,\alpha_{t})(x_{t}^{v}-x_{t})\big{)}dt$
$\displaystyle=$ $\displaystyle
E\int_{0}^{T}\left(\left(y^{\prime}_{t}B(t,\alpha_{t})+z^{\prime}_{t}D(t,\alpha_{t})\right)(v_{t}-u_{t})-x_{t}^{\prime}R(t,\alpha_{t})(x_{t}^{v}-x_{t})\right)dt.$
As $R$, $Q$ are nonnegative and $N$ is positive, we have
$\displaystyle J(v)-J(u)\geq$ $\displaystyle
E\int_{0}^{T}\left(y^{\prime}_{t}B(t,\alpha_{t})+z^{\prime}_{t}D(t,\alpha_{t})+u^{\prime}_{t}N(t,\alpha_{t})\right)(v_{t}-u_{t})dt$
$\displaystyle=$ $\displaystyle 0.$
So
$u(t)=-N^{-1}(t,\alpha_{t})\left(B^{\prime}(t,\alpha_{t})y_{t}+D^{\prime}(t,\alpha_{t})z_{t}\right)$
is an optimal control.
Uniqueness: Assume that $u^{1}$ and $u^{2}$ are both optimal controls with
$J(u^{1})=J(u^{2})=\gamma\geq 0$, and the corresponding trajectories are
$x^{1}$ and $x^{2}$. Due to the linear property of the system, the
trajectories corresponding to $\displaystyle\frac{u^{1}+u^{2}}{2}$ is
$\displaystyle\frac{x^{1}+x^{2}}{2}$.
From the classical parallelogram rule, $R$, $Q$ are nonnegative, and $N$ is
positive, there exists $\delta>0$ such that
$\displaystyle 2\gamma$ $\displaystyle=$ $\displaystyle\ J(u^{1})+J(u^{2})$
$\displaystyle=$ $\displaystyle\
2J\left(\frac{u^{1}+u^{2}}{2}\right)+E\bigg{(}\left(\frac{x_{T}^{1}-x_{T}^{2}}{2}\right)^{\prime}Q(\alpha_{T})\left(\frac{x_{T}^{1}-x_{T}^{2}}{2}\right)+\int_{0}^{T}\bigg{(}\left(\frac{x_{t}^{1}-x_{t}^{2}}{2}\right)^{\prime}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
R(t,\alpha_{t})\left(\frac{x_{t}^{1}-x_{t}^{2}}{2}\right)+\left(\frac{u_{t}^{1}-u_{t}^{2}}{2}\right)^{\prime}N(t,\alpha_{t})\left(\frac{u_{t}^{1}-u_{t}^{2}}{2}\right)\bigg{)}dt\bigg{)}$
$\displaystyle\geq$ $\displaystyle\
2J\left(\frac{u^{1}+u^{2}}{2}\right)+E\int_{0}^{T}\left(\frac{u_{t}^{1}-u_{t}^{2}}{2}\right)^{\prime}N(t,\alpha_{t})\left(\frac{u_{t}^{1}-u_{t}^{2}}{2}\right)dt$
$\displaystyle\geq$ $\displaystyle\
2\gamma+\frac{\delta}{4}E\int_{0}^{T}|u_{t}^{1}-u_{t}^{2}|^{2}dt$
Thus $E\int_{0}^{T}|u_{t}^{1}-u_{t}^{2}|^{2}dt\leq 0$ which yields that
$u^{1}=u^{2}$. ∎
### 2.2 BSDEs with Markov chains
The derivation of the optimal control for the above LQ problem (1) can be
regarded as one motivation for us to study BSDEs with Markov jumps. In this
subsection, we will study the solvability to the following BSDE with a Markov
chain firstly:
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s},\alpha_{s})ds-\int_{t}^{T}Z_{s}dB_{s}.$
(3)
Let ${\cal{N}}$ denote the class of all $P$-null sets of ${\cal{F}}$. For each
$t\in[0,T]$, we define
${\cal{F}}_{t}={\cal{F}}_{t}^{B}\vee{\cal{F}}_{t,T}^{\alpha}\vee{\cal{N}}$
where for any process $\\{\eta_{t};0\leq t\leq T\\}$,
${\cal{F}}_{t,T}^{\eta}=\sigma\\{\eta_{r};t\leq r\leq T\\}$ and
${\cal{F}}_{t}^{\eta}={\cal{F}}_{0,t}^{\eta}$. For convenience, we denote
$M^{2}(0,T;R^{n})=M_{\mathcal{F}_{t}}^{2}(0,T;R^{n})$ and $S^{2}(0,T;$
$R^{n})=$ $S_{\mathcal{F}_{t}}^{2}(0,$ $T;R^{n})$.
Thereinafter, we make the following assumption:
###### Assumption 2.1.
(i) $\xi\in L^{2}({\cal{F}}_{T};R^{k})$; (ii) $f:\Omega\times[0,T]\times
R^{k}\times R^{k\times d}\times\mathcal{M}\rightarrow R^{k}$ satisfies that
$\forall(y,z)\in R^{k}\times R^{k\times d}$, $\forall i\in\mathcal{M}$,
$f(\cdot,y,z,i)\in M^{2}_{\mathcal{F}_{t}^{B}}(0,T;R^{k})$, and
$\exists\mu>0$, such that $\forall i\in\mathcal{M}$,
$\forall(\omega,t)\in\Omega\times[0,T]$, $(y_{1},z_{1})$, $(y_{2},z_{2})\in
R^{k}\times R^{k\times d}$,
$|f(t,y_{1},z_{1},i)-f(t,y_{2},z_{2},i)|\leq\mu(|y_{1}-y_{2}|+|z_{1}-z_{2}|).$
Our main result in this section is in the following theorem.
###### Theorem 2.2.
Under Assumption 2.1, there exists a unique solution pair $(Y,Z)\in
S^{2}(0,T;R^{k})\times M^{2}(0,T;R^{k\times d})$ for BSDE $(\ref{eq:BSDE})$.
The proof of Theorem 2.2 consists of three steps.
Step 1: Extension of Itô’s representation theorem
It is noted that $\\{{\cal{F}}_{t};0\leq t\leq T\\}$ is neither increasing nor
decreasing, and it does not constitute a filtration. Inspired by the method
handling the BSDE with doubly Brown motions ([19]), we define a filtration
$({\mathcal{G}}_{t})_{0\leq t\leq T}$ by
${\cal{G}}_{t}\triangleq{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\vee\cal{N}$
For the filtration $({\mathcal{G}}_{t})_{0\leq t\leq T}$, we give the
following extension of Itô’s representation theorem. This result and its
corollary play key roles during the proof of Theorem 2.2.
###### Proposition 2.1.
For $N\in L^{2}(\mathcal{G}_{T};R^{k})$, there exist a unique random variable
$N_{0}\in L^{2}(\mathcal{F}_{T}^{\alpha};R^{k})$ and a unique stochastic
process $Z=\\{Z_{t};0\leq t\leq T\\}\in M^{2}_{\mathcal{G}_{t}}(0,T;R^{k\times
d})$ such that
$N=N_{0}+\int_{0}^{T}Z_{t}dB_{t},\quad 0\leq t\leq T.$ (4)
Actually, $N_{0}=E(N|\mathcal{F}_{T}^{\alpha})$.
During the derivation of Proposition 2.1, we need the following two lemmas.
###### Lemma 2.1.
([22]) If $X$, $Y$: $\Omega\rightarrow R^{d}$ are two given functions, $Y$ is
$\sigma(X)$-measurable if and only if there exists a Borel measurable function
$g$: $R^{d}\rightarrow R^{d}$ such that $Y=g(X)$.
###### Lemma 2.2.
(Doob’s martingale convergence theorem) Let $\\{\mathcal{F}_{t};t\geq 0\\}$ be
a filtration on the space $(\Omega,\mathcal{F},P)$, $X\in
L^{1}(\mathcal{F};R^{d})$, then
$E(X|\mathcal{F}_{t})\rightarrow E(X|\mathcal{F}_{\infty}),\quad\mbox{as
}t\rightarrow\infty,\textrm{ a.s. and in $L^{1}$ as well.}$
###### Proof.
Existence: Let $\\{t_{i}\\}_{i\geq 0}$, $\\{t^{\prime}_{j}\\}_{j\geq 0}$ be
two dense subsets of $[0,T]$ where $t_{0}=t^{\prime}_{0}=0$. For each integer
$n,m\geq 0$, let $\mathcal{G}_{n,m}$ be the $\sigma$-algebra generated by
$\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}},B_{t^{\prime}_{0}},B_{t^{\prime}_{1}},$
$\cdots,B_{t^{\prime}_{m}}$, i.e.,
$\mathcal{G}_{n,m}=\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}},B_{t^{\prime}_{0}},B_{t^{\prime}_{1}},$
$\cdots,B_{t^{\prime}_{m}}\\}$. Obviously,
$\mathcal{G}_{n,m}\subset\mathcal{G}_{n+1,m}$,
$\mathcal{G}_{n,m}\subset\mathcal{G}_{n,m+1}$,
$\mathcal{G}_{n,m}\subset\mathcal{G}_{n+1,m+1}$,
$\displaystyle\sigma(\cup_{m=1}^{\infty}\mathcal{G}_{n,m})=\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}}\\},$
and $\displaystyle\sigma(\cup_{n,m=0}^{\infty}$ $\mathcal{G}_{n,m})$
$=\mathcal{G}_{T}$.
For $N\in L^{2}(\mathcal{G}_{T};R^{k})$, $\forall n,m$, by Lemma 2.1 and Lemma
2.2, there exists a Borel measurable function $N_{n,m}:\mathcal{M}^{n+1}\times
R^{(m+1)\times d}\rightarrow R^{k}$ such that
$\begin{split}&E(N|\mathcal{G}_{n,m})=N_{n,m}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}},B_{t^{\prime}_{0}},B_{t^{\prime}_{1}},\cdots,B_{t^{\prime}_{m}})\\\
&N_{n,m}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}},B_{t^{\prime}_{0}},B_{t^{\prime}_{1}},\cdots,B_{t^{\prime}_{m}})\rightarrow
E\left(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}}\\}\right)\end{split}$
Denote $N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})\triangleq
E\left(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}}\\}\right)$,
it can be rewritten as
$N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}N_{n}(i_{0},i_{1},\cdots,i_{n})$
where $N_{n}(i_{0},i_{1},\cdots,i_{n})$ is $\mathcal{F}_{T}^{B}$-measurable.
For $(i_{0},i_{1},\cdots,i_{n})\in\mathcal{M}^{n+1}$, applying Itô’s
representation theorem,
$N_{n}(i_{0},i_{1},\cdots,i_{n})=N_{0}{(i_{0},i_{1},\cdots,i_{n})}+\int_{0}^{T}Z_{t}{(i_{0},i_{1},\cdots,i_{n})}dB_{t}$
where $N_{0}(i_{0},i_{1},$ $\cdots,i_{n})$ is a constant and
$Z{(i_{0},i_{1},\cdots,i_{n})}\in M^{2}_{\mathcal{F}_{t}^{B}}(0,T;R^{k\times
d})$.
Denote
$\displaystyle
N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}N_{0}{(i_{0},i_{1},\cdots,i_{n})}$,
and $\displaystyle
Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}Z_{t}{(i_{0},i_{1},\cdots,i_{n})}$.
Clearly, $N_{0}(\alpha_{t_{0}},$ $\alpha_{t_{1}},\cdots,\alpha_{t_{n}})$ $\in
L^{2}(\mathcal{F}_{T}^{\alpha};R^{k})$, $\\{Z_{t}(\alpha_{t_{0}},$
$\alpha_{t_{1}},\cdots,\alpha_{t_{n}});0\leq t\leq T\\}\in
M^{2}_{\mathcal{G}_{t}}(0,T;R^{k\times d})$, and we have
$\displaystyle
N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})+\int_{0}^{T}Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})dB_{t}.$
(5)
In the remained part, we will prove that as $n\rightarrow\infty$, both side of
(5) are Cauchy sequences.
For the left hand side, as $n\rightarrow\infty$, with the definition of
$N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})$ and Lemma 2.2, we
have
$N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})\rightarrow N.$
For the right hand side, since
$\displaystyle
E\left(\int_{0}^{T}Z(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})dB_{t}|\mathcal{F}_{T}^{\alpha}\right)$
(6) $\displaystyle=$
$\displaystyle\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}E\left(\int_{0}^{T}Z{(i_{0},i_{1},\cdots,i_{n})}dB_{t}|\mathcal{F}_{T}^{\alpha}\right)$
$\displaystyle=$ $\displaystyle 0.$
Thus
$N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})=E(N_{n}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})|\mathcal{F}_{T}^{\alpha})$.
As $n\rightarrow\infty$, we can conclude that
$N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})\rightarrow\
E(N|\mathcal{F}_{T}^{\alpha}).$
Now let us consider the sequence
$\\{Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}});0\leq t\leq
T\\}$. For $n,m\geq 0$,
$\begin{split}&E\int_{0}^{T}\left|Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})-Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}})\right|^{2}dt\\\
=\
&E\left(\int_{0}^{T}\left(Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})-Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}})\right)dB_{t}\right)^{2}\\\
=\
&E\big{(}E\left(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}}\\}\right)-E\left(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}}\\}\right)\\\
&-N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})+N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}})\big{)}^{2}\\\
\leq\
&2E\big{(}E(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}}\\})^{2}-E(N|\mathcal{F}_{T}^{B}\vee\sigma\\{\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}}\\})\big{)}^{2}\\\
&+2E(N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}})-N_{0}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{m}})\big{)}^{2}\\\
\rightarrow\ &0,\quad\textrm{ as }n,m\rightarrow\infty.\end{split}$
Thus $\\{Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}});0\leq
t\leq T\\}$ is a Cauchy sequence in $M^{2}_{\mathcal{G}_{t}}(0,T;$ $R^{k\times
d})$. Hence it converges to some $Z\in M^{2}_{\mathcal{G}_{t}}(0,T;R^{k\times
d})$.
Denote $N_{0}=E(N|\mathcal{F}_{T}^{\alpha})$, we can conclude that $(\ref{eq1:
formula 1 in extension of Ito})$ converges to the extended Itô’s
representation (4).
Uniqueness: By virtue of equation (6) and the fact that as
$n\rightarrow\infty$,
$\\{Z_{t}(\alpha_{t_{0}},\alpha_{t_{1}},\cdots,\alpha_{t_{n}});$ $0\leq t\leq
T\\}$ is a Cauchy sequence, we have
$E(\int_{0}^{T}Z_{t}dB_{t}|\mathcal{F}_{T}^{\alpha})$ $=0$. Then, for
$(N_{0},Z),(N_{0}^{\prime},Z^{\prime})$ satisfying the extended Itô’s
representation $(\ref{eq: extension of Ito representation})$, we get
$N_{0}=N^{\prime}_{0}$ by taking conditional expectation with respect to
$\mathcal{F}_{T}^{\alpha}$. Uniqueness of $Z$ follows easily from the fact
that
$E\int_{0}^{T}|Z_{t}-Z^{\prime}_{t}|^{2}dt=E\left(\int_{0}^{T}(Z_{t}-Z^{\prime}_{t})dB_{t}\right)^{2}=E\left(N_{0}-N_{0}^{\prime}\right)^{2}=0.$
∎
The following corollary is useful in the proof of Theorem 2.2 and its proof is
similar to Proposition 2.1.
###### Corollary 2.1.
For $t\leq T$, we consider the filtration $(\mathcal{N}_{s})_{t\leq s\leq T}$
defined by
$\mathcal{N}_{s}=\mathcal{F}_{s}^{B}\vee\mathcal{F}_{t,T}^{\alpha}$. For $N\in
L^{2}(\mathcal{N}_{T};R^{k})$, there exists a unique stochastic process
$Z=\\{Z_{s};t\leq s\leq T\\}\in M^{2}_{\mathcal{N}_{s}}(t,T;R^{k\times d})$
such that
$N=E(N|\mathcal{N}_{t})+\int_{t}^{T}Z_{s}dB_{s}.$
Step 2: The special case: the generator $f$ is independent of $y$ and $z$.
###### Proposition 2.2.
Under Assumption 2.1, the following BSDE
$Y_{t}=\xi+\int_{t}^{T}f(s,\alpha_{s})ds-\int_{t}^{T}Z_{s}dB_{s},\qquad 0\leq
t\leq T$ (7)
has a solution pair $(Y,Z)\in S^{2}(0,T;R^{k})\times M^{2}(0,T;R^{k\times
d})$.
###### Proof.
From Assumption 2.1 and Hölder inequality, we obtain
$E\left(\int_{0}^{T}f(s,\alpha_{s})ds\right)^{2}\leq
CE\int_{0}^{T}|f(s,\alpha_{s})|^{2}ds\leq
C\sum_{i=1}^{m}E\int_{0}^{T}|f(s,i)|^{2}ds<\infty$
which yields
$\xi+\int_{0}^{T}f(s,\alpha_{s})ds\in L^{2}({{\cal{G}}_{T}};R^{k})$
For the filtration $({\mathcal{G}}_{t})_{0\leq t\leq T}$ where
${\cal{G}}_{t}={\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\vee{\cal{N}}={\cal{F}}_{t}^{B}\vee{\cal{F}}_{t,T}^{\alpha}\vee{\cal{F}}_{t}^{\alpha}\vee{\cal{N}}={\cal{F}}_{t}\vee{\cal{F}}_{t}^{\alpha},$
we can define the following ${\cal{G}}_{t}$-measurable square integrable
martingale
$N_{t}=E\left(\xi+\int_{0}^{T}f(s,\alpha_{s})ds|{{\cal{G}}_{t}}\right),\qquad
0\leq t\leq T.$
By the extended Itô’s representation theorem (Proposition 2.1), there exist
$N_{0}\in L^{2}(\mathcal{F}_{T}^{\alpha};R^{k})$ and $Z=\\{Z_{t};0\leq t\leq
T\\}\in M^{2}_{{\cal{G}}_{t}}(0,T;R^{k\times d})$ such that
$N_{t}=N_{0}+\int_{0}^{t}Z_{s}dB_{s},\qquad 0\leq t\leq T.$
For $t\in[0,T]$, we define
$\begin{split}Y_{t}=N_{t}-\int_{0}^{t}f(s,\alpha_{s})ds,\textrm{ \ i.e.,
}Y_{t}=E\left(\xi+\int_{t}^{T}f(s,\alpha_{s})ds|{{\cal{G}}_{t}}\right).\end{split}$
(8)
It is easy to verify that the ${\cal{G}}_{t}$-measurable process $(Y,Z)$
satisfies BSDE $(\ref{eq:BSDE f independent of y and z})$ and $Y\in
M^{2}_{{\cal{G}}_{t}}(0,T;$ $R^{k})$. We refer interested reader to [17, 16]
for the detailed verification.
The left work is to show that the processes $Y=\\{Y_{t};0\leq t\leq T\\}$ and
$Z=\\{Z_{t};0\leq t\leq T\\}$ are ${\cal{F}}_{t}$-measurable, i.e.
${\cal{F}}_{t}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable. $\forall
t\in[0,T]$, we denote $\vartheta=\xi+\int_{t}^{T}f(s,\alpha_{s})ds$,
$\vartheta$ is ${\cal{F}}_{T}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable.
Let $\\{\bar{t}_{i}\\}_{i\geq 0}$, $\\{\bar{t}^{\prime}_{j}\\}_{j\geq 0}$ be
respectively dense subsets of $[t,T]$ and $[0,T]$, with $\bar{t}_{0}=t$ and
$\bar{t}^{\prime}_{0}=0$. For each integer $n,m\geq 0$, let
$\bar{\mathcal{G}}_{n,m}$ be the $\sigma$-algebra generated by
$\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}}$,
i.e.,
$\bar{\mathcal{G}}_{n,m}=\sigma\\{\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}},$
$B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}}\\}$.
Obviously, $\bar{\mathcal{G}}_{n,m}\subset\bar{\mathcal{G}}_{n+1,m+1}$,
$\bar{\mathcal{G}}_{n,m}\subset\bar{\mathcal{G}}_{n+1,m}$,
$\bar{\mathcal{G}}_{n,m}\subset\bar{\mathcal{G}}_{n,m+1}$, and
$\sigma(\cup_{n,m=0}^{\infty}\bar{\mathcal{G}}_{n,m})={\cal{F}}_{T}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$.
From Lemma 2.1, for each $n,m$, there exists a Borel measurable function
$\vartheta_{nm}:\mathcal{M}^{n+1}\times R^{(m+1)\times d}\rightarrow R^{k}$
such that
$\begin{split}E[\vartheta|\bar{\mathcal{G}}_{n,m}]&=\vartheta_{nm}(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}}).\end{split}$
Since
$I_{\\{(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}\in{\cal{F}}_{t,T}^{\alpha}\subset{\cal{F}}_{T}^{\alpha}$,
we have
$\begin{split}&E\left(\vartheta_{nm}(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\right)\\\
=&\
E\Bigg{(}\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}\\\
&\hskip
28.45274pt\vartheta_{nm}(i_{0},i_{1},\cdots,i_{n},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\Bigg{)}\\\
=&\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}\\\
&\hskip
28.45274ptE(\vartheta_{nm}(i_{0},i_{1},\cdots,i_{n},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}).\end{split}$
For the reason that
$\vartheta_{nm}(i_{0},i_{1},\cdots,i_{n},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})$
is $\mathcal{F}_{T}^{B}$-measurable, by Itô’s representation theorem, we know
that there exist $\nu_{0}^{n,m}(i_{0},i_{1},\cdots,i_{n})$ $\in R^{k}$ and
$Z^{n,m}(i_{0},i_{1},\cdots,i_{n})\in$ $L^{2}_{\mathcal{F}_{t}^{B}}(0,T;$
$R^{k\times d})$ such that
$\begin{split}&E\left(\vartheta_{nm}(i_{0},i_{1},\cdots,i_{n},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\right)\\\
=&E\left(\nu_{0}^{n,m}(i_{0},i_{1},\cdots,i_{n})+\int_{0}^{T}Z_{r}^{n,m}(i_{0},i_{1},\cdots,i_{n})dB_{r}|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\right)\\\
=&\nu_{0}^{n,m}(i_{0},i_{1},\cdots,i_{n})+\int_{0}^{t}Z_{r}^{n,m}(i_{0},i_{1},\cdots,i_{n})dB_{r}\end{split}$
is ${\cal{F}}_{t}^{B}$-measurable. Therefore
$\begin{split}&E\left(\vartheta_{nm}(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\right)\\\
=&\sum_{i_{0},i_{1},\cdots,i_{n}=1}^{m}I_{\\{(\alpha_{\bar{t}_{0}},\alpha_{\bar{t}_{1}},\cdots,\alpha_{\bar{t}_{n}})=(i_{0},i_{1},\cdots,i_{n})\\}}\\\
&\hskip
28.45274ptE\left(\vartheta_{nm}(i_{0},i_{1},\cdots,i_{n},B_{\bar{t}^{\prime}_{0}},B_{\bar{t}^{\prime}_{1}},\cdots,B_{\bar{t}^{\prime}_{m}})|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}\right)\end{split}$
is ${\cal{F}}_{t}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable. With Lemma 2.2,
as $n,m\rightarrow\infty$,
$E[\vartheta|\bar{\mathcal{G}}_{n,m}]\rightarrow
E[\vartheta|{\cal{F}}_{T}^{B}\vee{\cal{F}}_{t,T}^{\alpha}]=\vartheta.$
Thus $E[\vartheta|{\cal{F}}_{t}^{B}\vee{\cal{F}}_{T}^{\alpha}]$, i.e.,
$Y_{t}$, is also ${\cal{F}}_{t}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable.
Considering
$\int_{t}^{T}Z_{s}dB_{s}=-Y_{t}+\xi+\int_{t}^{T}f(s,\alpha_{s})ds,$
its right side is ${\cal{F}}_{T}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable.
With Corollary 2.1, we know $\forall t<s$, $Z_{s}$ is
${\cal{F}}_{s}^{B}\vee{\cal{F}}_{t,T}^{\alpha}$-measurable. Then, by the
continuous property of the Markov chain $\alpha$, we obtain that $Z_{s}$ is
${\cal{F}}_{s}^{B}\vee{\cal{F}}_{s,T}^{\alpha}$-measurable.
Together with the Burkholder-Davis-Gundy inequality and the form of BSDE
$(\ref{eq:BSDE f independent of y and z})$, we can conclude that
$\\{Y_{t};0\leq t\leq T\\}$ is continuous and satisfies $\displaystyle
E(\sup_{0\leq t\leq T}|Y_{t}|^{2})<\infty$. It yields that $Y\in
S^{2}(0,T;R^{k})$. ∎
Step 3: The general case: Proof of Theorem 2.2.
###### Proof.
Firstly, we define a mapping $I$ from $M^{2}(0,T;R^{k}\times R^{k\times d})$
into itself such that $(Y,Z)\in S^{2}(0,T;R^{k})\times M^{2}(0,T;R^{k\times
d})$ is the solution to BSDE $(\ref{eq:BSDE})$ iff it is a fixed point of $I$.
For a constant $\beta>0$, we introduce the following equivalent norm of
$M^{2}(0,T;R^{k}\times R^{k\times d})$
$\|v(\cdot)\|_{\beta}=\left(E\int_{0}^{T}|v_{s}|^{2}e^{\beta
s}ds\right)^{\frac{1}{2}}.$
For $(y,z)\in M^{2}(0,T;R^{k}\times R^{k\times d})$, we set
$Y_{t}=\xi+\int_{t}^{T}f(s,y_{s},z_{s},\alpha_{s})ds-\int_{t}^{T}Z_{s}dB_{s}.$
From Assumption 2.1 and H$\ddot{\textrm{o}}$lder’s inequality,
$\begin{split}&E\left(\int_{0}^{T}f(s,y_{s},z_{s},\alpha_{s})ds\right)^{2}\\\
\leq&\
2E\left(\int_{0}^{T}(f(s,y_{s},z_{s},\alpha_{s})-f(s,0,0,\alpha_{s}))ds\right)^{2}+2E\left(\int_{0}^{T}f(s,0,0,\alpha_{s})ds\right)^{2}\\\
\leq&\
C\left(E\int_{0}^{T}\left(|y_{s}|^{2}+|z_{s}|^{2}\right)ds+\sum_{i=1}^{m}E\int_{0}^{T}|f(s,0,0,i)|^{2}ds\right)<\infty\end{split}$
which yields that
$\xi+\int_{0}^{T}f(s,y_{s},z_{s},\alpha_{s})ds\in
L^{2}({{\cal{G}}_{T}};R^{k}).$
From Proposition 2.2, we can define the following contraction mapping under
the norm $\|\cdot\|_{\beta}$
$I((y,z))=(Y,Z):M^{2}(0,T;R^{k}\times R^{k\times d})\rightarrow
M^{2}(0,T;R^{k}\times R^{k\times d}).$
The proof of contraction property is similar to [17, 10, 16]. For the
compactness of the paper, the detail is omit here.
Together with the form of BSDE $(\ref{eq:BSDE})$ and Burkholder-Davis-Gundy
inequality, $Y\in S^{2}(0,T;R^{k})$. Thus, by the fixed point theorem, we know
that BSDE $(\ref{eq:BSDE})$ has a unique solution pair. ∎
## 3 BSDEs with Singularly Perturbed Markov Chains
In this section, after recalling several relevant results of singularly
perturbed Markov chains given by Zhang and Yin ([23]), we will consider the
asymptotic property of BSDE with a singularly perturbed Markov chain.
Following the averaging approach to aggregate the states according to their
jump rates and replace the actual coefficient with its average with respect to
the quasi stationary distributions of the singularly perturbed Markov chain,
we get the asymptotic probability distribution of the solution to the BSDE
with an limit averaged Markov chain which has a much smaller state space than
the original one.
### 3.1 Relevant results of singularly perturbed Markov chains
Focused on a continuous-time $\varepsilon$-dependent singularly perturbed
Markov chain $\alpha^{\varepsilon}=\\{\alpha^{\varepsilon}_{t};0\leq t\leq
T\\}$ which have the generator $\displaystyle
Q^{\varepsilon}=\frac{1}{\varepsilon}\tilde{Q}+\hat{Q},$ where $\tilde{Q}$ and
$\hat{Q}$ are time-invariant generators, with
$\tilde{Q}=\textrm{diag}(\tilde{Q}^{1},\cdots,\tilde{Q}^{l})$. The state space
can be decomposed as
${\cal{M}}=\\{1,2,\cdots,m\\}={\cal{M}}_{1}\cup\cdots\cup{\cal{M}}_{l},$
${\cal{M}}_{k}=\\{s_{k1},\cdots,s_{km_{k}}\\}$, and for
$k\in\\{1,\cdots,l\\}$, $\tilde{Q}^{k}$ is the weakly irreducible
generator333A generator $Q$ is called weakly irreducible if the system of
equations $\nu Q=0$ and $\sum_{i=1}^{m}\nu_{i}=1$ has a unique nonnegative
solution. This nonnegative solution $\nu=(\nu_{1},\cdots,\nu_{m})$ is called
the quasi-stationary distribution of $Q$. corresponding to the states in
${\cal{M}}_{k}$. The generator $\tilde{Q}$ dictates the fast motion of the
Markov chain and $\hat{Q}$ governs the slow motion, i.e., the underlying
Markov chain fluctuates rapidly in a single group $\mathcal{M}_{k}$ and jumps
less frequently among groups $\mathcal{M}_{k}$ and $\mathcal{M}_{j}$ for
$k\neq j$.
As shown in [23], when the states in $\mathcal{M}_{k}$ are lumped into a
single state, all such states are coupled by $\hat{Q}$. By defining
$\bar{\alpha}^{\varepsilon}_{t}=k$, when
$\alpha^{\varepsilon}_{t}\in{\cal{M}}_{k}$, we can obtain the aggregated
process $\bar{\alpha}^{\varepsilon}=\\{\bar{\alpha}^{\varepsilon}_{t};0\leq
t\leq T\\}$ containing $l$ states. The process $\bar{\alpha}^{\varepsilon}$ is
not necessarily Markovian, but it converges weakly to a continuous-time Markov
chain $\bar{\alpha}$.
###### Proposition 3.1.
([23]) (i) $\bar{\alpha}^{\varepsilon}$ converges weakly to $\bar{\alpha}$
generated by
$\bar{Q}=\textrm{diag}(\nu^{1},\cdots,\nu^{l})\hat{Q}\textrm{diag}(\mathbb{I}_{m_{1}},\cdots,\mathbb{I}_{m_{l}})$
as $\varepsilon\rightarrow 0$, where $\nu^{k}$ is the quasi-stationary
distribution of $\tilde{Q}^{k}$, $k=1,\cdots,l$, and
$\mathbb{I}_{k}=(1,\cdots,1)^{\prime}\in R^{k}.$
(ii) For any bounded deterministic function $\beta(\cdot)$,
$E\left(\int_{s}^{T}(I_{\\{\alpha^{\varepsilon}_{t}=s_{kj}\\}}-\nu_{j}^{k}I_{\\{\bar{\alpha}^{\varepsilon}_{t}=k\\}})\beta(t)dt\right)^{2}=O(\varepsilon),\forall\
k=1,\cdots,l,\forall\ j=1,\cdots,m_{k}.$
Here $I_{A}$ is the indicator function of a set $A$.
### 3.2 Weak convergence of BSDEs with singularly perturbed Markov chains
In this subsection, denote $D(0,T;R^{k})$ as the Skorohod space of
c$\grave{a}$dl$\grave{a}$g trajectories endowed with the Jakubowski S-topology
([9]) which is weaker than the Skorohod topology. As shown in the appendix of
[2], the tightness criteria under this S-topology is the same as the “Meyer-
Zheng tightness criteria” used in [14].
Here, we only consider the asymptotic property of the solution to the
following BSDE with a singularly perturbed Markov chain where the generator
$f$ does not depend on $Z^{\varepsilon}$,
$Y^{\varepsilon}_{t}=\xi+\int_{t}^{T}f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})ds-\int_{t}^{T}Z^{\varepsilon}_{s}dB_{s},$
(9)
For the difficulty to study the general case that the generator $f$ depends on
$Z^{\varepsilon}$, we refer interested reader to the explanation in section 6
of [16].
Firstly, we make the following assumption:
###### Assumption 3.1.
(i) $\xi\in L^{2}(\mathcal{F}_{T}^{B};R^{k}).$ (ii) For $f:[0,T]\times
R^{k}\times\mathcal{M}\rightarrow R^{k}$, there exists a constant $C>0$ such
that $\displaystyle\sup_{\begin{subarray}{c}0\leq t\leq T\\\ 1\leq i\leq
m\end{subarray}}|f(t,0,i)|\leq C$.
###### Theorem 3.3.
Under Assumption 2.1 and Assumption 3.1, the sequence of process
$(Y^{\varepsilon}_{t},$ $\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s})$ converges in
distribution to the process $(Y_{t},\int_{0}^{t}Z_{s}d\bar{B}_{s})$ as
$\varepsilon\rightarrow 0$, when probability measures on $D(0,T;R^{2k})$
equipped with the Jakubowski S-topology. Here $(Y,Z)$ is the solution pair to
the following BSDE with the limit averaged Markov chain
$Y_{t}=\xi+\int_{t}^{T}\bar{f}(s,Y_{s},\bar{\alpha}_{s})ds-\int_{t}^{T}Z_{s}d\bar{B}_{s},$
(10)
$\bar{B}=\\{\bar{B}_{t};0\leq t\leq T\\}$ with $\bar{B}_{0}=0$ is a
$d$-dimensional Brownian motion, $\bar{\alpha}$ is defined in subsection 3.1,
and $\displaystyle\bar{f}(s,y,i)=\sum_{j=1}^{m_{i}}\nu^{i}_{j}f(t,y,s_{ij})$
for $i\in\bar{{\cal{M}}}=\\{1,\cdots,l\\}$.
###### Remark 3.1.
It is obvious that the limit BSDE depends on the limit averaged Markov chain
$\bar{\alpha}$ with a state space much smaller than that of the original
singularly perturbed Markov chain $\alpha^{\varepsilon}$. Moreover, as
$\varepsilon\rightarrow 0$, the
$\mathcal{F}^{\alpha^{\varepsilon}}_{T}$-measurable random variables sequence
$(Y^{\varepsilon}_{0})$ converges in distribution to the random variable
$Y_{0}$ which is $\mathcal{F}^{\bar{\alpha}}_{T}$-measurable.
For the proof of Theorem 3.3, we follow a classical approach as in [20, 2] to
prove the weak convergence of BSDE: after showing the tightness and
convergence for $(Y_{t}^{\varepsilon},\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s})$,
we identify the limit.
Step 1: Tightness and convergence for
$(Y_{t}^{\varepsilon},\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s})$.
###### Proposition 3.2.
Under Assumption 2.1 and Assumption 3.1, BSDE $(\ref{eq:BSDE with perturbed})$
and BSDE $(\ref{eq:limit BSDE })$ have unique solutions
$(Y^{\varepsilon},Z^{\varepsilon})$ and $(Y,Z)\in S^{2}(0,T;R^{k})$ $\times
M^{2}(0,T;R^{k\times d})$. Moreover, there exists a positive constant $C$ such
that $\forall\varepsilon>0$,
$\begin{split}E\left(\sup_{0\leq t\leq
T}|Y^{\varepsilon}_{t}|^{2}+\int_{0}^{T}(Z^{\varepsilon}_{t})^{2}dt\right)&\leq
C,\\\ E\left(\sup_{0\leq t\leq
T}|Y_{t}|^{2}+\int_{0}^{T}(Z_{t})^{2}dt\right)&\leq C.\end{split}$
###### Proof.
For BSDE $(\ref{eq:BSDE with perturbed})$, by Theorem 2.2, the existence and
uniqueness of solution $(Y^{\varepsilon},Z^{\varepsilon})$ is obtained for all
$\varepsilon>0$.
Using Itô’s formula to $|Y^{\varepsilon}_{s}|^{2}$ on $[t,T]$, we get the
following from Schwartz’s inequality,
$\begin{split}&|Y^{\varepsilon}_{t}|^{2}+\int_{t}^{T}|Z^{\varepsilon}_{s}|^{2}ds\\\
=&\
|\xi|^{2}+2\int_{t}^{T}Y^{\varepsilon}_{s}f(s,Y^{\varepsilon}_{s},Z^{\varepsilon}_{s},\alpha_{s})ds-2\int_{t}^{T}Y^{\varepsilon}_{s}Z^{\varepsilon}_{s}dB_{s}\\\
\leq&\
|\xi|^{2}+2\int_{t}^{T}\left((1+\mu^{2})|Y^{\varepsilon}_{s}|^{2}+|f(s,0,0,\alpha^{\varepsilon}_{s})|^{2}\right)ds-2\int_{t}^{T}Y^{\varepsilon}_{s}Z^{\varepsilon}_{s}dB_{s}\end{split}$
here $\mu$ is the Lipschitz constant of $f$ which is independent of
$\varepsilon$. By taking expectation, we can deduce
$E\left(|Y^{\varepsilon}_{t}|^{2}+\frac{1}{2}\int_{t}^{T}|Z^{\varepsilon}_{s}|^{2}ds\right)\leq|\xi|^{2}+2\int_{t}^{T}((1+\mu^{2})|Y^{\varepsilon}_{s}|^{2}+|f(s,0,0,\alpha^{\varepsilon}_{s})|^{2})ds.$
From Gronwall’s lemma, we get
$\displaystyle
E\left(|Y^{\varepsilon}_{t}|^{2}+\int_{t}^{T}|Z^{\varepsilon}_{s}|^{2}ds\right)\leq
CE\left(|\xi|^{2}+\int_{0}^{T}|f(s,0,0,\alpha^{\varepsilon}_{s})|^{2}ds\right)\leq
C,$
and then the estimation for $(Y^{\varepsilon},Z^{\varepsilon})$ is obtained
from the Burkholder-Davis-Gundy inequality.
From the form of $\bar{f}$ presented in Theorem 3.3, we know that $\bar{f}$
also satisfies Assumption 2.1 and Assumption 3.1, thus the estimation about
$(Y,Z)$ can be obtained similarly. ∎
We set $M_{t}^{\varepsilon}=\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s}$ for the
convenience. Thus BSDE $(\ref{eq:BSDE with perturbed})$ can be rewritten as
$Y^{\varepsilon}_{t}=\xi+\int_{t}^{T}f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})ds-(M_{T}^{\varepsilon}-M_{t}^{\varepsilon}).$
(11)
###### Proposition 3.3.
The sequence of $(Y^{\varepsilon},M^{\varepsilon})$ is tight on the space
$D(0,T;$ $R^{k})\times D(0,T;R^{k}).$
###### Proof.
Let
${\cal{G}}_{t}^{\varepsilon}={\cal{F}}^{B}_{t}\vee{\cal{F}}_{T}^{\alpha^{\varepsilon}}\vee{\cal{N}},$
we define the conditional variation
$CV(Y^{\varepsilon})=\sup
E\left(\sum_{i}|E(Y^{\varepsilon}_{t_{i+1}}-Y^{\varepsilon}_{t_{i}}|{\cal{G}}^{\varepsilon}_{t_{i}})|\right)$
where the supreme is taken over all partitions of the interval $[0,T]$.
From the Proof of Proposition 2.2, we know that $M^{\varepsilon}$ is a
${\cal{G}}_{t}^{\varepsilon}$-martingale. It follows that
$CV(Y^{\varepsilon})\leq
E\int_{0}^{T}|f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})|ds.$
From $(ii)$ of Assumption 2.1, $(ii)$ of Assumption 3.1, and Proposition 3.2,
we know
$\sup_{\varepsilon}\left(CV(Y^{\varepsilon})+\sup_{0\leq t\leq
T}E|Y^{\varepsilon}_{t}|+\sup_{0\leq t\leq
T}E|M^{\varepsilon}_{t}|\right)<\infty.$
Thus the “Meyer-Zheng tightness criteria” ([2, 14]) is fully satisfied and the
result is followed. ∎
Together with the properties of $Y^{\varepsilon}$ obtained above, the
following proposition can be seen as an obvious result of Lemma 7.3 in [25].
###### Proposition 3.4.
Suppose $g(t,x)$ is a function defined on $[0,T]\times R^{m}$ satisfying that
$g(\cdot,\cdot)$ is Lipschitz continuous with $x$ and $\forall x\in R^{m}$,
either $|g(t,x)|\leq K(1+|x|)$ or $|g(t,x)|\leq K$. Denote
$\pi_{ij}^{\varepsilon}(t)=\pi_{ij}^{\varepsilon}(t,\alpha^{\varepsilon}_{t})$,
with
$\pi_{ij}^{\varepsilon}(t,\alpha)=I_{\\{\alpha=s_{ij}\\}}-\nu_{j}^{i}I_{\\{\alpha\in
M_{i}\\}}$, then for any $\ k=1,\cdots,l,j=1,\cdots,m_{k}$,
$\sup_{0<t\leq
T}E\left|\int_{0}^{t}g(t,Y_{s}^{\varepsilon})\pi_{ij}^{\varepsilon}(s,\alpha^{\varepsilon}_{s})ds\right|\rightarrow
0,\textrm{\quad as }\varepsilon\rightarrow 0.$
Step 2: Identification of the limit.
From Proposition 3.3, we know that there exists a subsequence of
$(Y^{\varepsilon},M^{\varepsilon})$, which we still denote by
$(Y^{\varepsilon},M^{\varepsilon})$, and which converges in distribution on
the space $D(0,T;R^{k})\times D(0,T;R^{k})$ toward a
c$\grave{a}$dl$\grave{a}$g process $(\bar{Y},\bar{M})$. Furthermore, there
exists a countable subset $D$ of $[0,T]$, such that
$(Y^{\varepsilon},M^{\varepsilon})$ converges in finite-distribution to
$(\bar{Y},\bar{M})$ on $D^{c}$.
###### Proposition 3.5.
For the limit process $(\bar{Y},\bar{M})$, we have
(i) For every $t\in[0,T]-D$,
$\bar{Y}_{t}=\xi+\int_{t}^{T}\bar{f}(s,\bar{Y}_{s},\bar{\alpha}_{s})ds-(\bar{M}_{T}-\bar{M}_{t}).$
(ii) For a $d$-dimensional Brownian motion $\bar{B}=\\{\bar{B}_{t};0\leq t\leq
T\\}$ with $\bar{B}_{0}=0$, $\bar{Y}$ is measurable with
${\cal{H}}_{t}={\cal{F}}^{\bar{B}}_{t}\vee{\cal{F}}^{\bar{\alpha}}_{T}$, then
$\bar{M}$ is a ${\cal{H}}_{t}$-martingale.
###### Proof.
From Proposition 3.4, as $\varepsilon\rightarrow 0$,
$\begin{split}&\sup_{0\leq t\leq
T}E\left|\int_{0}^{t}f(s,Y^{\varepsilon}_{s},s_{ij})\left(I_{\\{\alpha_{s}^{\varepsilon}=s_{ij}\\}}-\nu_{j}^{i}I_{\\{\alpha_{s}^{\varepsilon}\in{\cal{M}}_{i}\\}}\right)ds\right|\rightarrow
0.\end{split}$
Since $(Y^{\varepsilon},\bar{\alpha}^{\varepsilon})$ converge weakly to
$(\bar{Y},\bar{\alpha})$,
$\begin{split}&\int_{0}^{t}\bar{f}(s,Y^{\varepsilon}_{s},\bar{\alpha}^{\varepsilon}_{s})ds\textrm{
converges in distribution
to}\int_{0}^{t}\bar{f}(s,\bar{Y}_{s},\bar{\alpha}_{s})ds\textrm{ on
}C(0,T;R^{k}).\end{split}$
Thus
$\begin{split}&\int_{0}^{t}f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})ds\\\
=&\int_{0}^{t}\sum_{i=1}^{l}\sum_{j=1}^{m_{i}}f(s,Y^{\varepsilon}_{s},s_{ij})I_{\\{\alpha_{s}^{\varepsilon}=s_{ij}\\}}\\\
=&\int_{0}^{t}\sum_{i=1}^{l}\sum_{j=1}^{m_{i}}f(s,Y^{\varepsilon}_{s},s_{ij})\left(I_{\\{\alpha_{s}^{\varepsilon}=s_{ij}\\}}-\nu_{j}^{i}I_{\\{\alpha_{s}^{\varepsilon}\in{\cal{M}}_{i}\\}}\right)ds+\int_{0}^{t}\bar{f}(s,Y^{\varepsilon}_{s},\bar{\alpha}^{\varepsilon}_{s})ds.\end{split}$
As $\varepsilon\rightarrow 0$, passing to the limit in the backward component
of the BSDE $(\ref{eq: BSDE with perturbed in M form})$, we can derive
assertion (i).
Now, we prove assertion (ii).
For any $0\leq t_{1}\leq t_{2}\leq T$, $\Phi_{t_{1}}$ is a continuous mapping
from $C(0,t_{1};R^{d})\times D(0,t_{1};R^{k})\times D(0,T;\bar{{\cal{M}}})$.
$\forall\varepsilon>0$, since $M^{\varepsilon}$ is a martingale with respect
to
$\mathcal{G}_{t}^{\varepsilon}=\mathcal{F}_{T}^{\alpha^{\varepsilon}}\vee\mathcal{F}_{t}^{B}$,
$Y^{\varepsilon}$ and $\bar{\alpha}^{\varepsilon}$ are
$\mathcal{G}_{t}^{\varepsilon}$-adapted, we know
$E\left(\Phi_{t_{1}}(B,Y^{\varepsilon},\bar{\alpha}^{\varepsilon})\left(Y^{\varepsilon}_{t_{2}}-Y^{\varepsilon}_{t_{1}}+\int_{t_{1}}^{t_{2}}f(s,Y_{s}^{\varepsilon},\alpha_{s}^{\varepsilon})ds\right)\right)=0$
and
$E\left(\Phi_{t_{1}}(B,Y^{\varepsilon},\bar{\alpha}^{\varepsilon})\int_{0}^{\delta}(M_{t_{2}+r}^{\varepsilon}-M_{t_{1}+r}^{\varepsilon})dr\right)=0,$
here $B$ is the Brownian motion.
From the weak convergence of $(Y^{\varepsilon},\bar{\alpha}^{\varepsilon})$ to
$(\bar{Y},\bar{\alpha})$,
$\int_{0}^{t}\bar{f}(s,Y^{\varepsilon}_{s},\bar{\alpha}^{\varepsilon}_{s})ds$
converges in distribution to
$\int_{0}^{t}\bar{f}(s,\bar{Y}_{s},\bar{\alpha}_{s})ds$ on $C(0,T;R^{k})$. For
a $d$-dimensional Brownian motion $\bar{B}=\\{\bar{B}_{t};0\leq t\leq T\\}$
with $\bar{B}_{0}=0$, from the fact that $\bar{B}$ has the same probability
distribution with $B$ and $\displaystyle E(\sup_{0\leq t\leq
T}|M_{t}^{\varepsilon}|^{2})\leq C$, we obtain
$E\left(\Phi_{t_{1}}(\bar{B},\bar{Y},\bar{\alpha})\left(\bar{Y}_{t_{2}}-\bar{Y}_{t_{1}}+\int_{t_{1}}^{t_{2}}\bar{f}(s,\bar{Y}_{s},\bar{\alpha}_{s})ds\right)\right)=0$
and
$E\left(\Phi_{t_{1}}(\bar{B},\bar{Y},\bar{\alpha})\int_{0}^{\delta}(\bar{M}_{t_{2}+r}-\bar{M}_{t_{1}+r})dr\right)=0.$
Dividing the second identity by $\delta$, letting $\delta\rightarrow 0$, and
exploiting the right continuity, we obtain that
$E\left(\Phi_{t_{1}}(\bar{B},\bar{Y},\bar{\alpha})(\bar{M}_{t_{2}}-\bar{M}_{t_{1}})\right)=0.$
From the freedom choice of $t_{1}$, $t_{2}$, and $\Phi_{t_{1}}$, we deduce
that $\bar{M}$ is a ${\cal{H}}_{t}$-martingale. ∎
###### Proposition 3.6.
Let $\\{(Y_{t},Z_{t});0\leq t\leq T\\}$ be the unique solution of BSDE
$(\ref{eq:limit BSDE })$, then $\forall t\in[0,T]$,
$E|Y_{t}-\bar{Y}_{t}|^{2}+E\left([\bar{M}-\int_{0}^{\cdot}Z_{r}d\bar{B}_{r}]_{T}-[\bar{M}-\int_{0}^{\cdot}Z_{r}d\bar{B}_{r}]_{t}\right)=0.$
###### Proof.
Let $M_{t}=\int_{0}^{t}Z_{r}d\bar{B}_{r}$, by the proof of Proposition 2.2, we
know that $M_{t}$ is a
${\cal{F}}_{t}^{\bar{B}}\vee{\cal{F}}_{T}^{\bar{\alpha}}$-martingale.
From Itô’s formula and Proposition 3.5, we know that
$\begin{split}&E|Y_{t}-\bar{Y}_{t}|^{2}+E\left([M-\bar{M}]_{T}-[M-\bar{M}]_{t}\right)\\\
=\
&2E\int_{t}^{T}\left(\bar{f}(s,Y_{s},\bar{\alpha}_{s})-\bar{f}(s,\bar{Y}_{s},\bar{\alpha}_{s})\right)(Y_{s}-\bar{Y}_{s})ds\\\
\leq\ &CE\int_{t}^{T}|Y_{s}-\bar{Y}_{s}|^{2}ds.\end{split}$
From Gronwall’s lemma, we obtain $E|Y_{t}-\bar{Y}_{t}|^{2}=0$, $\forall
t\in[0,T]-D$, and the result follows. ∎
We come back to finish the Proof of Theorem 3.1:
Since $Y$ is continuous, $\bar{Y}$ is c$\grave{a}$dl$\grave{a}$g, and $D$ is
countable, we get $Y_{t}=\bar{Y}_{t}$, $P-$a.s., $\forall t\in[0,T]$.
Moreover, we can deduce that $M\equiv\bar{M}$. Hence, we get the result that
the sequence $(Y^{\varepsilon}_{t},\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s})$
converges in distribution to the process
$(Y_{t},\int_{0}^{t}Z_{s}d\bar{B}_{s})$, and the proof of Theorem 3.3 is
completed. ∎
### 3.3 Examples
###### Example 3.1.
Consider the case that $\tilde{Q}$ is weakly irreducible with the state space
$\mathcal{M}=\\{1,\cdots,m\\}$ and $\nu=(\nu_{1},\cdots,\nu_{m})$ is the quasi
stationary distribution, then $\alpha^{\varepsilon}$ can be considered as a
fast-varying noise process. As shown in the following, the noise is averaged
out with respect to the quasi stationary distribution. In this case, the
corresponding BSDE is
$Y^{\varepsilon}_{t}=\xi+\int_{t}^{T}f(s,Y^{\varepsilon}_{s},\alpha^{\varepsilon}_{s})ds-\int_{t}^{T}Z^{\varepsilon}_{s}dB_{s}.$
(12)
Under Assumption 2.1 and Assumption 3.1, from Theorem 3.3, as
$\varepsilon\rightarrow 0$, the sequence of process $(Y^{\varepsilon}_{t},$
$\int_{0}^{t}Z^{\varepsilon}_{s}dB_{s})$ converges in distribution to the
process $(Y_{t},\int_{0}^{t}Z_{s}d\bar{B}_{s})$, where $(Y,Z)$ is the unique
solution to the following BSDE
$Y_{t}=\xi+\int_{t}^{T}\sum_{i=1}^{m}\nu_{i}f(s,Y_{s},i)ds-\int_{t}^{T}Z_{s}d\bar{B}_{s}.$
(13)
It is noted that the generator of BSDE (13) depends on the quasi stationary
distribution of the Markov chain. Thus we can adopt the distribution of a
$\mathcal{F}_{t}^{\bar{B}}$-adapted process $Y$, the solution of BSDE (13), as
the asymptotic distribution for the solution of
$\mathcal{F}_{t}^{B}\vee\mathcal{F}^{\alpha^{\varepsilon}}_{t,T}$-adapted
process $Y^{\varepsilon}$.
In practical systems, the small parameter $\varepsilon$ is just a fixed
parameter and it separates different scales in the sense of order of magnitude
in the generator. It does not need to tend to 0. We give a detailed example
for interpretation.
###### Example 3.2.
Suppose the generator of the continuous-time Markov chain affected BSDE (3) is
$Q=\begin{pmatrix}{-22}&{20}&{2}\\\ {41}&{-42}&{1}\\\
{1}&{2}&{-3}\end{pmatrix}$, and the corresponding state space is
$\mathcal{M}=\\{s_{1},s_{2},s_{3}\\}$. It is obvious that the transition rate
between $s_{1}$ and $s_{2}$ is larger than the transition rate between $s_{3}$
and other states, i.e., the jumps between $s_{1}$ and $s_{2}$ are more
frequent than jumps between $s_{3}$ and other states. We can rewrite $Q$ as
following
$\displaystyle
Q=\frac{1}{0.05}\tilde{Q}+\hat{Q}=\frac{1}{0.05}\begin{pmatrix}{-1}&{1}&{0}\\\
{2}&{-2}&{0}\\\ {0}&{0}&{0}\end{pmatrix}+\begin{pmatrix}{-2}&{0}&{2}\\\
{1}&{-2}&{1}\\\ {1}&{2}&{-3}\end{pmatrix}$
It is noted that we choose suitable $\varepsilon$ to guarantee that
$\tilde{Q}$ and $\hat{Q}$ to be the generator with the same order of
magnitude.
Now, we introduce the continuous-time $\varepsilon$-dependent singularly
perturbed Markov chain $\alpha^{\varepsilon}=\\{\alpha^{\varepsilon}_{t};0\leq
t\leq T\\}$ which have the generator $\displaystyle
Q^{\varepsilon}=\frac{1}{\varepsilon}\tilde{Q}+\hat{Q}=\frac{1}{\varepsilon}\begin{pmatrix}{-1}&{1}&{0}\\\
{2}&{-2}&{0}\\\ {0}&{0}&{0}\end{pmatrix}+\begin{pmatrix}{-2}&{0}&{2}\\\
{1}&{-2}&{1}\\\ {1}&{2}&{-3}\end{pmatrix},$ and define the aggregated process
$\bar{\alpha}^{\varepsilon}=\\{\bar{\alpha}^{\varepsilon}_{t};0\leq t\leq
T\\}=\left\\{\begin{array}[]{ll}1,\
\alpha^{\varepsilon}_{t}\in\\{s_{1},s_{2}\\}\\\ 2,\
\alpha^{\varepsilon}_{t}\in\\{s_{3}\\}\end{array}\right.$
Proposition 3.1 yields that $\bar{\alpha}^{\varepsilon}$ converges in
distribution to a continuous-time Markov chain $\bar{\alpha}$ generated by
$\displaystyle\bar{Q}=\begin{pmatrix}-\frac{5}{3}&\frac{5}{3}\\\
{3}&-{3}\end{pmatrix}.$ By Theorem 3.3, we can adopt the probability
distribution of the solution to the following BSDE
$Y_{t}=\xi+\int_{t}^{T}\bar{f}(s,Y_{s},\bar{\alpha}_{s})ds-\int_{t}^{T}Z_{s}d\bar{B}_{s}$
as an asymptotic probability distribution of the solution to the original
BSDE. Here
$\displaystyle\bar{f}(t,y,1)=\frac{2}{3}f(t,y,s_{1})+\frac{1}{3}f(t,y,s_{2})$
and $\bar{f}(t,y,2)=f(t,y,s_{3}).$
Since the limit averaged Markov chain has two states and the original one has
three states, we have reduced the complexity of the model. This advantage will
be more clear when the state space of the original Markov chain is
sufficiently larger.
## 4 Homogenization of One System of PDEs
As an application of our results in previous section, we show the
homogenization of a sequence of semi-linear backward PDE with a singularly
perturbed Markov chain. In this section, after showing the relation between
BSDEs with Markov chain and one system of semi-linear PDE with Markov chain,
we derive the homogenization property of backward PDE with a singularly
perturbed Markov chain based on the weak convergence of the associated BSDE.
Here, we give some notations as follows: $C^{k}(R^{p};R^{q})$ is the space of
functions of class $C^{k}$ from $R^{p}$ to $R^{q}$, $C^{k}_{l,b}(R^{p};R^{q})$
is the space of functions of class $C^{k}$ whose partial derivatives of order
less than or equal to $k$ are bounded, and $C^{k}_{p}(R^{p};R^{q})$ is the
space of functions of class $C^{k}$ which, together with all their partial
derivatives of order less than or equal to $k$, grow at most like a polynomial
function of the variable $x$ at infinity.
### 4.1 Relation between BSDEs with Markov chains and semi-linear PDEs
systems with Markov chains
For $t\in[0,T]$, consider the following semi-linear backward PDE with a Markov
chain:
$u(t,x)=h(x)+\int_{t}^{T}\left(\mathcal{L}u(r,x)+f(r,x,u(r,x),(\nabla
u\sigma)(r,x),\alpha_{r})\right)dr,$ (14)
here $u:[0,T]\times R^{m}\rightarrow R^{k}$, and
$\displaystyle\mathcal{L}u=\left(Lu_{1},\cdots,Lu_{k}\right)^{\prime}$, with
$\allowdisplaybreaks\displaystyle
L=\frac{1}{2}\sum_{i,j=1}^{m}(\sigma\sigma^{\prime})_{ij}$
$\displaystyle(t,x)\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}+\sum_{i=1}^{m}b_{i}(t,x)\frac{\partial}{\partial x_{i}}.$
Firstly, we make the following assumption:
###### Assumption 4.1.
$b\in C^{3}_{l,b}(R^{m};R^{m})$, $\sigma\in C^{3}_{l,b}(R^{m};R^{m\times d})$,
$h\in C_{p}^{3}(R^{m};R^{k})$. For $f:[0,T]\times R^{m}\times R^{k}\times
R^{k\times d}\times\mathcal{M}\rightarrow R^{k}$, $\forall s\in[0,T]$,
$i\in\mathcal{M}$, $(x,y,z)\rightarrow f(s,x,y,z,i)$ is of class $C^{3}$.
Moreover, $f(s,\cdot,0,0,i)\in C^{3}_{p}(R^{m};R^{k})$, and the first order
partial derivatives in $y$ and $z$ are bounded on $[0,T]\times R^{m}\times
R^{k}\times R^{k\times d}\times\mathcal{M}$, as well as their derivatives of
order one and two with respect to $x$, $y$, $z$.
###### Definition 4.1.
A classical solution of PDE (14) is a $R^{k}$-valued stochastic process
$\\{u(t,x);$ $0\leq t\leq T,x\in R^{m}\\}$ which is in $C^{0,2}([0,T]\times
R^{m};R^{k})$ and satisfies that $u(t,x)$ is $\mathcal{F}_{t,T}$-measurable.
$\forall t\in[0,T]$, $x\in R^{m}$, we introduce the following FBSDE with a
Markov chain on $[t,T]$:
$X_{s}^{t,x}=x+\int_{t}^{s}b(X_{r}^{t,x})dr+\int_{t}^{s}\sigma(X_{r}^{t,x})dB_{r},$
(15)
$Y_{s}^{t,x}=h(X_{T}^{t,x})+\int_{s}^{T}f(r,X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x},\alpha_{r})dr-\int_{s}^{T}Z_{r}^{t,x}dB_{r}.$
(16)
The aim of this subsection is to show that, under above assumptions, the FBSDE
(15)-(16) provides both a probabilistic representation and the unique
classical solution for PDE (14).
For SDE (15), it is well known that under Assumption 4.1, it has a unique
solution $\\{X_{s}^{t,x};t\leq s\leq T\\}$ which has a version that is a.s. of
class $C^{2}$ in $x$, the function and its derivatives are a.s. jointly
continuous in $(t,s,x)$. Moreover,
$\sup_{t\leq s\leq T}\left(|X_{s}^{t,x}|+|\nabla
X_{s}^{t,x}|+|D^{2}X_{s}^{t,x}|\right)\in\bigcap_{p\geq
1}L^{p}(R),\forall(t,x)\in[0,T]\times R^{m}$
where $\nabla X_{s}^{t,x}$, $D^{2}X_{s}^{t,x}$ denote respectively the matrix
of first order and second order derivatives of $X_{s}^{t,x}$ with respect to
$x$.
For BSDE (16), denote $\tilde{f}(s,y,z,i)=f(s,X^{t,x}_{s},y,z,i)$, $\forall
i\in\mathcal{M}$, we know that $\tilde{f}$ satisfies Assumption 2.1 since $f$
satisfies Assumption 4.1. So there exists a unique solution pair
$\\{(Y_{s}^{t,x},Z_{s}^{t,x});t\leq s\leq T\\}$ to BSDE (16).
Define $X_{s}^{t,x}=X_{s\vee t}^{t,x}$, $Y_{s}^{t,x}=Y_{s\vee t}^{t,x}$, and
$Z_{s}^{t,x}=0$, for $s\leq t$. Then
$(X,Y,Z)=(X_{s}^{t,x},Y_{s}^{t,x},Z_{s}^{t,x})$ is defined on
$(s,t)\in[0,T]^{2}$.
###### Theorem 4.4.
Under Assumption 4.1, let $\\{u(t,x);0\leq t\leq T,x\in R^{m}\\}$ be a
classical solution of PDE $(\ref{Eq:SPDEs })$. Suppose that there exists a
constant $C$ such that,
$|u(t,x)|+|\partial_{x}u(t,x)\sigma(t,x)|\leq
C(1+|x|),\quad\forall(t,x)\in[0,T]\times R^{m},$ (17)
then
$(Y_{s}^{t,x}=u(t,X_{s}^{t,x}),Z_{s}^{t,x}=\partial_{x}u(t,X_{s}^{t,x})\sigma(t,X_{s}^{t,x});t\leq
s\leq T)$ is the unique solution of BSDE $(\ref{Eq:BSDE of FBSDE})$. Here
$(X_{s}^{t,x};t\leq s\leq T)$ is the solution to SDE $(\ref{Eq:SDE of
FBSDE})$.
Proof: $\forall t\leq s\leq T$, let $s=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=T$, with
Itô’s formula and PDE $(\ref{Eq:SPDEs })$, we get
$\begin{split}&Y_{s}^{t,x}-h(X_{T}^{t,x})\\\ =\ &u(s,x)-u(T,X_{T}^{t,x})\\\ =\
&\sum_{i=0}^{n-1}\left(u(t_{i},X_{t_{i}}^{t,x})-u(t_{i+1},X_{t_{i+1}}^{t,x})\right)\\\
=\
&\sum_{i=0}^{n-1}\left(u(t_{i},X_{t_{i}}^{t,x})-u(t_{i},X_{t_{i+1}}^{t,x})\right)+\sum_{i=0}^{n-1}\left(u(t_{i},X_{t_{i+1}}^{t,x})-u(t_{i+1},X_{t_{i+1}}^{t,x})\right)\end{split}$
$\begin{split}=\
&\sum_{i=0}^{n-1}\bigg{(}-\int_{t_{i}}^{t_{i+1}}\left(\mathcal{L}u(t_{i},X_{s}^{t,x})ds-(\nabla
u\sigma)(t_{i},X_{s}^{t,x})dB_{s}\right)\\\
&+\int_{t_{i}}^{t_{i+1}}\left(\mathcal{L}u(s,X_{t_{i+1}}^{t,x})+f(s,X_{t_{i+1}}^{t,x},u(r,X_{t_{i+1}}^{t,x}),(\nabla
u\sigma)(s,X_{t_{i+1}}^{t,x}),\alpha_{s})\right)ds\bigg{)}.\end{split}$
(17) yields that
$E\left(\sup_{t\leq s\leq
T}|u(t,X_{s}^{t,x})|^{2}+\int_{t}^{T}|\partial_{x}u\sigma(s,X_{s}^{t,x})|^{2}ds\right)<\infty,$
and the adaptability is obvious. The result is followed as
$\displaystyle\triangle=\sup_{0\leq i\leq n-1}|t_{i+1}-t_{i}|\rightarrow 0$. ∎
Now we deduce the converse side of Theorem 4.4.
###### Theorem 4.5.
Assume that for some $p>2$,
$E|\xi|^{p}+E\int_{0}^{T}|\tilde{f}(t,0,0,\alpha_{t})|^{p}dt<\infty,$ let
$b,\sigma,f,h,\alpha$ satisfy Assumption 4.1, then the process
$\\{u(t,x)=Y_{t}^{t,x};0\leq t\leq T,x\in R^{m}\\}$ is the unique classical
solution to PDE $(\ref{Eq:SPDEs })$.
As preliminaries for the proof, we give two propositions about the regularity
of the solution of BSDE (16) whose proofs are put in the Appendix.
###### Proposition 4.1.
Under the assumption of Theorem 4.5, $\\{Y_{s}^{t,x};(s,t)\in[0,T]^{2},x\in
R^{m}\\}$ has a version whose trajectories belong to
$C^{0,0,2}([0,T]^{2}\times R^{m})$. Hence $\forall t\in[0,T]$, $x\rightarrow
Y_{t}^{t,x}$ is of class $C^{2}$ a.s..
###### Proposition 4.2.
Under the assumption of Theorem 4.5, $\\{Z_{s}^{t,x};(s,t)\in[0,T]^{2},x\in
R^{m}\\}$ has an a.s. continuous version which is given by $Z_{s}^{t,x}=\nabla
Y_{s}^{t,x}(\nabla X_{s}^{t,x})^{-1}\sigma(X_{s}^{t,x})$. In particular,
$Z_{t}^{t,x}=\nabla Y_{t}^{t,x}\sigma(x)$. Here $\displaystyle\Big{(}\nabla
Y_{s}^{t,x}=\frac{\partial Y_{s}^{t,x}}{\partial x},$ $\displaystyle\nabla
Z_{s}^{t,x}=\frac{\partial Z_{s}^{t,x}}{\partial x}\Big{)}$ is the unique
solution of
$\begin{split}\nabla Y_{s}^{t,x}=&h^{\prime}(X_{T}^{t,x})\nabla
X_{T}^{t,x}+\int_{s}^{T}\big{(}f^{\prime}_{x}(r,X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x},\alpha_{r})\nabla
X_{r}^{t,x}+f^{\prime}_{y}(r,X_{r}^{t,x},\\\
&Y_{r}^{t,x},Z_{r}^{t,x},\alpha_{r})\nabla
Y_{r}^{t,x}+f^{\prime}_{z}(r,X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x},\alpha_{r})\nabla
Z_{r}^{t,x}\big{)}dr-\int_{s}^{T}Z_{r}^{t,x}dB_{r}.\end{split}$
Proof of Theorem 4.5: Let $t=t_{0}<t_{1}<\cdots<t_{n}=T$, we have
$\begin{split}&h(x)-u(t,x)\\\ =\ &u(T,x)-u(t,x)\\\
=&\sum_{i=0}^{n-1}\left(u(t_{i+1},x)-u(t_{i},x)\right)\\\
=&\sum_{i=0}^{n-1}\left(u(t_{i+1},x)-u(t_{i+1},X_{t_{i+1}}^{t_{i},x})+u(t_{i+1},X_{t_{i+1}}^{t_{i},x})-u(t_{i},x)\right).\end{split}$
Since
$u(t_{i+1},X_{t_{i+1}}^{t_{i},x})=Y_{t_{i+1}}^{t_{i+1},X_{t_{i+1}}^{t_{i},x}}=Y_{t_{i+1}}^{t_{i},x}$,
we obtain the following from BSDE (16)
$\begin{split}&u(t_{i+1},X_{t_{i+1}}^{t_{i},x})-u(t_{i},x)\\\ =\
&Y_{t_{i+1}}^{t_{i},x}-Y_{t_{i}}^{t_{i},x}\\\ =\
&-\int_{t_{i}}^{t_{i+1}}f(r,X_{r}^{t_{i},x},Y_{r}^{t_{i},x},Z_{r}^{t_{i},x},\alpha_{r})dr+\int_{t_{i}}^{t_{i+1}}Z_{r}^{t,x}dB_{r}.\end{split}$
It is known that $u(t,\cdot)\in C^{2}(R^{m})$ from Proposition 4.1. Then, with
Itô’s formula, we get
$\displaystyle\begin{split}&h(x)-u(t,x)\\\ =\
&\sum_{i=0}^{n-1}\bigg{(}\int_{t_{i}}^{t_{i+1}}\mathcal{L}u(t_{i+1},X_{r}^{t_{i},x})dr-\int_{t_{i}}^{t_{i+1}}(\nabla
u\sigma)(t_{i+1},X_{r}^{t_{i},x})dB_{r}\\\
&-\int_{t_{i}}^{t_{i+1}}f(r,X_{r}^{t_{i},x},Y_{r}^{t_{i},x},Z_{r}^{t_{i},x},\alpha_{r})dr+\int_{t_{i}}^{t_{i+1}}Z_{r}^{t,x}dB_{r}\bigg{)}\\\
=&-\sum_{i=0}^{n-1}\int_{t_{i}}^{t_{i+1}}\left(\mathcal{L}u(t_{i+1},X_{r}^{t_{i},x})+f(r,X_{r}^{t_{i},x},Y_{r}^{t_{i},x},Z_{r}^{t_{i},x},\alpha_{r})\right)dr\\\
&+\sum_{i=0}^{n-1}\int_{t_{i}}^{t_{i+1}}\left(Z_{r}^{t_{i},x}-(\nabla
u\sigma)(t_{i+1},X_{r}^{t_{i},x})\right)dB_{r}.\end{split}$
From Proposition 4.1 and Proposition 4.2, letting
$\displaystyle\triangle=\sup_{0\leq i\leq n-1}|t_{i+1}-t_{i}|\rightarrow 0$,
we have
$u(t,x)=h(x)+\int_{t}^{T}\left(\mathcal{L}u(r,x)+f(r,x,u(r,x),(\nabla
u\sigma)(r,x),\alpha_{r})\right)dr,$
here $u\in C^{0,2}([0,T]\times R^{m};R^{k})$. The uniqueness property is
followed from Theorem 4.4 and the uniqueness of the solution of BSDE (16). ∎
### 4.2 Homogenization of PDEs system with a singularly perturbed Markov
chain
Now we can give the application of our theoretical result in previous section
(Theorem 3.1): homogenization of PDEs system with a singularly perturbed
Markov chain.
Consider the following sequence of semi-linear backward PDE with a singularly
perturbed Markov chain, indexed by $\varepsilon>0$, for $t\in[0,T],x\in
R^{m},$
$u^{\varepsilon}(t,x)=h(x)+\int_{t}^{T}\left(\mathcal{L}u^{\varepsilon}(r,x)+f(r,x,u^{\varepsilon}(r,x),\alpha_{r}^{\varepsilon})\right)dr,\quad$
(18)
Here $\alpha^{\varepsilon}$ is the singularly perturbed Markov chain which is
stated in subsection 3.1. We have the following homogenization result.
###### Theorem 4.6.
Under Assumption 3.1 and Assumption 4.1, PDE $(\ref{Eq:SPDEs with parameter})$
has a classical solution $\\{u^{\varepsilon}(t,x);0\leq t\leq T,x\in
R^{m}\\}$. As $\varepsilon\rightarrow 0$, the sequence of $u^{\varepsilon}$
converges in distribution to a process $u$, where $u(t,x)$ is the classical
solution of the following PDE with the limit averaged Markov chain
$\bar{\alpha}$
$u(t,x)=h(x)+\int_{t}^{T}\left(\mathcal{L}u(r,x)+\bar{f}(r,x,u(r,x),\bar{\alpha}_{r})\right)dr,\quad
0\leq t\leq T.$ (19)
Here $\bar{f}$ is the average of $f$ defined as
$\displaystyle\bar{f}(t,x,u,i)=\sum_{j=1}^{m_{i}}\nu_{j}^{i}f(t,x,u,s_{ij})$,
for $i\in\bar{\mathcal{M}}=\\{1,\cdots,l\\}$.
###### Proof.
From Theorem 4.5, we know that
$\\{u^{\varepsilon}(t,x)=Y_{t}^{\varepsilon,t,x};0\leq t\leq T,x\in R^{m}\\}$
is the unique classical solution of PDE (18) where
$\\{Y_{s}^{\varepsilon,t,x};t\leq s\leq T\\}$ satisfies
$Y_{s}^{\varepsilon,t,x}=h(X_{T}^{t,x})+\int_{s}^{T}f(r,X_{r}^{t,x},Y_{r}^{\varepsilon,t,x},\alpha_{r}^{\varepsilon})dr-\int_{s}^{T}Z_{r}^{\varepsilon,t,x}dB_{r},$
(20)
and $\\{X_{s}^{t,x};t\leq s\leq T\\}$ satisfies SDE (15).
$\forall(t,x)\in[0,T]\times R^{m}$, from Theorem 3.3, we obtain that
$Y_{t}^{\varepsilon,t,x}$ converges in distribution to $Y_{t}^{t,x}$ as
$\varepsilon\rightarrow 0$ where $\\{Y_{s}^{t,x};t\leq s\leq T\\}$ satisfies
$Y_{s}^{t,x}=h(X_{T}^{t,x})+\int_{s}^{T}\bar{f}(r,X_{r}^{t,x},Y_{r}^{t,x},\bar{\alpha}_{r})dr-\int_{s}^{T}Z_{r}^{t,x}d\bar{B}_{r},$
(21)
Again from Theorem 4.5, we know that $u(t,x)=Y_{t}^{t,x}$ is the unique
classical solution to PDE (19), and the results are followed. ∎
## 5 Conclusion
In this paper, stemmed from the adjoint equation for deriving the optimal
control of stochastic LQ control problem with Markovian jumps, we study the
solvability of one kind of BSDE with the generator depending on a Markov
switching. Then, we consider the case that the Markov chain has a large state
space. To reduce the complexity, we adopt a hierarchical approach and study
the asymptotic property of BSDE with a singularly perturbed Markov chain.
Also, as an application, we present the homogenization property of one system
of PDE with a singularly perturbed Markov chain.
It is noted that in this paper, we only give the homogenization result of PDEs
system with Markov chains when there exists classical solution under smooth
assumptions. In the successive work, we will study the Sobolev space weak
solution for the related PDEs system and homogenization problem by virtue of
BSDEs with Markov chain. Some applications of this kind of BSDEs in optimal
control and mathematics financial problems would also be interesting to
investigate in our future research.
## Acknowledgements
It is our great pleasure to express the thankfulness to Professor Qing Zhang
in University of Georgia for many useful discussions and suggestions.
## Appendix A Proof of Proposition 4.1 and Proposition 4.2
The proof of Proposition 4.1 and Proposition 4.2 follow a classic approach as
shown in [18, 19]. Here, we will give a sketch of the proof. Firstly, we
present a higher order moment estimation to the solution of BSDE (3).
###### Corollary A.1.
Assume that for some $p>2$, $E|\xi|^{p}+E\int_{0}^{T}|f(t,0,$
$0,\alpha_{t})|^{p}dt<\infty,$ under Assumption 2.1, we have the following
estimation for BSDE $(\ref{eq:BSDE})$,
$\displaystyle E\left(\sup_{0\leq s\leq
t}|Y_{s}|^{p}+(\int_{0}^{t}Z_{s}^{2}ds)^{\frac{p}{2}}\right)<\infty,\quad\forall
0\leq t\leq T.$
###### Proof.
Applying Itô’s formula to $|Y_{t}|^{p}$ from $t$ to $T$, we can get
$\begin{split}&|Y_{t}|^{p}+\frac{p(p-1)}{2}\int_{t}^{T}|Y_{s}|^{p-2}|Z_{s}|^{2}ds\\\
=&\
|\xi|^{p}+p\int_{t}^{T}|Y_{s}|^{p-2}Y_{s}f(s,Y_{s},Z_{s},\alpha_{s})ds-p\int_{t}^{T}|Y_{s}|^{p-2}Y_{s}Z_{s}dB_{s}.\end{split}$
By the same technique as that in Lemma 2.1 of Pardoux and Peng [18], we obtain
that
$\begin{split}&E|Y_{t}|^{p}+\frac{p(p-1)}{2}E\int_{t}^{T}|Y_{s}|^{p-2}|Z_{s}|^{2}ds\\\
\leq&\
E|\xi|^{p}+pE\int_{t}^{T}|Y_{s}|^{p-2}Y_{s}f(s,Y_{s},Z_{s},\alpha_{s})ds.\end{split}$
From Assumption 2.1, using H$\ddot{\textrm{o}}$lder and Young’s inequalities,
there exist $K>0$ and $C$ such that
$\begin{split}&E|Y_{t}|^{p}+KE\int_{t}^{T}|Y_{s}|^{p-2}|Z_{s}|^{2}ds\\\ \leq&\
E|\xi|^{p}+CE\int_{t}^{T}\left(|Y_{s}|^{p}+|f(s,0,0,\alpha_{s})|^{p}\right)ds.\end{split}$
It follows from Gronwall’s lemma that
$\sup_{0\leq t\leq
T}E|Y_{t}|^{p}+E\int_{0}^{T}|Y_{t}|^{p-2}|Z_{t}|^{2}dt<\infty.$
Since
$\begin{split}|Y_{t}|^{p}\leq\
|\xi|^{p}+p\int_{t}^{T}|Y_{s}|^{p-2}Y_{s}f(s,Y_{s},Z_{s},\alpha_{s})ds-p\int_{t}^{T}|Y_{s}|^{p-2}Y_{s}Z_{s}dB_{s},\end{split}$
Burkholder-Davis-Gundy inequality yields that $E(\sup_{0\leq t\leq
T}|Y_{t}|^{p})<\infty$.
Now we prove $E(\int_{0}^{t}Z_{s}^{2}ds)^{\frac{p}{2}}<\infty$. Since
$\int_{0}^{t}Z_{s}dB_{s}=Y_{t}-Y_{0}+\int_{0}^{t}f(s,Y_{s},Z_{s},\alpha_{s})ds,$
$\sup_{0\leq t\leq T}|\int_{0}^{t}Z_{s}dB_{s}|\leq 2\sup_{0\leq t\leq
T}|Y_{t}|+\int_{0}^{T}|f(s,Y_{s},Z_{s},\alpha_{s})|ds,$
the result is followed from Assumption 2.1 and Burkholder-Davis-Gundy
inequality. ∎
###### Lemma A.1.
(Lemma 2.7 in [18]) For any $p>2$, there exists a constant $c_{p}$ such that
for any $t,t^{\prime}\in[0,T]$, $x,x^{\prime}\in R^{m}$,
$i\in\\{1,\cdots,d\\}$, $h,h^{\prime}\in R\backslash\\{0\\}$,
$E(\sup_{0\leq s\leq T}|X_{s}^{t,x}|^{p})\leq c_{p}(1+|x|^{p}),$
$E(\sup_{0\leq s\leq T}|X_{s}^{t,x}-X_{s}^{t^{\prime},x^{\prime}}|^{p})\leq
c_{p}(1+|x|^{p})(|x-x^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}),$
$E(\sup_{0\leq s\leq T}|\triangle_{h}^{i}X_{s}^{t,x}|^{p})\leq c_{p},$
$E(\sup_{0\leq s\leq
T}|\triangle_{h}^{i}X_{s}^{t,x}-\triangle_{h^{\prime}}^{i}X_{s}^{t^{\prime},x^{\prime}}|^{p})\leq
c_{p}(|x-x^{\prime}|^{p}+|h-h^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}).$
Here $\displaystyle\triangle_{h}^{i}g(x)=\frac{g(x+he_{i})-g(x)}{h}$, $1\leq
i\leq d$, where $e_{i}$ denotes the $i$th vector of an arbitrary orthonormal
basis of $R^{m}$.
Proof of Proposition 4.1: Since
$E(\sup_{0\leq s\leq T}|X_{s}^{t,x}|^{p})\leq c_{p}(1+|x|^{p}),$
from the proof of Corollary A.1, $\forall p>2$, there exist $C_{p}$ and $q$
such that
$E\left(\sup_{0\leq s\leq
t}|Y_{s}^{t,x}|^{p}+(\int_{0}^{t}|Z_{s}^{t,x}|^{2}ds)^{\frac{p}{2}}\right)\leq
C_{p}(1+|x|^{q}).$
Note that for $t\vee t^{\prime}\leq s\leq T$
$\begin{split}&Y_{s}^{t,x}-Y_{s}^{t^{\prime},x^{\prime}}\\\ =&\
\bigg{(}\int_{0}^{1}h^{\prime}(X_{T}^{t,x}+\lambda(X_{T}^{t,x}-X_{T}^{t^{\prime},x^{\prime}}))d\lambda\bigg{)}(X_{T}^{t,x}-X_{T}^{t^{\prime},x^{\prime}})\\\
&+\int_{s}^{T}\int_{0}^{1}\bigg{(}f^{\prime}_{x}(\Xi_{r,\lambda}^{t,x,t^{\prime},x^{\prime}},\alpha_{r})(X_{r}^{t,x}-X_{r}^{t^{\prime},x^{\prime}})+f^{\prime}_{y}(\Xi_{r,\lambda}^{t,x,t^{\prime},x^{\prime}},\alpha_{r})(Y_{r}^{t,x}-Y_{r}^{t^{\prime},x^{\prime}})\\\
&+f^{\prime}_{z}(\Xi_{r,\lambda}^{t,x,t^{\prime},x^{\prime}},\alpha_{r})(Z_{r}^{t,x}-Z_{r}^{t^{\prime},x^{\prime}})\bigg{)}d\lambda
dr-\int_{s}^{T}(Z_{r}^{t,x}-Z_{r}^{t^{\prime},x^{\prime}})dB_{r},\end{split}$
where
$\displaystyle\Xi_{r,\lambda}^{t,x,t^{\prime},x^{\prime}}=(r,X_{r}^{t^{\prime},x^{\prime}}+\lambda(X_{r}^{t,x}-X_{r}^{t^{\prime},x^{\prime}}),Y_{r}^{t^{\prime},x^{\prime}}+\lambda(Y_{r}^{t,x}-Y_{r}^{t^{\prime},x^{\prime}}),Z_{r}^{t^{\prime},x^{\prime}}+\lambda(Z_{r}^{t,x}-Z_{r}^{t^{\prime},x^{\prime}})).$
Since
$E(\sup_{0\leq s\leq T}|X_{s}^{t,x}-X_{s}^{t^{\prime},x^{\prime}}|^{p})\leq
c_{p}(1+|x|^{p})(|x-x^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}),$
combing with the proof of Corollary A.1, we can deduce that $\forall p\geq 2$,
there exist $C_{p}$ and $q$ such that
$\displaystyle E\left(\sup_{0\leq s\leq
T}|Y_{s}^{t,x}-Y_{s}^{t^{\prime},x^{\prime}}|^{p}+\left(\int_{t}^{T}|Z_{s}^{t,x}-Z_{s}^{t^{\prime},x^{\prime}}|^{2}ds\right)^{\frac{p}{2}}\right)$
$\displaystyle\leq\ \ $ $\displaystyle
C_{p}(1+|x|^{q})(|x-x^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}).$
Then using Kolmogorov’s lemma, we know that
$\\{Y_{s}^{t,x};(s,t)\in[0,T]^{2},x\in R^{m}\\}$ has an a.s. continuous
version.
Next, we have
$\begin{split}\triangle_{h}^{i}Y_{s}^{t,x}=&\int_{0}^{1}h^{\prime}(X_{T}^{t,x}+\lambda
h\triangle_{h}^{i}X_{T}^{t,x})\triangle_{h}^{i}X_{T}^{t,x}d\lambda+\int_{s}^{T}\int_{0}^{1}\big{(}f^{\prime}_{x}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}X_{r}^{t,x}\\\
&+f^{\prime}_{y}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}Y_{r}^{t,x}+f^{\prime}_{z}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}Z_{r}^{t,x}\big{)}d\lambda
dr-\int_{s}^{T}\triangle_{h}^{i}Z_{r}^{t,x}dB_{r}\end{split}$
where $\Theta_{r,\lambda}^{t,x,h}=(r,X_{r}^{t,x}+\lambda
h\triangle_{h}^{i}X_{r}^{t,x},Y_{r}^{t,x}+\lambda
h\triangle_{h}^{i}Y_{r}^{t,x},Z_{r}^{t,x}+\lambda
h\triangle_{h}^{i}Z_{r}^{t,x})$.
Since for each $p\geq 2$, there exists $c_{p}$ such that
$E(\sup_{0\leq s\leq T}|\triangle_{h}^{i}X_{s}^{t,x}|^{p})\leq c_{p}.$
We can have the following estimation
$\displaystyle E\left(\sup_{t\leq s\leq
T}|\triangle_{h}^{i}Y_{s}^{t,x}|^{p}+(\int_{t}^{T}|\triangle_{h}^{i}Z_{s}^{t,x}|ds)^{\frac{p}{2}}\right)\leq
c_{p}(1+|x|^{q}+|h|^{q}).$
Then we consider
$\begin{split}&\triangle_{h}^{i}Y_{s}^{t,x}-\triangle_{h^{\prime}}^{i}Y_{s}^{t^{\prime},x^{\prime}}\\\
=&\int_{0}^{1}h^{\prime}(X_{T}^{t,x}+\lambda
h\triangle_{h}^{i}X_{T}^{t,x})\triangle_{h}^{i}X_{T}^{t,x}d\lambda-\int_{0}^{1}h^{\prime}(X_{T}^{t^{\prime},x^{\prime}}+\lambda
h\triangle_{h^{\prime}}^{i}X_{T}^{t^{\prime},x^{\prime}})\triangle_{h^{\prime}}^{i}X_{T}^{t^{\prime},x^{\prime}}d\lambda\\\
&+\int_{s}^{T}\int_{0}^{1}(f^{\prime}_{x}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}X_{r}^{t,x}-f^{\prime}_{x}(\Theta_{r,\lambda}^{t^{\prime},x^{\prime},h^{\prime}},\alpha_{r})\triangle_{h^{\prime}}^{i}X_{r}^{t^{\prime},x^{\prime}})d\lambda
dr\\\
&+\int_{s}^{T}\int_{0}^{1}(f^{\prime}_{y}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}Y_{r}^{t,x}-f^{\prime}_{y}(\Theta_{r,\lambda}^{t^{\prime},x^{\prime},h^{\prime}},\alpha_{r})\triangle_{h^{\prime}}^{i}Y_{r}^{t^{\prime},x^{\prime}})d\lambda
dr\\\
&+\int_{s}^{T}\int_{0}^{1}(f^{\prime}_{z}(\Theta_{r,\lambda}^{t,x,h},\alpha_{r})\triangle_{h}^{i}Z_{r}^{t,x}-f^{\prime}_{z}(\Theta_{r,\lambda}^{t^{\prime},x^{\prime},h^{\prime}},\alpha_{r})\triangle_{h^{\prime}}^{i}Z_{r}^{t^{\prime},x^{\prime}})d\lambda
dr\\\
&-\int_{s}^{T}(\triangle_{h}^{i}Z_{r}^{t,x}-\triangle_{h^{\prime}}^{i}Z_{r}^{t^{\prime},x^{\prime}})dB_{r}.\end{split}$
It is noted that
$E(\sup_{0\leq s\leq
T}|\triangle_{h}^{i}X_{s}^{t,x}-\triangle_{h^{\prime}}^{i}X_{s}^{t^{\prime},x^{\prime}}|^{p})\leq
c_{p}(|x-x^{\prime}|^{p}+|h-h^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}).$
$\forall i\in\mathcal{M}$, using similar arguments, we can show that
$\begin{split}&E\left(\sup_{0\leq s\leq
T}|\triangle_{h}^{i}Y_{s}^{t,x}-\triangle_{h^{\prime}}^{i}Y_{s}^{t^{\prime},x^{\prime}}|^{p}+\left(\int_{t\wedge
t^{\prime}}^{T}|\triangle_{h}^{i}Z_{s}^{t,x}-\triangle_{h^{\prime}}^{i}Z_{s}^{t^{\prime},x^{\prime}}|^{2}ds\right)^{{\frac{p}{2}}}\right)\\\
\leq&\
c_{p}(1+|x|^{q}+|x^{\prime}|^{q}+|h|^{q}+|h^{\prime}|^{q})\times(|x-x^{\prime}|^{p}+|h-h^{\prime}|^{p}+|t-t^{\prime}|^{\frac{p}{2}}).\end{split}$
The existence of a continuous derivative of $Y_{s}^{t,x}$ with respect to $x$,
and a mean-square derivative of $Z_{s}^{t,x}$ with respect to $x$ follow from
this estimation. And the existence of a continuous second derivative of
$Y_{s}^{t,x}$ with respect to $x$ can be proved in a similar scheme. Using
similar arguments as in the proof of Theorem 2.9 in [18], we can show that
$\\{Y_{s}^{t,x};(s,t)\in[0,T]^{2},x\in R^{m}\\}$ has an a.s. continuous
version. ∎
Proof of Proposition 4.2: For any random variable $F$ of the form
$F=f(\varphi,B(h_{1}),\cdots,$ $B(h_{n}))$ with $f\in C_{0}^{\infty}(R^{n})$,
$\varphi\in L_{\mathcal{F}_{T}^{\alpha}}^{2}$, $h_{1},\cdots,h_{n}\in
L^{2}_{\mathcal{F}_{t}}(0,T;R^{d})$ and $B(h_{i})=\int_{0}^{T}h_{i}(t)dB_{t}$,
where $\mathcal{F}_{t}=\mathcal{F}^{\alpha}_{t,T}\vee\mathcal{F}_{t}^{B}$, let
$\displaystyle D_{t}F=\sum_{i=1}^{n}f^{\prime}_{i}(B(h_{1}),\cdots,$
$B(h_{n}))h_{i}(t),$ $0\leq t\leq T$. For such $F$, we define its norm as
$\|F\|_{1,2}=\left(E\left(F^{2}+\int_{0}^{T}|D_{t}F|^{2}dt\right)\right)^{\frac{1}{2}}.$
Denote $S$ as the set of random variables of the above form, we can define
sobolev space: $D^{1,2}=\bar{S}^{\|\cdot\|_{1,2}}.$ Using the same argument in
Proposition 2.3 in [19], we can obtain the result.∎
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|
arxiv-papers
| 2010-09-26T08:49:22 |
2024-09-04T02:49:13.100339
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huaibin Tang and Zhen Wu",
"submitter": "Huaibin Tang Dr.",
"url": "https://arxiv.org/abs/1009.5074"
}
|
1009.5089
|
# Detection and quantification of inverse spin Hall effect from spin pumping
in permalloy/normal metal bilayers
O. Mosendz New address: San Jose Research Center, Hitachi Global Storage
Technologies, San Jose, California 95135, USA Materials Science Division,
Argonne National Laboratory, Argonne, IL 60439, USA V. Vlaminck Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439, USA J. E.
Pearson Materials Science Division, Argonne National Laboratory, Argonne, IL
60439, USA F. Y. Fradin Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA G. E. W. Bauer Kavli Institute of
NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands
S. D. Bader Materials Science Division, Argonne National Laboratory, Argonne,
IL 60439, USA Center for Nanoscale Materials, Argonne National Laboratory,
Argonne, IL 60439, USA A. Hoffmann hoffmann@anl.gov Materials Science
Division, Argonne National Laboratory, Argonne, IL 60439, USA
###### Abstract
Spin pumping is a mechanism that generates spin currents from ferromagnetic
resonance (FMR) over macroscopic interfacial areas, thereby enabling sensitive
detection of the inverse spin Hall effect that transforms spin into charge
currents in non-magnetic conductors. Here we study the spin-pumping-induced
voltages due to the inverse spin Hall effect in permalloy/normal metal
bilayers integrated into coplanar waveguides for different normal metals and
as a function of angle of the applied magnetic field direction, as well as
microwave frequency and power. We find good agreement between experimental
data and a theoretical model that includes contributions from anisotropic
magnetoresistance (AMR) and inverse spin Hall effect (ISHE). The analysis
provides consistent results over a wide range of experimental conditions as
long as the precise magnetization trajectory is taken into account. The spin
Hall angles for Pt, Pd, Au and Mo were determined with high precision to be
$0.013\pm 0.002$, $0.0064\pm 0.001$, $0.0035\pm 0.0003$ and $-0.0005\pm
0.0001$, respectively.
###### pacs:
72.25.Rb, 75.47.-m, 76.50.+g
## I Introduction
Information in semiconductor electronic devices and data storage technologies
is mainly transported and manipulated by charge currents. With advancing
miniaturization, heat dissipation and power consumption become significant
obstacles to further technological advances. Alternative technologies that
solve or at least circumvent these problems are needed. One promising
candidate to replace existing charge-based technologies is based on using spin
currents; an effort referred to as spintronicsBader and Parkin (2010).
Magnetoelectronic devices employing spin-polarized charge currents are already
actively in use in hard drive read-heads and non-volatile magnetic random
access memories (MRAMs).
Pure spin currents that are not accompanied by a net charge current, may offer
additional advantages in applications,Chappert and Kim (2008); Hoffmann (2007)
such as reduced power dissipation, absence of stray Oersted fields, and
decoupling of spin and charge noise. Furthermore, undisturbed by charge
transport, pure spin currents can provide more direct insights into the basic
physics of spin-dependent effects. Pure spin currents can be created by, for
instance: (i) Non-local electrical injection from ferromagnetic contacts in
multi-terminal structures; (ii) optical injection using circularly polarized
light; (iii) spin pumping from a precessing ferromagnet; and (iv) spin Hall
effect. The last possibility is particularly interesting, since ferromagnets
are not involvedDyakonov and Perel (1971); Hirsch (1999); Zhang (2000). The
spin Hall effect is caused by the spin-orbit interactions of defect scattering
potentials or the host electronic structure. The efficiency of this spin-
charge conversion can be quantified by a single material-specific parameter,
viz. the spin Hall angle $\gamma$, which is defined as the ratio of the spin
Hall and charge conductivities Dyakonov and Khaetskii (2008) and can be
measured by magnetotransport measurements Fert et al. (1981); Valenzuela and
Tinkham (2006); Kimura et al. (2007); Seki et al. (2008); Morota et al.
(2009). Previous experimental studies report quite different $\gamma$ values
for nominally identical materials. For example, for Au a giant $\gamma=0.113$
was reported Seki et al. (2008), while subsequent experiments found values
that are one or even two orders of magnitude smallerMihajlović et al. (2009);
Mosendz et al. (2010a). Similarly, for Pt different experiments Kimura et al.
(2007); Mosendz et al. (2010a); Ando et al. (2008a) resulted in $\gamma$
values that vary between 0.0037 and 0.08.
Recently we demonstrated a robust technique Mosendz et al. (2010a) to measure
spin Hall angles with high accuracy in arbitrary conductors. Our approach is
based on the combination of spin pumping, which generates pure spin currents,
and measurements of electric voltages due to the inverse spin Hall effect
(ISHE)Saitoh et al. (2006). Here we present a detailed discussion of the
measurements in Ref. Mosendz et al., 2010a and examine the validity of the
theoretical model used to describe the voltages induced in the
Ni80Fe20(Py)/normal metal (N) bilayers. In particular we measure the ISHE
voltage as a function of angle of the applied magnetic field, and microwave
frequency and power. We find excellent agreement between model calculations
and experimental results. Accounting for the proper magnetization trajectory
is important for a quantitative interpretation of the results. Good agreement
between the theoretical model and experiments for a wide range of controlled
experimental parameters implies that our approach is robust and can be used to
determine the magnitude and sign of spin Hall effects in more conductors than
included in the present study.
## II Coupling between spin and charge currents
### II.1 Spin pumping in Py/N bilayers
Spin pumping generates pure spin currents in normal metals (N), when they are
in contact with a ferromagnet with time-dependent magnetization induced, e.g.,
by ferromagnetic resonance (FMR)Heinrich et al. (2003); Woltersdorf et al.
(2007); Mosendz et al. (2009); Kardasz et al. (2008); Kardasz and Heinrich
(2010). The instantaneous spin-pumping current $j_{s}^{0}$ at the Py/N
interface is given by:Tserkovnyak et al. (2002a); Tserkovnyak et al. (2005)
$j_{s}^{0}\vec{s}=\frac{\hbar}{8\pi}Re(2g^{\uparrow\downarrow})\left[\vec{m}\times\frac{\partial\vec{m}}{\partial
t}\right],$ (1)
where $\vec{m}$ is the unit vector of the magnetization, $\vec{s}$ is the unit
vector of the spin current polarization in N, and $Re(g^{\uparrow\downarrow})$
is the real part of the spin-mixing conductance. The spin current generated by
spin pumping is polarized perpendicular to the instantaneous magnetization
direction $\vec{m}$ and its time derivative $\partial\vec{m}/\partial t$ (see
Fig. 1). Note that this spin current always has a polarization component along
$\vec{H}_{dc}$ and propagates into N normal to the interface.
The spin current generated at the Py/N interface accumulates a spin density
$\vec{\mu}_{N}$ inside the N layer. In the ballistic limit (i.e., no spin
relaxation in N) the spin current reaching the N/vacuum interface is fully
reflected and reabsorbed upon returning to the Py/N interface, without
influencing the magnetization dynamics of the bilayer system.
Figure 1: (Color online) Schematic model of spin pumping in Py/N bilayer.
$\vec{s}$ shows that the polarization of the spin current is oscillating in
time with the frequency of the magnetization precession. The polarization of
the spin current is perpendicular to the instantaneous magnetization direction
$\vec{m}$, and the rate of magnetization change $\partial\vec{m}/\partial t$.
In real systems, pure spin currents are not conserved, since spins relax over
length scales given by the spin diffusion length $\lambda_{sd}$ in N, and the
accumulated spin density moves across the N layer via spin diffusion limited
by momentum scattering (leading to electrical resistance) and spin-flip
scattering (leading to loss of spin angular momentum) by spin orbit coupling
or magnetic impurities. The spin diffusion equation describes the dissipative
propagation of the spin accumulation (difference in local electrochemical
potentials of up and down spins) $\vec{\mu}_{N}$ in the N layer:
$i\omega\vec{\mu}_{N}=D\frac{\partial^{2}\vec{\mu}_{N}}{\partial
z^{2}}-\frac{1}{\tau_{sf}}\vec{\mu}_{N},$ (2)
where $\omega$ is the angular frequency, $\tau_{sf}$ is the spin-flip time,
$z$ is the coordinate normal to the interface, and $D=v_{F}^{2}\tau_{el}/3$ is
the electron diffusion constant, with $\tau_{el}$ the electron momentum
relaxation timeTserkovnyak et al. (2002b). The solutions of Eq. (2) depend on
the boundary conditions. For a single magnetic layer structure Py/N the
boundary condition at the Py/N interface is given by Tserkovnyak et al. (2005)
$j_{s}^{0}\vec{s}(z=0)=-D\frac{\partial\vec{\mu}_{N}}{\partial
z}\Bigm{|}_{z=0},$ (3)
while for the outer interface we use the free magnetic moment condition (full
spin current reflection)
$\frac{\partial\vec{\mu}_{N}}{\partial z}\Bigm{|}_{z=L}=0.$ (4)
Equations (2)–(4) can be solved analytically to yield the decay of the spin
accumulation as a function of the distance from the Py/N interface. This decay
results in spin accumulation profile in the N layer, which decays as a
function of the distance from the interface, thus driving a spin current with
a dc contribution:
$j_{s}(z)=j_{s}^{0}\frac{\sinh[(z-t_{N})/\lambda_{sd}]}{\sinh(t_{N}/\lambda_{sd})},$
(5)
where $t_{N}$ is the thickness of the N layer. The spin accumulation in N
gives rise to spin backflow into the ferromagnet, which effectively reduces
the spin pumping current, which can be accounted for by replacing
$g^{\uparrow\downarrow}$ in Eq. (1) with an effective spin mixing conductance
$g^{\uparrow\downarrow}_{eff}$.Tserkovnyak et al. (2002b)
In FMR experiments the absorption of the microwave field that excites the
magnetization is monitored. The magnetization dynamics in ferromagnetic films
can be described by the Landau-Lifshitz-Gilbert (LLG) equation of motion:
$\frac{1}{\gamma_{g}}\frac{\partial\vec{m}}{\partial
t}=-\left[\vec{m}\times\vec{H}_{eff}\right]+\frac{\alpha_{G}}{\gamma_{g}}\left[\vec{m}\times\frac{\partial\vec{m}}{\partial
t}\right],$ (6)
where $\gamma_{g}=ge/2mc$ is the absolute value of the gyromagnetic ratio, and
$\alpha_{G}$ is the dimensionless Gilbert damping parameter. The first term on
the right-hand side represents the precessional torque due to the effective
internal field $\vec{H}_{eff}$, which for the case of permalloy with small
anisotropy is approximately equal to the externally applied magnetic field
$H_{dc}$. The second term in Eq. (6) represents the Gilbert damping torque
Heinrich and Cochran (1993); Heinrich (2004). The spin pumping can be
accounted for in the LLG equation of motion by adding a spin pumping
contribution $\alpha_{sp}$ to $\alpha_{G}$, i.e., the effective damping
becomes $\alpha_{eff}=\alpha_{G}+\alpha_{sp}$. The damping can be quantified
by measuring the FMR line width $\Delta H$, half width at half maximum (HWHM),
of the imaginary part of the rf susceptibility $\chi^{{}^{\prime\prime}}$,
which is commonly measured at a constant microwave frequency by sweeping the
dc magnetic field $H_{dc}$. In case of Gilbert damping, $\Delta H$ depends
linearly on the microwave angular frequency $\omega_{f}$, i.e., $\Delta
H=\alpha_{eff}\omega_{f}/\gamma_{g}$. The difference in the damping parameter,
determined by the FMR line width, for samples without capping layer and
samples in which the capping N layer is sufficiently thick to fully dissipate
the pumped magnetic moment, is attributed to the loss of spin momentum in Py
due to relaxation of the spin accumulation in N. This permits the
determination of the additional interface damping due to spin pumping Urban et
al. (2001), which in turn fixes the interfacial spin-mixing conductance to:
$g^{\uparrow\downarrow}_{eff}=\frac{4\pi\gamma_{g}M_{s}t_{Py}}{g\mu_{B}\omega_{f}}(\Delta
H_{Py\mid N}-\Delta H_{Py}),$ (7)
where $t_{Py}$ is the Py layer thickness, $M_{s}$ is the Py saturation
magnetization, and $\mu_{B}$ is the Bohr magneton. Note that Eq. (7) is only
applicable when the damping is governed by the Gilbert phenomenology or
$\Delta H\propto\omega_{f}$, i.e., when inhomogeneous linewidth broadening is
negligible. Otherwise $g^{\uparrow\downarrow}_{eff}$ can still be determined
from the additional Gilbert-like damping contribution
$\alpha_{sp}=\alpha_{Py\mid N}-\alpha_{Py}$, where the latter two are obtained
from the linewidth difference that scales linear with frequency, i.e., $\Delta
H=\Delta H_{ih}+\alpha_{eff}\omega_{f}/\gamma_{g}$, where $\Delta H_{ih}$ is
the sample-dependent inhomogeneous linewidth, measured as the zero-frequency
intercept.
The dc component of the spin current pumped into N is polarized parallel to
the equilibrium magnetization and has previously been detected via a dc
voltage normal to the Py/N interface Costache et al. (2006a). Under a simple
circular precession of the Py magnetization the time-averaging of the spin
current from Eq. (1) for small precession cone angles $\theta$ reads:
$j_{s,dc}^{0,circ}=\frac{\hbar\omega_{f}}{4\pi}Reg^{\uparrow\downarrow}_{eff}\sin^{2}\theta.$
(8)
In thin magnetic films the trajectory of the magnetization precession is not
circular but elliptic due to the strong demagnetizing fields, which force the
magnetization into the film-plane. The time-dependent cone angle $\theta$
modifies the dc component of the pumped spin current by an ellipticity
correction factor $P$ as derived and measured by Ando et al..Ando et al.
(2009) For an in-plane equilibrium magnetization $j_{s,dc}^{eff}=P\ast
j_{s,dc}^{circ}$ with:
$P=\frac{2\omega_{f}\left[\gamma_{g}4\pi M_{s}+\sqrt{(\gamma_{g}4\pi
M_{s})^{2}+(2\omega_{f})^{2}}\right]}{(\gamma_{g}4\pi
M_{s})^{2}+(2\omega_{f})^{2}}.$ (9)
Equation (9) is a non-monotonic function of $\omega_{f}$, and $P$ can become
slightly larger than 1, but tends towards 1 for high frequencies, i.e., large
applied fields.
### II.2 Inverse spin Hall effect
Spin-orbit coupling or magnetic impurities give rise to different scattering
directions for electrons with opposite spin. In their presence, a spin current
in N induces a transverse Hall voltage. This ISHE transforms spin currents
into electrical voltage differences over the sample edges. Spin pumping
generates dc and ac components to the spin current: $j_{s,dc}^{eff}$ and an rf
component transverse to the equilibrium magnetization direction. In this paper
we address only the ISHE effect generated by the dc component
$j_{s,dc}^{eff}$.
The dc ISHE transverse charge current reads:
$\vec{j}_{c}^{ISH}(z)=\gamma(2e/\hbar)j_{s,dc}^{eff}[\vec{n}\times\langle\vec{s}\rangle],$
(10)
where $\gamma$ is the spin Hall angle, $\vec{n}$ is the unit vector normal to
the interface and $\langle\vec{s}\rangle$ is the polarization vector of the dc
spin current. For $j_{s,dc}^{eff}$ the spin-polarization
$\langle\vec{s}\rangle$ is along the equilibrium magnetization direction in
Py. The dc electric field lies in N in the plane of the films and
perpendicular to the equilibrium magnetization of PyMosendz et al. (2010a);
Saitoh et al. (2006); Ando et al. (2008b).
## III Experimental results
Here we elucidate our method to obtain voltage signals due to the ISHE in
various Py/N combinations under FMR conditions, thereby determining the spin
Hall angle $\gamma$ with high accuracy. The measured voltage signals scale
with the sample length and, therefore, can be increased readily by making the
samples longer. We identify two contributions to the dc voltage: one stems
from the anisotropic magnetoresistance (AMR) and the second from the ISHE,
which can be distinguished by their symmetries with respect to the field-
offset from the resonance field. Furthermore, we present a theoretical model
for the spin Hall angle contribution and test its functional dependence of
several parameters that can be controlled experimentally.
### III.1 Experimental set-up
Figure 2: (Color online) Experimental setup: (a) Optical image of the Py/Pt
bilayer integrated into the coplanar waveguide. (b) Contacts are added at the
end of the bilayer to measure the voltage along the waveguide direction.
The Py/N bilayers were integrated into coplanar waveguides with additional
leads in order to measure the dc voltage along the sample. This is shown in
Fig. 2 for a Py/Pt bilayer, with lateral dimensions of 2.92 mm $\times$ 20
$\mu$m and 15-nm thick individual layers. The bilayer was prepared by optical
lithography, sputter deposition, and lift-off on a GaAs substrate.
Subsequently, we prepared Ag contacts for the voltage measurements, covered
the whole structure with 100 nm of MgO (for dc insulation between bilayer and
waveguide), and defined a 30-$\mu$m wide and 200-nm thick Au coplanar
waveguide on top of the bilayer. Similar samples were prepared with Pd, Au and
Mo layers replacing Pt.
The high bandwidth of the coplanar waveguide setup enabled us to carry out
measurements with microwave excitations in the frequency interval of 4–11 GHz.
The power of the rf excitation was varied from 15 to 150 mW. For a given
frequency, experiments were carried out as a function of external magnetic
field $\vec{H}_{dc}$, with an in-plane orientation that could be rotated to
arbitrary angles $\alpha$ with respect to the central axis of the coplanar
wave guide. While the FMR signal was determined from the impedance of the
waveguide Mosendz et al. (2008), the dc voltage was measured simultaneously
with a lock-in modulation technique as a function of $\vec{H}_{dc}$.
### III.2 FMR measurements
The FMR frequency vs. peak position for the Py/Pt sample is shown in Fig.
3(a). Fitting the data to the Kittel formula:
$\left(\omega_{f}/\gamma_{g}\right)^{2}=H_{dc}(H_{dc}+4\pi M_{s}),$ (11)
results in the saturation magnetization for Py of $M_{s}=852$ G. Figure 3(b)
shows the FMR line width as a function of frequency. The linear behavior of
the FMR line width indicates that damping in Py is governed by the intrinsic
Gilbert phenomenology and any extrinsic effects are small.
Figure 3: (Color online) Experimental data (symbols) for FMR peak positions
and FMR linewidths of a Py/Pt bilayer are shown as a function of rf frequency
in (a) and (b), respectively. The solid line in (a) represents the fit to Eq.
(11) and results in $M_{s}=852$ G. The solid line in (b) represents a linear
fit to the linewidth vs. frequency dependence.
Figure 4 shows FMR spectra for a Py/Pt bilayer and a Py single layer at 4-GHz
excitation frequency. The FMR peak positions for the two samples are similar.
The main difference between the spectra is their FMR linewidth. The FMR line
widths (HWHM) extracted from fits to Lorentzian absorption functions are
$\Delta H_{Pt/Py}=16.9$ Oe for Py/Pt and $\Delta H_{Py}=12.9$ Oe for Py. The
difference in $\Delta H$ can be attributed to the loss of pumped spin momentum
in the Pt layer. The thickness of the Pt layer is 15 nm, which is larger than
$\lambda_{sd}^{Pt}=10\pm 2$ nm Kurt et al. (2002). Thus, all pumped spin
momentum is dissipated in the Pt layer and we can extract the value of the
spin mixing conductance $g^{\uparrow\downarrow}_{eff}$ from the increased
linewidth. Using Eq. (7) we calculate a spin mixing conductance
$g^{\uparrow\downarrow}_{eff}=2.1\times 10^{19}$ m-2 at the Py/Pt interface.
This experimental value is somewhat smaller than the previously reported
$2.58\times 10^{19}$ m-2Mizukami et al. (2001); Tserkovnyak et al. (2002b).
Cao et al. Cao et al. (2009) showed that for high power rf excitation, the
spin mixing conductance can be reduced due to the loss of coherent spin
precession in the ferromagnet. This could be the case here, since the cone
angle for the FMR at 4 GHz is relatively large, and a slightly larger mixing
conductance for the smaller precession angles at 11 GHz would lead to more
consistent frequency dependent values of the spin Hall angles as discussed
below.
Figure 4: (Color online) Derivative FMR spectra at 4 GHz for (a) Py and (b)
P/Pt. Solid lines are Lorentzian line shape fits.
### III.3 dc Voltage due to ISHE and AMR effects
Figure 5 shows the dc voltage measured along the samples with an external
field applied at 45 degrees from the coplanar waveguide axis. For the Py/Pt
sample we observe a resonant increase in the dc voltage along the sample at
the FMR position. However, the lineshape is complex: below the resonance field
the voltage is negative, it changes sign just below the FMR resonance field,
and has a positive tail in the high field region. In contrast, the single
layer Py sample, which is not affected by spin pumping, shows a voltage signal
that is purely antisymmetric with respect to the FMR position and thus mirrors
the derivative FMR signal shown in Fig. 4(b). The voltage generated by the
ISHE depends only on the cone angle $\theta$ of the magnetization precession
[see Eq. (8)] and thus must be symmetric with respect to the FMR resonance
position. This means that the voltage measured in the Py/Pt sample has two
contributions: (i) a symmetric signal due to the ISHE, and (ii) an
antisymmetric signal of the same origin as that in the Py control sample.
Figure 5: Voltage measured along the samples vs. field $H_{dc}$ for Py and
Py/Pt at 4 GHz is shown with symbols in (a) and (b) respectively. Only the AMR
contribution is present in the Py sample. Solid line in (a) shows a fit to Eq.
(12). Both AMR and ISHE effects are observed in the Py/Pt. Dotted and dash-
dotted lines show the AMR and ISHE contributions, respectively, which are
extracted from fitting the data to Eqs. (12) and (18); the solid line in (b)
shows the combined fit for the Py/Pt sample.
The antisymmetric voltages observed in both Py and Py/Pt depend on the cone
angle $\theta$ of the magnetization dynamics, since they vary rapidly around
the FMR resonance position. This suggests that the antisymmetric signal
originates from the AMR.Costache et al. (2006b); Gui et al. (2007a, b)
Although the MgO provides dc insulation between the sample and the waveguide,
see Fig. 6(a), the strong capacitive coupling allows some leakage of the rf
driving current $I_{rf}=I_{rf}^{m}\sin\omega_{f}t$ into the sample,
$I_{rf,S}$, which flows along the waveguide direction. Its magnitude can be
estimated from the ratio between the waveguide resistance $R_{wg}$ and the
sample resistance $R_{S}$: $I_{rf,S}=I_{rf}R_{wg}/R_{S}$, since the capacitive
coupling impedance is negligible for $\omega_{f}\gtrsim 4$ GHz. Furthermore,
due to the strong capacitive coupling (50 pF) between sample and wave guide,
both rf currents in the sample and waveguide are for all practical purposes in
phase, i.e., the relative phase shift is expected to be at most $10^{-3}\pi$.
Indeed, experiments with single layer Py samplesMosendz et al. (2010a) and
with a MgO layer inserted between the Py and Pt layersMosendz et al. (2010b)
are consistent with a pure AMR signal, as described below, without any
appreciable phase shift.
The precessing magnetization in the Py [see Fig. 6(b)] results in a time-
dependent $R_{S}[\psi(t)]=R_{0}-\Delta R_{AMR}\sin^{2}\psi(t)$ due to the AMR
given by $\Delta R_{AMR}$, where $R_{0}$ is the sample resistance with the
magnetization along the waveguide axis, and $\psi$ is the angle between the
instantaneous magnetization $\vec{m}$ and the waveguide axis [see Fig.
6(b)]Costache et al. (2006b). $\Delta R_{AMR}$ can be experimentally
determined by static magnetoresistance measurements under rotation of an in-
plane field sufficiently large to saturate the magnetization. Since the AMR
contribution to the resistance oscillates at the same frequency as the rf
current, but phase-shifted, a homodyne dc voltage develops and is given
by:Mosendz et al. (2010a)
$V_{AMR}=I_{rf}^{m}\frac{R_{wg}}{R_{S}}\Delta
R_{AMR}\frac{\sin(2\theta)}{2}\frac{\sin(2\alpha)}{2}\cos\varphi_{0}\;,$ (12)
where $\varphi_{0}$ is the phase angle between magnetization precession and
driving rf field, and the relation between $\theta$, $\alpha$ and $\psi$ is
illustrated in Fig. 6(b). Well below the FMR resonance the phase angle
$\varphi_{0}$ is zero, it becomes $\pi/2$ at the peak, and is $\pi$ far above
the resonance Guan et al. (2006). Thus, $\cos\varphi_{0}$ changes sign upon
going through the resonance, which gives rise to an antisymmetric $V_{AMR}$,
as is observed in both the Py and Py/Pt samples. Following Guan et al. Guan et
al. (2006) we calculate the cone angle $\theta$ and $\sin\varphi_{0}$ as a
function of the applied field $H_{dc}$, FMR resonance field $H_{r}$, FMR
linewidth $\Delta H$ and rf driving field $h_{rf}$:
$\theta=\frac{h_{rf}\cos\alpha}{\Delta
H\sqrt{1+\left(\frac{(H_{dc}-H_{r})(H_{dc}+H_{r}+4\pi M_{s})}{\Delta H4\pi
M_{s}}\right)^{2}}},\mathrm{and}$ (13)
$\sin\varphi_{0}=\frac{1}{\sqrt{1+\left(\frac{(H_{dc}-H_{r})(H_{dc}+H_{r}+4\pi
M_{s})}{\Delta H4\pi M_{s}}\right)^{2}}}\;.$ (14)
The anisotropic magnetoresistance was determined by dc magnetoresistance
measurements with fields applied along the hard-axis as $\Delta
R_{AMR}=0.95$%. This allows us to fit the Py data [see Fig. 5(a)] with only
one adjustable parameter $h_{rf}=4.5$ Oe using Eqs. (12–14).
Figure 6: (Color online) Splitting of rf current due to capacitive coupling is
schematically shown in (a) together with the directions of the applied dc
magnetic field $\vec{H}_{dc}$ and the rf driving field $\vec{h}_{rf}$ with
respect to the bilayer and waveguide. (b) Schematic of $\vec{m}$ precessing in
Py. $\vec{m}$ precesses around its equilibrium direction given by
$\vec{H}_{dc}$ at the driving frequency $\omega_{f}$ and with a phase delay
$\varphi_{0}$ with respect to $h_{rf}$. $\alpha$ is the angle between
$\vec{H}_{dc}$ and the waveguide axis (along $y$), $\theta$ is the cone angle
described by $\vec{m}$ and $\psi$ is the angle between $\vec{m}$ and the
waveguide axis. Due to the strong capacitive coupling part of $I_{rf}$ flows
through the Py given by $I_{rf,S}$. (c) Geometry of the dc component of the
pumped spin current with polarization direction $\langle\vec{s}\rangle$ along
the equilibrium magnetization direction $\vec{m}_{||}$. The charge current due
to ISHE $\vec{j}_{c}^{ISH}$ is orthogonal to the spin current direction
(normal to the interface) and $\langle\vec{s}\rangle$. The voltage due to the
ISHE is measured along $y$ (waveguide axis). Solid arrows indicate the spin
accumulation inside N, which decays with the spin diffusion length
$\lambda_{sd}$.
In order to understand the symmetric contribution to the Py/Pt voltage data we
have to include an additional voltage due to the ISHE. In principle an
inductive coupling (if any) could result in a symmetric voltage contribution
to the signal. However, this type of coupling is unlikely in our samples due
to the fact that sample and transmission line are prepared as a stack with a
thin insulator in the middle. Furthermore, our recent work Mosendz et al.
(2010b) showed that if spin pumping is suppressed by inserting a 3-nm MgO
layer at the Py/Pt interface, then the symmetric part of the voltage vanishes.
This unambiguously shows that the symmetric part of the measured voltage is
related to spin accumulation in N, which appears due to the ISHE. The absence
of a symmetric contribution for Py alone also suggests that inductive effects
are negligible.
In an open circuit, an electric field $\vec{E}$ is generated leading to a
total current density
$\vec{j_{c}}(z)=j_{c}^{ISH}(z)(\hat{x}\cos\alpha+\hat{y}\sin\alpha)+\sigma\vec{E},$
(15)
where $\sigma$ is the charge conductivity and $\hat{x}$, $\hat{y}$ are defined
in Fig. 6. Since there is no current flowing in the open-circuit
$\int_{-t_{Py}}^{t_{N}}\vec{j_{c}}(z)dz=0.$ (16)
When the wire is much longer than thick, the electric field in the wire is
constant. On the other hand, voltage generation occurs only in the Pt layer
(more precisely in a skin depth of the spin-flip length in which the ISHE emf
is generated), while the Py layer acts as a short, which decreases the voltage
difference at the sample terminals. Solving the system of Eqs. (15) and (16),
we obtain the component of the electric field along the measurement direction
$y$ as
$E_{y}=-\frac{Pg^{\uparrow\downarrow}_{eff}\sin\alpha\sin^{2}\theta\gamma
e\omega_{f}\lambda_{sd}}{2\pi(\sigma_{N}t_{N}+\sigma_{Py}t_{Py})}\tanh\left(\frac{t_{N}}{2\lambda_{sd}}\right),$
(17)
where $\sigma_{N}$ and $\sigma_{Py}$ are the charge conductivities in the N
layer (e.g., Pt) and Py, and $t_{N}$ and $t_{Py}$ are the thicknesses of the N
and Py layers. Using Eq. (17) we calculate the voltage due to the ISHE
generated along the sample with length $L$:
$V_{ISH}=-\frac{\gamma
eLP\omega_{f}\lambda_{sd}g^{\uparrow\downarrow}_{eff}\sin\alpha\sin^{2}\theta}{2\pi(\sigma_{N}t_{N}+\sigma_{Py}t_{Py})}\tanh\left(\frac{t_{N}}{2\lambda_{sd}}\right).$
(18)
Note that this voltage is proportional to $L$ and, thus, sufficiently large
voltage signals can be measured even for small $\gamma$ values by increasing
the sample length. Furthermore, note that for the case of the normal layer
thickness $t_{N}$ being comparable to the spin diffusion length $\lambda_{sd}$
the measured voltage depends only very weakly on either value, since
$t_{N}/\lambda_{sd}\tanh(t_{N}/2\lambda_{sd})$ is approximately constant.
One of the input parameters in $V_{ISH}$ is the ellipticity correction factor
$P$. At 4 GHz excitation, FMR occurs at $H_{dc}\approx 200$ Oe. Therefore, the
magnetization precession trajectory is highly elliptical and a correction to
the dc voltage component due to the ellipticity is significant. Figure 7 shows
$P$ as a function of microwave frequency as calculated using Eq. (9). In the
range from 4 to 13 GHz, $P$ changes almost by a factor of 3 and, therefore,
has to be taken into account. At frequencies above 10 GHz $P$ becomes larger
than 1 and reaches a maximum value of 1.3 at $\approx 28$ GHz before it slowly
decreases towards 1 for higher frequencies. This means that the most effective
pumping of dc component of spin current is achieved not for circular
precession, but rather for some elliptical trajectory of magnetization
precession.
Figure 7: Elliptical precession trajectory results in a time-dependent cone
angle of magnetization precession, that modifies the dc component of pumped
spin current. The ellipticity correction factor $P$ for the dc component of
spin current is calculated as a function of microwave frequency according to
Eq. (9).
We used Eqs. (18) and (12) to fit the voltage measured for the Py/Pt sample
[see the solid line in Fig. 5(b)]. The dashed and dotted lines in Fig. 5(b)
are the AMR and ISHE contributions, respectively. By using a literature value
for Pt of $\lambda_{sd}=10\pm 2$ nm Kurt et al. (2002), the only remaining
adjustable parameters are the rf driving field $h_{rf}$ and the spin Hall
angle $\gamma\approx 0.011\pm 0.002$. Note that through the cone angle
$\theta$, $h_{rf}$ affects both the AMR and ISHE contributions. In fact, as
seen from the fit to the control Py sample, $h_{rf}$ is already determined by
the negative and positive tails of the AMR part.
Figure 8: (Color online) (a) and (b) show voltages measured at 4 and 11 GHz.
Voltages measured at 11 GHz are smaller due to a decreased precession cone
angle. Note that the ratio between the ISHE and AMR contributions change due
to a faster decrease of the AMR voltage at high frequencies. (c) Spin Hall
angle in Pt as a function of frequency. The slightly decreased values at lower
frequencies may be due to non-linear effects and a concomitant decrease of
spin pumping at large angles of magnetization precession.
We carried out additional measurements of the spin Hall angle as a function of
the microwave frequency. Since the spin pumping is proportional to the time
derivative of the magnetization [as manifested by the factor $\omega_{f}$ in
Eq. (18)] and the ellipticity correction factor $P$, which increases with
frequency (see Fig. 7), the voltages due to the ISHE are expected to increase
at higher microwave frequencies. However, the cone angle of magnetization
precession for a constant power of rf excitation decreases due to higher
resonance fields. Since spin pumping is proportional to $\sin^{2}\theta$ an
overall decrease in the voltage due to the ISHE is observed. However, the
relative strength of the antisymmetric (AMR) and symmetric (ISHE) parts of the
signal change since the AMR decreases faster than the ISHE contribution as a
function of the frequency. This effect is illustrated in Figs. 8(a) and (b).
Figure 8(c) shows the values of the spin Hall angle $\gamma$ extracted from
the fits, which are essentially constant for all frequencies, except for a
slight decrease of $\gamma$ at lower frequencies. Equation (9) is strictly
valid only for small precession cones and constant power of rf excitation.
From the fitting of the data we can extract the cone angles of the
magnetization precession. At 4 GHz the fitted value $\theta\approx 10^{\circ}$
at the resonance, while at 11 GHz $\theta\approx 2.5^{\circ}$. For 4-GHz
excitation non-linear effects may start to play a role, possibly slightly
changing the estimated value of $\gamma$.
Figure 9: (Color online) (a) Voltage measured at 4 GHz as a function of angle
$\alpha$ of the external magnetic field with respect to the coplanar waveguide
axis. Experimental data and fits are shown with symbols and solid lines,
respectively. (b) Spin Hall angle extracted from the fits. The theoretical
model correctly takes into account the angular dependence for the ISHE and AMR
contributions.
Our model was further tested by varying the angle $\alpha$ of the applied
field with respect to the microwave transmission line. Note that both Eqs.
(12) and (18) have besides the explicit dependence on $\alpha$ an additional
dependence through the implicit $\alpha$-dependence of $\theta$ given by Eq.
(13). For small cone angles $\theta$ this results in both $V_{AMR}$ and
$V_{ISH}$ being proportional to $\sin\alpha\cos^{2}\alpha$. The dependence on
the dc magnetic field direction is shown in Fig. 9. The measured voltage
profile is consistent with the theoretical model, and results in a
consistently constant fitted value of the spin Hall angle in Pt. Due to the
specific geometry of the sample we were not able to measure at angles close to
$\alpha=90^{\circ}$, at which the magnetization dynamics cannot be excited,
because the component of $h_{rf}$ perpendicular to the magnetization vanishes
and FMR cannot be excited. But in the range of angles from $-5^{\circ}$ to
$45^{\circ}$, excellent agreement between experiment and theory was achieved.
Figure 10: (Color online) Power dependence of the symmetric ISHE voltage
contribution measured at 4 GHz. The inset shows that the maximum of the ISHE
signal is linear vs. rf excitation power. The highest power (150 mW) deviates
from the linear behavior due to excitation of non-uniform modes. This
deviation is also observed in the corresponding FMR spectra.
The other adjustable parameter in our measurements is the power of the
microwave excitation. The rf microwave field amplitude increases as a square
root of the power. According to Eq. (13) the cone angle $\theta$ of the
magnetization precession increases linearly with driving field. The voltage
due to the ISHE is quadratic in $\theta$ and, thus, is expected to be
proportional to the power. After fitting the data, we extracted the symmetric
part due to the ISHE, which is shown in Fig. 10. The maximum values of the
measured voltage depend linearly on power, as expected by theory, except for
the highest power of about 150 mW, at which the system is driven into the non-
linear regime. We observe a deviation of the FMR peak position as a well as a
deviation of the FMR spectra from a Lorentzian shape. It is known that at high
rf power other modes beside the uniform FMR mode are excited. In this case one
expects a substantial line broadening and even saturation of the FMR
absorption, as observed.
Figure 11: (Color online) Voltages measured at 11 GHz for (a) Py/Pd, (b)
Py/Au, and (c) Py/Mo. Shown are data (symbols), combined fits (black lines)
and individual AMR and ISHE contributions, with dotted green and dash-dotted
blue lines, respectively. Note the opposite sign of the ISHE contribution for
Mo compared to those for Au and Pd. Table 1: Spin Hall angle $\gamma$
determined using $\lambda_{sd}$ and $\sigma_{N}$
from data measured at 11 GHz. Normal metal $\lambda_{sd}$ (nm) $\sigma_{N}$
$1/(\Omega m)$ $\gamma$ Pt 10$\pm$2 (2.4$\pm$0.2)$\times 10^{6}$
0.013$\pm$0.002 Pd 15$\pm$4 (4.0$\pm$0.2)$\times 10^{6}$ 0.0064$\pm$0.001 Au
35$\pm$3 (2.52$\pm$0.13)$\times 10^{7}$ 0.0035$\pm$0.0003 Mo 35$\pm$3
(4.66$\pm$0.23)$\times 10^{6}$ -0.0005$\pm$0.0001
### III.4 Spin Hall angle in Pd, Au and Mo
Since the sample structure in our experiments is just a bilayer of Py with the
non-magnetic material of interest, this technique can be readily applied to
determine $\gamma$ in any conducting material. In Fig. 11 we show voltages
measured for Py/Pd, Py/Au, and Py/Mo measured at 11 GHz excitation. The spin
Hall contributions in Au and Mo are smaller than in Pt, and note that for Mo
the spin Hall contribution has the opposite sign. Fitting of the data enabled
us to extract the values of $\gamma$ for Pd, Au and Mo (see Table 1). The
effective mixing conductance at the intermetallic Py/N interface is governed
by N and we adopt the value obtained by the experimentally measured increased
damping in Py/N. Note that the determination of $\gamma$ furthermore requires
$\sigma_{N}$ and $\lambda_{sd}$ as input parameters. $\sigma_{N}$ was obtained
using four-probe measurements for all samples. Reported values for
$\lambda_{sd}$ vary considerably. We choose literature values for Pt and Pd
from Ref. Kurt et al., 2002 and Au from Ref. Mosendz et al., 2009, and for Mo
we assumed that $\lambda_{sd}$ is comparable to that for Au. Even though this
latter assumption may not necessarily hold, the sign change is consistent with
earlier measurements Morota et al. (2009). We furthermore note that the
$\gamma$ values in Table 1 differ from the previously reported ones in Ref.
Mosendz et al., 2010a, where we assumed circular precession and therefore
underestimated $\gamma$ by a factor of roughly 2. In addition, Table 1 is
based on 11 GHz data, which due to the smaller cone angles is less susceptible
to deviations stemming from non-linear effects, and therefore should be more
reliable.
Our values for $\gamma$ are in good agreement with values reported by Otani et
al. Vila et al. (2007); Morota et al. (2009) from measurements in lateral spin
valves, but conflict with values reported by other groupsSeki et al. (2008);
Ando et al. (2008a). We note that in lateral spin valves it is important to
also understand the charge current distribution in order to rule out or
correct for additional non-local voltage contributions Mihajlović et al.
(2009). A distinct advantage of our approach is that the measured voltage
signal scales with the sample dimension, and no additional charge current is
directly applied to the sample that could result in unwanted spurious voltage
signals.
We also gain insights into spin orbit coupling in non-magnetic metals, which
ultimately give rise to spin Hall effects. Even for non-magnetic materials,
that are next to each other in the periodic table (Pt and Au) the spin Hall
angle differs almost by a factor of 4. On the other hand, Mo has a
significantly smaller spin Hall angle with opposite sign. This sign change can
be rationalized by a simple $s$-$d$ hybridization model and is supported by
first-principles calculations,Kontani et al. (2009) indicating that the spin
Hall angle should be negative for less than half-filled $d$-bands, and
positive for more than half-filled ones, consistent with our experimental
results. Pd in spite of being a lighter element than Au has a spin Hall angle
which is almost 2 times larger. First-principles calculations are again
consistent with the experimental observation of $\gamma$ being larger for Pt
and Pd compared to Au.Tanaka et al. (2008)
## IV Conclusions
We presented a spin pumping technique that enables measuring spin Hall angles
in various materials, which has clear advantages over standard dc electrical
spin injection in Hall bar microstructures. Our results for Pt, Pd, Au and Mo
show that spin Hall angles are rather small, with the largest value found to
be 0.013 for Pt. Our approach provides a uniform spin current across a
macroscopic sample. The voltage signal from the inverse spin Hall effect can
readily be increased via the use of longer samples, since $V_{ISH}\propto L$.
Furthermore, by using an integrated coplanar waveguide architecture we can
control parameters, such as the rf driving field distribution, microwave
frequency and power of rf excitation. This enabled a quantitative analysis of
the data and a test of the theoretical model under various experimental
conditions. Our model accounts for both the anisotropic magnetoresistance and
the inverse spin Hall effect contributions and agrees with experiments for a
wide range of controllable parameters. We demonstrated the existence of
symmetric (ISHE) and antisymmetric (AMR) voltages and could model the
frequency, magnetic field direction, and excitation power dependence well. The
AMR voltage in our experiments originates from capacitive coupling between the
waveguide and the sample and is consistent with the parameters characteristic
for the ferromagnetic resonance. Our method will enable additional studies of
spin Hall effects in other materials, and, therefore, will be useful to
further understand the spin-orbit coupling mechanism in metals. This is
necessary in order to develop and optimize the spin Hall effect as a method to
generate and detect spin currents in various circumstances, such as in the
spin Seebeck effectUchida et al. (2008).
## V Acknowledgements
We would like to thank R. Winkler and G. Mihajlović for valuable discussions.
Furthermore, we would like to thank C.-M. Hu for pointing out that the angular
dependence of the AMR and ISHE contributions are identical. This work was
supported by the U.S. Department of Energy - Basic Energy Sciences under
Contract No. DE-AC02-06CH11357 and EU-IST through project MACALO (no. 257159).
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|
arxiv-papers
| 2010-09-26T12:49:57 |
2024-09-04T02:49:13.112191
|
{
"license": "Public Domain",
"authors": "O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\n S. D. Bader, and A. Hoffmann",
"submitter": "Axel Hoffmann",
"url": "https://arxiv.org/abs/1009.5089"
}
|
1009.5131
|
# Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements
Pierre-Olivier Quirion1,2,3, and Jørgen Christensen-Dalsgaard2,3, Torben
Arentoft2,3 1 Canadian Space Agency, 6767 Route de l’Aéroport, Saint-Hubert,
QC, J3Y 8Y9 Canada
2 Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,
Denmark
3 Danish AsteroSeismology Center pierre-olivier.quirion@asc-
csa.gc.ca,jcd@phys.au.dk,ta@vucaarhus.dk
###### Abstract
Space-based projects are providing a wealth of high-quality asteroseismic
data, including frequencies for a large number of stars showing solar-like
oscillations. These data open the prospect for precise determinations of key
stellar parameters, of particular value to the study of extra-solar planetary
systems. Given the quantity of the available and expected data it is important
to develop efficient and reliable techniques for analyzing them, including the
determination of stellar parameters from the observed frequencies. Here we
present the SEEK package developed for the analysis of asteroseismic data from
the Kepler mission. A central goal of the package is to obtain a fast and
automatic determination of the stellar radius and other parameters, in a form
that is statistically well-defined. The algorithms are tested by comparing the
results of the analysis with independent measurements of stellar radius and
mass, for a sample of well-observed stars. We conclude that the SEEK package
fixes stellar parameters with accuracy and precision.
methods: statistical – stars: fundamental parameters – stars: oscillations –
stars: solar-type
††slugcomment: The Astrophysical Journal; Accepted
## 1 ASTROPHYSICAL CONTEXT
Solar-like oscillation occurs in stars with convective envelopes and are seen
as acoustic modes (p modes) with low angular degree ${l}\lesssim 3$ and
intermediate to high radial order $10\lesssim n\lesssim 30$. After early work
by Ando & Osaki (1975) and Goldreich & Keeley (1977a, b), evidence has been
found that the excitation of solar-like oscillation is caused by turbulent
convection in the uppermost part of the envelopes of stars (see for example
Balmforth, 1992; Goode et al., 1992; Goldreich et al., 1994). An important
aspect of the driving caused by turbulent convection is that the excitation of
the mode occurs at random times, and hence the process is known as stochastic
excitation. One of the effects of this stochastic behavior is that the
eigenmode phases are changing with time. In the Sun, the modes with the
largest amplitudes, around 3000 $\mu$Hz, have a coherence time of less than 12
days (Gelly et al., 2002; Chaplin et al., 1997). We can contrast this value to
classical pulsators where the coherence time is of the order of the stellar
evolution time scale. Short but recent introductions to the topic of
stochastic modes excitation, and references to modern approaches to that
problem, can be found in Houdek (2006) and Samadi et al. (2008).
The first star other than the Sun showing evidence for individual frequencies
of solar-like oscillation, $\eta$ Boötis, was observed by Kjeldsen et al.
(1995), using a technique measuring the equivalent widths of the stellar
Balmer lines over time. Most subsequent observations of solar-like oscillation
have been made with Doppler-velocity measurements using a laboratory reference
(typically an iodine cell) to measure the shift in a stellar spectrum. This
technique finally permitted the first unambiguous detection of solar-like
oscillation in $\alpha$ Centauri A (Bouchy & Carrier, 2001, 2002). Since then,
p-mode excitation has been detected with ground-based observations in more
than a dozen main-sequence and subgiant stars. Lists of these solar-like
oscillators can be found in Bedding & Kjeldsen (2008) and Bruntt et al.
(2010).
Doppler-velocity measurement is the technique of choice to observe solar-like
oscillation, providing the least interference from stellar background
contributions to the ‘noise’ in the observations. The ratio between the signal
and the stellar background noise, that is non-periodic signals coming, for
instance, from granulation, is far higher in Doppler-velocity than in
photometric intensity measurements. Solar-like oscillations can also be
observed in photometric intensity measurement, albeit with a higher stellar
background noise, but with the advantage of allowing simultaneous observation
of a large number of stars. However, owing to noise induced by the Earth’s
atmosphere such observations are essentially only possible from space (Harvey,
1988). Space-based photometric data for solar-like oscillations have started
to be obtained with the help of the CoRoT (Baglin et al., 2006) and Kepler
(Borucki et al., 2010) missions. Results coming from these satellites, in
particular the Kepler satellite (Gilliland et al., 2010), are indeed a
motivation for the work undertaken in the present study. The development of an
objective technique to interpret results from asteroseismology, more than
being intrinsically desirable, is now made necessary by the era of high-speed
space photometry, where data bases containing large numbers of light curves
ready for asteroseismic study are made available. By objective
asteroseismology we mean a technique that is not directed at every step by
subjective choice of a scientist and that results in an estimate of global
stellar parameters along with realistic, if not completely objective,
uncertainties on these parameters. The need for efficient and reliable,
automatic analysis of asteroseismic data will be further emphasized with the
ESA PLATO mission (Catala, 2009) which, if finally selected, will be launched
around 2018.
The Sun is evidently the canonical example of solar-like oscillation. As shown
in detail in the review on helioseismology by Christensen-Dalsgaard (2002) the
study of the Sun’s oscillations has proven that physical information can be
extracted from the understanding of its vibration modes. The extraction of
information from other solar-like stars has started to occur and the sustained
modeling efforts by several groups has yielded many asteroseismic studies
(e.g. Thoul et al., 2003; Eggenberger et al., 2004; Miglio & Montalbán, 2005;
Carrier et al., 2005; Eggenberger et al., 2005, 2008; Mosser et al., 2008).
This has positively confirmed the promise of asteroseismology: to dramatically
improve knowledge about stellar parameters. This prospect, especially the
promise to determine stellar radii within a 3 % error, is very attractive for
the exoplanet groups involved in the Kepler mission. For example, Baines et
al. (2007) showed how the interferometric radius measurement of HD 189733 has
dramatically improved and clarified the value for the radius a nd mean density
of its planetary companion. We expect radius measurement from asteroseismology
to have similar effect. An early indication of the power of Kepler
asteroseismology to characterize exoplanet hosts was provided by Christensen-
Dalsgaard et al. (2010).
Photometric measurement of planetary transits, given that a precise modeling
of the limb darkening is achieved (Prsa et al. 2010, in preparation), permits
to identify four parameters of the star-planet systems: its orbital period
$\Pi_{\rm orb}$, its inclination $i$ toward the observer and the fractional
radii of the star and planet, which are defined as
$\displaystyle r_{\rm A}=\frac{R_{\rm A}}{a}{\rm,}$ $\displaystyle{\rm and}$
$\displaystyle r_{\rm b}=\frac{R_{\rm b}}{a}\;,$ (1)
respectively. We clearly see that the knowledge of the stellar host radius
$R_{\rm A}$ and its associated error directly leads to a determination of the
desired planet radius $R_{\rm b}$, as well as the orbital semi-major axis $a$.
Further details on the influence of the precision of stellar parameters on the
inference of physical properties of planetary system were provided by
Southworth (2009). One of the goals of the present study is to test the
reliability of asteroseismology in determining the radius of stars by
comparing the results with available interferometric measurements.
This work presents an objective procedure to extract information from solar-
like oscillations. The procedure, named SEEK, determines fundamental stellar
parameters with the use of quantities inspired by the p-mode asymptotic
asteroseismic relations, namely the large and small separations, $\Delta\nu$
and $\delta\nu$, respectively (Tassoul, 1980). It can also make use of
standard astrophysical input, the parallax $\pi$, the $V$ band magnitude and
the reddening $E_{B-V}$, as well as input derived from atmosphere modeling,
the effective temperature $T_{\rm eff}$, the gravity $\log g$ and finally the
iron to hydrogen ratio [Fe/H]. Therefore, SEEK is not a purely asteroseismic
procedure that only uses the frequency spectra of a pulsating star to
constrain its fundamental parameters, but rather a hybrid procedure that uses
both traditional and seismic input.
The procedure is part of the Kepler Asteroseismic Science Consortium (KASC)
pipeline that provides in priority stellar parameters to be used by exoplanet
seekers of the Kepler team. Its aims are also to analyze every star in which
$\Delta\nu$ and/or $\delta\nu$ has been fixed and to provide an extended and
homogeneous overview of the disk population observed by Kepler. A description
of SEEK and of its main characteristics is made in the following sections. We
tested SEEK on a set of well-studied stars. Of course the Sun is the golden
standard, but we also tested the procedure against stars with diameters
determined through interferometric measurements and/or with masses derived
from Kepler’s third law, thus testing our procedure against the closest we
have to absolute quantities. The stars studied comprise $\beta$ Hydri (HD
2151), $\tau$ Ceti (HD 10700), Procyon A (HD 61421), $\eta$ Boötis (HD
121370), $\alpha$ Centauri A and B (HD 128620 and HD 128621), and $70$
Ophiuchi A (HD 165341).
## 2 THE SEEK PROCEDURE
### 2.1 Stellar models
The SEEK procedure makes use of a large grid of stellar models computed with
the Aarhus Stellar Evolution Code (ASTEC). It compares an observed star with
every model of the grid and makes a probabilistic assessment, with the help of
Bayesian statistics, about the structure of that star.
Details of ASTEC and its input physics were described by Christensen-Dalsgaard
(2008a). Out of the possible options offered by ASTEC, we used, for all our
computations: the OPAL equation of state (Swenson et al., 1996) along with the
OPAL plus Ferguson & Alexander opacity tables (Iglesias & Rogers, 1996;
Alexander & Ferguson, 1994), the element to element ratios in the metallic
mixture of Grevesse & Sauval (1998), and convection treated with the mixing-
length formulation of Böhm-Vitense (1958); the mixing length to pressure scale
height ratio $\alpha$, characterizing the convective efficacy, was treated as
a variable parameter in the SEEK fits. Neither diffusion nor overshooting was
included.
A brief introduction to a beta version of SEEK was presented along with a
series of tests using synthetic observations in $\S$ 5 of Stello et al.
(2009). The beta version was built with an ad hoc probabilistic approach that
is quite different from the one used in the present version. There is a major
difference in the way errors on stellar parameters is obtained. This
difference lies in the fact that the new grids have been extended and are
dense enough to permit the use of Bayesian statistic. While SEEK yields
similar results with both approaches, the Bayesian method used here is more
rigorous, more robust and more comprehensive than the technique developed for
the beta version of SEEK.
The core of SEEK is a grid of models used to fit observations. We have
calculated 7,300 evolution tracks or 5,842,619 individual models. Each track
calculation was stopped after the track had reached the giant branch or an age
of $\tau=15\times 10^{9}$ yr. The tracks are separated into 100 subsets using
a different combination of metallicity $Z$, initial hydrogen content $X_{0}$
and mixing-length parameter $\alpha$. These combinations are separated into
two regularly spaced and interlaced subgrids. The first subgrid comprises
tracks with $Z=[0.005,0.01,0.015,0.02,0.025,0.03]$,
$X_{0}=[0.68,0.70,0.72,0.74]$, and $\alpha=[0.8,1.8,2.8]$ while the second
subset has $Z=[0.0075,0.0125,0.0175,0.0225,0.0275]$, $X_{0}=[0.69,0.71,0.73]$,
$\alpha=[1.3,2.3]$. Every subset is composed of 73 tracks spanning from 0.6 to
3.0 $M_{\rm\odot}$. The spacing in mass between the tracks is 0.02
$M_{\rm\odot}$ from 0.6 to 1.8 $M_{\rm\odot}$ and 0.1 from 1.0 to 3.0
$M_{\rm\odot}$. A slice of the grid is presented in Figure reffig:slice for
$Z=0.015$, $X_{0}=0.72$ and $\alpha=1.8$ in a $\log g-\log T_{\rm eff}$
diagram.
The structure of the SEEK procedure is quite flexible. It will be easy in
future to expand the grid with regularly or irregularly spaced tracks. In
addition, we have started to test the possibility of interpolating models
along the age-mass plane of the grid and between grid values of $Z$ and
$X_{0}$. The results are quite promising and it seems that the grid is dense
enough with regards to these parameters to realize precise interpolation.
However, interpolation tests in the $\alpha$ direction were not convincing and
it seems that the grid would have to be refined significantly with respect to
that parameter to permit precise interpolation.
Some care has been required in defining the parameters of the grid, including
the specific range and spacing we have chosen. Several factors influenced the
definition of the composition. Recent analyses have led to substantial
revisions of the solar surface composition (Asplund, 2005; Grevesse et al.,
2007; Asplund et al., 2009); since the solar composition is used as reference
to relate the observed [Fe/H] to $Z/X$ this potentially affects the assumed
stellar composition. Here we have chosen to use the older value of Grevesse &
Sauval (1998). This can be justified based on the fact that solar models using
the old values are closer to the helioseismic inferences than those using the
more recent abundances (see, for example Basu et al., 2007; Basu & Antia,
2008). We also emphasize that for most stars, atmospheric parameters in the
literature are still only available with older metallic mixtures. Homogeneity
of our analysis requires the use of these mixtures. The metallicity range used
in the grid, $-0.61\lesssim{\rm[Fe/H]}\lesssim 0.20$, covers most ($\gtrsim
90\%$) of the stellar population of the disk if the Hipparcos color-magnitude
diagram is taken as a reference (Reid, 1999; Chabrier, 2001). This should
permit the analysis of a large majority of the Kepler stars.
The careful reader will also remark that relatively high values of
$Y_{\odot}=0.2713$ and $Z_{\odot}=0.0196$ have been used in the standard
definition of [Fe/H]. This value is used to calibrate solar models of ASTEC to
the right luminosity (Christensen-Dalsgaard, 1998). This value is used in all
conversions done here.
The mass step of 0.02 $M_{\rm\odot}$ from 0.6 to 1.8 $M_{\rm\odot}$ has first
been chosen to be small enough to permit interpolation in the $\log g-\log
T_{\rm eff}$ diagram. However, we realized that this small step in mass has
the virtue of making the grid sufficiently dense that we do not have to rely
on interpolation to make our probabilistic estimate of the stellar parameters.
The lower limit to the range results from the fact that molecules start to
play an important role in these stars and that our evolution code does not
include them. This should not be a handicap to SEEK since the low absolute
magnitude of the stars not included in the grid, with $M<0.5$ $M_{\rm\odot}$,
makes them improbable Kepler targets. The upper limit of 1.8 $M_{\rm\odot}$ is
approximately the highest mass of the grid at which a model on the main
sequence has at least 1% of its radius as a convective envelope. We wanted to
keep a high density of models up to that limit where it is most likely to
observe stars with stochasticall y excited oscillations. The models at higher
masses are put as a safeguard permitting us to pin down a star with a very
thin convective zone showing solar-like oscillations. We also remark that
typically models with higher $\alpha$ and/or higher $X_{0}$ and/or higher $Z$
will tend to keep a convective envelope at slightly higher masses. Similarly,
these models develop a convective core, leading to the typical hook path in
the H-R diagram at higher masses.
The grid mid-value of $\alpha=1.8$ was chosen in accordance with the solar
calibration. The span of $\alpha$ is a trade-off between trying to get a wide
range of values, from 0.8 to 2.8, and to limit the quantity of computed models
in the grid. We had a bias towards testing a wide range of convective
efficiency since it is not clear how the mixing-length theory extends from the
Sun to other main-sequence stars. However, this choice is not ideal since, as
mentioned, the steps are too large to permit interpolation between the grid
points.
The role of SEEK is more to measure the size of the “valley of good solutions”
produced by Kepler observations than it is to probe the deepest abyss of that
valley. Specifically, the quality of a solution is measured by how well the
computed observables in the grid, $\\{q_{i}^{\rm g}\\}$, match the observed
values $\\{q_{i}^{\rm obs}\\}$; this is quantified in terms of
$\chi^{2}={\displaystyle\sum_{i=0}^{n}}\left(\frac{{q}_{i}^{\rm
obs}-{q}_{i}^{\rm g}}{\sigma_{i}}\right)^{2}\;,$ (2)
where $\sigma_{i}$ is the estimated error for the observation $q_{i}^{\rm
obs}$ and $n$ the number of observables. SEEK does not claim to find the best
model for an observed star; its aim is to draw the contour of good solutions
which is located around $\chi^{2}_{r}=\chi^{2}/P\sim 1$, $P$ being the number
of degrees of freedom of the problem. SEEK also outputs the stellar parameters
with reliable error bars. We do not want to underestimate the errors by
restricting the size of our grid. The approach is especially well suited to
our problem since our solutions coming from the observation of $\Delta\nu$ are
always heavily degenerate. This means that we do not expect a single and small
region of the parameter space to yield a result that is much better than any
other in terms of $\chi^{2}$. Instead we have, as illustrated in Figure 2 for
the star $\alpha$ Centauri A, a large region of the parameter space with
$\chi^{2}_{r}\sim 1$. In Figure 2, we present for drawing convenience
$\log\chi^{2}_{r}$ as a function of the mass $M$ and the normalized age
defined as
$\tau_{\rm N}=\frac{\tau}{\min(\tau_{\rm RGB},14{\rm Gyr})}\;,$ (3)
where $\tau$ is the age of a model on a specific track and $\tau_{\rm RGB}$ is
the age of the model following the same track when it reaches the red-giant
branch (RGB). In Figure 2, this parameterization is useful for visual purposes
as it permits a smooth transition of the $\chi^{2}$ values from the lower-mass
models not reaching the RGB to the models reaching it. The exact position of
$\tau_{\rm RGB}$ on the RGB is not critical since the evolution time scale on
the RGB is much faster than on the main sequence. We stress that this
parameterization was used for cosmetic purposes and is not involved in the
SEEK computations. The models showing higher $\chi_{r}^{2}$ to the left of the
valley are caused by the appearance of convective cores in heavier models;
this makes the automatic computation of the small separation somewhat
difficult and creates these artifacts.
Two works presenting pipelines to do asteroseismology of solar-like stars have
recently been published, Metcalfe et al. (2009) and Basu et al. (2010). In the
former, the authors combine a pipeline using a genetic algorithm to find the
best model in the parameter space and a local analysis, based on linearization
around this solution, to refine the fit and estimate the error in that
solution. Basu et al. (2010) find the best model with a grid of stellar
models, not unlike SEEK but using a smaller grid, while the errors are
determined with a frequentist approach, using a Monte Carlo simulation of
synthetic observations. We certainly recognize the desirability of a genetic
algorithm, or a similar technique aiming at the absolute minimum in the
parameter space, while also acknowledging the large computation power needed
for such techniques. However, we note that a local error estimator used by
Metcalfe et al. (2009) assumes that the solution is linear within the error
bars and that no nearby local minima can be deep enough to contribute to the
error analysis; this can lead to underestimation of the errors. In our view,
only an error estimator, Bayesian or frequentist, looking at various models
around the best solutions may provide fully reliable results. This is
particularly true in a problem, as the present, where the valley of solution
is extended. This preference for Bayesian statistics has motivated our use of
it in the development of SEEK.
### 2.2 The Observables
The grid permits the mapping of the model physical input parameters
$\boldsymbol{p}\equiv\\{M,\tau,Z,X_{0},\alpha\\}$ into the grid of observable
quantities $\boldsymbol{q}^{\rm g}\equiv\\{\Delta\nu,\delta\nu,T_{\rm
eff},\log g,{\rm[Fe/H]},M_{V},...\\}$, defining the transformation
$\boldsymbol{q}^{\rm g}=\mathcal{K}(\boldsymbol{p}).$ (4)
We use these quantities and compare them to the actual observed quantities
$\boldsymbol{q}^{\rm obs}$.
Some of the observables used for the subsequent fit are easily extracted from
the grid like $T_{\rm eff}$, $\log g$, [Fe/H]. However, $M_{V}$, $\Delta\nu$,
and $\delta\nu$ need some extra attention. We compute absolute magnitude
$M_{V}$, in the Johnson $V$ band, of our model by applying the bolometric
correction $B_{\rm c}(V)$ of the VandenBerg & Clem (2003) tables to our
luminosity $L$,
$M_{V}=M_{\rm Bol,\odot}-2.5\log\frac{L}{L_{\rm\odot}}-B_{\rm c}(V),$ (5)
where $M_{\rm Bol,\odot}$ is the VandenBerg & Clem (2003) prescription of 4.75
for the Sun’s bolometric magnitude. We can then compare this value to its
observed counterpart defined as
$M^{\rm obs}_{V}=5\log\pi-(V-A_{v})+5.$ (6)
The conversion is done using the following relation between the absorption
$A_{V}$ and the interstellar reddening $E_{B-V}$: $A_{V}=3.1E_{B-V}$.
We computed p-mode frequencies for all models on the main sequence and
subgiant branch using the ADIPLS code (Christensen-Dalsgaard, 2008b); the
models on the red-giant branch are not included in this asteroseismic study.
The large separation $\Delta\nu$ is computed with modes of $l=0$ only,
$\Delta\nu=\langle\nu_{n,0}-\nu_{n-1,0}\rangle\;,$ (7)
while $\delta\nu$ is the combination of modes with $l=0$, $2$ only,
$\delta\nu=\langle\nu_{n,0}-\nu_{n-1,2}\rangle\;;$ (8)
here $\nu_{n,l}$ is the frequency of the pulsation mode with angular degree
$l$ and radial order $n$. We know that these asymptotic values change with the
radial order $n$ in models. This is especially true in the case of
$\delta\nu$. To compare observed separations with model values we have
computed the average separations with up to 8 modes having different $n$ and
centered on a predetermined value of $\nu_{\rm central}$. The quantity
$\nu_{\rm central}$ is related to the maximum power $\nu_{\rm max}$ seen in
the Fourier transform of light curves of solar-like stars (see Stello et al.,
2009, for comments on $\nu_{\rm max}$). Thus, in our grid the large and small
separations are functions of the observed $\nu_{\rm max}$.
The computation of the average large and small separations is done by finding
the maximum of the cross-correlation function of the frequencies selected
around a suitable value $\nu_{\rm central}$ near $\nu_{\rm max}$. Figure 3
shows how the small separation is computed for a specific model of the grid.
In panel (a), the function $\psi_{0}$ corresponding to the $l=0$
eigenfrequencies is shown in blue and the function $\psi_{2}$ for $l=2$ in
red. These functions are convolved with the triangular distribution $T$ of
panel (b). The base of the triangular distribution is one fifth of the
smallest distance between any combination of the $l=0$ eigenfrequencies. The
convolution
$\displaystyle D_{i}(x)\equiv\int_{-\infty}^{\infty}\psi_{i}(s)T(x-s)ds$
$\displaystyle i=0,2$ (9)
is presented in panel (c) while the resulting cross-correlation,
$\displaystyle(D_{0}\ast
D_{2})(x)\equiv\int_{-\infty}^{\infty}D_{0}(s)D_{2}(x+s)ds$ , (10)
is in panel (d). The maximum of the correlation is at $\delta\nu$. It is easy
to see that the extra $l=2$ mode around $\nu\sim 750$ $\mu$Hz is only
producing a small bump in the correlation function and has no effect on the
value of $\delta\nu$. For $\Delta\nu$, the location of the maximum of the
auto-correlation $\left(D_{0}\ast D_{0}\right)$, apart from the maximum at
zero frequency shift, is taken. This technique is more robust than using the
mean value of Equation (7) and especially (8) over different values of $n$.
The cross-correlation is not influenced by missing or extra eigenmodes,
usually nonradial, produced by ADIPLS or by ambiguous determination of the
radial order $n$ of a mode. The method is especially robust when it comes to
computing $\delta\nu$ for models having a convective core, or for models with
avoided crossing. Robustness is very desirable since it is not possible to
control the ADIPLS output for the 6 million models included in th e grid.
In the present version of SEEK, we computed all adiabatic modes with $l=0$,
$2$ in a fixed range of the dimensionless frequency, $10\leq\sigma^{2}\leq
2800$, where $\sigma$ is related to the angular mode frequency $\omega$ by
$\sigma^{2}=\frac{R^{3}}{GM}\ \omega^{2}\;,$ (11)
where $G$ is the gravitational constant. This range makes it possible to cover
observed solar-like pulsation in known stars on the main sequence and subgiant
branch. These pulsations cluster around the value of $\nu_{\rm max}$ which
tends to follow the scaling relation of Kjeldsen & Bedding (1995)
$\nu_{\rm max}\propto\frac{M}{R^{2}T^{1/2}}.$ (12)
To accelerate the SEEK procedure and for models showing eigenmodes in these
region, we precalculated 25 different sets of $\Delta\nu$ and $\delta\nu$ for
values of $\nu_{\rm central}$ equally spaced on a logarithmic scale, ranging
from the Nyquist frequency of the short cadence of Kepler, 8330 $\mu{\rm Hz}$
(corresponding to a period of 2 min) to 27.8 $\mu{\rm Hz}$ (corresponding to a
period of 10 h). (For any given model the actual range in $\nu_{\rm central}$
was restricted to values corresponding to acoustic modes for that model.) If,
for example, we want to compare the grid value of $\delta\nu$ with the solar
value of $\delta\nu_{\odot}$ at $\nu_{\rm max,\odot}\sim 3333$ $\mu{\rm Hz}$,
our grid automatically chooses the set $\delta\nu(\nu_{\rm central})$ which is
the closest to $\nu_{\rm max,\odot}$ at $\nu_{\rm central}=3220$ $\mu{\rm
Hz}$.
## 3 BAYESIAN APPROACH
The Bayesian statistics method of SEEK and the notation used here were
inspired mainly by the work of Pont & Eyer (2004) and Jørgensen & Lindegren
(2005). These investigations compared the Bayesian approach to other means
used for the determination of stellar age. We refer the reader to these papers
and to the textbook by Gregory (2005) for technical questions involved in the
use of Bayesian statistics.
Let us first define the maximum likelihood function
$\mathcal{L}=\left({\displaystyle\prod_{i=0}^{n}}\frac{1}{\sqrt{2\pi}\sigma_{i}}\right)\exp(-\chi^{2}/2),$
(13)
with $\chi^{2}$ defined in Equation (2). A maximum-likelihood estimate of
stellar parameters can be obtained by finding the maximum of $\mathcal{L}$,
which, in case of Gaussian errors, is equivalent to minimizing $\chi^{2}$. It
can be argued that a maximum likelihood estimator is often enough to estimate
stellar parameters. However, the highly non-linear mapping function
$\mathcal{K}$ makes it necessary to provide a set of priors, especially if a
realistic estimate of the errors on the stellar parameters is to be made. The
prior can be seen as a specific weight attached to every point of the grid
determining the probability of that point to be observed. The most obvious
example of the utility of priors is well studied in Jørgensen & Lindegren
(2005). It arises when one has to choose between different stellar models at
different evolutionary stages but showing the same observables. In their
example, one model is slowly evolving on the main sequence while the other is
rushing up the subgiant branch. An experienced astronomer would naturally
choose the slowly evolving star as the most probable model even if the
resulting $\mathcal{L}$ value is the same for both models. For the astronomer
using Bayesian tools, a larger prior weight $f_{0}(\boldsymbol{p})$ given to
the slowly evolving model automatically makes it a better candidate because
the resulting probability, or posterior $f$ defined at each point $j$ as
$f^{j}\propto f_{0}^{j}\mathcal{L}^{j}\;,$ (14)
is bigger for that model. The grid overall posterior can be described as a
density function,
$f(\boldsymbol{p})\propto
f_{0}(\boldsymbol{p})\mathcal{L}(\mathcal{K}(\boldsymbol{p}))\;.$ (15)
¿From the grid presented in the previous section, the computation of
$\mathcal{L}$ at every point is straightforward. On the other hand, the
definition of the priors $f$ requires some insight in the problem.
The most natural way to write the prior density is as a function of
$\boldsymbol{p}$, the model parameters,
$f_{0}=\Psi(\tau)\Phi(Z|\tau)\zeta(X_{0}|Z,\tau)\xi(M|X_{0},Z,\tau)\beta(\alpha|X_{0},Z,\tau,M)$
(16)
where $\Psi(\tau)$ is the star formation rate (SFR) through time, and
$\Phi(Z|\tau)$ and $\zeta(X_{0}|Z,\tau)$ describe the metallicity and initial
hydrogen mixture as a function of age and as a function of age and
metallicity, respectively. The initial mass function (IMF)
$\xi(M|X_{0},Z,\tau)$ is function of the element mixture as well as a function
of time, and the prior related to the mixing-length parameter
$\beta(\alpha|X_{0},Z,\tau,m)$ can also depend on the other stellar
parameters.
The distribution and the correlations can be built with the help of
assumptions made from observations and from stellar and galactic models. In
our case, since we are mainly interested in the Kepler satellite field of view
which is well documented through the Kepler Input Catalog (KIC), we could use
Equation (4) to map a color-color distribution, or any other relation found in
the KIC, into the prior of Equation (16). However, we shall keep here a more
conservative approach, or as Jørgensen & Lindegren (2005) put it, a non-
committal approach and assume that $\tau$, $Z$, $X_{0}$, $M$ and $\alpha$ are
independent. Thus we write
$f_{0}=\Psi(\tau)\Phi(Z)\zeta(X_{0})\xi(M)\beta(\alpha)\;,$ (17)
and also take $\Psi(\tau)$, $\Phi(Z)$, $\zeta(X_{0})$, and $\beta(\alpha)$ to
be flat. This means that the prior density is constant through these
dimensions. We only make an assumption on the IMF and choose the so-called
IMF1 model from Chabrier (2001) where
$\displaystyle\xi(M)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0.019\ M^{-1.55}&\text{if }M\leq
1.0M_{\rm\odot}\\\ 0.019\ M^{-2.70}&\text{if
}M>1.0M_{\rm\odot}.\end{array}\right.$ (20)
In a grid, flat priors can be easily illustrated by $N$ equally spaced points,
every point having an equal probability $1/N$. Of course, if the grid is not
regularly spaced the weight assignment can get a bit more complicated. For a
non-flat prior, we could distribute the $N$ grid points so that they follow
the density prescribed for example by Equation (20) and give every point a
weight of $1/N$ and meet the non-flat IMF prior prescription. However, and
this is especially true in the $\tau$ direction, we have an irregularly spaced
grid. Also, we do not wish to interpolate our grid for the sake of precision
and for the same reason it is not necessarily reasonable to make the grid
sparser in the region where the prior is relatively small. A star observed in
that sparser region, even if less likely, would not be analyzed in the same
detail by SEEK.
We have solved this problem by dividing the parameter space into a number of
small 5-D bins of (penta-)volume $V^{i}$ containing each a small number of
point $n^{i}\geq 1$. Then, a prior weight $f^{i}_{0}$, where
$f^{i}_{0}=\int_{V^{i}}\Psi(\tau)\Phi(Z)\zeta(X_{0})\xi(M)\beta(\alpha)dV^{i}\;,$
(21)
is distributed through the $n^{i}$ point contained in the bin such that the
weight $f_{0k}^{i}$ given to model $k$ is inversely proportional to the
evolution speed of that model,
$\displaystyle
f_{0k}^{i}=\frac{\Delta\tau_{k}}{\sum_{k^{\prime}}\Delta\tau_{k^{\prime}}}f_{0}^{i}$
$\displaystyle{\rm where}$ $\displaystyle k=1,2,3,...,n^{i}\;.$ (22)
Here $\Delta\tau_{k}$ is the time taken by model $k$ to evolve to the next
point in its evolution track. This forces the slowly evolving models of the
bin to have larger priors. These two equations are central pieces of SEEK.
They ensure that the probability given to a model reflects the size of the
parameter space where no other models are computed around it. These empty
regions of the parameter space will be seen as gaps in some of the probability
distributions of the next section.
## 4 RESULTS
In practice, once $f$ is fixed, the computation of the posterior probability
$\mathcal{G}$ on an arbitrary set of parameters is easily done. In one
dimension, the grid is sliced along the desired parameter and all models
entering a bin from $s$ to $s+\Delta s$ are added
$\mathcal{G}(s_{i})=C{\displaystyle\sum_{k=0}^{K}}f^{k}(s_{i})\;,$ (23)
where the $f^{k}(s_{i})$ are the $K$ points of the grid lying from $s_{i}$ to
$s_{i+1}=s_{i}+\Delta s$, and $C$ is a constant ensuring that the total
probability of the problem is 1 (see Equation 24). The results of this
computation is a probability distribution $\mathcal{G}(s)$ that is best seen
as a histogram. The results for four parameters $R$, $M$, $\tau$, and $T_{\rm
eff}$, computed for 70 Ophiuchi A, are presented in Figure 4. For convenience,
there and in the following figures, we have normalized the histograms so that
${\rm max}(\mathcal{G}(s))=1$.
Another important quantity is the running integral of $\mathcal{G}$ seen as
the solid red line in Figure 4. The running integral over the entire parameter
space is the total probability of the grid. This should be 1 since we assume
that the observed star lies within the grid. We use that relation to fix the
constant $C$ in Equation (23)
${\displaystyle\sum_{i}}\mathcal{G}(s_{i})=1\;.$ (24)
Similarly, we can integrate over a section of the distribution to get a $1$,
$2$, or $3\,\sigma$ probability range for $s$. The error interval
$[s^{(1)},s^{(2)}]$ output by SEEK is by default at the $1\,\sigma$ level and
is determined based on the probability integral of $\mathcal{G}$.
Specifically,
$\int_{s_{0}}^{s^{(1)}}\mathcal{G}(s)ds=0.1585\;,\qquad\int_{s_{0}}^{s^{(2)}}\mathcal{G}(s)ds=0.8415\;,$
(25)
where $s_{0}$ is the initial value of the parameter, and the integrals are
represented by suitable sums, as in Equation (24). (Note that
$0.8415-0.1585=0.6830$ corresponds to $1\,\sigma$ for a normal distribution.)
The solution SEEK outputs is the mid-point $(s^{(1)}+s^{(2)})/2$ of this
interval. In this way the error bars are symmetric; this would not be the case
if the median value $s^{(m)}$, with
$\int_{s_{0}}^{s^{(m)}}\mathcal{G}(s)ds=0.5\;,$ (26)
had been chosen. This $1\,\sigma$ section of the running integral is depicted
in blue (between the dots) on the running integral of Figure 4.
These histograms along with the net numbers output by SEEK are useful to
interpret the results. In the case of 70 Oph A, the solutions $R=0.86\pm 0.01$
[$R_{\rm\odot}$] and $T_{\rm eff}=5306\pm 36$ [K] are well constrained and
quasi Gaussian. The histograms visually confirm the validity of the numerical
solutions. The result for the mass $M=0.90\pm 0.04$ $M_{\rm\odot}$ reveals
some of the grid’s limitation as there are gaps in the otherwise well defined
structure of $\mathcal{G}(M)$. These gaps are artifacts of the 0.02
$M_{\rm\odot}$ mesh used in the grid below 1.8 $M_{\rm\odot}$. However, since
the prior density is properly weighted through Equations (23) and (24), the
overall value of the mass, with its error bars, is correct. For the age,
$\tau=8.65\pm 3.64$ Gyr, a large error is revealed by the output and the
histogram shows the probability distribution to be populated all along $\tau$
from 2 Gyr all the way to 13 Gyr. In that case a more reasonable answer is a
lower limit of $\tau\gtrsim 2$ Gyr. The histogram for $\tau$ also shows gaps
that are due to the finite resolution of the grid.
Table Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements shows
our selection of solar-like oscillators and observables found in the
literature. These observables are processed by SEEK and produce the outputs in
Table Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements and
Table 3. In Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements, we compared SEEK results with direct measurements of radius via
interferometry and of mass via Kepler’s third law. For SEEK, the fits have
been made for two different posterior sets, the full set
$\boldsymbol{q}^{\rm g,f}=\\{\Delta\nu,\delta\nu,T_{\rm eff},\log
g,{\rm[Fe/H]},V,\pi,E_{B-V}\\}\;,$ (27)
and one using only a selected subset of the same observables,
$\boldsymbol{q}^{\rm g,s}=\\{\Delta\nu,T_{\rm eff},{\rm[Fe/H]}\\}\;.$ (28)
The subset may be more representative of what we can expect from a typical
Kepler observation when no ground-based follow-up has been done for a star,
and where the asteroseismic data have not yielded a reliable determination of
$\delta\nu$. This can occur in a stochastically oscillating star when the
width of observed modes is larger than the small separation itself. In that
case, we have $\Delta\nu$ from the satellite observations, while $T_{\rm eff}$
and [Fe/H] can be obtained from the KIC.
The results of Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements show that the 3% promise on the radius precision mentioned in the
introduction is generally reached with both the full set and the subset of
observables. More striking and certainly more important than the precision
itself is the very good accuracy reached by SEEK. It is on this basis that our
method, or any other technique, should be judged, provided that independent
measurements are available. SEEK has pinned down all stellar radii at the 3%
level when the extended set of observable was used. The same level of accuracy
is generally reached if the subset of observables (Eq. 28) is used.
We note that in well-posed problems, where parameters are actually strongly
constrained, the value of $\mathcal{L}$ is much larger than $f_{0}$ close to
the best solution. It is only when we get away from the best parameter values
that the prior gets more important. The details of the priors used here only
have an influence on the wing of the $\mathcal{G}$ distribution. Thus, the
choice of priors has an effect on the errors, especially if the errors are
large, but little effect on the solution itself.
## 5 DISCUSSION
We clearly need to understand the exceptions to the general success of SEEK;
in particular, we note that $\beta$ Hyi was a bit further from the direct
observations with an offset of 6%. An extra tool offered by SEEK can be used
to study the $\beta$ Hyi radius offset when compared to direct measurements.
Figure 5 shows a two-dimensional projection $\mathcal{G}(R,Z/X)$ of the
posterior computed for $\beta$ Hyi with only $\Delta\nu$ and $T_{\rm eff}$ as
input observables. The metallicity is not included as an observable in the
computation as a trick to have a clearer view of the correlation between that
observable and $R$. Several possibilities are offered to explain the
discrepancy between SEEK’s radius $R/R_{\odot}=1.92\pm 0.05$ and the direct
measurement of $1.814\pm 0.017$. The first possibility is that the observed
$Z/X$ is too high. Reducing [Fe/H] by $\sim 0.2$ dex, to $Z/X\sim 0.0163$,
would reconcile the asteroseismic and interferometric radii. This can be
concluded from a visual examination of Figure 5. In fact, lowering the
metallicity by this amount would put it back to the level measured in $\beta$
Hyi by Dravins et al. (1998). We have made an a posteriori test using the
value of metallicity and temperature published by Dravins et al. (1998),
$T_{\rm eff}=5800\pm 100$ K, and [Fe/H]$=0.2\pm 0.1$ instead of the more
recent values of Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements (da Silva et al., 2006). These inputs processed by SEEK place the
radius $R/R_{\odot}=1.85\pm 0.03$ in much better agreement (2%) with the
observations. We note that Dravins et al. (1998) used the older Noels &
Grevesse (1993) metal mixtures in the computation. This underlines that
accuracy can only be reached if the input metallicity is selected with some
care.
The offset in radius for $\beta$ Hyi can be explained otherwise. The extension
of the grid to lower values of $X_{0}$ would to some extent reconcile SEEK and
the interferometric measurement. It can be seen in Figure 5 that the left-hand
side of the correlation function $\mathcal{G}$ is in fact the edge of the grid
where $X_{0}=0.68$. The extension of the grid would likely expand the existing
ridge of solutions to the left in Figure 5, at lower radii. A simple eye-ball
estimate in the figure can be made to indicate that the combination, $Z\sim
0.0165$, $X_{0}\sim 0.64$ would yield the right metallicity and the right
radius. However, it seems that a hydrogen content that low is not reasonable
and perhaps this possibility should not be considered seriously. The effect of
the hydrogen content, or should we say helium content, on the radius is in
fact the main factor compromising the precision of $R$ in SEEK or any other
technique using the large sepa ration to determine the radii. The error on the
radius is influenced by the size of the grid in the $X_{0}$ direction. The
imperative we had not to underestimate the errors on stellar parameters
influenced our choice of having a large range of values for that parameter. It
also means that raising the precision on the metallicity ratio $Z/X$ to fix
the radius with better precision has some intrinsic limitations, unless the
helium abundance can be fixed independently. We also note that the gaps
between the islands of Figure 5, like the one for the 1D histograms, are
caused by the finite resolution of the grid. The gaps in this figure also
reflect that if the grid is regularly spaced in $X_{0}$ and $Z$ it is not in
$Z/X$ since the gaps are not regularly spaced along that direction.
One other result obtained by SEEK is worth examining in more detail. For
Procyon A, if the subset of observables is used to do the fit, the values of
$M$ and $R$ are less precise, which is in accordance with expectation, but
more accurate than the case where all observables are used. This could reveal
some inconsistency in the observables as well as some of the limitations of
the SEEK procedure. Indeed forcing SEEK to look for a model of Procyon A with
the right astrometry, $V=0.363$ and $\pi=134.07$ mas, reduces the accuracy of
the results, compared with using only $T_{\rm eff}$, [Fe/H] and $\Delta\nu$.
This discrepancy could come from inhomogeneity between the tool used in the
spectroscopic study giving the observables and in our models. The metallicity
and temperature taken as input for Procyon A are from 3D modeling (Allende
Prieto et al., 2002), while we used the conversion of the VandenBerg & Clem
(2003) tables using 1D models to get our value of $M_{V}$. Using older
atmospheric parameters, or a f ully consistent conversion, could perhaps
improve the fit.
We also remark that the behavior of our results for the mass of $\alpha$ Cen B
presents similarity to those of Procyon A. The results are slightly more
accurate if $V$, $\pi$ and $\delta\nu$ are dropped from the fit. It seems that
this reveals some of the limitation of our grid when extremely precise
measurements are available for a star. The small box defined within $\sim
1-\sigma$ of all $\alpha$ Cen B observables only includes a few models. This
number is too small and can only produce a weak probabilistic assessment over
the value of the mass. Computing more models to cover with more precision the
parameters space of $\alpha$ Cen B would most likely resolve the small
deviation over the value of the mass. This underscores the fact that modeling
dedicated to individual stars becomes useful once classical observables of a
star are very precise.
In any case, Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements shows that SEEK is certainly capable of pinning down the radius
of solar-like oscillators with a minimum effort and that it is also able to
fix the mass of these stars with an accuracy not attainable when only non
asteroseismic inputs are available. This improvement in accuracy is especially
potent in case the distance to the star is not known.
In Table 3, all primary stellar parameters $\boldsymbol{p}$ of the modeled
stars are shown as well as some selected secondary values. We address here the
question of the age of stars fixed by SEEK, and especially the age of the Sun,
since it is the only star where an independent determination of the age is
available. A fairly good accuracy, 3.96$\pm$0.41 Gyr (13%), compared to the
meteoritic value of 4.57$\pm$0.02 Gyr (Wasserburg, in Bahcall & Pinsonneault,
1995), is reached in the case where $\Delta\nu$ and $\delta\nu$ are known. A
look at the histogram in the top panel of Figure 6 confirms visually that the
solution is well constrained. This is quite promising in the case where the
small separation is known. It seems that we can expect a precision of 5 to 20%
for the age when $\Delta\nu$, $T_{\rm eff}$, [Fe/H], and especially
$\delta\nu$ are known with reasonable precision.
If only the three observables in the subset are used, the result varies from
star to star. For the Sun, the lower panel of Figure 6 shows for the age two
distinct and very probable solutions. This specific example shows how useful
the histograms are since the output of Table 3 does not reveal the presence of
two solutions. One thing not shown here, but that we were able to see from the
computation of the probability distribution $\mathcal{G}(\tau,\alpha)$, is
that the two islands of solution for the age are at two different value of
$\alpha$, the young solution corresponding to the standard value of
$\alpha=1.8$ and the older one to $\alpha=2.8$. As seen in the lower panel of
Figure 6 higher values of $\alpha$ are acceptable if only the subset of
observables is used.
Figure 7 illustrates another type of effect on the age when the subset of
observables is used. The precision on the age of $\beta$ Hyi is degraded by
20%, but the overall answer, or accuracy, is the same. As mentioned previously
in the description of Figure 4, 70 Oph A shows a third type of scenario where
not much can be concluded on the age of the star as the posterior is showing
solutions at almost any age. Since the relation between the age and the
observables is highly non-linear, it is very difficult to estimate beforehand
what will be the precision obtained on the age depending on the type of star
and/or on the precision of the observables, especially when $\delta\nu$ is not
known.
We also note that any age determination based on stellar modeling is
susceptible to changes in basic aspects of the modeling. In the solar case
helioseismic fits to low-degree modes result in a solar age very close to the
meteoritic value, for models including diffusion and settling of helium and
heavy elements, and when the ‘old’ solar composition is used, whereas use of
the Asplund et al. (2009) composition induces a significant shift (see
Christensen-Dalsgaard, 2009, for a review). Also, the age determination based
on fits to the Kepler data for HAT-P-7, which has a convective core, was
understandably rather sensitive to the extent to which convective-core
overshoot was included (Christensen-Dalsgaard et al., 2010). Such potential
systematic effects must clearly be taken into account in the interpretation of
results of fits such as those carried out with SEEK. Of course the long-term
goal is to reduce these effects through an improved understanding of stellar
structure and evolution based on more detailed asteroseismic investigations.
## 6 CONCLUSIONS
We have shown the details involved in the SEEK procedure and we have tested
the validity of the approach with all independent measurement known to us.
This is the first step before Kepler observations of solar-like stars, now
becoming available in large quantity (Gilliland et al., 2010), are analyzed.
We hope that SEEK will be able to process these observation with ease and lead
to the publication of a catalog that includes a homogeneous sample of stars
revealing their radius, mass, age, and other stellar parameters.
We expect that the results of SEEK can be a useful starting point in more
detailed asteroseismic analyses in the, likely frequent, cases where extensive
sets of oscillation frequencies are available. Such detailed investigations
can further refine the basic stellar parameters, including the age which, as
shown above, is sometimes not significantly constrained by SEEK. Also,
detailed analyses of the frequencies are likely to uncover evidence for the
need for improvements in stellar modeling.
The SEEK model grid extends to the base of the red-giant branch and hence SEEK
can be used to determine properties of stars on both the main sequence and the
subgiant branch. In fact, the results obtained for the subgiants $\eta$ Boo
and $\beta$ Hyi indicate the potential of SEEK for the investigation of
subgiants, common amongst the Kepler asteroseismic targets. On the other hand
we acknowledge the substantially different diagnostic potential between
centrally hydrogen burning stars and subgiants. In particular, $\delta\nu$
provides a direct measure of stellar age in the former case, while the
diagnostic potential of $\delta\nu$ for subgiants is more subtle. In the
latter case, detailed analysis of individual frequencies of mixed modes may
provide much more stringent constraints (e.g., Metcalfe et al., 2010). These
issues of optimizing the asteroseismic diagnostics certainly require further
investigations.
In parallel, further tests and development of SEEK are required. A first step
will be to test the level of systematic errors that are introduced by the
neglect of relevant physical effects in the stellar modeling. An obvious
example is diffusion and settling which have been demonstrated to have a
substantial effect on solar modeling, as tested with helioseismology. Also,
effects of convective core overshoot could influence the results for stars
slightly more massive than the Sun. New grids of models including such effects
can then be computed for inclusion in SEEK.
A related issue, with less obvious solutions, concerns the systematic effects
of the frequency errors introduced by the failure to model properly the
outermost layers of the star, which are known to dominate the difference
between observed and modeled frequencies in the solar case (e.g., Christensen-
Dalsgaard et al., 1996) and which also have potentially significant effects on
the large and small separations. It is important at least to obtain an
estimate of the extent to which these effects may influence the results
obtained with SEEK. One may hope that improvements in the modeling of the
relevant effects, combined with detailed analyses of Kepler data for a broad
range of stars, will eventually allow us to reduce or eliminate such potential
systematic errors.
POQ would like to thank H. Bruntt, J. Rowe, G. Doğan, T. Campante and J.
Dupuis for their interest in this work. This project was supported by the
European Helio- and Asteroseismology Network (HELAS), a major international
collaboration funded by the European Commission’s Sixth Framework Programme,
by the Danish Natural Science Research Council and by the Natural Sciences and
Engineering Research Council of Canada through a Canadian Space Agency
Visiting Fellow grant.
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Figure 1: Slice of the SEEK grid projected in the $\log g-\log T_{\rm eff}$
diagram for $Z=0.015$ $X_{0}=0.72$ and $\alpha=1.8$. Three isochrones are also
drawn. The computation was stopped after the star had reached 15 billion years
or an arbitrary point on the RGB branch. The low-mass models are separated by
steps of 0.02 $M_{\rm\odot}$, the more massive ones by 0.10 $M_{\rm\odot}$.
Figure 2: Slice of the solution valley obtained by SEEK for $\alpha$ Cen A at
$Z=0.025$, $X_{0}=0.70$, and $\alpha=1.8$. Note the logarithmic color scale
showing the value of $\chi^{2}_{r}$ as a function of $M$ and $\tau_{\rm N}$
(see Eq. 3). The solution is centered on the constant radius line of
$R=1.25\,R_{\rm\odot}$, shown as a solid red line. The solid black lines are
also radius isocurves. Models older than 13 Gyr, on the upper right region of
the figure, and the models on the ZAMS, with $\tau_{\rm N}=0$, were excluded
from the fit and given values of $\log\chi^{2}_{r}=6.0$. The slight deviation
in the constant-radius isocurves with $R\geq 1.5\,R_{\rm\odot}$ and $\tau_{\rm
N}\gtrsim 0.7$ marks the position of the hook in the evolution tracks. Figure
3: Description of the $\delta\nu$ computation. The functions
$\psi_{0},\psi_{2}$ corresponding to the computed frequencies for $l=0,2$,
shown in blue and red respectively in panel (a), are convolved with the
distribution $T$ in panel (b) to create the distribution $D_{0},2$ in panel
(c). The resulting cross-correlation function ($D_{0}\ast D_{2}$) fixing the
$\delta\nu$ at its maximum is shown in panel (d). The extra bump of the cross-
correlation is caused by the extra $l=2$ mode at $\nu\sim 760$ $\mu$Hz and has
no influence on the computed separation. Figure 4: SEEK’s posterior for 70
Oph A. All the observables of Table Automatic Determination of Stellar
Parameters via Asteroseismology of Stochastically Oscillating Stars:
Comparison with Direct Measurements are used in the fit. The probability
distribution $\mathcal{G}(p)$ for $R$, $M$, $\tau$ and $T_{\rm eff}$,
(normalized to a maximum of 1, black histogram) is plotted along with the
probability integral (solid red line). The delimitation of the $1\,\sigma$
probability of the distribution, determined by the interval
$[s^{(1)},s^{(2)}]$ (cf. Eq. 25), is plotted between the blue dots. Figure 5:
Correlation between $Z/X$ and $R$ for $\beta$ Hyi. The two thick solid curves
through peaks in the correlation function $\mathcal{G}(R,Z/X)$ follow the
constant metallicity ridges for $Z=0.025$ and 0.015. Along these curves the
hydrogen mass ratio varies from the upper left peak to the lower right one as
$X_{0}=0.68$, 0.70, 0.72, and 0.74. A clear general trend is that $R$
increases with $Z$ and that it also increases with increasing $X_{0}$. The
solid horizontal line is the $\beta$ Hyi metallicity ratio of $Z/X=0.0258\pm
0.003$ (da Silva et al., 2006), with its error bars as dashed lines, while the
solid and dashed vertical lines show the direct interferometric measurement of
$R=1.814\pm 0.107$. Note that the merging of the constant metallicity ridges
at lower values is an artifact of the drawing procedures.
$\begin{array}[]{c}\includegraphics[angle={0},width=341.43306pt]{./fig6a.eps}\\\
\\\ \\\ \includegraphics[angle={0},width=341.43306pt]{./fig6b.eps}\end{array}$
Figure 6: SEEK’s posterior for the Sun. The top 9 panels represent the
probability distribution (normalized to 1) as well as the probability integral
for the model using all the observables of Table Automatic Determination of
Stellar Parameters via Asteroseismology of Stochastically Oscillating Stars:
Comparison with Direct Measurements. The lower panels are similar but use a
subset of observables ($\Delta\nu$, $T_{\rm eff}$, and [Fe/H]).
$\begin{array}[]{c}\includegraphics[angle={0},width=341.43306pt]{./fig7a.eps}\\\
\\\ \\\ \includegraphics[angle={0},width=341.43306pt]{./fig7b.eps}\end{array}$
Figure 7: Same as Figure 6 for $\beta$ Hyi. Table 1: Observables
star | | $\Delta\nu$ $[\mu\text{Hz}]$ | $\delta\nu$ $[\mu\text{Hz}]$ | $\nu_{\rm max}$ $[\mu\text{Hz}]$ | $T_{\rm eff}$ [K] | [Fe/H] | $\log g$ [cgs] | $V$ | $\pi$ [mas] | $E_{B-V}$ | Reference
---|---|---|---|---|---|---|---|---|---|---|---
$\beta$ Hyi | | $57.2\pm 0.5$ | $5.3\pm 0.5$ | $1000$ | $5964\pm 100$ | $-0.03\pm 0.05$ | $-$ | $2.8\pm 0.01$ | $134.07\pm 0.11$ | $0.01$ | 1 ; a
$\tau$ Cet | | $169.6\pm 0.5$ | $12.7\pm 1.2$ | $4500$ | $5264\pm 100$ | $-0.5\pm 0.03$ | $4.36\pm 0.15$ | $3.39\pm 0.01$ | $-$ | $0$ | 2 ; b
Procyon A | | $55.5\pm 0.5$ | $-$ | $1000$ | $6514\pm 27$ | $-0.05\pm 0.05$ | $-$ | $0.363\pm 0.003$ | $285.93\pm 0.88$ | $0$ | 3 ; c
$\eta$ Boo | | $39.9\pm 0.9$ | $4.0\pm 1.0$ | $750$ | $6007\pm 255$ | $0.09\pm 0.01$ | $-$ | $2.68\pm 0.03$ | $88.17\pm 0.75$ | $0$ | 4 ; d
$\alpha$ Cen A | | $105.5\pm 0.5$ | $5.6\pm 0.7$ | $2410$ | $5847\pm 27$ | $0.24\pm 0.03$ | $4.34\pm 0.12$ | $-0.01\pm 0.01$ | $747.1\pm 1.2$ | $0$ | 5 ; e
$\alpha$ Cen B | | $161.5\pm 0.5$ | $10.1\pm 0.6$ | $4100$ | $5316\pm 28$ | $0.25\pm 0.04$ | $4.44\pm 0.15$ | $1.35\pm 0.01$ | $747.1\pm 1.2$ | $0$ | 6 ; e
70 Oph A | | $161.7\pm 0.8$ | $-$ | $4500$ | $5300\pm 50$ | $0.04\pm 0.05$ | $-$ | $4.19\pm 0.014$ | $194.2\pm 1.2$ | $0$ | 7 ; f
Sun | | $134.8\pm 0.5$ | $9.1\pm 0.2$ | $3034$ | $5778\pm 20$ | $0\pm 0.01$ | $4.44\pm 0.01$ | $-$ | $-$ | $-$ | 8 ; g
References. — Asteroseismology (1) Bedding et al. 2007; (2) Teixeira et al.
2009; (3) Eggenberger et al. 2005; (4)Carrier et al. 2005; (5)Bouchy & Carrier
2002; (6) Kjeldsen et al. 2005; (7) Carrier & Eggenberger 2006; (8) Thiery et
al. 2000
References. — Other observables (a) da Silva et al. 2006; (b) Soubiran et al.
1998; (c) Allende Prieto et al. 2002; (d) Morossi et al. 2002; (e) Porto de
Mello et al. 2008; (f) Eggenberger et al. 2008 ; (g) Grevesse & Sauval 1998
Table 2: Comparison With Direct Measurements
| | $R$ $[R_{\rm\odot}]$ | | $M$ $[M_{\rm\odot}]$ |
---|---|---|---|---|---
| | SEEK | | | | SEEK | | |
Star | | Selected | All | Direct/Interferometry | Reference | | Selected | All | Direct/Kepler’s Law | Reference |
$\beta$ Hyi | | $1.92\pm 0.05$ | $1.87\pm 0.01$ | $1.814\pm 0.017$ | 1 | | $1.27\pm 0.11$ | $1.16\pm 0.02$ | | |
$\tau$ Cet | | $0.79\pm 0.01$ | $0.81\pm 0.01$ | $0.790\pm 0.005$ | 2 | | $0.77\pm 0.03$ | $0.83\pm 0.01$ | | |
Procyon A | | $2.04\pm 0.05$ | $2.12\pm 0.02$ | $2.048\pm 0.025$ | 3 | | $1.45\pm 0.10$ | $1.59\pm 0.05$ | $1.497\pm 0.037$ | 4 |
$\eta$ Boo | | $2.65\pm 0.08$ | $2.69\pm 0.06$ | $2.672\pm 0.028$ | 5 | | $1.62\pm 0.13$ | $1.71\pm 0.08$ | | |
$\alpha$ Cen A | | $1.23\pm 0.04$ | $1.25\pm 0.01$ | $1.224\pm 0.003$ | 6 | | $1.09\pm 0.09$ | $1.11\pm 0.01$ | $1.105\pm 0.007$ | 6 |
$\alpha$ Cen B | | $0.87\pm 0.01$ | $0.87\pm 0.01$ | $0.863\pm 0.005$ | 6 | | $0.92\pm 0.04$ | $0.89\pm 0.01$ | $0.935\pm 0.006$ | 6 |
70 Oph A | | $0.86\pm 0.02$ | $0.86\pm 0.01$ | | | | $0.89\pm 0.06$ | $0.89\pm 0.05$ | $0.890\pm 0.020$ | 7 |
Sun | | $1.02\pm 0.03$ | $1.01\pm 0.01$ | $1$ | | | $1.03\pm 0.08$ | $1.01\pm 0.02$ | $1$ | |
References. — (1)North et al. 2007; (2)di Folco et al. 2007; (3)Kervella et
al. 2004; (4)Girard et al. 2000 ;(5)van Belle et al. 2007; (6)Miglio &
Montalbán 2005; (7)Eggenberger et al. 2008;
Table 3: SEEK Output | | | Stellar Parameters
---|---|---|---
| | Primary | | Secondary |
Star | Observables | | $M$ $[M_{\rm\odot}]$ | $\tau$ [Gyr] | $Z$ | $X_{0}$ | $\alpha$ | | $R$ $[R_{\rm\odot}]$ | $\log(L/L_{\rm\odot})$ | $\log g$ [cgs] | $T_{\rm eff}$ [K] | $Y$
$\beta$ Hyi | All | | $1.16\pm 0.02$ | $5.41\pm 0.89$ | $0.018\pm 0.001$ | $0.708\pm 0.022$ | $1.74\pm 0.60$ | | $1.87\pm 0.01$ | $0.58\pm 0.02$ | $3.964\pm 0.004$ | $5845\pm 38$ | $0.274\pm 0.023$
| Selected | | $1.27\pm 0.11$ | $3.94\pm 1.45$ | $0.019\pm 0.002$ | $0.710\pm 0.022$ | $1.69\pm 0.62$ | | $1.92\pm 0.05$ | $0.62\pm 0.05$ | $3.97\pm 0.02$ | $5959\pm 111$ | $0.271\pm 0.024$
$\tau$ Cet | All | | $0.83\pm 0.01$ | $0.75\pm 0.03$ | $0.007\pm 0.001$ | $0.730\pm 0.003$ | $1.30\pm 0.17$ | | $0.81\pm 0.01$ | $-0.32\pm 0.02$ | $4.547\pm 0.006$ | $5450\pm 34$ | $0.263\pm 0.004$
| Selected | | $0.77\pm 0.03$ | $6.31\pm 2.18$ | $0.007\pm 0.001$ | $0.729\pm 0.004$ | $1.30\pm 0.17$ | | $0.79\pm 0.01$ | $-0.36\pm 0.03$ | $4.53\pm 0.01$ | $5286\pm 71$ | $0.264\pm 0.005$
Procyon A | All | | $1.59\pm 0.05$ | $1.83\pm 0.24$ | $0.021\pm 0.003$ | $0.731\pm 0.012$ | $2.42\pm 0.46$ | | $2.12\pm 0.02$ | $0.88\pm 0.02$ | $3.991\pm 0.010$ | $6548\pm 37$ | $0.248\pm 0.015$
| Selected | | $1.45\pm 0.10$ | $2.25\pm 0.64$ | $0.018\pm 0.002$ | $0.710\pm 0.023$ | $1.93\pm 0.62$ | | $2.04\pm 0.05$ | $0.82\pm 0.02$ | $3.98\pm 0.02$ | $6513\pm 64$ | $0.272\pm 0.025$
$\eta$ Boo | All | | $1.71\pm 0.08$ | $1.90\pm 0.38$ | $0.025\pm 0.001$ | $0.729\pm 0.013$ | $1.32\pm 0.60$ | | $2.69\pm 0.06$ | $0.97\pm 0.02$ | $3.81\pm 0.02$ | $6166\pm 80$ | $0.246\pm 0.014$
| Selected | | $1.62\pm 0.13$ | $2.24\pm 0.62$ | $0.025\pm 0.001$ | $0.728\pm 0.013$ | $1.33\pm 0.64$ | | $2.65\pm 0.08$ | $0.88\pm 0.08$ | $3.80\pm 0.02$ | $5974\pm 280$ | $0.247\pm 0.014$
$\alpha$ Cen A | All | | $1.11\pm 0.01$ | $4.95\pm 0.04$ | $0.030\pm 0.001$ | $0.680\pm 0.003$ | $1.80\pm 0.17$ | | $1.25\pm 0.01$ | $0.18\pm 0.02$ | $4.299\pm 0.001$ | $5850\pm 34$ | $0.290\pm 0.004$
| Selected | | $1.09\pm 0.09$ | $8.04\pm 4.05$ | $0.030\pm 0.001$ | $0.698\pm 0.020$ | $2.25\pm 0.59$ | | $1.23\pm 0.04$ | $0.20\pm 0.03$ | $4.30\pm 0.01$ | $5850\pm 37$ | $0.272\pm 0.021$
$\alpha$ Cen B | All | | $0.89\pm 0.01$ | $6.95\pm 0.03$ | $0.030\pm 0.001$ | $0.680\pm 0.003$ | $1.80\pm 0.17$ | | $0.87\pm 0.01$ | $-0.32\pm 0.02$ | $4.521\pm 0.001$ | $5250\pm 34$ | $0.290\pm 0.004$
| Selected | | $0.92\pm 0.04$ | $6.99\pm 4.45$ | $0.029\pm 0.001$ | $0.709\pm 0.030$ | $2.23\pm 0.52$ | | $0.87\pm 0.01$ | $-0.26\pm 0.03$ | $4.53\pm 0.01$ | $5309\pm 66$ | $0.262\pm 0.020$
70 Oph A | All | | $0.89\pm 0.05$ | $8.65\pm 3.70$ | $0.023\pm 0.003$ | $0.716\pm 0.018$ | $2.06\pm 0.34$ | | $0.86\pm 0.01$ | $-0.27\pm 0.02$ | $4.525\pm 0.006$ | $5311\pm 65$ | $0.261\pm 0.021$
| Selected | | $0.89\pm 0.06$ | $7.54\pm 4.54$ | $0.022\pm 0.003$ | $0.710\pm 0.026$ | $1.99\pm 0.33$ | | $0.86\pm 0.02$ | $-0.28\pm 0.03$ | $4.52\pm 0.01$ | $5304\pm 71$ | $0.268\pm 0.029$
Sun | All | | $1.01\pm 0.02$ | $3.96\pm 0.41$ | $0.020\pm 0.001$ | $0.707\pm 0.011$ | $1.80\pm 0.17$ | | $1.01\pm 0.01$ | $0.02\pm 0.02$ | $4.432\pm 0.005$ | $5753\pm 36$ | $0.273\pm 0.012$
| Selected | | $1.03\pm 0.08$ | $6.80\pm 5.43$ | $0.020\pm 0.001$ | $0.729\pm 0.013$ | $2.18\pm 0.53$ | | $1.02\pm 0.03$ | $0.03\pm 0.03$ | $4.44\pm 0.01$ | $5757\pm 39$ | $0.251\pm 0.014$
|
arxiv-papers
| 2010-09-26T21:35:49 |
2024-09-04T02:49:13.122005
|
{
"license": "Public Domain",
"authors": "P.-O. Quirion, J. Christensen-Dalsgaard and T. Arentoft",
"submitter": "Pierre-Olivier Quirion Dr.",
"url": "https://arxiv.org/abs/1009.5131"
}
|
1009.5144
|
11institutetext: Key Laboratory of Solar Activities of National Astronomical
Observatories, Chinese Academy of Sciences,Beijing 100012, China;
22institutetext: State Key Laboratory of Space Weather, Chinese Academy of
Sciences, Beijing 100080, China
# Monte Carlo simulations of a diffusive shock with multiple scattering
angular distributions
Xin Wang NAOC fellow: . This work was funded in part by CAS-NSFC grant
10778605 and NSFC grant 10921303 and the National Basic Research Program of
the MOST (Grant No. 2011CB811401).1122 wangxin@nao.cas.cn Yihua Yan 11
###### Abstract
Aims. We independently develop a simulation code following the previous
dynamical Monte Carlo simulation of the diffusive shock acceleration under the
isotropic scattering law during the scattering process, and the same results
are obtained.
Methods. Since the same results test the validity of the dynamical Monte Carlo
method for simulating a collisionless shock, we extend the simulation toward
including an anisotropic scattering law for further developing this dynamical
Monte Carlo simulation. Under this extended anisotropic scattering law, a
Gaussian distribution function is used to describe the variation of scattering
angles in the particle’s local frame.
Results. As a result, we obtain a series of different shock structures and
evolutions in terms of the standard deviation values of the given Gaussian
scattering angular distributions.
Conclusions. We find that the total energy spectral index increases as the
standard deviation value of the scattering angular distribution increases, but
the subshock’s energy spectral index decreases as the standard deviation value
of the scattering angular distribution increases.
###### Key Words.:
acceleration of particles– shock wave–solar energetic particle
## 1 Introduction
It is well known that the diffusive acceleration model has been popular for
more than five decades since Fermi (1949) first proposed that cosmic rays
could be produced via diffusive processes. Until now, diffusive shock
acceleration (DSA) (i.e. first-order Fermi acceleration) has been extensively
applied to many physical systems, such as shocks in the solar system, in our
Galaxy, and throughout the Universe (Baring 1997; Bell 2004).
It is also well known that the nonlinear interactions in plasma usually
include such things as the turbulence of scattering wave field, cosmic ray
(CR) injection, and “back reaction” by CR pressure. These complex behaviors
have held back comprehensive understanding of the DSA and nonlinear DSA
theory. Therefore, to study the properties of the acceleration process and
dynamical behavior of the CR’s “back reaction” on the background flow,
choosing numerical simulation methods has been a primary and essential
problem. (Ostrowski 1991; Kang & Jones 1991; Malkov 1997; Ellison, Baring &
Jones 1995, 1996; Knerr, Jokipii & Ellison 1996; Berezhko et al. 1994;
Shalchi, Li & Zank 2010; Berezhko & Völk 2000). The main simulation methods
are introduced here, and a more detailed review can be seen in Kang (2001).
Monte Carlo method. In Monte Carlo simulations, the scattering processes
between individual particles with the collective background flow are simulated
around a one-dimensional parallel shock. The particle scattering process is
based on a prescribed scattering law, and collecting moments based on the
background computational grid for scattering calculation is done by particle-
in-cell (PIC) techniques. Knerr, Jokipii & Ellison (1996) successfully
developed the dynamical Monte Carlo simulations for the Earth’s bow shock with
important results in the maximum energetic particles achieving greater than
1MeV accelerated by the shock. Before the dynamical Monte Carlo simulation,
Ellison, Möbius & Paschmann (1990) had developed stationary Monte Carlo
simulations with the result that the cutoff energy accelerated by the shock
only reached 100 keV owing to the limitation on the size of bow shock. Baring,
Ellison & Jones (1995) also used the stationary Monte Carlo method for
simulating the oblique interplanetary shocks, and the calculated results are a
good fit to the observed data. There are many works that use the Monte Carlo
method: Ellison, Baring & Jones (1996), Ellison & Double (2002), Vladimirov,
Ellison & Bykov (2006, 2008), and others.
Hybrid simulation. The particles’ motion equations are solved explicitly based
on the electromagnetic field of the background plasma. Since the proton mass
is about 2000 times that of the electron mass, the total plasma is assumed to
be a coupling of two components: one component (e.g. electrons) is treated as
a massless fluid and the other component(e.g. protons,ions) is treated as
individual particles (Leroy et al. 1982). This method also employs the PIC
techniques based on computational grids. However, the limited computational
resources imply that the extensive calculation of the electromagnetic field
uses unrealistic parameters and are unable to follow the shock for long enough
(Giacalone et al. 1993; Giacalone & Jokipii 2009).
Two-fluid model. The two-fluid model uses the diffusive-convection equation,
coupled with the gas dynamic equations, to simulate the CR’s acceleration as a
gas component and an accelerated particle component (Drury & Falle 1986; Dorfi
1990; Jones & Kang 1992). Since the CR energy density is solved instead of the
particle’s distribution function in this model, the simulation results are
based on some assumptions, such as the CR’s adiabatic index, averaged
diffusive coefficient, and injection rate. The averaged diffusive coefficient
needs to be inferred from the diffusive-advection equation. The acceleration
efficiency dependens on the assumption of the injection rate. Under the
reasonable a priori models of these free parameters, a lot of simulations have
tested the acceleration efficiency and the shock structure, and both agree
with those derived from the diffusion-advection method. And these semi-
analytic solutions that have been extensively used can be seen in recent
works: Malkov, Diamond & Völk (2000), Bednarz & Ostrowski (2001), Blasi (2002,
2004), Amato & Blasi (2005, 2006), Blasi, Amato & Caprioli (2007), Caprioli et
al. (2009a), Caprioli, Amato & Blasi (2010a).
Kinetic simulation. Within this full numerical simulation, the diffusion-
convection equation for the distribution function is solved with a momentum-
dependent diffusion coefficient and a suitable assumption of injection rate
(Kang & Jones 1991; Berezhko et al. 1994; Berezhko & Völk 2000). Unlike the
two-fluid model, the kinetic model should not assume the CR’s adiabatic index,
in addition to using the momentum-dependent diffusion coefficient instead of
the averaged diffusion coefficient. Berezhko and collaborators have studied
numerous DSA models for supernova remnants (SNRs) with the kinetic model and
conclude that about 20 % of the total SN energy transferred to the CR’s
population in the SNRs are more than the results calculated from the two-fluid
model. As for the energy spectrum, the CR spectrum in this model shows a basic
power-law spectrum in the total energy range with a concave curve at some
energy range because of the precursor structure. More detailed new studies of
the kinetic model can be referred to in these papers (Kang, Jones & Gieseler
2002; Kang & Jones 2007; Amenomori et al. 2008; Li et al. 2009; Zirakashvili &
Aharonian 2010).
In an effort to follow and extend the previous dynamical Monte Carlo
simulation (Knerr, Jokipii & Ellison 1996), we independently developed a
simulation code based on the Matlab platform using multiple scattering laws.
Our multiple scattering angular distributions consist of three Gaussian
distributions and one isotropic distribution for the scattering angles during
the scattering process. The aim of the isotropic scattering angular
distribution is to check the dynamical Monte Carlo method independently.
Besides this, we want to know how the Gaussian distributions affect the
scattering angular distribution function and the shock wave’s evolution and
propagation; even more, we expect to find the relationships between the
multiple scattering law and the shock compression ratio. To validate the
multiple scattering angular distributions, we followed the parallel-plane
collisionless shock and the particle’s acceleration using the same parameters
and data as from Earth’s bow shock, which was used in previous dynamical Monte
Carlo simulation.
In Section 2, we outline the motivations for performing these four cases with
four different scattering angular distributions. In Section 3, the specific
simulation techniques are described. In Section 4, we present the shock
simulation results and different cases with four assumptions for scattering
angle distributions. Section 5 includes a summary and the conclusions.
## 2 The model
The dynamical Monte Carlo simulation has been developed by Knerr, Jokipii &
Ellison (1996) to study Earth’s bow shocks. It gives good results for the
higher than 1MeV cutoff in energy particles and the power-law energy tail in
the energy spectra. The dynamical Monte Carlo simulation method uses the
prescribed scattering law instead of the complex electromagnetic field
calculation like in the hybrid model. In addition, the dynamical Monte Carlo
simulation need not assume the CR’s injection rate and the associated
diffusive coefficient as do the two-fluid and kinetic models. For the above
reasons, we consider that developing a simulation code by following the
previous dynamical simulation is necessary. Although the previous results
successfully agree with observed data, the authors mention that their results
show that the total compression ratio of the shock is more than 4, which
should be less than the ratio of the standard value for a nonrelativistic
shock (Pelletier 2001). The Rankine-Hugoniot (RH) jump conditions allow
deriving the relation of the compression ratio with the Mach number:
$r=(\gamma_{a}+1)/(\gamma_{a}-1+2/M^{2})$, for a nonrelativistic shock, the
adiabatic index $\gamma_{a}$ = 5/3 , if the Mach number $M\gg 1$, then the
maximum compression ratio should be 4. To validate these consistent results
from the previous model and extend this study to find what might be
responsible for the shock compression ratio, we extend the previous isotropic
scattering angular law by including an anisotropic scattering angular law.
This prescribed multiple scattering law consists of an isotropic scattering
angular distribution and an anisotropic scattering angular distribution. The
scattering angles consist of two variables of pitch angle and azimuthal angle.
Once a particle has a collision with the massive scattering centers, its pitch
angle becomes $\theta^{\prime}$=$\theta$+$\delta\theta$, and the azimuthal
angle becomes $\phi^{\prime}$=$\phi$+$\delta\phi$, where $\delta\theta$ is the
variation in the pitch angle $\theta$, and $\delta\phi$ is the variation in
the azimuthal angle $\phi$. The pitch angles $\theta$ and $\theta^{\prime}$
are both in the range $0\leq\theta,\theta^{\prime}\leq\pi$, and azimuthal
angles $\phi$ and $\phi^{\prime}$ are both in the range
$0\leq\phi,\phi^{\prime}\leq 2\pi$ on the unit sphere. The variation in the
pitch angle $\delta\theta$ and azimuthal angle $\delta\phi$ are composed of
the scattering angle, and its anisotropic character is described by the
Gaussian function $f(\delta\theta,\delta\phi)$.
Under the multiple scattering angular distribution law, four cases are
calculated with three Gaussian distributions and one isotropic random
distribution for the scattering angles. Here, the sign $\sigma$ is used to
represent the standard deviation of the Gaussian function, and the sign $\mu$
is used to represent the statistical average or expected value of Gaussian
function for the scattering angles ($\delta\theta,\delta\phi$). We catalog the
four cases as follows.
(1) Case A: the scattering angles ($\delta\theta,\delta\phi$) are distributed
with a standard deviation $\sigma=\pi$/4 and an average value $\mu=0$.
(2) Case B: the scattering angles ($\delta\theta,\delta\phi$) are distributed
with a standard deviation $\sigma=\pi$/2 and an average value $\mu=0$.
(3) Case C: the scattering angles ($\delta\theta,\delta\phi$) are distributed
with a standard deviation $\sigma=\pi$ and an average value $\mu=0$.
(4) Case D: the scattering angles ($\delta\theta,\delta\phi$) are distributed
with a standard deviation $\sigma=\infty$ and an average value $\mu=0$, with
$\delta\theta$ varying from $-\pi/2$ to $\pi/2$, and $\delta\phi$ varying from
$-\pi$ to $\pi$ isotropically.
We performed four simulations according to the four different assumptions of
the scattering angular distributions algorithm. We also assumed the scattering
time (i.e., the mean time between two scattering events) is the same constant
in the four cases as in the previous model. The idea that such a simple law
can be used to describe the entire scattering process was postulated by
Eichler (1979), based on the two-stream instability work done by Parker
(1961). Put simply, it is assumed that the turbulence generated by both
energetic particles streaming in front of the shock and by thermal particles
produces nearly elastic scattering for particles for all energies in diffusive
shocks.
## 3 Description of the method
For simulating the total properties of shocks as they evolve from formation to
a final steady state as energy increases via Fermi acceleration, we used the
dynamical Monte Carlo model which employed the PIC techniques. Because there
is no assumption of the injection rate or transparency function in PIC
techniques, the shock-heated downstream ions can freely scatter back across
the subshock into the upstream region without being thermalized, and the
superthermal particles are produced in the thermal background self-
consistently. In addition, unlike the hybrid simulation, there is no
complicated electromagnetic field calculation for individual particles,
because it is replaced by the prescribed scattering law (Ellison et.al. 1993).
To reproduce these acceleration and scattering processes, a similar simulation
box and the same parameters (see Appendix A) as the previous dynamical Monte
Carlo method are used in these new codes. As described in the previous
simulation (see Figure 8), the particles with an initial bulk velocity $U_{0}$
and a Maxwellian thermal velocity $V_{L}$ in their local frame are moving
along a parallel magnetic field $B_{0}$ in a one-dimensional box. To maintain
a continuous flux-weighted flow upstream, a new particle fluid with the same
density needs to be injected into the simulation box from the left boundary.
For a shock initialization, the reflecting wall on the right boundary is used
to reflect the incoming particles, and forms a piston shock. The model also
includes the escape of energetic particles at the upstream free escape
boundary (FEB). The FEB phenomenologically models a finite shock size or the
lack of sufficient scattering far upstream to turn particles around. Once ions
cross the FEB, they are assumed to decouple from the shock system, and are
taken as the energy losses (Jones & Ellison 1991). The size of the foreshock
region (the distance from the shock front to the FEB) thus sets a limit on the
maximum energy a particle can obtain.
As shown in Figure 8, one particle’s box frame velocity $V$ is a total
velocity, which is composed of the local thermal velocity $V_{L}$ and the bulk
fluid speed $U$ (i.e. $V=V_{L}+U$, for upstream $U=U_{0}$, for downstream
$U=0$). After one particle arrives in the downstream region, its kinetic
energy is converted into random thermal energy by dissipation processes. With
the development of these many processes, the bulk fluid speed of downstream
flow becomes zero, and the length of downstream region is extended
dynamically.
As listed in Table 2, all of the specific parameters are used in our
simulations, considering PIC techniques. The total length ($X_{max}=300$) is
divided into the number of grids ($n_{x}=600$) with a grid length ($\Delta
x=1/2$). Initial grid density of the particles ($n_{0}=650$) is set. The total
time ($T_{max}=2400$) is divided into the number of time steps ($N_{t}=72000$)
with an increment of time ($dt=1/30$). In summary, these new codes consist of
the following three substeps like the previous simulation, except for the
third substep employing the extended multiple scattering laws.
(i) Individual particles move. Particles with their velocities move along the
one-dimensional $x$ axis:
$X^{t}_{k}=X^{t-1}_{k}+(V_{x})^{t}_{k}dt,t\in[1,t_{max}],k\in[1,k_{max}],$ (1)
where
$(V_{x})_{k}=(V_{Lx})_{k}+(U_{k})_{x},$ (2)
$(U_{k})_{x}=\frac{1}{n_{k}}\displaystyle\sum_{i=1}^{n_{k}}(V_{x})_{i}.$ (3)
here $t_{max}$=72000, $k_{max}$=600, and “k” represents the index of the
computational grid, $(U_{k})_{x}$ represents the bulk fluid speed of the
computational grid along to the $\hat{x}$ direction, and the value of $U_{k}$
is obtained from substep (ii). Since we are simulating a diffusive shock based
on a one-dimensional parallel magnetic field, the fluid quantities only vary
in the $\hat{x}$ direction.
(ii) Mass collection. Summation of particle masses and velocities are
calculated at the center of each computational grid:
$P_{k}=\displaystyle\sum_{i=1}^{n_{k}}m_{p}(V_{x})_{i},k=1,2,...k_{max}$ (4)
$U_{k}=\frac{1}{n_{k}}\displaystyle\sum_{i=1}^{n_{k}}(V)_{i},k=1,2,...k_{max},$
(5)
where $n_{k}$ is the number density of particles in the “k” grid, representing
the mass of the computational grid. Here, $P_{k}$ is the total momentum of the
protons in the “k” grid, $m_{p}$ is the mass of an individual proton, and
$U_{k}$ is the average bulk fluid speed of the grid (i.e. the velocity of the
scattering center). The collected grid-based mass and momentum densities will
directly decide the velocity of the scattering center $U_{k}$. The particle’s
total velocity $V$ in the box frame is decided by Equation (2). Once the value
of $U_{k}$ becomes zero, the shock front is decided by the position of the
corresponding grid, and it means that the shock is formed and the length of
the downstream region is extended dynamically. Similarly, if the value of
$U_{k}$ is between $U_{0}$ and zero, it means that the foreshock region or
precursor (i.e. between the FEB and the shock front) is formed by the “back
pressure.” The FEB and the shock front both dynamically move away from the
reflective wall with a shock velocity $v_{sh}$.
(iii) Applying multiple scattering laws. A certain fraction of the particles
are chosen to scatter the background scattering center with their
corresponding scattering angles according to the prescribed scattering angular
distributions. The average number of scattering events occurring in an
increment of time $dt$ depends on the scattering time scale $\tau$, and the
scattering rate is presented by
$R_{s}=dt/\tau,$ (6)
where $R_{s}$ is the probability of the scattering events occurring in an
increment of time. The candidates with their local velocities and scattering
angles scatter off the grid-based scattering centers. These individual
particles do not change their routes until they are selected to scatter once
again. So the particle’s mean free path is proportional to the local thermal
velocities in the local frame with
$\lambda\propto V_{L},$ (7)
for simplicity, we take its formula as
$\lambda=V_{L}\cdot\tau.$ (8)
For the individual protons, the grid-based scattering center can be seen as a
sum of individual momenta. So these scattering processes can be taken as the
elastic collisions. In an increment of time, once all of the candidates
complete these elastic collisions, the momentum of the grid-based scattering
center is changed. In turn, the momentum of the grid-based scattering center
will affect the momenta of the individual particles in their corresponding
grid in the next increment time. One complete time step consists of the above
three substeps. The total simulation temporally evolves forward by repeating
this time step sequence. To calculate the scattering processes accurately and
produce an exponential mean free path distribution, the time step should be
less than the scattering time (i.e. $dt<\tau$).
The presented multiple scattering law simulations are developed on the Matlab
platform. Any one of the four cases can occupy the CPU time for about seven
weeks on a 3. 4GHZ (MF) CPU per core. To speed up the running programs, the
parallel algorithm should be used on a high performance computer (HPC).
## 4 Results & discussions
We present all of the shock profiles for the shock simulations of the four
cases in Figure 1, and we present all aspects of simulation results including
the density and velocity profiles, compression ratios, analysis of the heating
and acceleration, and energy spectrum, as well as the correlations between the
shock compressions with the energy spectral index. For the convenience of
comparison and discussion, we list the specific calculated items in Table 1.
Here, $\sigma$ is the standard deviation of the scattering angular
distribution function $f(\delta\theta,\delta\phi)$, and $\mu$ is the average
value of scattering angles $(\delta\theta,\delta\phi)$. The subshock
compression ratios $r_{sub}$ are calculated from the velocity profiles in the
same shock frame reference. The compression ratios $r_{u}$ and $r_{\rho}$ are
calculated from velocity profiles and density profiles, respectively. The
total energy spectral index $\Gamma_{tot}$ and the subshock’s energy spectral
index $\Gamma_{sub}$ are calculated from compression ratios $r_{u}$ and
$r_{sub}$, respectively. The last two rows are shown as scaled values.
Table 1: Results of calculation with an initial energy of $E_{0}$=1.3105keV. Items | Case A | Case B | Case C | Case D
---|---|---|---|---
$f(\delta\theta,\delta\phi)$ | Gaussian | Gaussian | Gaussian | Isotropy
$\sigma$ | $\pi/4$ | $\pi/2$ | $\pi$ | $\infty$
$\mu$ | 0 | 0 | 0 | 0
$v_{sub}$ | 0\. 1118 | 0\. 1465 | 0\. 1733 | 0\. 2159
$v_{sh}$ | -0. 0433 | -0. 0535 | -0. 0617 | -0. 0733
$X_{sh}$ | 196 | 171\. 5 | 152 | 124
$FEB$ | 106 | 81\. 5 | 62 | 34
$\rho_{2}$ | 5000 | 4225 | 3759 | 3303
$\rho_{1}$ | 650 | 650 | 650 | 650
$r_{sub}$ | 2\. 5421 | 3\. 0207 | 3\. 3903 | 3\. 9444
$r_{u}$ | 7\. 9231 | 6\. 6031 | 5\. 8649 | 5\. 0909
$r_{\rho}$ | 7\. 6936 | 6\. 5001 | 5\. 7836 | 5\. 0815
$\Gamma_{tot}$ | 0\. 7167 | 0\. 7677 | 0\. 8083 | 0\. 8667
$\Gamma_{sub}$ | 1\. 4727 | 1\. 2423 | 1\. 1275 | 1\. 0094
$V_{Lmax}$ | 10\. 8251 | 16\. 0001 | 17\. 7824 | 20\. 5286
$E_{cutoff}$ | 1\. 10MeV | 2\. 41MeV | 2\. 98MeV | 4\. 01MeV
$|V_{sh}|$ | 58\. 16km/s | 71\. 87km/s | 82\. 88km/s | 98\. 46km/s
As shown in Figure 1, the present isotropic Case D largely appears similar to
the results from previous dynamical simulations by Knerr, Jokipii & Ellison
(1996). In addition, all aspects of the shock wave structure, density and
velocity profiles, compression ratio and energy spectra present in isotropic
Case D also give similar results to the previous outcome. The specific results
of the present isotropic Case D are shown in the fifth column of Table 1:
$v_{sh}$=-0.0733, $X_{sh}$=124, $FEB$=34, $r_{sub}$=3.9444, $r_{u}$=5.0909,
$r_{\rho}\sim r_{u}$ with a difference of 0.18%, $\Gamma_{sub}$=1.0094,
$\Gamma_{tot}$=0.8667, $|V_{sh}|$=98.46km/s, and $E_{cutoff}$=4.01MeV. As a
comparison, the corresponding results of the previous simulation are also
given: $v_{sh}$=-0.0720, $X_{sh}$=127.5, $FEB$=37.5, $r_{sub}$=3.20,
$r_{u}$=5.20, $r_{\rho}\sim r_{u}$ with a difference of 2.3% ,
$\Gamma_{sub}$=1.1818, $\Gamma_{tot}$=0.8571, $|V_{sh}|$=96.9 km/s, and
$E_{cutoff}\sim$4.00MeV. Among comparable results, a slightly larger
differences in the values of the $r_{sub}$ and $\Gamma_{sub}$ between the two
isotropic simulations would be distributed between different sizes of the
subshock region, decided differently. As seen from the comparison of the
results coming from two independent simulation codes, the present simulation
code successfully produced good agreement in the results with those in
previous dynamical Monte Carlo simulation. Therefore, the present simulation
code is based on the Matlab platform without using a supercomputer that can
independently validate the previous dynamical simulation method using a
completely different code for that supercomputer. Next, we offer a series of
discussions about the different cases considering the specific aspects of the
simulation results for diffusive shock.
Figure 1: Four cases of density profiles for the entire simulation box vs the
time. The dashed line represents the position of the FEB in each plot.
Figure 2: The velocity and density profiles (in the box frame) vs the position
at the end of the simulation ($T_{max}$=2400) in the four cases. The vertical
dashed line in each plot represents the position of the FEB, from case A to
case D.
### 4.1 Shock Profiles
Figure 1 shows time sequences of the density profiles of four cases. In each
plot, a shock forms and moves away from the reflective wall, and the dashed
line represents the FEB position with the time in each case parallel to the
shock front position. We can see that both the shock position and the FEB
position are moving with a virtually constant velocity from the beginning of
the simulation to the end of the simulation (i.e. $T_{max}$=2400) in each
case. Simultaneously, as far as the positions of the FEB are concerned, we can
see that the FEB position at the end of the simulation is significantly
different in four different cases. As for the average density fluctuation in
the downstream region, there are also apparent changes in different cases, and
case A has the slowest shock propagation speed among these four cases. Case D
has the lowest average density profile in the downstream region among these
four cases. Because from Cases A to D the only difference is the prescribed
scattering angular distribution, we conclude that these differences of the
results for shock propagation speed and density profiles are decided by the
standard deviation value $\sigma$ of the scattering angular distribution.
Figure 2 shows four cases of density and velocity profiles at the end of the
simulations. From Cases A to D, the position of the FEB approaches zero as the
value of the standard deviation $\sigma$ increases. The effects of the
accelerated particles are clearly seen in the upstream smoothing of the
velocity profiles in each case. In the simulations, when high-energy particles
cross the shock front and diffuse upstream, they contribute negatively to the
velocity profile. This reduces the grid-based velocity in the zones upstream
of the shock, which in turn affects particles that are scattering in that
region. In fact, the accelerated particles slow and heat the incoming flow and
smooth the shock transition by their “back reaction.” As is obvious to see
from the velocity and density plots, different scattering angular
distributions produce different effects on the shock wave evolution. For the
examples presented here, we consider that a difference of approximately 40.93%
of the shock velocity is contributed by the scattering angle distribution.
### 4.2 Compression ratios
Here, we compare the compression ratios calculated from the velocity profiles
with those from the density relationships. First, the value of the total
compression ratio can simply be calculated from the formula
$r_{u}=u_{1}/u_{2}$ (9)
where $u_{1}=u_{0}+|v_{sh}|$ , $u_{2}=|v_{sh}|$, and $u_{1}(u_{2})$ is the
upstream (downstream) velocity in the shock frame. The shock velocity at the
end of the simulation ($T_{max}$) can be derived from the formula
$v_{sh}=(X_{max}-X_{sh})/T_{max}$ (10)
where $X_{max}$=300, $T_{max}$=2400, and $X_{sh}$ is the position of the shock
at the end of the simulation (see Table 1). The specific calculated results
are shown in Table 1.
But in terms of the specific shock structure as seen in Figure 3, an accurate
subshock compression ratio calculation should be more complicated. In any one
of the cases in Figure 3 (plotted in the box reference frame), we show the
specific aspects of a shock modified by an energetic particle population whose
mean-free-path is an increasing function of momentum. The shock structure in
each plot consists of three main parts: precursor, subshock, and downstream.
The smooth precursor is on the largest length scale between the FEB and near
shock position $X_{sh}$, where the fluid speed gradually decreases from value
$U_{0}$ to $v_{sub}$. The size of this precursor is almost the mean-free-path
length of the maximum energy accelerated particles. One of the smallest scales
is the subshock region with a sharp deflection of the fluid speed decreasing
from $v_{sub}$ to $v_{box}=0$. The downstream region changes after the fluid
speed becomes $v_{box}=0$ by microphysical dissipation processes. The gas
subshock is just an ordinary discontinuous classical shock embedded in the
comparably larger scale energetic particle shock (Berezhko & Ellison 1999).
The value of $v_{sub}$ is determined by a sharp deflection of smooth curves in
velocity profiles near the shock front, and the value of the subshock velocity
increases from cases A, B, and C to Case D (i.e.
$(v_{sub})_{A}<(v_{sub})_{B}<(v_{sub})_{C}<(v_{sub})_{D}$). All of the
velocity profiles are based on the box frame. That value of the box frame’s
velocity is zero ($v_{box}=0$) in all cases. The subshock compression ratio
$r_{sub}$ is calculated from the formula $r_{sub}=(v_{sub}+|v_{sh}|)/|v_{sh}|$
. For the sake of the comparison of the values of $r_{sub}$ in different
cases, the subshock compression ratios are calculated in the same shock frame
reference, and the calculated results of $r_{sub}$ are shown in Table 1.
Figure 3: Velocity profiles in the shock region at the end of the simulation
($T_{max}$=2400) in the four cases; the vertical solid and dotted lines
indicate the shock front and FEB in each plot, respectively; the horizontal
solid, dotted, dot-dashed and dashed lines show the values of the shock
velocity $v_{sh}$, velocity of box frame $v_{box}$, subshock velocity
$v_{sub}$ and initial bulk velocity $U_{0}$, respectively. Two vertical bars
in each plot represent the two deflections of velocity, the upper bar
represents the part of the shock precursor and the lower one represents the
subshock. Figure 4: Density profiles in an entire simulation box at the end of
the simulation ($T_{max}$=2400) in the four cases. The vertical solid line is
located in the position of the shock front, the upper horizontal dot-dashed
line represents the value of the downstream density $\rho_{2}$, and the lower
horizontal dashed line indicates the value of the upstream density $\rho_{1}$
in each plot.
We then have the calculations of the compression ratio from the density
relationships between the upstream and downstream flows:
$r_{\rho}=\rho_{2}/\rho_{1}$ (11)
where $\rho_{1}=n_{0}$ is the upstream density, and $\rho_{2}$ is the
downstream density. This value is presented by which is the average value of
$n_{k}$ over the downstream region.
$\rho_{2}=\frac{1}{(k_{max}-k_{sh})}\displaystyle\sum_{k=k_{sh}}^{k_{max}}(n_{k})$
(12)
where $n_{k}$ is the number density of particles in the “k” grid,
$k_{sh}=x_{sh}/dx$ is the grid index of the shock at the end of the simulation
($T_{max}=2400$), and $k_{max}=600$ is the grid index of the $X_{max}$.
Figure 4 shows the complete density plots of the four cases at the end of the
simulation. The value of the upstream density $\rho_{1}$ is the same constant
value, which is equal to the initial density $n_{0}$ in each case. The value
of the downstream density $\rho_{2}$ decreases from cases A, B, and C to Case
D (i.e. $(\rho_{2})_{A}>(\rho_{2})_{B}>(\rho_{2})_{C}>(\rho_{2})_{D}$).
Similarly, the detailed calculation results of the compression ratios
$r_{\rho}$ are listed in Table 1. As listed in Table 1, the values of the
subshock compression ratios, $r_{sub}$=2.5421, $r_{sub}$=3.0207,
$r_{sub}$=3.3903, and $r_{sub}$=3.9444, corresponding to cases A, B, C, and D,
respectively, are all lower than the standard value of $r=4$. Unfortunately,
the values of total compression ratio $r_{u}$ and $r_{\rho}$ in each case are
both higher than the standard value of $r=4$. But Knerr, Jokipii & Ellison
(1996) consider that, if energy is lost from the system (e.g. by particles
escaping via FEB), it is possible to produce a shock with a total compression
ratio that is higher than the standard value predicted by the Rankine-Hugoniot
(RH) conditions. We have examined the mass and energy losses via the FEB in
each case. The results definitely show that the case with more energy losses
would produce a higher total compression ratio than those in the case with
less energy loss. Consequently, we consider that the energy loss rates would
be affected by the prescribed scattering law. In any case, the energy losses
are always an important and interesting problem in the nonlinear diffusive
shock acceleration theory, so we will perform more precise research focusing
on these problems in later papers. In addition, although the values of $r_{u}$
are correspondingly slightly higher than the value of $r_{\rho}$ in each case,
all these differences are less than 3%, and the specific difference in each
case is 2.9%,1.5%,1.3% and 0.18% corresponding to cases A, B, C and D. As seen
from Figs. 3 and 4, the value of the total compression ratio $r_{tot}$,
determined from the velocity profiles, is more consistent with the density
profiles in each case (i.e. the total compression ratios $r_{tot}$ in all
cases are satisfied by the Rankine-Hugoniot (RH) conditions :
$u_{1}/u_{2}=\rho_{2}/\rho_{1}$). Therefore, it is not difficult for us to
conclude that the total compression ratios $r_{u}$ and $r_{\rho}$ decrease as
the value of the standard deviation $\sigma$ of the scattering angular
distribution increases, but the subshock compression ratio $r_{sub}$ increases
as the value of the standard deviation $\sigma$ of the scattering angular
distribution increases.
### 4.3 Heating & acceleration
Figure 5: The scatter plots of the particle’s thermal velocities in the local
frame vs its position at the end of the simulations ($T_{max}$=2400), and the
vertical dashed line and solid line in each plot indicate the approximate
position of the FEB and shock front, respectively. Only the ratio of 1/12 of
the total number of particles are plotted.
Here we contrast between two important aspects of the heating and acceleration
processes in the diffusive shock acceleration. Figure 5 shows the particles’
scatter plots in four cases at the end of the simulations ($T_{max}$=2400). In
each case, particles with local velocity scatter into the simulation box’s
position. A large amount of particles that do not get injected into the Fermi
acceleration mechanism and that have lower thermal velocities stay in the
downstream region, and a few particles with higher energy via multiple shock
encounters can move far away from the shock front and even escape from the
FEB. From Cases A to D, more and more particles are injected into the Fermi
acceleration mechanism, and they gain greater and greater maximum energy.
Obviously, the maximum thermal velocity in Case D would be several times that
of the ones in Case A. We supposed that this difference is mainly contributed
by the scattering angular distribution function $f(\delta\theta,\delta\phi)$.
In short, the majority of particles, which flow toward the shock, cross the
shock only once, and (after scattering) remain fairly stationary in the
downstream region, which would consist of the “heated” elements, and a few
high-energy particles represent the “power-law” part of the simulated
particles flows. Actually, the “back pressure” from the accelerated particles
via their back reaction reduces the incoming fluid speed, leading to a
smoothed precursor heating. Therefore, an anisotropic scattering angular
distribution in the shocks dominate the “gas heating” process in the simulated
plasma, and isotropic scattering angular distribution in the shocks play an
important role in nonlinear acceleration of the energetic particles via the
Fermi mechanism, and they dominate the “precursor heating”.
### 4.4 Energy spectra
Figure 6: The final energy spectrum in the four cases and the initial energy
spectrum plots. The thick solid line with a narrow peak at $E_{0}=$1.3105keV
represents the initial Maxwell energy distribution. The solid, dashed, dot-
dashed, and dotted extended curves indicate the simulated particles’ energy
spectral distribution, averaged over the entire downstream region, at the end
of the simulations ($T_{max}$=2400), corresponding to Cases A, B, C, and D,
respectively. Most particles cross the shock only once, producing the large
broad peak centered at E${}_{A}\sim$ 0.05keV, E${}_{B}\sim$0\. 1keV,
E${}_{C}\sim$ 0.15keV, and E${}_{D}\sim$0.20keV in Cases A, B, C, and D,
respectively. However, some particles gain enough energy via the Fermi
acceleration mechanism to produce the “power-law” tail in the energy spectrum
with the cutoff at $E_{A}$=1.10 MeV, $E_{B}$=2.41 MeV, $E_{C}$=2.98 MeV, and
$E_{D}$=4.01 MeV corresponding to Cases A, B, C, and D. All these energy
spectra are plotted in the same shock frame.
Spectra calculated in the shock frame from the initial and final particle
(proton) energy distributions in all cases are shown in Figure 6. The energy
units in this plot are derived from the scaling parameters presented in Table
2 (see Appendix A). Initially, all particles move toward the wall with a
certain thermal spread in energy. A narrow peak at E=1.3105keV represents the
initial Maxwell energy distribution. The four extended curves indicate the
simulated particle energy spectral distribution, averaged over the entire
downstream region, at the end of the simulations, corresponding to the four
cases, respectively. The majority of the particles cross the shock only once,
producing an expanded energy spectrum with a central peak at E${}_{A}\sim$
0.05keV, E${}_{B}\sim$0.1keV, E${}_{C}\sim$ 0.15keV, and E${}_{D}\sim$0.20keV
in Cases A, B, C, and D, respectively. However, as is shown in Figure 6, the
minority of the particles gain enough energy via the Fermi acceleration
mechanism to produce the “power-law” tail in the energy spectrum with the
cutoff at $E_{A}$=1.10 MeV, $E_{B}$=2.41 MeV, $E_{C}$=2.98 MeV and
$E_{D}$=4.01 MeV corresponding to Cases A, B, C and D, respectively. For more
details about the calculated results, see Table 1. It is evident from Figure 6
that the values of the central peak of the extended energy spectra in the four
cases are far from the initial energy peak in their respective order. This
means the values of the central peak in each case increase as the value of the
standard deviation of the scattering angular distribution increases, and each
extended curve shows a harder power-law slope in its high-energy tail as the
expand energy range increases, respectively. Therefore, we can see that the
case of applying an anisotropic scattering angular distribution function will
produce a slightly softer energy spectrum, and the case of applying an
isotropic scattering angular distribution will produce a slightly harder
energy spectrum.
### 4.5 Spectral index & compression ratios
Usually, we could predict the power-law energy spectral index from diffusive
shock acceleration theory:
$dJ/dE\propto E^{-\Gamma}$ (13)
where $dJ/dE$ is the energy flux and $\Gamma$ is the energy spectral index,
and
$\Gamma=(r+2)/(2\times(r-1)).$ (14)
According to Equation 14, we substituted the values of the compression ratio
$r$ with two group values of $r_{tot}$ and $r_{sub}$ obtained in each case.
Then, the two group energy spectral indices $\Gamma_{tot}$ and $\Gamma_{sub}$
in each case are calculated. Two groups’ spectral index values are listed in
Table 1 as $\Gamma_{A}$= 0.7167, $\Gamma_{B}$ =0.7677, $\Gamma_{C}$ =0.8083,
and $\Gamma_{D}$=0.8667 in the total group $\Gamma_{tot}$, and $\Gamma_{A}$=
1.4727, $\Gamma_{B}$ =1.2423, $\Gamma_{C}$ =1.1275, and $\Gamma_{D}$=1.0094 in
subshock group $\Gamma_{sub}$, corresponding to the cases A, B, C, and D,
respectively. As shown in Figure 7, from Cases A to D, all of the values of
the subshock’s energy spectral index $\Gamma_{sub}>1$ and show that a slightly
harder power-law slope in the respective order. From Cases A to D, all of the
values of the total energy spectral index $\Gamma_{tot}<1$ show a slightly
decreasingly deviation from the power-law slope in the respective energy
spectrum.
Figure 7: The correlation of the deviation value of the Gaussian distribution
vs the energy spectral index. The triangles represent the total energy
spectral index in each case. The circles indicate the subshock’s energy
spectral index in each case. From Cases A to D, all of the values of the
subshock’s energy spectral index $\Gamma_{sub}>1$ show a slightly harder
power-law slope in the respective order. From Cases A to D, all of the values
of the total energy spectral index $\Gamma_{tot}<1$ show a slightly decreasing
deviation from the power-law slope in the respective energy spectrum.
## 5 Summary and conclusions
We followed the previous dynamical Monte Carlo simulation by using a new code
based on the Matlab platform independently, and presented the same results as
the outcome from the previous simulation. In addition, we successfully
extended the simulation include the multiple scattering angular distributions
using these new codes to study the diffusive shock acceleration mechanism
further.
In conclusion, the comparison of the calculated results come from different
extensive cases, we find that the total energy spectral index increases as the
standard deviation value of the scattering angular distribution increases, but
the subshock’s energy spectral index decreases as the standard deviation value
of the scattering angular distribution increases. In these multiple scattering
angular distribution simulation cases, the prescribed scattering law dominates
the shock structure and plays an important role in balancing whether the
particles have more heating or more acceleration. In other words, the cases
with anisotropic scattering distribution give the overall velocity-deflection
precursor sizes, which are larger than the isotropic case, and give a
relatively greater “heating” effect or less “acceleration” effect on
background flows than does the isotropic case.
As a result, the shock compression ratio and the energy spectral index are
both modified naturally by the prescribed scattering law. Specifically, the
cases of applying an anisotropic scattering distribution function will produce
a slightly softer subshock’s energy spectral index, and the case of applying
an isotropic scattering angular distribution will produce a slightly harder
subshock’s energy spectral index. Simultaneously, from the isotropic case to
the anisotropic case, the total energy spectrum shows an increasing deviation
from the “power-law” distribution.
In addition, although we find no case producing the total compression ratio
which should be less than the standard value 4 according to the Rankine-
Hugoniot (RH) jump conditions, the fact is clear that the prescribed
scattering angular distribution function would have an effect on the total
compression ratio. If there is a suitably prescribed scattering law that leads
to much less energy loss, it is possible to constrain the total compression
ratio to be less than 4.
###### Acknowledgements.
The authors would like to thank Doctors G. Li, Hongbo Hu, Siming Liu, Xueshang
Feng, and Gang Qin for many useful and interesting discussions concerning this
work. In addition, we also appreciate Profs. Qijun Fu and Shujuan Wang, as
well as other members of the solar radio group at NAOC.
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## Appendix A Simulation box & parameters
With respect to the validity and consistency of verifying the previous dynamic
Monte Carlo simulation method, the present simulation program uses the same
simulation box and identical parameters as the previous dynamical Monte Carlo
simulation (Knerr, Jokipii & Ellison 1996). The schematic diagram of the
simulation box is shown in Figure 8 and all the simulation parameters are
listed in Table 2.
Figure 8: Shock is produced by supersonic flow toward the stationary
reflective wall to the right. Continuous inflow of new particles occurs at the
left boundary. Inflow velocity is $U_{0}$, a particle’s total velocity is $V$,
the local frame particle velocity is $V_{L}$, and thermal velocity
$<V_{L}>=v_{0}$. The two circles represent one typical particle in the
upstream region and one in the downstream region, respectively. The vertical
solid line represents the shock front, the vertical dashed line represents the
FEB, the velocity of the shock is $V_{shock}$, size of the foreshock region is
$Xfeb=90$, the upstream flow velocity $U=U_{0}$, and the downstream flow
velocity is $U=0$. The magnetic field $B_{0}$ and inflow velocity $U_{0}$ are
both normal to the shock front (see Knerr, Jokipii & Ellison 1996). Table 2:
The parameters of the simulated cases
Inflow velocity | $u_{0}$=0. 3 | 403km/s
---|---|---
Thermal speed | $\upsilon_{0}$=0. 02 | 26\. 9km/s
Scattering time | $\tau$=0. 833 | 0\. 13s
Box size | $X_{max}$=300 | $10R_{e}$
Total time | $T_{max}$=2400 | 6\. 3minutes
Time step size | $dt$=1/30 | 0\. 0053s
Number of zones | $nx$=600 | . . .
Initial particles per cell | $n_{0}$=650 | . . .
FEB distance | $X_{feb}$=90 | $3R_{e}$
111Scaling used a box size = 10$R_{e}$ (where $R_{e}$ represents the Earth’s
radius) and the box frame inflow velocity $u_{0}$= 403km/s. This implies the
following scale factors for distance, velocity, and time:
$X_{scale}$=10$R_{e}$/300, $v_{scale}$=403km $s^{-1}$/0.3, and
$t_{scale}$=$x_{scale}$/$v_{scale}$. Here, the Mach number M=11.6.
Dimensionless or normalized numbers are used in the text to describe our
simulations, except for specifically highlighted examples (see Knerr, Jokipii
& Ellison 1996).
|
arxiv-papers
| 2010-09-27T02:42:43 |
2024-09-04T02:49:13.134299
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Wang and Yihua Yan",
"submitter": "Xin Wang Mr.",
"url": "https://arxiv.org/abs/1009.5144"
}
|
1009.5332
|
††thanks: Present address: Physics Department, Northwestern College, Orange
City, Iowa 51041 USA
# Discovering the Value of Multidisciplinary Approaches to Research: Insights
from a Sabbatical
Frank W. Bentrem sciguy137@yahoo.com Marine Geosciences Division, Naval
Research Laboratory, Stennis Space Center, Mississippi 39525 USA
(December 2009)
###### Abstract
In this informal report, I outline my research efforts, collaborations, and
other experiences while participating in the Naval Research Laboratory (NRL)’s
Advanced Graduate Research Program (AGRP), aka Sabbatical, from October 2008
through September 2009. This report is in no way intended to present the
technical details of the various research projects, but rather a broad
overview of the small ways my efforts may have contributed to ongoing
research. I wish to convey to the reader the value of multidisciplinary
approaches to scientific research and how the AGRP facilitates these
opportunities.
Disclaimer: The views expressed in this article are those of the author and do
not represent opinion or policy of the US Navy or Department of Defense.
###### pacs:
###### Contents
1. I Why Pursue the AGRP?
1. I.1 Polymer Physics Background
2. I.2 Role of Polymers in Marine Sediments
3. I.3 Opportunities in Polymer Science
1. I.3.1 Tulane’s PolyRMC
2. I.3.2 NRL-DC’s CBMSE
4. I.4 Broad Support
2. II Polymer Physics with the Tulane Green Wave
1. II.1 Hello, Mate!
2. II.2 Delivering Drugs (Legally)
3. II.3 Building a Better Mousetrap–Polymer Simulations
4. II.4 A Distinguished Gentleman
3. III Researching Soft Matter at NRL-DC
1. III.1 Micro- and Nano-scale Plastic Spaghetti
2. III.2 Artificial Muscles
4. IV New Directions with Nano-material
1. IV.1 It’s a Bird, It’s a Plane, It’s Super-Carbon!
2. IV.2 Hey, We Can Do That
5. V Final Thoughts
1. V.1 Multidisciplinary Magic
2. V.2 Bean Counting
## I Why Pursue the AGRP?
What is the purpose of the Advanced Graduate Research Program (AGRP) and why
would a researcher at the Naval Research Laboratory (NRL) consider applying
for the program? The answer to the first question can be found in NRL’s Human
Resources instruction document NRL (1996):
> It is NRL policy to maintain a highly competent corps of professional
> personnel by providing opportunities for employees to keep abreast of
> advances in their fields. The purpose of the Advanced Graduate Research
> Program is to permit selected employees to pursue collaborative research in
> their own or related fields on a full-time basis.
To answer the latter question, I will provide my personal motivation for
embarking on this year-long program.
### I.1 Polymer Physics Background
My career with NRL began in May 2000 while a part-time graduate student in
computational polymer physics. Although my efforts with NRL were focused on
acoustic imagery Bentrem _et al._ (2006); Bentrem (2009b) and sediment
classification Brown _et al._ (2001); Bentrem _et al._ (2002a); Bentrem _et
al._ (2006); Harris _et al._ (2008), I completed my doctoral research in
simulations for polymer electrocoatings Bentrem _et al._ (2000); Bentrem _et
al._ (2002c, b); Bentrem and Pandey (2005). (See Fig. 1). In recent years, I
have searched for ways to use my background and expertise in molecular
simulations to best serve the missions of both the Marine Geosciences Division
and the laboratory as a whole.
Figure 1: Simulation snapshot Bentrem _et al._ (2000) for polymer
electrocoating with uniform electric field $E$. Inset shows surface roughness
from a cross section with thickness $h$.
### I.2 Role of Polymers in Marine Sediments
Much progress is being made towards understanding the physical properties of
geologic sediments in terms of the chemical constituents. However, much
remains to be understood regarding the influence of organic matter on
mechanical and electrical properties of marine sediments. In particular, how
do the polymer components (polysaccharides, biomolecules, etc.) of organic
matter affect the flocculation, aggregation, and shear strength of muddy
sediment on the seafloor? The Marine Geosciences division is increasing
efforts to understand these important issues, which impact Navy interests,
such as trafficability, mine burial, and beach morphology, to name a few.
### I.3 Opportunities in Polymer Science
#### I.3.1 Tulane’s PolyRMC
In August 2007, the physics department at Tulane University announced the
opening of the Center for Polymer Reaction Monitoring and Characterization
(PolyRMC) with the mission “To be the world’s premier center for R&D in
polymerization reaction monitoring.” Since Tulane is a mere 45-minute drive
from Stennis Space Center, I soon began interacting with the Center’s Director
and Assistant Director, Professors Wayne Reed and Alina Alb, to discover
potential topics for collaboration.
#### I.3.2 NRL-DC’s CBMSE
About the same time (summer 2007) I became aware of NRL’s Center for
Bio/Molecular Science and Engineering (CBMSE) and their interests in the
theory and simulations for polymer systems. In discussions with some CBMSE
researchers, we identified areas where my expertise could complement the work
they were doing. I was named as an investigator on a proposal to the NRL
Nanoscience Institute. The proposal for an innovative type of body armor was
one of fifteen selected for presentation before the nanoscience committee,
however, it was not selected to go before the Research Advisory Council.
### I.4 Broad Support
Although armed with the background and opportunities for polymer research, I
realized that for substantial impact to NRL, I would need to arrange for
intensive study and research in the most up-to-date applications and
techniques both in polymer physics and materials science in general. The AGRP
seemed to fit the bill precisely, and I began to discuss this possibility with
my supervisors at NRL. Only the Assistant Director of Research (ADOR) need
write a letter of recommendation for the AGRP candidate, yet I received
tremendous support within my division with first-level supervisor, branch
head, and division superintendent all kindly recommending my participation in
the program in addition to that of the ADOR. Also, I received letters of
invitation from Tulane University and NRL’s CBMSE. In spring 2008, to my great
delight, I was notified that I had been selected for the AGRP.
## II Polymer Physics with the Tulane Green Wave
### II.1 Hello, Mate!
On the morning of October 1, 2008, I arrived at the PolyRMC and received a
staff ID card, a shared office, and a brand-new computer. My office mate,
Tomasz Kreft, was in the last year of his doctoral work, and made me feel most
welcome. His newest experimental results in polymerization kinetics led me to
alter my original computational research plans. In April 2009, after his
dissertation defense, Tomasz accepted my invitation to present his research as
a seminar at NRL-Stennis.
### II.2 Delivering Drugs (Legally)
One particular Ph.D. student, Colin McFaul, had been researching a thermo-
responsive polymer, that is, one that changes its properties with a change in
temperature. Colin demonstrated a sharp change in the polymer size for a
Poly(N-isopropylacrylamide), or simply PNIPAm, solution when temperature
crossed $33^{\circ}$C. Since this is very close to body temperature
($37^{\circ}$C), there is much interest in applications for drug delivery. At
temperatures lower than $33^{\circ}$C, the swollen PNIPAm polymer chains can
exist as a gel which encapsulates a given drug. After ingesting the PNIPAm
capsule, the warming polymer collapses and the gel dissolves releasing its
contents. The mechanism for (and hence the ability to control) this collapse
is still under investigation. Much of my research at Tulane focused on
developing a computer simulation technique for understanding this temperature
dependent transition.
### II.3 Building a Better Mousetrap–Polymer Simulations
The bond-fluctuation model (BFM) is a computationally efficient simulation
method for researching polymer systems at time and length scales not
accessible to other methods. After appropriately assigning the interaction
potentials, the BFM has proved successful in reproducing much of the measured
polymer structures and dynamics. However, accuracy from the BFM suffers in the
highly constrained geometries of dense polymer melts and tightly collapsed
chains (as with collapsed PNIPAm globules). I developed an enhancement to the
BFM which greatly increased the flexibility in the polymer chains, yet
retained the computational efficiency. The resulting polymer coil-globule
transition demonstrates the compact collapse (shown in Fig. 2) expected for
PNIPAm. This enhanced BFM provides broader capability for the simulation of
polymer systems. Colin was taking a computational physics course and asked me
for a project idea. He helped me modify the enhanced BFM to simulate a dense
polymer melt and analyze the dynamics.
Figure 2: Collapse of a self-attracting polymer chain. Chain at high
temperature (left) has a large radius of gyration, but at low temperature
(right) the same chain has collapsed with a small radius of gyration.
### II.4 A Distinguished Gentleman
On one occasion, while briefing Wayne and Alina (the PolyRMC directors) on my
recent progress at the hallway blackboard, a senior gentleman approached and
entered the discussion. His insights into the polymer coil-globule transition
research were most welcome. He later visited my office for further discussion
on my current simulation technique, and mentioned that years earlier he had
worked on polymer simulations with Walter Stockmayer–a well known pioneer in
polymer simulations whose research was a forerunner to my current simulation
method. A friendly and engaging man, he had taken a keen interest in my work.
Later, I searched the internet for his name, Dr. Hyuk Yu. It turns out that he
is the Walter H. Stockmayer Professor of Chemistry at the University of
Wisconsin-Madison, an American Physical Society (APS) fellow, and had received
the APS High Polymer Physics Prize! We continue to correspond and plan to meet
at the 2010 APS March meeting.
## III Researching Soft Matter at NRL-DC
The second phase of my sabbatical was spent at the Center for Bio/Molecular
Science and Engineering in Washington, DC (NRL). I drove to DC with my wife,
Amelia, and 3-year-old daughter, Abby, in mid-May 2009 to settle into a modest
furnished apartment for a 3-month stay.
### III.1 Micro- and Nano-scale Plastic Spaghetti
I continued my work with the center’s senior scientist, and the “nano-Play-Doh
group” on the polymerization kinetics in microfluid channels. NRL has
developed a microfluidic system with remarkable control of the shape of the
microfluid channels. Polymerizing the material in the microfluid channel
produces polymer microfibers (Fig. 3) of predetermined shapes. A potential
application for the microfibers is in producing high-strength materials such
as body armor. My results for the kinetics of the photopolymerization in the
microfluid device were intended to be used as a guide for the design of the
apparatus in producing micro- and nano-scale fibers.
Figure 3: Polymerized microfibers under magnification Thangawng _et al._
(2009).
### III.2 Artificial Muscles
Another group I worked with at CBMSE was the liquid-crystal (LC) group. I
became interested in an electroclinic LC experiment that they had recently
performed. An electroclinic LC is one for which the aligned LC components tilt
in either direction under the influence of a switching electric field. The
tilting causes the electroclinic LC to contract so that it behaves as an
artificial muscle controlled by the applied electric field. Experiments had
been performed both with the ordinary electroclinic LC and an electroclinic LC
elastomer, where polymer backbones attached to the aligned LC components. The
data showed a puzzling difference between the two experiments presumed to be
caused by the polymer dynamics. Since I had experience in this area, I
performed an analysis on the tilt dynamics and related this to the theory of
dynamics for entangled polymer. Interaction among the LC group members is
quite active and I had regular discussions and input from various group
members. Key parts of this work were included in a recent paper Spillmann _et
al._ (2010).
Figure 4: An electroclinic LC elastomer with cross-linking Spillmann _et al._
(2010).
## IV New Directions with Nano-material
### IV.1 It’s a Bird, It’s a Plane, It’s Super-Carbon!
While listening to some short proposal presentations at CBMSE, my ears perked
up at a proposal for experimental research on graphene. Graphene is a 1-atom
thick, pure-carbon sheet (See Fig.5) with extraordinary electronic,
mechanical, and optical properties. It is the strongest material ever tested
(200 times stronger than steel), the most sensitive chemical sensor possible
(shown to detect single molecules), and the thinnest possible membrane (1-atom
thick sheet). Assumed not to exist (based on prior theory), researchers were
surprised to discover graphene sheets Novoselov _et al._ (2004), and, due to
its unique properties and vast assortment of potential applications
(particularly in carbon-based nano-electronics and bio/chemical sensing), it
has become one of the “hottest” topics in physics. One of the great challenges
in using graphene as a semiconductor in nano-electronics and bio/chemical
sensing is understanding and controlling the ripples that spontaneously form
throughout the sheets, which can be seen with electron microscopy.
Figure 5: A flat graphene sheet Wikipedia (2009) highlighting the
characteristic honeycomb pattern.
### IV.2 Hey, We Can Do That
I imagined extending the (1-dimensional) polymer model from Sec. II.3 to a
2-dimensional graphene model. Indeed, my doctoral research adviser has already
researched clay platelets using a similar technique. The key to modeling
graphene would be in using a lattice-in-a-lattice approach to access the fine
(nano-)scales appropriate to the carbon bonds. This ultra-efficient technique
should allow graphene-ripple simulations at unprecedented time and length
scales. The CBMSE researcher (whose proposal sparked my interest in graphene
simulations) and I began discussing the possibility for me to lead a 6.1 New
Start proposal with the goal of understanding graphene ripples and the
adsorption of biomolecules onto graphene sheets with the potential application
to marine biosensing.
## V Final Thoughts
### V.1 Multidisciplinary Magic
Prior to my sabbatical year, I had already experienced the benefits of
multidisciplinary approaches to research by applying my skills in statistical
physics to problems in underwater acoustics Bentrem _et al._ (2002a); Bentrem
_et al._ (2006); Bentrem (2009b). My sabbatical experience, however, amplified
my appreciation for multidisciplinary and collaborative science. It was a
privilege to work with the many diverse, enthusiastic, and creative
researchers at both Tulane and NRL-DC. I agree wholeheartedly with the
following statement by our recently retired distinguished NRL colleague.
> One of the most important considerations concerning the support of research
> with public funds is the relation of basic research to applied research or
> technology. Technology flourishes when it is stimulated by the results of
> basic research. To illustrate this, refer to a statement made during the
> 1950’s when the attack on poliomyelitus was bearing fruit: “With technology
> one can build the best iron lung in the world, but from basic research you
> have a vaccine”. The administration in charge of research at the NRL clearly
> understands the relationship between basic research and technology.
Dr. Jerome Karle
Nobel Laureate
Chief Scientist in NRL’s Lab for Structure of Matter
### V.2 Bean Counting
As a direct result of my efforts in the AGRP, I have:
* •
1 accepted refereed journal article Bentrem (2009b) (sole author)
* •
1 submitted journal article (coauthor)
* •
1 manuscript in progress (lead author)
* •
1 international conference presentation (coauthor)
* •
2 future international conference presentations (March 2010, lead author)
* •
1 Provisional Patent Application filed and approved by NRL’s Invention
Evaluation Board Bentrem (2009a)
* •
1 DTRA proposal submitted (not funded)
* •
1 planned proposal
* •
Enhanced polymer simulation model ideally suited for the Marine Geosciences
Division’s ARI (starting in FY10).
My partaking in the AGRP (sabbatical) was time and effort well spent. As the
old adage goes, “A wood chopper never wastes time when he takes the time to
sharpen his ax”. By broadening and deepening my skills in science, math, and
computation, I believe I am well-positioned to creatively solve the scientific
and engineering problems I will encounter at NRL.
## References
* Bentrem and Pandey (2005) Bentrem, F., and R. Pandey, 2005, Macromolecules 38(3), 992.
* Bentrem _et al._ (2002a) Bentrem, F., J. Sample, M. Kalcic, and M. Duncan, 2002a, in _OCEANS ’02 MTS/IEEE_ , volume 1, pp. 7–11 vol.1.
* Bentrem (2009a) Bentrem, F. W., 2009a, 4-level spin-glass image segmentation with energy minimization, Provisional Patent Application, navy Case No. 99,755.
* Bentrem (2009b) Bentrem, F. W., 2009b, Cent. Eur. J. Phys. , in pressURL http://www.springerlink.com/content/xn763nv69jk3725t.
* Bentrem _et al._ (2006) Bentrem, F. W., W. E. Avera, and J. Sample, 2006, Sea Technol. 47, 37.
* Bentrem _et al._ (2000) Bentrem, F. W., R. B. Pandey, and F. Family, 2000, Phys. Rev. E 62(1), 914.
* Bentrem _et al._ (2002b) Bentrem, F. W., J. Xie, and R. B. Pandey, 2002b, Journal of Molecular Structure: THEOCHEM 592(1-3), 95 , ISSN 0166-1280, URL http://www.sciencedirect.com/science/article/B6TGT-45Y6GJP-6/%2/1a1a8fc55eb0cbd9f1e43006d536378e.
* Bentrem _et al._ (2002c) Bentrem, F. W., J. Xie, and R. B. Pandey, 2002c, Phys. Rev. E 65(4), 041606.
* Brown _et al._ (2001) Brown, W., D. Newcomb, M. Barlett, G. Rayborn, and F. Bentrem, 2001, in _OCEANS, 2001. MTS/IEEE Conference and Exhibition_ , volume 3, pp. 1672–1677 vol.3.
* Harris _et al._ (2008) Harris, M. M., W. E. Avera, A. Abelev, F. W. Bentrem, and L. D. Bibee, 2008, in _OCEANS 2008_ , volume 2008-Supplement, pp. 1–11.
* Novoselov _et al._ (2004) Novoselov, K., A. Geim, S. Morozov, D. Jiang, Y. Zhang, S. Dubonos, I. Grigorieva, and A. Firsov, 2004, Science 306(5696), 666.
* NRL (1996) NRL, 1996, _NRL HRO INSTRUCTION_ , NRLHROINST 12410.2, Naval Research Laboratory, Washington, D.C.
* Spillmann _et al._ (2010) Spillmann, C. M., A. V. Kapur, F. W. Bentrem, J. Naciri, and B. R. Ratna, 2010, Phys. Rev. Lett. 104(22), 227802\.
* Thangawng _et al._ (2009) Thangawng, A. L., P. B. Howell Jr, J. J. Richards, J. S. Erickson, and F. S. Ligler, 2009, Lab Chip 9, 3126.
* Wikipedia (2009) Wikipedia, 2009, Graphene — wikipedia, the free encyclopedia, [Online; accessed 29-November-2009], URL http://en.wikipedia.org/w/index.php?title=Graphene&oldid=3707%00655.
|
arxiv-papers
| 2010-06-29T20:02:39 |
2024-09-04T02:49:13.153285
|
{
"license": "Public Domain",
"authors": "Frank W. Bentrem",
"submitter": "Frank Bentrem",
"url": "https://arxiv.org/abs/1009.5332"
}
|
1009.5462
|
# Rapidity distribution dependence of the transition energy in heavy-ion
collisions
Sanjeev Kumar1 Suneel Kumar1 suneel.kumar@thapar.edu 1School of Physics and
Materials Science, Thapar University, Patiala - 147004, (Punjab) INDIA
## Introduction
The relation between the nuclear equation of state (EOS) and flow phenomena
has been explored extensively in the simulations as well as by the
experimentalist. Recently the analysis of transverse-momentum dependence of
elliptical flow has also been put forwardedLuka055 ; Andr055 ; Dani00 . The
elliptical flow pattern of participant matter is affected by the presence of
cold spectators Dani00 . During the expansion, the particles emitted towards
the reaction plane can encounter the cold spectator pieces and, hence, get
redirected. In contrast, the particles emitted essentially perpendicular to
the reaction plane are largely unhindered by the spectators. Thus, for the
beam energies leading to rapid expansion in the vicinity of the spectators,
elliptic flow directed out of the reaction plane (squeeze-out) is expected.
This squeeze-out is related with the pace at which expansion develops, and is,
therefore, related to the EOS. This contribution of the participant and
spectator matter Dani00 in the intermediate energy heavy-ion collisions
motivated us to perform a detailed analysis of the excitation function of
elliptical flow over different regions of participant and spectator matter in
term of rapidity distribution bins. Attempts shall also be made to
parameterize the transition energy for the same.
## The IQMD Model
The model is modified version of QMD modelHart98 . In this model, the nucleons
of target and projectile interact via two and three body Skyrme forces, Yukawa
and Coulomb potential. A symmetry potential between protons and neutrons
corresponding to Bethe-Weizsacker mass formula has been included. The detail
of the model is discussed in ref.Hart98 by us and others.
## Results and Discussion
For the present analysis, simulations are carried out for thousand of events
for the reaction of ${}_{79}Au^{197}~{}+~{}_{79}Au^{197}$ at semi-central
geometry using a hard equation of state. The whole of the analysis is
performed for light charged particles (LCP’s)[1$\leq$ A $\leq$ 4].
Figure 1: The incident energy dependence of elliptical flow for LCP’s
collectively for projectile as well as target matter including mid-rapidity
region. The different lines are at different size of the rapidity bin, which
includes the participant as well as spectator matter. Figure 2: The
dependence of rapidity distribution on the transition energy. The figure is
parameterized with the straight line interpolation $Y=mX~{}+~{}C$, where m is
the slope.
In Fig.1, we present the excitation function of elliptical flow for the
different conditions mentioned in the figure, where
$Y_{c.m.}/Y_{beam}~{}=~{}Y^{red}$. The incident energy dependence of
elliptical flow is well explained in the literatureLuka055 ; Andr055 . On the
other hand, the transition energy increases with rapidity region from
$|Y^{red}|~{}\leq~{}0.1$ and $|Y^{red}|~{}\leq~{}1.5$. With the increase in
the rapidity region, dominance of the spectator matter from projectile as well
as target takes place that will further result in the dominance of the mean
field up to higher energies. After the transition energy, the collective
expansion is found to have less squeeze out with an increase in the rapidity
region. As we know, the passing time for the spectator is very less compared
to expansion time of the participant zone, leading to the decreasing effect of
the spectator shadowing on the participant zone. Due to this, the chances of
the participant to move in-plane increases with increase in the rapidity bin.
If one see carefully, no transition is observed after
$|Y^{red}|~{}\leq~{}1.1$. The inset in the figure shows interesting results:
One can have a good study of elliptical flow below and above this particular
incident energy with variation in the rapidity distribution.
Extracting the transition energy values from Fig.1, we have displayed the
rapidity distribution dependence of transition energy in Fig.2. The curve is
fitted with straight line equation $Y~{}=~{}mX~{}+~{}C$, where m is slope of
line and C is a constant. The transition energy is found to be sensitive
towards the different bins of rapidity distributions. It is observed that
transition energy is found to increase with the size of the rapidity bin for
light charged particles.
In conclusion, the transition from in-plane to out-of-plane is observed only
when the mid-rapidity region is included in the rapidity bin otherwise no
transition is observed. The transition energy is found to be strongly
dependent on the size of the rapidity bin. The transition energy is
parameterized with a straight line interpolation.
## References
* (1) J. Lukasik, G. Auger, and M. L. Begemann-Blaich et al., Phys. Lett. B 608, 223 (2005).
* (2) A. Andronic et al., Phys. Lett. B 612, 173 (2005); S. Kumar, S. Kumar, and R. K. Puri, Phys. Rev. C 81, 014611 (2010).
* (3) P. Danielewicz, Nucl. Phys. A673, 375 (2000).
* (4) C. Hartnack et al., Eur. Phys. J. A1, 151 (1998); S. Kumar, Rajni, and S. Kumar, Phys. Rev. C 82, 024610 (2010).
|
arxiv-papers
| 2010-09-28T06:39:03 |
2024-09-04T02:49:13.163383
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sanjeev Kumar and Suneel Kumar",
"submitter": "Sanjeev Kumar M.Sc.",
"url": "https://arxiv.org/abs/1009.5462"
}
|
1009.5464
|
arxiv-papers
| 2010-09-28T06:50:15 |
2024-09-04T02:49:13.166971
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sanjeev Kumar, Rajeev Chugh",
"submitter": "Sanjeev Kumar",
"url": "https://arxiv.org/abs/1009.5464"
}
|
|
1009.5469
|
# On the role of different Skyrme forces and surface corrections in exotic
cluster-decay
Narinder K. Dhiman narinder.dhiman@gmail.com Govt. Sr. Sec. School, Summer
Hill, Shimla -171005, (India)
Ishwar Dutt Department of Physics, Panjab University, Chandigarh -160014,
(India)
###### Abstract
We present cluster decay studies of 56Ni∗ formed in heavy-ion collisions using
different Skyrme forces. Our study reveals that different Skyrme forces do not
alter the transfer structure of fractional yields significantly. The cluster
decay half-lives of different clusters lies within $\pm$10% for PCM and
$\pm$15% for UFM.
Heavy-ion reactions; cluster decay
###### pacs:
25.70Jj,23.70+j,24.10-i,23.60+e.
## I Introduction
Recently, a renewed interest has emerged in nuclear physics research. This
includes low energy fusion process id , intermediate energy phenomena qmd as
well as cluster-decay and/or formation of super heavy nuclei gupta ; kp . In
the last one decade, several theoretical models have been employed in the
literature to estimate the half-life times of various exotic cluster decays of
radioactive nuclei. These outcome have also been compared with experimental
data. Most of these models applied to study exotic cluster decay can be
classified into two categories: In the first category, only barrier
penetration probabilities are considered. Such models have been labeled as
unified fission models (UFM) 7Poen ; 7Buck ; 7Sand . In the second category,
clusters are assumed to be formed well before penetration. This is done by
including the preformation probability in the calculations. These models have
been dubbed as preformed cluster models (PCM) rkg88 ; mal89 ; kum97 . In
either of these approach, one needs complete knowledge of the potential.
This problem is tackled in the literature in two different manners: One tries
to adjust various parameters of model to known experimental data 8MS ; 8CW ;
8Blocki77 ; 8Bass . Alternatively, one starts from a basic fundamental
approach free from such adjustable parameters 8Deni02 ; 8Stancu ; 8Puri92 ;
8Dob ; 8Brack ; 2Skyrme ; VB72 . It remain to be seen how particular set of
model parameters influence the cluster decay process. We plan to address this
question in this paper. We shall work out the above problem with potential
obtained from the Skyrme interactions. The Skyrme interactions are well used
to describe the fusion process at low incident energies as well as in
subthreshold, collective flow, and multifragmentation at intermediate
energies.
The Skyrme force is an effective interaction, which parameterizes the _G-
matrix_ by a zero range, density and momentum dependent ansatz. The Skyrme
force consists of two-body as well as three-body parts as 2Skyrme :
$V=\sum_{i<j}v_{ij}+\sum_{i<j<k}v_{ijk}.$ (1)
Using a short-range expansion of the two-body interaction, the matrix elements
in momentum space can be written as:
$\langle\vec{k}\mid
v_{12}\mid\vec{k}^{\prime}\rangle=t_{0}\left(1+x_{0}P_{\sigma}\right)+\frac{1}{2}t_{1}\left(k^{2}+k^{\prime
2}\right)+t_{2}\vec{k}\cdot\vec{k}^{\prime}+iW_{0}\left(\vec{\sigma}_{1}+\vec{\sigma}_{2}\right)\cdot\vec{k}\times\vec{k}^{\prime},$
(2)
where $\vec{k}$ and $\vec{k}^{\prime}$ are the relative wave vectors of the
nucleons. $P_{\sigma}$ is spin exchange operator and $\vec{\sigma}$ are Pauli
spin matrices. To deal with such interaction, it is convenient to write the
matrix elements in configuration space as:
$\displaystyle v_{12}$ $\displaystyle=$ $\displaystyle
t_{0}\left(1+x_{0}P_{\sigma}\right)\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)+\frac{1}{2}t_{1}\left[\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)k^{2}+k^{\prime
2}\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)\right]$ (3)
$\displaystyle+t_{2}\vec{k}^{\prime}\cdot\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)\vec{k}+iW_{0}\left(\vec{\sigma}_{1}+\vec{\sigma}_{2}\right)\cdot\vec{k}^{\prime}\times\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)\vec{k},$
here $\vec{k}(=(\vec{\nabla}_{1}-\vec{\nabla}_{2})/2\iota)$ denotes the
relative momentum operators acting on the right and
$\vec{k}^{\prime}(=-(\vec{\nabla}_{1}-\vec{\nabla}_{2})/2\iota)$, acting on
left, respectively.
The three-body term of the Skyrme force can be written as:
$v_{123}=t_{3}\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)\delta\left(\vec{r}_{2}-\vec{r}_{3}\right).$
(4)
For the Hartree-Fock calculations of even-even nuclei, this force is shown to
be equivalent to a two-body density dependent interaction:
$v_{12}=\frac{1}{6}t_{3}\left(1+P_{\sigma}\right)\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)\rho\left(\frac{\vec{r}_{1}+\vec{r}_{2}}{2}\right).$
(5)
The above form, Eq. (5), provides a simple phenomenological representation of
many body effects describing the way, in which the interaction between two
nucleons is influenced by the presence of others. The Skyrme interaction is an
approximate representation of the effective nucleon force which is valid only
for the low relative momentum. In Eqs. (2) to (5), we see several
constants/parameters like $t_{0}$, $t_{1}$, $t_{2}$, $t_{3}$, $x_{0}$, and
$W_{0}$ that need to be fitted. These parameters have been fitted by various
authors from time to time to get better description of various ground state
properties of nuclei 8Puri92 ; 2Skyrme ; VB72 ; shen09 . A particular set
comprising these parameters is known as _Skyrme force_. Till to-date, large
number of Skyrme forces are available in the literature shen09 . These
different Skyrme forces constituting different equation of state at
intermediate energiessk ; rk . All the conventional (i.e. with the three body
term replaced by a density dependent two body term), generalized (adjusting
the effective mass $m^{*}$ and compressibility $K$) and modified Skyrme forces
(adjusting the density parameter $t_{3}$ to fit the spectra) are unified in a
single form by Zhuo Li91 as an extended Skyrme force.
$V_{ES}=\sum_{i<j}v_{ij}.$ (6)
Our aim here is to study the role of various Skyrme forces and surface
corrections in the exotic cluster decay process. This study is still missing
in the literature.
In recent years, there have been a number of experimental and theoretical
studies Sanders99 ; Sanders88 ; Sanders91 ; Nouicer99 ; Beck01 ; Thumm01 ;
Bhatt02 ; Sanders86 ; Betts ; Sanders89 ; Sanders94 ; nkd03 ; MKS00 aimed at
understanding the decay of light compound nucleus formed through heavy-ion
reactions. In most of the reactions studied, whereas the general conclusion
about the formation probability for the compound nucleus and characteristic
features of its decay are debated in terms of either fusion-fission mechanism
Sanders99 ; Sanders91 ; mat97 , which may be considered as the emission of
complex (or intermediate mass) fragments, or a deep inelastic (DI) orbiting
shiva87 mechanism behaviour.
One of such system is the doubly magic 56Ni, which is studied by using several
entrance channels (16O + 40Ca, 32S \+ 24Mg, 28Si + 28Si) and at different
incident energies (1.5 to 2.2. times Coulomb barrier) Sanders99 ; Sanders88 ;
Sanders91 ; Nouicer99 ; Beck01 ; Thumm01 ; Bhatt02 ; Sanders86 ; Betts ;
Sanders89 ; Sanders94 . At these incident energies, the incident flux get
trapped that results in the formation of compound nucleus, which is in
addition to a significant large angle scattering cross-sections. For light
masses (A$<$44), the compound nucleus decays by the emission of light
particles and $\gamma$-rays. An experimental measure of this so called
particle evaporation residue is the compound nucleus fusion cross-section. For
heavier systems, such as 56Ni, a significant decay strength to heavier
fragments is also observed which could apparently not arise from a direct
reaction mechanism because of large mass asymmetry differences between the
entrance and exit channels. The measured angular distributions and energy
spectra are consistent with fission like decays of the respective compound
systems.
The measured mass distribution for 56Ni shows a preferential decays to
channels comprising $\alpha$-nuclei 16O, 20Ne, 24Mg and 28Si, and their
complimentary fragments Betts ; Sanders89 ; Sanders94 , independent of the
entrance channel nuclei and centre-of-mass energy $E_{cm}$. Such an
$\alpha$-structure is associated with the shell effects in the potential
energy surface of the compound nucleus Sanders89 , though these are almost
zero at the compound nucleus excitation energies involved. Such an
$\alpha$-nucleus structure in the measured mass distribution of 56Ni has its
origin in the macroscopic energy MKS00 .
Cluster decay is studied for 56Ni, when formed as an excited compound system
in heavy-ion collisions. Since 56Ni has negative $Q_{out}$, and hence stable
against both fission and cluster decay processes. However, if is is produced
in heavy-ion reactions depending on the incident energy and angular momentum,
the excited compound system could either fission, decay via cluster emissions
or results in resonance phenomenon. The negative $Q_{out}$ is different for
various exit channels and hence would decay only if it were produced with
sufficient compound nucleus excitation energy
$E^{\ast}_{CN}~{}(=E_{cm}+Q_{in})$, to compensate for negative $Q_{out}$, the
deformation energy of the fragments $E_{d}$, their total kinetic energy
($TKE$) and the total excitation energy ($TXE$), in the exit channel as:
$E^{\ast}_{CN}=\mid Q_{out}\mid+E_{d}+TKE+TXE.$ (7)
(see Fig. 1, where $E_{d}$ is neglected because the fragments are considered
to be spherical). Here $Q_{in}$ adds to the entrance channel kinetic energy
$E_{cm}$ of the incoming nuclei in their ground states.
Section II gives some details of the Skyrme energy density model and preformed
cluster model and its simplification to unified fission model. Our
calculations for the decay half-life times of 56Ni∗ compound system and a
discussion of the results are presented in Section III. Finally, the results
are summarized in Section IV.
## II Model
### II.1 Skyrme Energy Density Model
In the Skyrme Energy Density Model (SEDM), the real part of interaction
potential $V_{N}(R)$ is defined as difference between energy expectation value
$E$ of the whole system calculated at a finite distance $R$ and at infinity
8Puri92 ; VB72 .
$V_{N}\left(r\right)=E\left(r\right)-E\left(\infty\right),$ (8)
with
$E=\int H\left(\vec{r}\right)\vec{dr}.$ (9)
In this formalism, the energy density functional $H\left(\vec{r}\right)$ read
as;
$\displaystyle H(\rho,\tau,\vec{J})$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\tau+\frac{1}{2}t_{0}[(1+\frac{1}{2}x_{0})\rho^{2}-(x_{0}+\frac{1}{2})(\rho_{n}^{2}+\rho_{p}^{2})]+\frac{1}{4}(t_{1}+t_{2})\rho\tau$
(10)
$\displaystyle+\frac{1}{8}(t_{2}-t_{1})(\rho_{n}\tau_{n}+\rho_{p}\tau_{p})+\frac{1}{16}(t_{2}-3t_{1})\rho\nabla^{2}\rho$
$\displaystyle+\frac{1}{32}(3t_{1}+t_{2})(\rho_{n}\nabla^{2}\rho_{n}+\rho_{p}\nabla^{2}\rho_{p})+\frac{1}{4}t_{3}\rho_{n}\rho_{p}\rho$
$\displaystyle-\frac{1}{2}W_{0}(\rho\vec{\nabla}\cdot\vec{J}+\rho_{n}\vec{\nabla}\cdot\vec{J}_{n}+\rho_{p}\vec{\nabla}\cdot\vec{J}_{p}).$
Here $\rho=\rho_{n}+\rho_{p}$ is the nucleon density taken to be two-parameter
Fermi density and $\vec{J}=\vec{J}_{n}+\vec{J}_{p}$ is the spin density which
was generalized by Puri et al. 8Puri92 , for spin-unsaturated nuclei. The
remaining term is the kinetic energy density $\tau=\tau_{n}+\tau_{p}$. The
Coulomb effects are neglected in the above energy density functional, but will
be added explicitly. In Eq. (10), six parameters $t_{0}$, $t_{1}$, $t_{2}$,
$t_{3}$, $x_{0}$, and $W_{0}$ are fitted by different authors to obtain the
best description of the various ground state properties for a large number of
nuclei. As discussed in the introduction, these different parameterizations
have been labeled as S, SI, SII, SIII etc..
The evaluation of kinetic energy density term was done within the Thomas-Fermi
(TF) approximation which is a well known alternative to the Hartree-Fock
method. As shown by various authors gupta85 , the kinetic energy density
$\tau$ can be separated into volume term $\tau_{0}$ and surface term plus
reminder. In other words,
$\tau=\tau_{0}+\tau_{\lambda}+......$ (11)
In the first order approximation, one can limit to $\tau_{0}$ term only. The
volume term $\tau_{0}$ in this approximation is given by
$\tau_{0}=\frac{3}{5}\left(\frac{3}{2}\pi^{2}\right)^{\frac{2}{3}}\rho^{\frac{5}{3}}.$
(12)
The kinetic energy density $\tau$ gupta85 ; 2Von , after including additional
surface effects is
$\tau=\tau_{0}+\lambda\frac{\left(\vec{\nabla}\rho\right)^{2}}{\rho},$ (13)
here, $\lambda$ is a constant whose value has been a point of controversy and
different authors have suggested different values, lying between $1/36$ and
$9/36$. The above Thomas-Fermi approximation for $\tau$ reduces the dependence
of energy density $H(\vec{r})$ to nucleon density $\rho$ only. The exchange
effects due to anti-symmetrization can be assimilated to reasonable extent
when Eq. (13) is used 8Puri92 . We apply the standard Fermi mass density
distribution for nucleonic density:
$\rho_{i}\left(R\right)=\frac{\rho_{0i}}{1+\exp\left\\{\frac{R-R_{0i}}{a_{i}}\right\\}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\infty\leq
R\leq\infty$ (14)
The average central density $\rho_{0i}$ given by 8Stancu
$\rho_{0i}=\frac{3A_{i}}{4\pi
R^{3}_{0i}}\frac{1}{\left[1+\frac{\pi^{2}a^{2}_{i}}{R^{2}_{0i}}\right]},$ (15)
$R_{0i}$ and ai are, respectively, the half-density radii and surface
diffuseness parameters taken from Refs.8Puri92 ; Elton . For the details of
the model, reader is referred to Ref. 8Puri92 .
### II.2 The Preformed Cluster Model
For the cluster decay studies, we use the Preformed Cluster Model (PCM) rkg88
; mal89 ; kum97 . This model, based on the quantum mechanical fragmentation
theory 7Puri ; rkg75 ; 7SNG ; 7Maruhn74 , uses the decoupled approximation to
$\eta$\- and $R$-motions. The decay constant ($\Lambda$) in the PCM is defined
as,
$\Lambda=\nu_{0}PP_{0},\qquad\qquad\qquad\left(~{}{\rm
or}~{}~{}T_{1/2}=\frac{\ln 2}{\Lambda}~{}\right),$ (16)
here $\nu_{0}$ is the assault frequency with which the cluster hits the
barrier, $P$ is the probability of penetrating the barrier and $P_{0}$ is the
preformation probability. Thus, in contrast to the unified fission models
7Poen ; 7Buck ; 7Sand , the two fragments in PCM are considered to be formed
at a relative separation co-ordinate $R$ before the penetration of the
potential barrier with probability $P_{0}$. The Schrödinger equation in terms
of $\eta$ and $R$ coordinates as:
$H(\eta,R)\psi(\eta,R)=E\psi(\eta,R),$ (17)
The above equation can be solved in a decoupled approximation rkg88 ; mal89 ,
for which the Hamiltonian takes the form:
$H=-\frac{\hbar^{2}}{2\sqrt{B_{\eta\eta}}}\frac{\partial}{\partial\eta}\frac{1}{\sqrt{B_{\eta\eta}}}\frac{\partial}{\partial\eta}-\frac{\hbar^{2}}{2\sqrt{B_{RR}}}\frac{\partial}{\partial
R}\frac{1}{\sqrt{B_{RR}}}\frac{\partial}{\partial R}+V(\eta)+V(R).$ (18)
Since the potentials are calculated within the Strutinsky re-normalization
procedure ($V=V_{Macro}+\delta U$) by using an appropriate liquid drop model
potential $V_{Macro}$ and asymmetric two center shell model for shell
corrections $\delta U$, are nearly independent of the relative separation
coordinate $R$, $R$ can be taken as a time independent parameter. For the
Hamiltonian Eq. (18), the Schrödinger Eq. (17) can be separated in two co-
ordinates $\eta$ and $R$ as follows:
$\left[-\frac{\hbar^{2}}{2\sqrt{B_{\eta\eta}}}\frac{\partial}{\partial\eta}\frac{1}{\sqrt{B_{\eta\eta}}}\frac{\partial}{\partial\eta}+V(\eta)\right]\psi(\eta)=E_{\eta}\psi(\eta),$
(19)
and
$\left[-\frac{\hbar^{2}}{2\sqrt{B_{RR}}}\frac{\partial}{\partial
R}\frac{1}{\sqrt{B_{RR}}}\frac{\partial}{\partial
R}+V(R)\right]\psi(R)=E_{R}\psi(R),$ (20)
with $\psi(\eta,R)=\psi(\eta)\,\psi(R)$ and $E=E_{\eta}+E_{R}.$
The fragmentation potential (or collective potential energy) $V(\eta)$,
appearing in Eq. (19), is calculated as,
$V(\eta)=-\sum^{2}_{i=1}\left[V_{Macro}(A_{i},Z_{i})+\delta
U_{i}\exp\left(-\frac{T^{2}}{T_{0}^{2}}\right)\right]+\frac{Z_{1}\cdot
Z_{2}e^{2}}{R}+V_{N}(R)+V_{\ell},$ (21)
where the theoretical binding energies ($V=V_{Macro}+\delta U$) are taken from
Möller et al. mol95 . The charges $Z_{i}$ in Eq. (21) are fixed by minimizing
the potential $V(\eta_{Z})$, defined by Eq. (21) without $V_{N}(R)$ in
$\eta_{Z}$ co-ordinates. The shell corrections $\delta U$ are considered to
vanish exponentially for $E^{\ast}_{CN}\geq 60$ MeV, giving $T=1.5$ MeV. At
higher excitation energies, the shell corrections vanish completely and only
the liquid drop part of energy is present. The additional attraction due to
nuclear interaction potential $V_{N}(R)$ is calculated within SEDM potential.
The rotational energy due to angular momentum effects
$V_{\ell}~{}(=\hbar^{2}\ell(\ell+1)/2\mu R^{2})$ is not added here since its
contribution to the structure yields is shown to be small for lighter systems
7SNG . The nuclear temperature $T$ (in MeV), is related approximately to the
excitation energy $E^{\ast}_{CN}$, as:
$E^{\ast}_{CN}=\frac{1}{9}A{T}^{2}-T\qquad\qquad{(\rm in~{}MeV)}.$ (22)
The kinetic energy part of the Hamiltonian in Eq. (19) comes through the mass
parameter $B_{\eta\eta}$ which is calculated using the classical mass
parameter of Kröger and Scheid kro80 , based on the hydrodynamical flow. The
mass parameter $B_{\eta\eta}$ reads as:
$B_{\eta\eta}=\frac{AmR^{2}_{min}}{4}\left[\frac{v_{t}(1+\beta)}{v_{c}(1+\delta^{2})-1}\right],$
(23)
with
$\beta=\frac{R_{c}}{2R_{min}}\left[\frac{1}{1+\cos\theta_{1}}\left(1-\frac{R_{c}}{R_{1}}\right)+\frac{1}{1+\cos\theta_{2}}\left(1-\frac{R_{c}}{R_{2}}\right)\right],$
(24)
$\delta=\frac{1}{2R_{min}}\left[(1-\cos\theta_{1})(R_{1}-R_{c})+(1-\cos\theta_{2})(R_{2}-R_{c})\right],$
(25) $v_{c}=\pi
R^{2}_{c}R_{min},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{c}=0.4R_{2},$
(26)
and $v_{t}=v_{1}+v_{2}$, is the total conserved volume.
Solving Eq. (19) numerically, $\mid\psi(\eta)\mid^{2}$ gives the probability
of finding the mass fragmentation $\eta$ at a fixed position $R$, on the decay
path. Normalizing and scaling $\mid\psi(\eta)\mid^{2}$ to give the fractional
mass yield for each fragment in the ground state decay as:
$P_{0}(A_{i})=\mid\psi(\eta)\mid^{2}\sqrt{B_{\eta\eta}(\eta)}\left(\frac{4}{A_{i}}\right),\,\,\,\,\,\,\,\,\,(i=1~{}{\rm
or}~{}2).$ (27)
The nuclear temperature effects in Eq. (27) are also included through a
Boltzmann-like function,
$\mid\psi(\eta)\mid^{2}=\sum_{\nu=0}^{\infty}\mid\psi(\eta)\mid^{2}\exp\left(-\frac{E_{\eta}}{T}\right).$
(28)
For $R$-motion, instead of solving the stationary Schrödinger Eq. (20), the
WKB action integral was solved for the penetration probability $P$ 7Rkg94 .
For each $\eta$-value, the potential $V(R)$ is calculated by using SEDM for
$R\geq R_{d}$, with $R_{d}=R_{min}+\Delta R$ and for $R\leq R_{d}$, it is
parameterized simply as a polynomial of degree two in $R$:
$V(R)=\left\\{\begin{array}[]{ll}\mid
Q_{out}\mid+{a_{1}}(R-R_{0})+{a_{2}}(R-R_{0})^{2}&\mbox{for \quad$R_{0}\leq
R\leq R_{d}$},\\\ V_{N}(R)+Z_{1}\cdot Z_{2}e^{2}/R&\mbox{for $\quad R\geq
R_{d}$},\end{array}\right.$ (29)
where $R_{0}$ is the parent nucleus radius and $\Delta R$ is chosen for smooth
matching between the real potential and the parameterized potential (with
second-order polynomial in $R$). A typical scattering potential, calculated by
using Eq. (29) is shown in Fig. 1, with tunneling paths and the characteristic
quantities also marked. Here we choose the first (inner) turning point $R_{a}$
at the minimum configuration i.e. $R_{a}=R_{min}$ (corresponding to $V_{min}$)
with potential at this $R_{a}$-value as $V(R_{a}=R_{min})=\overline{V}_{min}$
(displayed in Fig. 1) and the outer turning point $R_{b}$ to give the
$Q_{eff}$-value of the reaction ($Q_{eff}=\mid Q_{out}\mid+TKE$) i.e.
$V(R_{b})=Q_{eff}$. This means that the transmission probability $P$ with the
de-excitation probability, $W_{i}=\exp(-bE_{i})$ taken as unity, can be
written as:
$P=P_{i}P_{b},$ (30)
where $P_{i}$ and $P_{b}$ are calculated by using WKB approximation, as:
$P_{i}=\exp\left[-\frac{2}{\hbar}\int\limits_{R_{a}}^{R_{i}}\\{2\mu[V(R)-V(R_{i})]\\}^{1/2}dR\right],$
(31)
and
$P_{b}=\exp\left[-\frac{2}{\hbar}\int\limits_{R_{i}}^{R_{b}}\\{2\mu[V(R)-Q_{eff}]\\}^{1/2}dR\right],$
(32)
here $R_{a}$ and $R_{b}$ are, respectively, the first and second turning
points. This means that the tunneling begins at $R=R_{a}~{}(=R_{min})$ and
terminates at $R=R_{b}$, with $V(R_{b})=Q_{eff}$. The integrals of Eqs. (31)
and (32) are solved analytically by parameterizing the above calculated
potential $V(R)$.
The assault frequency or the barrier impinging frequency $\nu_{0}$ in Eq.
(16), is given simply as,
$\nu_{0}=\frac{v}{R_{0}}=\frac{(2E_{2}/\mu)^{1/2}}{R_{0}},$ (33)
where $E_{2}=\frac{A_{1}}{A}Q_{eff}$ is the kinetic energy of the emitted
cluster, with $Q_{eff}$ shared between the two fragments and
$\mu=m(\frac{A_{1}A_{2}}{A})$ is the reduced mass.
The PCM can be simplified to unified fission model (UFM), if preformation
probability $P_{0}=1$ and the penetration path is straight to $Q_{eff}$-value.
## III Results and Discussions
The calculations are made in two steps: In the first steps, we studied the
role of different Skyrme forces in the cluster decay of 56Ni∗ and in the
second step, effect of surface correction term $\lambda$ is analyzed.
Fig. 1 shows the characteristic scattering potential for the cluster decay of
56Ni∗ into 16O + 40Ca channel as an illustrative example. In the exit channel
for the compound nucleus to decay, the compound nucleus excitation energy
$E_{CN}^{\ast}$ goes in compensating the negative $Q_{out}$, the total
excitation energy $TXE$ and total kinetic energy $TKE$ of the two outgoing
fragments as the effective Q-value (i.e. $TKE=Q_{eff}$ in the cluster decay
process). In addition, we plot the penetration paths for PCM and UFM. For PCM,
we begin the penetration path at $R_{a}=R_{min}$ with potential at this
$R_{a}$-value as $V(R_{a}=R_{min})=\overline{V}_{min}$ and ends at $R=R_{b}$,
corresponding to $V(R=R_{b})=Q_{eff}$, whereas for UFM, we begin at $R_{a}$
and end at $R_{b}$ both corresponding to $V(R_{a})=V(R_{b})=Q_{eff}$. We have
chosen only the case of different $Q_{eff}$ (listed in Table 1), for different
cluster decay products to satisfy the arbitrarily chosen relation
$Q_{eff}=0.4(28-\mid Q_{out}\mid)$ MeV, as it is more realistic MKS00 .
### III.1 Role of Different Skyrme Forces
Figs. 2(a) and (b) shows the fragmentation potential $V(\eta)$ and fractional
yield at $R=R_{min}$ with $V(R_{min})=\overline{V}_{min}$. The classical
hydrodynamical mass parameter $B_{\eta\eta}$ of Kröger and Scheid kro80 used
in the calculation of preformation probability. The fractional yields are
calculated within PCM at $T$ = 3.0 MeV using different Skyrme forces for
56Ni∗. From the figure, we observe that different Skyrme forces do not alter
the transfer structure of fractional yields. The Skyrme force parameters have
marginal role to play. Some variations in the absolute values are however
visible NkConf . The fine structure is not at all disturbed for different sets
of Skyrme forces.
The results for the cluster decay half-lives in 56Ni∗ are quantified by the
following quantity as:
$\left[\log T_{1/2}\right]\%=\frac{(\log T_{1/2})^{i}-(\log
T_{1/2})^{SIII}}{(\log T_{1/2})^{SIII}}\times 100,$ (34)
where $i$ stands for different sets of Skyrme force parameters and SIII for
one set of Skyrme force parameters, which is widely used. Here, the strength
parameter of surface correction is taken as zero (i.e. $\lambda=0$).
In Fig. 3(a) and (b), we display the quantified results using Eq. (34) for
$\log T_{1/2}$ within PCM and UFM models as a function of cluster mass
$A_{2}$. The role of temperature $T$ (or excitation energy $E_{CN}^{\ast}$)
enters only in the PCM via preformation probability $P_{0}$. These variation
in the cluster decay half-lives for different clusters lies within $\pm$10%
for PCM and $\pm$15% for UFM. This amount is significant once we understand
cluster decay probabilities can be measured with great accuracy in the
literature.
### III.2 Role of Strength Parameter of Surface Correction ($\lambda$)
The effect of different $\lambda$-values for the heavy-ion nuclear potential
is analyzed in Refs. 8Puri92 ; 8Puri06 , suggesting that different
$\lambda$-value, can alter the depth of the nuclear potential $V_{N}$
significantly. In Ref. 8Puri92 , it was shown that the barrier heights gets
lowered whereas the fusion barrier position shifts outward where stronger role
of $\lambda$ is taken into account. The effect of this strength parameter
$\lambda$ for additional surface effects in the decay calculations has yet not
been studied in the literature. In this subsection, we plan to study the
effect of strength parameter of surface correction on cluster decay half-lives
by taking different $\lambda$-values (equal to
$0,~{}1/36,~{}2/36,~{}3/36,~{}4/36$, and $5/36$) in SEDM for the compound
system 56Ni∗.
In Fig. 4, the scattering potential for different values of surface correction
factor $\lambda$ is plotted as a function of internuclear distance $R$. One
observes from the figure that variation in the $\lambda$-value changes the
interior part of the scattering potential thereby changing the penetration
probability.
In Fig 5(a) and (b), we show the fragmentation potential $V(\eta)$and
fractional mass distribution yield at $R=R_{min}$ with
$V(R_{min})=\overline{V}_{min}$. The fractional yields are calculated within
PCM at $T$ = 3.0 MeV using different values of surface correction factor for
56Ni∗. From figure, we observe that different values of $\lambda$ changes the
fractional yield to large extent but do not alter its transfer structure. The
fine structure is not at all disturbed for different values of surface
correction factor.
The results for the cluster decay half-lives in 56Ni∗ are quantified by the
following quantity as:
$\left[\log T_{1/2}\right]\%=\frac{(\log T_{1/2})^{i}-(\log
T_{1/2})^{\lambda=0}}{(\log T_{1/2})^{\lambda=0}}\times 100,$ (35)
where $i$ stands for different $\lambda$-values of the strength parameter of
surface correction. Skyrme force SIII is employed for these calculations. In
Fig. 6, we display the quantified results using Eq. (35) for the percentage
variation of $\log T_{1/2}$ within PCM and UFM as a function of cluster mass
$A_{2}$. The variation in the cluster decay half-lives for different clusters
lies within $\pm$10% for both PCM and UFM. Together with the effect of
different Skyrme forces, one can see that the net effect of different Skyrme
forces as well as surface corrections has sizable effect on the cluster decay
half-life times.
## IV Summary
We here reported the role of different Skyrme forces as well as surface
corrections in the cluster decay constant calculations. Our studies revealed
that the effect of different Skyrme forces on the cluster decay half-life
times is about $\pm$15%, whereas it is $\pm$10% in the case of surface
corrections.
This work was supported by a research grant from the Department of Atomic
Energy, Government of India.
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Table 1: The calculated characteristic quantities for cluster decay of 56Ni∗
compound system for fragment masses $A_{2}\geq 16$, with excitation energies
$E^{\ast}=Q_{eff}+\mid Q_{out}\mid$.
Cluster + | $\mid Q_{out}\mid$ | $Q_{eff}$ | $E^{\ast}$
---|---|---|---
Daughter | (MeV) | (MeV) | (MeV)
16O + 40Ca | -14.12 | 5.55 | 19.67
18Ne + 38Ar | -22.23 | 2.31 | 24.54
20Ne + 36Ar | -17.12 | 4.35 | 21.47
22Mg + 34S | -24.58 | 1.37 | 25.95
24Mg + 32S | -16.57 | 4.57 | 21.14
26Si + 30Si | -23.57 | 1.77 | 25.34
28Si + 28Si | -12.20 | 6.32 | 18.52
Figure 1: The scattering potential $V(R)$ (in MeV) for cluster decay of 56Ni∗
into 16O + 40Ca channel using Skyrme force SIII, with $\lambda=0$. The
distribution of compound nucleus excitation energy E${}_{CN}^{*}$ at both the
initial ($R=R_{0}$) and asymptotic ($R\to\infty$) stages and $Q$-values are
shown. The decay path for both PCM and UFM models is also displayed. Figure 2:
(a) The fragmentation potential $V(\eta)$ and (b) calculated fission mass
distribution yield with different Skyrme forces at $T$ = 3.0 MeV. Figure 3:
Percentage variation of $\log T_{1/2}$ for different Skyrme forces w.r.t. SIII
force. Figure 4: Same as Fig 1, but for different values of surface correction
factor ($\lambda$). Figure 5: Same as Fig 2, but for different values of
surface correction factor ($\lambda$). Figure 6: Same as Fig 3, but for
different values of strength parameter of surface correction ($\lambda\neq 0$)
w.r.t. surface correction ($\lambda=0$).
|
arxiv-papers
| 2010-09-28T07:02:44 |
2024-09-04T02:49:13.170584
|
{
"license": "Public Domain",
"authors": "Narinder K. Dhiman and Ishwar Dutt",
"submitter": "Ishwar Dutt",
"url": "https://arxiv.org/abs/1009.5469"
}
|
1009.5470
|
# Experimental balance energies and isospin-dependent nucleon-nucleon cross-
sections
Sanjeev Kumar Rajni Suneel Kumar suneel.kumar@thapar.edu School of Physics
and Materials Science, Thapar University, Patiala-147004, Punjab (India)
###### Abstract
The effect of different isospin-dependent cross-section on directed flow is
studied for variety of systems(for which experimental balance energies are
available) using an isospin-dependent Quantum Molecular Dynamic (IQMD) model.
We show that balance energies are sensitive towards isospin-dependent cross-
sections for light systems, while nearly no effect exist for heavier nuclei. A
reduced cross-section $\sigma=0.9\sigma_{NN}$ with stiff equation of state is
able to explain experimental balance energies in most of systems. A power law
behaviour is also given for the mass dependence of balance energy, which also
follow N/Z dependence.
###### pacs:
25.70.-z, 25.75.Ld
††preprint: APS/123-QED
## I Introduction
The heavy-ion physics branch has been renewed interest very recently. This
ranges from the fusion probabilities Puri92 to symmetry energy dependence at
intermediate energies as well as fragmentation of colliding matter Singh00 .
One observable that has been used extensively for extracting information from
heavy-ion collisions is the collective in-plane flow of various particles
Pan93 ; Sorge97 ; Ollitrauet98 ; Westfall93 ; Pak96 ; Pak97 ; Li96 ; Sood04 ;
Leiwen98 . Apart from transverse in-plane flow, one has also proposed, e.g
differential Li99 and elliptical flow Zhang99 etc.
In general, collective flow in heavy-ion collisions is affected by both the
nuclear mean field potential and nucleon-nucleon (NN) cross-sections. One
should also keep in the mind that reaction dynamics depends also on the
incident energy as well as on the impact parameter of the reaction Westfall93
; Pak96 ; Pak97 ; Li96 ; Sood04 . At low incident energies, reaction dynamics
is dominated by the attractive nuclear mean field potential which results in
deflection to negative angle. Worth mention, at these enegies the phenomenon
fusion, and cluster radioactivity are dominated RGupta93 ; Puri92 . With
increasing incident energy, repulsive nucleon-nucleon scattering becomes
important and results in reduced negative flow caused by the attractive mean
field potential. As a result, at a certain incident energy, called the balance
energy, the in-plane flow vanishes as a result of cancellation between these
two competing effects [10-14]. The composite dependence of the $E_{bal}$ on
the mean field and nucleon-nucleon cross-sections ($\sigma_{NN}$) can be
sorted out by noticing the sensitivity of $E_{bal}$ on the system size, impact
parameter as well as isospin degree of freedom of the reaction [6-10].
Experimentally, balance energy is observed for different systems ranging from
${}^{12}C+^{12}C$ to ${}^{197}Au+^{197}Au$ Westfall93 ; Pak96 ; Pak97 ;
Sullivain90 ; Cussol02 . The very accurate measurement of the $E_{bal}$ in
${}^{197}Au+^{197}Au$ Cussol02 has generated a renewed interest in the field.
Unfortunately, these studies have not provided any significant contribution of
isospin effects towards the balance energy. Later on, Pak et al. Pak97
demonstrated the isospin effect on the collective flow and balance energy at
central and peripheral geometries. These findings were limited only for
${}^{58}Fe+^{58}Fe$ and ${}^{58}Ni+^{58}Ni$ systems. Theoretically, the
disappearance of directed flow is studied using the Boltzmann Uehling
Uehlenbeck (BUU) model. Westfall93 ; Pak96 ; Li99 ; Sullivain90 and Quantum
Molecular Dynamics (QMD) model [3-5,10-21]. Different theoretical attempts
considered either a stiff or soft equation of state along with a variety of NN
cross-sections. Very recently, Puri and co-workers Sood04 , conducted a very
detailed analysis on the balance energy over entire periodic table with masses
between 24 and 394. These study shed light on various aspect of nuclear
dynamics. Unfortunately, this study along with all other studies reported in
literature are limited to Central/Semi-Central Collisions only Sood04 .
Following this work, the detailed analysis on the the semi-central and
peripheral collision is performed by the Puri and co-workers in 2010 Chugh10 .
All these studies indicated enhanced cross-section of 40-55mb with stiff
equation of state to verify the balance energy in intermediate energy heavy-
ion collisions. All these studies were independent of isospin effects. The
first study showing the isospin effects on the collective flow and balance
energy was reported by Li et al.Li96 using the isospin dependent Boltzmann
Uehling Uehlenbeck (IBUU) model, where strong dependence of isospin effects
was observed. In another contribution Zhang99 , they suggested the demand of
reduced isospin dependent cross-section($\sigma=0.88\sigma_{NN}$) to better
explain the experimental data. Chen et al.Leiwen98 studied the effect of
isospin degree of freedom on the balance energy using isospin dependent
Quantum Molecular Dynamics (IQMD) model, which was an improved version of
original QMD model Sood04 ; Aichelin91 ; Singh00 . The calculated results were
found to differ from the data at all colliding geometries. Recently, Gautam et
al. Gautam10 also studied the isospin effect on the balance energy by using
IQMD model Hartnack98 , and also compared their findings with the other
theoretical and experimental findings. They demanded to take care the Gaussian
width(L) and cross-section ($\sigma=0.88\sigma_{NN}$), while studying the
isospin effects in intermediate energy heavy-ion collisions.
From the above, it is cleared that one is demanding an enhanced constant
cross-section Sood04 in QMD model. On the other hand, it is also observed
that the reduced isospin dependent cross-section is valid for the soft as well
as for soft momentum dependent equation of state Zhang99 ; Gautam10 within
IQMD or IBUU model. The systematic concept of enhanced and reduced isospin
dependent cross-sections in the presence of hard equation of state is missing
throughout the literature. Moreover, the effect of isospin dependent cross-
sections are studied on the limited systems experimentally as well as
theoretically Pak97 ; Li96 ; Leiwen98 ; Gautam10 ; Hartnack98 . We plan to
study the effect of enhanced (30 % of $\sigma_{NN}$) as well as reduced (30 %
of $\sigma_{NN}$) isospin dependent cross-sections in the presence of hard
equation of state on the systems for which the experimental finding energy is
predicted in the literature and then will compare the results with
experimental findings. For this study, we will employ isospin dependent
Quantum Molecular Dynamics(IQMD) model which is discussed in sec. II. The
results are discussed in sec. III, followed by conclusion in sec. IV.
## II ISOSPIN-Dependent QUANTUM MOLECULAR DYNAMICS (IQMD) MODEL
The isospin-dependent quantum molecular dynamics (IQMD) Hartnack98 model
treats different charge states of nucleons, deltas and pions explicitly
Hartnack98 , as inherited from the VUU model Krus85 . The IQMD model has been
used successfully for the analysis of large number of observables from low to
relativistic energies Gautam10 . The isospin degree of freedom enters into the
calculations via symmetry potential, cross-sections and Coulomb interactions
Krus85 . The details about the elastic and inelastic cross-sections for
proton-proton and neutron-neutron collisions can be found in Ref. Hartnack98 .
These cross-sections follow the data published by particle data group (PDG)
for proton-neutron and proton-proton scattering Amsler08 . In this model,
baryons are represented by Gaussian-shaped density distributions
$f_{i}(\vec{r},\vec{p},t)=\frac{1}{\pi^{2}\hbar^{2}}\cdot
e^{-(\vec{r}-\vec{r_{i}}(t))^{2}\frac{1}{2L}}\cdot
e^{-(\vec{p}-\vec{p_{i}}(t))^{2}\frac{2L}{\hbar^{2}}}.$ (1)
Where L is the Gaussian Width. As mentioned in Ref. Hartnack98 , in IQMD the
value of Gaussian width L depends on the size of the system. This system size
dependence of L in IQMD has been introduced in order to obtain the maximum
stability of the nucleonic density profile. Therefore, in the present study,
by checking the stability, we have taken the value from 0.5L to L. Its earlier
version QMD has been very successful in explaining the multifragmentation
Gossiaux97 , temprature and density Khoa92 , flow Kumar98 , multifragments
Singh00 and particle production Huang93 .
Nucleons are initialized in a sphere with radius $R=1.12A^{1/3}$ fm, in
accordance with the liquid drop model. Each nucleon occupies a volume of
$h^{3}$, so that phase space is uniformly filled. The initial momenta are
randomly chosen between 0 and Fermi momentum($\vec{P}_{F}$). The nucleons of
target and projectile interact via two and three-body Skyrme forces, Yukawa
potential, Coloumb interactions. In addition to the use of explicit charge
states of all baryons and mesons a symmetry potential between protons and
neutrons corresponding to the Bethe- Weizsacker mass formula has been
included.
The hadrons propagate using Hamilton equations of motion:
$\frac{d\vec{r_{i}}}{dt}~{}=~{}\frac{d\it{\langle~{}H~{}\rangle}}{d{p_{i}}}~{}~{};~{}~{}\frac{d\vec{p_{i}}}{dt}~{}=~{}-\frac{d\it{\langle~{}H~{}\rangle}}{d{r_{i}}},$
(2)
with
$\displaystyle\langle~{}H~{}\rangle$ $\displaystyle=$
$\displaystyle\langle~{}T~{}\rangle+\langle~{}V~{}\rangle$ (3)
$\displaystyle=$
$\displaystyle\sum_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i}\sum_{j>i}\int
f_{i}(\vec{r},\vec{p},t)V^{\it ij}({\vec{r}^{\prime},\vec{r}})$
$\displaystyle\times
f_{j}(\vec{r}^{\prime},\vec{p}^{\prime},t)d\vec{r}d\vec{r}^{\prime}d\vec{p}d\vec{p}^{\prime}.$
The baryon-baryon potential $V^{ij}$, in the above relation, reads as:
$\displaystyle V^{ij}(\vec{r}^{\prime}-\vec{r})$ $\displaystyle=$
$\displaystyle V^{ij}_{Skyrme}+V^{ij}_{Yukawa}+V^{ij}_{Coul}+V^{ij}_{sym}$ (4)
$\displaystyle=$
$\displaystyle\left(t_{1}\delta(\vec{r}^{\prime}-\vec{r})+t_{2}\delta(\vec{r}^{\prime}-\vec{r})\rho^{\gamma-1}\left(\frac{\vec{r}^{\prime}+\vec{r}}{2}\right)\right)$
$\displaystyle+~{}t_{3}\frac{exp(|\vec{r}^{\prime}-\vec{r}|/\mu)}{(|\vec{r}^{\prime}-\vec{r}|/\mu)}~{}+~{}\frac{Z_{i}Z_{j}e^{2}}{|\vec{r}^{\prime}-\vec{r}|}$
$\displaystyle+t_{6}\frac{1}{\varrho_{0}}T_{3}^{i}T_{3}^{j}\delta(\vec{r_{i}}^{\prime}-\vec{r_{j}}).$
Here $Z_{i}$ and $Z_{j}$ denote the charges of $i^{th}$ and $j^{th}$ baryon,
and $T_{3}^{i}$, $T_{3}^{j}$ are their respective $T_{3}$ components (i.e. 1/2
for protons and -1/2 for neutrons). Meson potential consists of Coulomb
interactions only. The parameters $\mu$ and $t_{1},.....,t_{6}$ are adjusted
to the real part of the nucleonic optical potential. For the density
dependence of nucleon optical potential, standard Skyrme-type parametrization
is employed. The choice of equation of state (or compressibility) is still
controversial one. Many studies advocate softer matter, whereas, much more
believe the matter to be harder in nature Krus85 ; Mage00 . For the present
analysis, a hard (H) equation of state, has been employed along with standard
energy dependent cross-sections. Note that the relativistic effects are
neglisible at these enegies Lehmann93 .
## III Results and Discussion
We study the directed flow using a stiff equation of state along with enhanced
and reduced isospin dependent cross-sections ($\sigma$= 0.7 to 1.3
$\sigma_{NN}$), by simulating various reactions. The time evolution of
reaction is follow upto 200 fm/c. This is the time at which transverse in-
plane flow saturates for lighter as well as for heavier systems. For this
study, the reactions of ${}^{12}C_{6}~{}+~{}^{12}C_{6}$ ($\hat{b}=0.4$,
L=0.5L) where L=8.66 $fm^{2}$, ${}^{20}Ne_{10}~{}+~{}^{27}Al_{11}$
($\hat{b}=0.4$, L=0.5L), ${}^{40}Ar_{18}~{}+~{}^{45}Sc_{21}$ ($\hat{b}=0.4$,
L=0.5L), ${}^{40}Ar_{18}~{}+~{}^{51}V_{23}$ ($\hat{b}=0.3$, L=0.5L),
${}^{86}Kr_{36}~{}+~{}^{93}Nb_{41}$ ($\hat{b}=0.4$, L=0.6L),
${}^{64}Zn_{30}~{}+~{}^{58}Ni_{28}$ ($\hat{b}=2fm$, L=0.6L),
${}^{93}Nb_{41}~{}+~{}^{93}Nb_{41}$ ($\hat{b}=0.3$, L=0.7L),
${}^{129}Xe_{54}~{}+~{}^{118}Sn_{50}$ ($\hat{b}=0-3fm$, L=0.7L),
${}^{139}La_{57}~{}+~{}^{139}La_{57}$($\hat{b}=0.3$, L=0.8L),
${}^{197}Au_{79}~{}+~{}^{197}Au_{79}$($\hat{b}=2.5fm$, L=L) are simulated. The
choice of impact parameter is guided by the experimentally extracted
information Westfall93 ; Pak96 ; Pak97 ; Sullivain90 ; Cussol02 . The above
reactions were simulated between 45 and 200 MeV/nucleon using the hard
equation of state along with different isospin dependent cross-sections. We
have attempted to fit the reduced isospin dependent cross-sections in the
presence of stiff equation of state with experimental findings, as is
performed in the literature with soft equation of state with and without
momentum dependent interactions.
There are two methods in the literature used to find the balance energy Sood04
. In the first case, the balance energy is extracted from the $\langle
P_{x}/A\rangle$ Plots, where $\langle P_{x}/A\rangle$ is to plotted as a
function of rapidity distribution $Y_{c.m.}$/$Y_{beam}$, which is given as.
$Y(i)=\frac{1}{2}~{}ln\frac{E(i)+P_{z}(i)}{E(i)-P_{z}(i)}$ (5)
where E(i) and $P_{z}(i)$ are respectively, the total energy and longitudinal
momentum of $i^{th}$ particle. Naturally, the energy at which this flow passes
through zero is called balance energy. The second method is to study the
incident energy dependence of the directed transverse in-plane flow $\langle
P_{x}^{dir}\rangle$, which is defined as Sood04
$\langle P_{x}^{dir}\rangle=\frac{1}{A}\sum_{i}^{A}sign\\{Y(i)\\}P_{x}(i)$ (6)
where Y(i) is the rapidity distribution as discussed above and $P_{x}(i)$ is
the transverse momentum of the $i^{th}$ particle in x-direction. This $\langle
P_{x}^{dir}\rangle$ is defined over entire rapidity region and therefore
expected to present an easier way of measuring the in-plane flow rather than
complicated $\langle P_{x}/A\rangle$ plots. In the present study, we have
tried to study the effect of isospin dependent cross-sections on the flow or
alternatively on the balance energy by using both of the parameter and then
the detailed study is extended with later one.
Figure 1: The averaged $\langle P_{x}/A\rangle$ as function of the rapidity
distribution. Here we display the result for Kr+Nb system at different
incident energies and different isospin-dependent cross-sections. Figure 2:
As in 1, but for different system and at particular cross-section
$\sigma=0.9\sigma_{NN}$. Diffrent panels are at diffrent incident energies.
Figure 3: Time evolution of $\langle P_{x}^{dir}\rangle$ for different
systems at E=60 MeV/nucleon. The different lines in figure representing the
directed fow at different cross-sections. Figure 4: Time evolution of
$\langle P_{x}^{dir}\rangle$ for different systems at E=200MeV/nucleon. The
different lines in figure representing the directed fow at different cross-
sections. Figure 5: Energy dependence of the directed nuclear flow $\langle
P_{x}^{dir}\rangle$ for different systems. The lines have same meaning as that
in fig 3 and fig 4. Figure 6: same as in fig 5, but for different systems.
Figure 7: The shift in balance energy due to cross section as a function of
combined mass of the system. Figure 8: Balance energy as a function of
combined mass of the system. The experimental points are represented with
stars, QMD+40mb with crossed triangle and present with solid square. Figure
9: N/Z dependence of balance energy at $\sigma=0.9\sigma_{NN}$ The curve is
parametrized with power law.
In Fig. 1, we display the change in the transverse momentum $\langle
P_{x}/A\rangle$ as a function of the rapidity distribution at different
incident energies from 60 to 200 MeV/nucleon for
${}^{86}Kr_{36}~{}+~{}^{93}Nb_{41}$ systems. The different lines in the figure
are showing the variation with different cross-section values. From the
figure, we see that slope becomes less negative or more positive with increase
in the incident energy. On the other hand, with reduction in the cross-section
($\sigma_{NN}$), slope is getting more negative or less positive, while,
becoming more positive with enhanced cross-section. This indicates that we see
a change in the slope with incident energy and reduced isospin dependent
cross-section. The figure is indicating two values of balance energy i.e.
E=80MeV/nucleon (at $\sigma=0.7\sigma_{NN}$) and around E=60MeV/nucleon (at
$\sigma=0.9\sigma_{NN}$) for ${}^{86}Kr_{36}~{}+~{}^{93}Nb_{41}$ system. As
the experimental balance energy for ${}^{86}Kr_{36}~{}+~{}^{93}Nb_{41}$ is in
the range of 55-60 MeV/nucleon, So one is expecting to follow the whole
dynamics at $\sigma=0.9\sigma_{NN}$.
Further, the detailed analysis of rapidity distribution of transverse momentum
$\langle P_{x}/A\rangle$ for different systems at different energies with
$\sigma=0.9\sigma_{NN}$ is displayed in Fig. 2. The slope is becoming more
positive or less negative with increase in the composite mass of system,
indicating that lighter systems remain in the environment of mean field
compared to NN collisions at any given incident energy. The contribution of
the mean field the and collisions is discussed in detail in Ref. Sood04 . One
also notice that a higher incident energy is needed in lighter cases to
balance the attractive and repulsive forces. This energy is supposed to
decrease with increase in the system mass. Similar findings are also published
by Puri and co-workerSood04 . Note that their study did not take isospin
effects into consideration.
Figures 3 and 4 are displaying the time evolution of the second parameter
$\langle P_{x}^{dir}\rangle$ below (60MeV/nucleon) and above (200MeV/nucleon)
the experimental balance energy, respectively. The results in the figure are
displayed for five different systems and at reduced as well as enhanced
isospin dependent cross-sections. The figures are indicating the similar
scenerio with $\langle P_{x}^{dir}\rangle$ as is depicted with $\langle
P_{x}/A\rangle$. Below the balance energy (in Fig. 3), the directed in-plane
flow is negative during the initial phase of reaction for all the systems
under consideration. This becomes positive at sufficient high incident energy
say E=200 MeV/nucleon (in Fig. 4). These results shows that interaction among
nucleons are attractive during the initial phase of the reaction, which turns
out to be repulsive with increase in the incident energy. These interactions
remain either attractive or repulsive throughout the time evolution depends on
the incident energy, isospin dependent cross-sections as well as composite
mass of the system. It is clear from the figure that directed flow is becoming
more positive or less negative with incident energy, isospin dependent cross-
sections as well as with size of system. There is a sharp transition for each
system from negative to positive directed flow at a particular cross-section.
This particular transition is not possible for ${}^{12}C_{6}~{}+~{}^{12}C_{6}$
system indicates the requirement of other variable like momentum dependent
interaction as well as enhancement in cross-section more then 30 %. If one
compares the figs. 1-4, the same physics of balance energy elaborates with the
$\langle P_{x}/A\rangle$ as well as with $\langle P_{x}^{dir}\rangle$. Out of
these, as discussed earlier, $\langle P_{x}^{dir}\rangle$ is the simple and
more useful quantity, because it is summed over entire rapidity distribution,
that is why, $\langle P_{x}^{dir}\rangle$ is elaborated in detail for the
further study.
To study the influence of reduced as well as enhanced cross-sections on
directed flow $\langle P_{x}^{dir}\rangle$ or alternatively on the balance
energy, in Figs 5 and 6, incident energy dependence of directed flow is
displayed for different systems. The different lines in figure represent the
variation with different cross-sections. The studies with enhanced and reduced
cross-sections are also available in the literature Sood04 ; Zhang99 ;
Gautam10 . The experimental data are represented by stars. The directed flow
goes from negative to positive value with increase in the incident energy.
This is the general trend and is explained many times in the litrature by
taking the concept of mean field and NN cross-sections. On the other hand, the
role of different cross-sections is consistent through the present mass range.
By finding the evidence of reduced cross-sections from Fig. 5, the results are
displayed between (0.7-0.9 $\sigma_{NN}$)values in Fig. 6. The enhanced cross-
section (1.3 $\sigma_{NN}$) gives more positive value followed by the cugnon
cross-section ($\sigma_{NN}$) towards the reduced cross-section (0.7
$\sigma_{NN}$). In other words, with increase in the cross-section value from
(0.7$\sigma_{NN}$-1.3$\sigma_{NN}$), the directed flow is becoming more
positive or less negative. This is due to the reason that with increase in the
cross-section value, probability of reaction to take place increases that
further results increase in the NN collisions and hence more positive value of
the directed flow. This is resulting decrease in the balance energy. The
balance energy is also found to decrease with increase in the composite mass
of the system. This is due to dominance of Coulomb repulsion with an increase
in the composite mass of system. Except for some lighter systems, the cross-
section $\sigma=0.9\sigma_{NN}$ is found to explain the experimental balance
energy nicely. Similar parametrization was also performed by Sood et al.
Sood04 within QMD model. They also found that enhanced cross-section
($\sigma=40mb$) can best explain the data. In contrary, calculation in IQMD
model demand reduced value of cross-sections. The difference is due to the
additional effect of isospin dependent cross-sections in IQMD model Hartnack98
, which were absent in QMD model. As in QMD model, the strength of nn, pp, pp
cross-section is taken equal, while in IQMD,
$\sigma_{np}~{}=~{}3\sigma_{pp}\approx 3\sigma_{nn}$ Hartnack98 . Due to the
different strength of np, pp, nn cross-section in IQMD, additional repulsion
is produced compared to QMD model. This addition repulsion will force the
directed flow to take earlier transition from negative to positive value and
hence will lower the balance energy in IQMD model as compared to QMD for same
cross-section value. That is why, the balance energy that was obtained with
QMD at $\sigma=40mb$ are at $\sigma=0.9\sigma_{NN}$ in IQMD model. This is
first ever parametrization of balance energy with hard equation of state in
the presence of reduced cross-sections with experimental available balance
energy.
By taking the Ref. Sood04 into account, which depicts, that for heavier
colliding nuclei $E_{bal}$ is independent of the cross-section one is
choosing, we have plotted in Fig. 7 the $\Delta E_{bal}$ =
$(E_{bal})_{0.7\sigma_{NN}}-(E_{bal})_{0.9\sigma_{NN}}$ with composite mass of
system. Our findings are also supporting the findings of Ref. Westfall93 ;
Li99 . $\Delta E_{bal}$ is maximum for lighter systems and it goes on
decreasing with system mass. It is also indicating theoretically that balance
energy is almost independent of the nucleon-nucleon cross-section for the
heavier system such as Au+Au, U+U etc.
In Fig. 8, we display the energy of vanishing flow or balance
energy($E_{bal}$) as a function of composite mass of system that ranges from
${}^{40}Ar_{18}+^{45}Sc_{21}$ to ${}^{197}Au_{79}+^{197}Au_{79}$. In this
figure, $E_{bal}$ is showed for the experimental data (open stars), QMD+40mb
(crossed triangle) and IQMD+0.9 $\sigma_{NN}$ (solid squares). All the curves
are fitted with power law of the form $C(A_{TOT})^{\tau}$. The experimental
data are fitted by $\tau=-0.33\pm 0.06$ The balance energy is found to
decrease with the composite mass of the system, which is a well known trend
discussed many times in literature Sood04 . The difference is in the $\tau$
values obtained by different theoretical model. The BUU model report $\tau$
between $-0.28\leq\tau^{th}\leq-0.32$. In another study Westfall93 again with
BUU model $\tau^{th}=-0.41\pm 0.03$. The present calculation depicts the
$\tau$ value $(-0.29\pm 0.06)$, which is close to the experimental $\tau$
value $(-0.33\pm 0.06)$ as compared to QMD+40mb calculation having $\tau$
value $(-0.27\pm 0.17)$. In other words, the present IQMD model with a stiff
equation of state along with $\sigma=0.9\sigma_{NN}$ can explain the data much
better than any other theoretical calculations. The $\sigma=0.9\sigma_{NN}$
explains the data for all nuclei, except for some lighter nuclei. The lighter
nuclei, when checked out, demand for an enhanced cross-sections Sood04 ;
Kumar10 along with momentum dependent interactions Gossiaux97 . Our
calculations about the strength of reduced NN cross-section is in agreement
with earlier calculation, where disappearance of transverse in-plane flow
Gautam10 as well as elliptical flow is parametrized with experimental data
Zhang99 .
We have also tried to fit the balance energy in terms of other parameter such
as the neutron to proton ratio of colliding nuclei. This attempt is shown in
Fig. 9, where balance energy is plotted as a function of N/Z. The $E_{bal}$ is
parametrized with power law of the form $(N/Z)^{\tau}$. The $\tau$ value in
N/Z dependence is $-2.39\pm 0.40$, while in $A_{TOT}$ dependence in Fig. 8 is
$-0.29\pm 0.06$. The $\tau$ value in this case is larger compared to the mass
dependence. The difference in the slopes may be due to different charge to
mass ratio in heavier colliding nuclei.
## IV Conclusion
By using the IQMD model, we have studied the effect of reduced as well as
enhanced isospin dependent cross-sections on the directed flow and balance
energy. A large number of reactions were studied having mass range from 24 to
394, where experimental balance energy is available. Our calculation with
stiff equation of state and reduced cross-section ($\sigma=0.9\sigma_{NN}$)
are in good agreement with the experimental findings, except for
${}^{12}C_{6}+^{12}C_{6}$. The dependence of isospin dependent cross-sections
get weakens with increase in the size of system. The balance energy is
parametrized with N/Z ratio in terms of power law, which is to be quite
similar with the parametrization of composite mass of system, but the $\tau$
values are different in both of the cases. One could try the balance energy
prediction with enhanced isospin dependent cross-section in the presence of
momentum dependent interaction for ${}^{12}C_{6}+^{12}C_{6}$, which is earlier
studied by Sood et al. Sood04 by using QMD in the presence of momentum
dependent interactions.
###### Acknowledgements.
This work has been supported by the Grant no. 03(1062)06/ EMR-II, from the
Council of Scientific and Industrial Research (CSIR) New Delhi, Govt. of
India.
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|
arxiv-papers
| 2010-09-28T07:03:36 |
2024-09-04T02:49:13.177094
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sanjeev Kumar, Rajni, and Suneel Kumar",
"submitter": "Sanjeev Kumar M.Sc.",
"url": "https://arxiv.org/abs/1009.5470"
}
|
1009.5546
|
11institutetext: Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1,
D-85748 Garching, Germany 22institutetext: Ludwig-Maximilians-Universität
München,
Am Coulombwall 1,D-85748 Garching, Germany
22email: justin.gagnon@mpq.mpg.de
# The direct evaluation of attosecond chirp from a streaking measurement
Justin Gagnon 1122 Vladislav S. Yakovlev 1122
(Received: date / Revised version: date)
###### Abstract
We derive an analytical expression that relates the breadth of a streaked
photoelectron spectrum to the group-delay dispersion of an isolated attosecond
pulse. Based on this analytical expression, we introduce a simple, efficient
and robust procedure to instantly extract the attosecond pulse’s chirp from
the streaking measurement. We show that our method is robust against
experimental artifacts.
###### Keywords:
attosecond – streaking – electron – trajectory
††journal: ar$\chi$iv
## 1 Introduction
The characterization of isolated attosecond pulses has played an important
role in the development of attosecond science Hentschel:2001 ; Baltuska:2003 ;
Krausz:2009 . The generation and application of ever shorter attosecond
extreme-ultraviolet (XUV) pulses Nisoli:2006 ; Goulielmakis:2008 relies on
knowledge of their time-domain properties, which can be obtained by means of
attosecond streaking measurements Itatani:2002 ; Kienberger:2004 .
So far, the main functions of attosecond streaking are (i) to characterize the
field of an attosecond pulse and (ii) to temporally resolve a physical process
on the attosecond scale. Here, we are concerned with the former application of
attosecond streaking, that of characterizing an attosecond pulse. Much effort
has been exerted on the development of methods for extracting physical
information from the streaking measurement Mairesse:2005 ; Ge:2008 ;
Chini:2010 , with the current state-of-the-art being the FROG retrieval
algorithm Trebino:1997 . The FROG algorithm has already been used to
characterize the shortest attosecond pulses Goulielmakis:2008 , and to uncover
a measured delay of $20\,\textrm{as}$ between photoemissions from the $2s$ and
$2p$ sub-shells of neon Schultze:2010 . It is relatively robust Gagnon:2009 ,
and provides a wealth of information about the temporal characteristics of the
attosecond and laser fields Gagnon:2008 .
However, the application of FROG to attosecond streaking requires quite
stringent experimental requirements, such as a sufficient amount of recorded
spectra, with a delay step between them on the order of the attosecond pulse’s
duration. These experimental parameters become unwieldy as the duration of
attosecond pulses approaches the atomic unit of time. Moreover, the FROG
algorithm is a somewhat complicated numerical optimization procedure, whose
output (the attosecond field and the laser field) is not transparently related
to the input (the set of streaked spectra). Thus, errors in the reconstructed
pulses are difficult to interpret due to the FROG algorithm’s black-box
nature. Although FROG provides a complete characterization of the attosecond
XUV field, the _duration_ of the attosecond pulse is the primordial quantity
that will be interrogated as attosecond streaking continues to expand beyond
its original scope into various research fields.
In this article, we introduce a simple and robust method for quantifying the
chirp of an attosecond pulse based on an analytical formula we derive from
laser-dressed photoelectron trajectories. Using this formula, we develop a
method that _directly_ evaluates the attosecond pulse’s group-delay dispersion
from a sequence of streaked spectra, which in turn sets the pulse’s duration
provided its spectrum is known. Our method avoids the stringent experimental
conditions required for the attosecond FROG technique, and provides accurate
results with very few electron spectra in a matter of seconds. We begin this
article with the derivation of the analytical expression for the change in
photoelectron bandwidth due to the streaking effect, and then introduce our
method with a numerical example. All quantities are expressed in atomic units
unless otherwise stated.
## 2 Classical electron trajectory analysis of the streaking effect
Let us first consider an attosecond XUV pulse with electric field
$F_{\mathrm{X}}(t)$ given by
$\displaystyle F_{\mathrm{X}}(t)$
$\displaystyle=|F_{\mathrm{X}}(t)|\mathrm{e}^{\mathrm{i}\left(\Omega_{\mathrm{X}}t+\varphi_{\mathrm{X}}(t)\right)},$
(1a) $\displaystyle\varphi_{\mathrm{X}}(t)$
$\displaystyle=\frac{1}{2}\beta_{1}t^{2}+\frac{1}{6}\beta_{2}t^{3}+\ldots,$
(1b)
where the spectrum of the attosecond pulse is centered at
$\Omega_{\mathrm{X}}$ with small variations in frequency due to the higher-
order temporal phase $\varphi_{\mathrm{X}}(t)$. The attosecond pulse launches
electron trajectories that are parameterized with an initial time $t$ as well
as an electron energy $\varepsilon=p^{2}/2$. Due to the attosecond pulse’s
finite _bandwidth_ , we consider the energy $\varepsilon$ as an independent
variable, while the independent variable $t$ is a result of the finite
_duration_ of the attosecond pulse. Thus, the set of trajectories is described
by a time-energy distribution with respect to $\\{t,\varepsilon\\}$.
The final energy $\varepsilon_{\mathrm{S}}$ of an electron, launched at some
moment $t$ in a continuum permeated by a near-infrared (NIR) laser field, is
then
$\displaystyle\varepsilon_{\mathrm{S}}$
$\displaystyle=\frac{1}{2}\left(\sqrt{2\big{(}\varepsilon+\omega_{\mathrm{X}}(t)\big{)}}-A_{\mathrm{L}}(t)\right)^{2}$
(2a) $\displaystyle\approx\varepsilon-
pA_{\mathrm{L}}(t)+\frac{1}{2}A_{\mathrm{L}}^{2}(t)+\left(1-\frac{A_{\mathrm{L}}(t)}{p}\right)\omega_{\mathrm{X}}(t),$
(2b)
where we define the instantaneous frequency
$\omega_{\mathrm{X}}(t)=\dot{\varphi}_{\mathrm{X}}(t)$ due to the chirp of the
attosecond pulse, and $A_{\mathrm{L}}(t)$ is the vector potential of the laser
field. Since the change in frequency over the temporal profile of the
attosecond pulse is much smaller than the central frequency
$\Omega_{\mathrm{X}}$, the last term in (2b) is comparably small and can be
dropped, leading to the simple relation
$\varepsilon_{\mathrm{S}}\approx\varepsilon-
pA_{\mathrm{L}}(t)+A_{\mathrm{L}}^{2}(t)/2$ for the shift of the photoelectron
spectrum.
It is known Kienberger:2004 that the spectral shift alone is not sufficient
to obtain information about the attosecond pulse’s chirp because the final
energy $\varepsilon_{\mathrm{S}}$ is hardly sensitive to the temporal phase
$\varphi_{\mathrm{X}}(t)$ of the attosecond pulse. The main manifestation of
the attosecond chirp in the streaking measurement is the change in breadth of
the streaked photoelectron spectrum. To describe this effect, we interpret
(2a) as a mapping of the initial time and energy of an electron trajectory to
a final energy (e.g. measured at the detector). To describe the effect of
chirp, it is useful to consider small changes
$\mathrm{d}\varepsilon_{\mathrm{S}}$ in the final energy with respect to small
changes in the initial energy $\mathrm{d}\varepsilon$ and time $\mathrm{d}t$
of the trajectory. The total differential of (2a) is then
$\displaystyle\mathrm{d}\varepsilon_{\mathrm{S}}$
$\displaystyle\approx\left(1-\frac{A_{\mathrm{L}}(t)}{p}\right)\Big{(}\left(\beta_{\mathrm{X}}(t)+pF_{\mathrm{L}}(t)\right)\mathrm{d}t+\mathrm{d}\varepsilon\Big{)},$
(3)
where we again neglect the small terms containing $\omega_{\mathrm{X}}(t)$.
The temporal phase of the attosecond pulse appears in (3) as
$\beta_{\mathrm{X}}(t)=\ddot{\varphi}_{\mathrm{X}}(t)$, which defines the
_chirp_ of the attosecond pulse. We have also introduced the electric field of
the laser pulse $F_{\mathrm{L}}(t)=-\dot{A}_{\mathrm{L}}(t)$. Thus, the chirp
of the attosecond pulse and the electric field of the laser pulse both
influence the spread in final energies resulting from the streaking effect.
To proceed further, we interpret the effects of the NIR field on the time-
energy distribution of electron trajectories, as described by (3), in a
straightforward manner. Initial inspection of (3) shows that the NIR field
imparts an additional energy sweep of $pF_{\mathrm{L}}(t)$ to the
photoelectron, resulting in a total chirp
$\beta_{\mathrm{S}}(t)=\beta_{\mathrm{X}}(t)+pF_{\mathrm{L}}(t)$. Furthermore,
the NIR field re-scales the energy spread by a factor
$(1-A_{\mathrm{L}}(t)/p)$. As a result, both the NIR electric field
$F_{\mathrm{L}}(t)$ and the NIR vector potential $A_{\mathrm{L}}(t)$ have a
role in modifying the breadth of the photoelectron spectrum.
In order to account for the effects of the streaking field, we recall that the
attosecond electron wave packet can be viewed as a replica Itatani:2002 of
the attosecond pulse $F_{\mathrm{X}}(t)$. We model this photoelectron replica
as
$\displaystyle\chi(t)$
$\displaystyle=\mathrm{e}^{-\frac{1}{2}(t/\tau_{\mathrm{X}})^{2}}\mathrm{e}^{\mathrm{i}\left(\varepsilon_{\mathrm{C}}t+\frac{1}{2}\beta_{\mathrm{X}}t^{2}\right)},$
(4)
where $\varepsilon_{\mathrm{C}}$ is the central photoelectron energy.
Naturally, since the electron trajectories are launched by the attosecond
pulse, the duration $\tau_{\mathrm{X}}$ of the electron wave packet
Yakovlev:2010 should be nearly the same as that of the attosecond pulse; and
as $\chi(t)$ is a replica of $F_{\mathrm{X}}(t)$, its chirp
$\beta_{\mathrm{X}}$ is the same as that of the attosecond pulse. For
simplicity, we assume $\beta_{\mathrm{X}}$ to be constant and we also assume
that the attosecond pulse is shorter than any relevant time scale of the NIR
field, so that $F_{\mathrm{L}}(t)$ and $A_{\mathrm{L}}(t)$ are evaluated at
the central time $t_{0}$ of the attosecond pulse.
Now, in order to include the effects of the streaking field, we first consider
the shift of the photoelectron spectrum due to $A_{\mathrm{L}}(t_{0})$ and the
change in bandwidth due to the chirp induced by $F_{\mathrm{L}}(t_{0})$. To
this end, we modify the wave packet’s central energy
$\varepsilon_{\mathrm{C}}=p^{2}_{\mathrm{C}}/2$ and chirp $\beta_{\mathrm{X}}$
as follows:
$\displaystyle\varepsilon_{\mathrm{C}}$
$\displaystyle\longrightarrow\varepsilon_{\mathrm{S}}=\varepsilon_{\mathrm{C}}-p_{\mathrm{C}}A_{\mathrm{L}}(t_{0})+\frac{1}{2}A^{2}_{\mathrm{L}}(t_{0})$
(5a) $\displaystyle\beta_{\mathrm{X}}$
$\displaystyle\longrightarrow\beta_{\mathrm{S}}=\beta_{\mathrm{X}}(t)+p_{\mathrm{C}}F_{\mathrm{L}}(t).$
(5b)
With these substitutions, the _streaked_ photoelectron wave packet is modeled
as
$\displaystyle\chi_{\mathrm{S}}(t)$
$\displaystyle=\mathrm{e}^{-\frac{1}{2}(t/\tau_{\mathrm{X}})^{2}}\mathrm{e}^{\mathrm{i}\left(\varepsilon_{\mathrm{S}}t+\frac{1}{2}\beta_{\mathrm{S}}t^{2}\right)}.$
(6)
To obtain an expression for the bandwidth of the streaked photoelectron
spectrum, we note that the streaked photoelectron spectrum is just a Fourier-
transform of $\chi_{\mathrm{S}}(t)$ Kitzler:2002 . Since the streaked wave
packet is a Gaussian, the Fourier transform of $\chi_{\mathrm{S}}(t)$ can be
carried out analytically, yielding the following expression for the bandwidth
of the streaked spectrum:
$\displaystyle\delta_{\mathrm{S}}(t_{0})$
$\displaystyle=\frac{\delta_{\mathrm{X}}}{\eta_{\mathrm{X}}}\left(1-\frac{A_{\mathrm{L}}(t_{0})}{p_{\mathrm{C}}}\right)\sqrt{\left(\eta_{\mathrm{X}}^{(0)}\right)^{2}+\Big{(}\delta^{2}_{\mathrm{X}}\gamma_{\mathrm{S}}(t_{0})\Big{)}^{2}},$
(7a) $\displaystyle\gamma_{\mathrm{S}}(t_{0})$
$\displaystyle=\gamma_{\mathrm{X}}+\left(\frac{\eta_{\mathrm{X}}}{\delta^{2}_{\mathrm{X}}}\right)^{2}p_{\mathrm{C}}F_{\mathrm{L}}(t_{0}),$
(7b)
where $\delta_{\mathrm{X}}$ and $\gamma_{\mathrm{X}}$ represent the bandwidth
and group-delay dispersion (GDD)—defined as the second derivative of the
spectral phase—of the attosecond pulse. The quantity
$\eta_{\mathrm{X}}=\sqrt{\left(\eta_{\mathrm{X}}^{(0)}\right)^{2}+\left(\delta^{2}_{\mathrm{X}}\gamma_{\mathrm{X}}\right)^{2}}$
is the attosecond pulse’s time-bandwidth product, with a Fourier-limited time-
bandwidth product $\eta_{\mathrm{X}}^{(0)}$ ($\eta_{\mathrm{X}}^{(0)}=1/2$ for
a Gaussian spectrum). The quantities $\tau_{\mathrm{X}}$,
$\delta_{\mathrm{X}}$ and $\delta_{\mathrm{S}}(t_{0})$ are all taken as
standard deviations of their respective distributions. According to (7),
$\gamma_{\mathrm{X}}$ determines the width $\delta_{\mathrm{S}}(t_{0})$ of the
streaked spectrum as a function of $t_{0}$. Provided that the characteristics
of the field-free spectrum ($\Omega_{\mathrm{X}}$, $\delta_{\mathrm{X}}$ and
$\eta_{\mathrm{X}}^{(0)}$) as well as those of the laser field
($A_{\mathrm{L}}(t)$ and $F_{\mathrm{L}}(t)$) are known, $\gamma_{\mathrm{X}}$
remains the only free parameter.
In writing (7), we also explicitly included the energy re-scaling pre-factor
$(1-A_{\mathrm{L}}(t_{0})/p_{\mathrm{C}})$. Similar but less general
expressions for the streaked photoelectron bandwidth were previously derived
in Itatani:2002 ; Gagnon:2009 from the semi-classical expression for
streaking Kitzler:2002 . These expressions consider photoionization at the
zero-crossing of the vector potential, $A_{\mathrm{L}}(t_{0})=0$, where there
is no spectral shift but only a change in spectral bandwidth due to the NIR
field. These expressions therefore do not contain the bandwidth re-scaling
factor $(1-A_{\mathrm{L}}(t_{0})/p_{\mathrm{C}})$, which is needed to
accurately represent the bandwidth of the streaked spectra at arbitrary delay
times $t_{0}$, when the NIR field simultaneously shifts the photoelectron
spectrum and changes its bandwidth.
Although (7) was deduced assuming a Gaussian wave packet, it actually applies
to more general pulse shapes owing to the fact that the relation
$\eta_{\mathrm{X}}^{2}=\left(\eta_{\mathrm{X}}^{(0)}\right)^{2}+\delta^{4}_{\mathrm{X}}\gamma_{\mathrm{X}}^{2}$
holds for arbitrary spectra with a constant GDD (see Appendix A). The
following section presents numerical examples in further support of this
claim.
## 3 A method to directly extract the attosecond chirp from a set of streaked
photoelectron spectra
Equation (7) serves as the basis for our method to extract the attosecond
chirp from a streaking measurement. Our procedure is very straightforward: we
evaluate the first moments ($\varepsilon_{\mathrm{S}}$) of the streaked
spectra to obtain the laser field’s vector potential $A_{\mathrm{L}}(t)$,
which in turn gives us the laser’s electric field $F_{\mathrm{L}}(t)$. We also
compute a curve $\delta_{\mathrm{S}}^{(\mathrm{M})}(t_{0})$ of standard
deviations of the measured streaked spectra as a function of the XUV-NIR delay
$t_{0}$. Lastly, we find the attosecond chirp $\gamma_{\mathrm{X}}$—the only
free parameter in (7)—which minimizes the discrepancy between the widths
$\delta_{\mathrm{S}}^{(\mathrm{M})}(t_{0})$ obtained from the set of streaked
spectra and those given by the model (7). To compare these two, we define a
figure of merit
$\displaystyle M$
$\displaystyle=\frac{\sum_{j}\left(\delta_{\mathrm{S}}(t_{j})-\delta_{\mathrm{S}}^{(\mathrm{M})}(t_{j})\right)^{2}}{\sum_{j}\Big{(}\delta_{\mathrm{S}}(t_{j})\Big{)}^{2}+\sum_{j}\Big{(}\delta_{\mathrm{S}}^{(\mathrm{M})}(t_{j})\Big{)}^{2}},$
(8)
where the sums range over the XUV-NIR delays $t_{j}$. The goal of our
procedure is to find $\gamma_{\mathrm{X}}$ that best reproduces the measured
curve $\delta_{\mathrm{X}}^{(\mathrm{M})}(t_{0})$ according to model (7).
Figure 1: Panels (a) and (b) show sets of 101 streaked photoelectron spectra
evaluated by solving the TDSE, using streaking fields with $\phi_{0}=0$ and
$\phi_{0}=\pi/2$, respectively. Panel (c) shows the attosecond pulse’s
spectrum (solid line) and phase (dotted line), while panel (d) displays its
temporal intensity profile (solid line) and temporal phase (dotted line).
As an example, we consider the case of a non-Gaussian $\sim 226\,\textrm{as}$
XUV pulse. This pulse has a constant GDD of $\sim 5885\,\textrm{as}^{2}$.
However, since its spectrum (Figure 1-b) is irregular
($\eta_{\mathrm{X}}^{(0)}\approx 0.5515$), i.e. it is asymmetric and contains
some fine structure, its chirp $\beta_{\mathrm{X}}$ is time-dependent. The
streaking field is a NIR pulse given by
$\displaystyle A_{\mathrm{L}}(t)$
$\displaystyle=A_{0}\cos^{4}(t/\tau_{\mathrm{L}})\sin(\omega_{\mathrm{L}}t+\phi_{0})$
(9)
with $\tau_{\mathrm{L}}\approx 5.743\,\textrm{fs}$, yielding a
$3\,\textrm{fs}$ full width at half maximum (FWHM) duration,
$\omega_{\mathrm{L}}\approx 2.355\,\textrm{rad/fs}$ corresponding to a central
wavelength of $800\,\textrm{nm}$ and with
$A_{0}\approx-0.41915\,\textrm{a.u.}$, giving a peak intensity of
$20\,\textrm{TW}/\textrm{cm}^{2}$. For this example, we consider carrier-
envelope phase values of $\phi_{0}=0$ (Figure 1-a) and $\phi_{0}=\pi/2$
(Figure 1-b).
The simulated streaking measurements, shown in Figure 1-a and Figure 1-b, are
composed of a sequence of streaked spectra computed for different delays
between the XUV and NIR fields by propagating the time-dependent Schrödinger
equation (TDSE) using a split-step FFT scheme. The Hamiltonian is that of a
single electron in one dimension, assuming a soft-core potential with an
ionization energy $W\approx 59\,\textrm{eV}$.
The results of our analytical chirp evaluation (ACE) procedure, applied to the
spectrograms shown in Figures 1-a and 1-b, are shown in Figures 2 and 3. In
both cases, we have applied ACE to different subsets of streaked spectra, by
considering a varying number $N$ of spectra about the central delay value
$t_{0}=0$.
For the case $\phi_{0}=0$, Figure 2-a shows a false-color plot of the figure
of merit $M$ as defined in (8). Darker areas correspond to a smaller value of
$M$. When too few spectra are considered, Figure 2-a shows a local minimum
near $\gamma_{\mathrm{X}}=12\,500\,\textrm{as}^{2}$ which disappears as more
spectra ($N\gtrsim 13$) are considered. Nonetheless, Figure 2-b shows that we
recover the exact GDD (the dashed line) from the global minimum to within
$\sim 4\%$ with as few as three spectra. As $N$ increases, the global minimum
eventually stabilizes around the red dashed line representing the exact GDD,
and ACE converges nearly to the exact value
$\gamma_{\mathrm{X}}=5885\,\textrm{as}^{2}$. Figure 2-c shows that the model
(7) reproduces the correct curve $\delta_{\mathrm{S}}(t_{0})$ for the exact
GDD. In contrast, we found that the attosecond FROG retrieval Gagnon:2008
fails to converge when fewer than 25 spectra are included, for which it
recovers a GDD $\gamma_{\mathrm{X}}=5740\,\textrm{as}^{2}$.
For $\phi_{0}=\pi/2$, Figure 3-ashows that the figure of merit has only one
minimum as a function of GDD. This minimum quickly converges to the correct
GDD as more spectra are considered in the evaluation, as displayed in Figure
3-b, and is already accurate to within $0.7\%$ for $N=13$ spectra. Figure 3-c
shows that the model (7) once again reproduces the correct curve (hollow
circles) of streaked breadths for the exact GDD
$\gamma=5885\,\textrm{as}^{2}$.
The main advantage of the ACE procedure is that it requires very few spectra.
As long as $A_{\mathrm{L}}(t_{0})$ is properly sampled by the delay step
between the spectra, there is enough information for ACE to recover the GDD of
the attosecond pulse. In contrast, FROG requires the delay step to be on the
order of the attosecond pulse’s duration. To illustrate this point, we apply
ACE to a subset of the spectra shown in Figure 1-a and 1-b. Specifically, we
consider $17$ spectra over the interval $[-2\,\textrm{fs},1.84\,\textrm{fs}]$
(containing $1.5$ cycles of the streaking field), with a delay step of
$240\,\textrm{as}$ between them, i.e. a third of the original spectra in
$[-2\,\textrm{fs},2\,\textrm{fs}]$. Even with so few spectra, ACE still
recovered accurate GDD’s of $6150\,\textrm{as}^{2}$ and
$6110\,\textrm{as}^{2}$ for $\phi_{0}=0$ and $\phi_{0}=\pi/2$, respectively.
On the other hand, FROG failed to converge to anything meaningful in both
cases, most likely because the delay step was too large.
To further demonstrate ACE’s robustness against a non-Gaussian spectrum, we
consider a clipped version of the XUV spectrum shown in Figure 1-c, for which
we remove energy components above $175\,\textrm{eV}$. Experimentally, such a
sharp edge in the XUV spectrum might result from the beam’s transmission
through a metallic filter. Using the clipped XUV spectrum, we compute sets of
$101$ streaked photoelectron spectra, with the same parameters as those
displayed in Figure 1-a and 1-b. In spite of this heavy clipping, ACE recovers
GDD’s of $5940\,\textrm{as}^{2}$ and $5960\,\textrm{as}^{2}$ for $\phi_{0}=0$
and $\phi_{0}=\pi/2$, respectively. In comparison, FROG recovers a GDD of
$5620\,\textrm{as}^{2}$ for both $\phi_{0}=0$ and $\phi_{0}=\pi/2$.
As previously mentioned, these examples assume a constant GDD over an
irregular spectral distribution, resulting in a chirp $\beta_{\mathrm{X}}$
that depends on time. Since expression (7)—which is at the core of the ACE
procedure—assumes a constant chirp in time, then the chirp parameter
$\beta_{\mathrm{X}}$ is interpreted as the _average_ chirp over the attosecond
pulse’s temporal profile. Conversely, if a non-uniform GDD was considered,
then ACE would have recovered the _average_ GDD over the spectral profile.
Figure 2: The analytical chirp evaluation (ACE) is applied to the streaking
example shown in Figure 1-a. Panel (a) is a false-color logarithmic plot of
the figure of merit $M$, defined by (8), versus the number of spectra ($N$)
considered for the ACE procedure. Panel (b) plots the retrieved GDD (squares)
at the global minimum of $M$ as a function of $N$. In panels (a) and (b), the
dotted red line represents the exact GDD. Panel (c) shows the energy
$\varepsilon_{\mathrm{S}}$ (dotted line) and breadth
$\delta_{\mathrm{S}}^{(\mathrm{M})}(t_{0})$ (solid line) evaluated from the
streaked spectra. The hollow circles represent the breadths
$\delta_{\mathrm{S}}(t_{0})$ computed from (7) with the exact
$\gamma_{\mathrm{X}}=5885\,\textrm{as}^{2}$.
Figure 3: The analytical chirp evaluation (ACE) is applied to the streaking
example shown in Figure 1-b. The data shown here are presented in the same
manner as in Figure 2.
As an additional verification of ACE’s robustness, we investigate the effect
of noise in the streaked spectra. To this end, we add noise to the sets of
$101$ spectra shown in Figure 1-a and 1-b. We assume that the number of counts
$n$ in a spectral bin follows a Poisson distribution
$P(n;\mu)=\mu^{n}\mathrm{e}^{-\mu}/n!$, with an expectation value $\mu$
proportional to the spectral intensity (we set $\mu=1$ for the peak of the
spectrogram, corresponding to a very low count rate). From these
considerations, we compute the noisy spectra which are shown in Figure 4.
Figure 4: Panels (a) and (b) shows sets of streaked spectra, computed by
adding Poisson noise to those of Figures 1-a and 1-b, respectively. ACE
recovers GDD’s of $5590\,\textrm{as}^{2}$ and $6040\,\textrm{as}^{2}$ from the
spectra in panels (a) and (b), respectively.
Even under such nefarious conditions, ACE recovers accurate values of the GDD:
$5480\,\textrm{as}^{2}$ from the spectrogram shown in Figure 4-a, and
$5830\,\textrm{as}^{2}$ from the one in Figure 4-b. In comparison, FROG
recovers GDD’s of $6250\,\textrm{as}^{2}$ and $6100\,\textrm{as}^{2}$,
respectively. This example demonstrates that ACE can tolerate very noisy
spectra, and moreover that it is robust against errors in the vector potential
$A_{\mathrm{L}}(t_{0})$, as determined from the streaked spectra.
## 4 Conclusion
In conclusion, we have derived a general analytical expression (7) for the
change in spectral breadth due to the streaking effect by considering the
trajectories of a photoelectron ejected by an isolated attosecond pulse in a
laser field. We have used this equation as a basis for a method to directly
extract the attosecond chirp from a sequence of streaked spectra. In contrast
to the attosecond FROG retrieval, the ACE procedure does not require streaked
spectra to be recorded with a delay step on the order of the attosecond pulse
duration: it only requires the delay step to properly sample the streaking
field. This alleviates many of the experimental constraints related to the
current approaches to characterize isolated attosecond pulses. In addition,
the ACE procedure is simple to implement, robust against experimental
artifacts, and fast—taking seconds to execute and requiring very few
($\lesssim 10$) streaked spectra. This makes ACE ideal for real-time
diagnostics in attosecond streaking measurements.
###### Acknowledgements.
The authors are grateful for discussions with F. Krausz. This work was
supported by the Max Planck Society and the DFG Cluster of Excellence: Munich
Centre for Advanced Photonics (MAP). The final publication is available at
www.springerlink.com
## Appendix A A relation between duration, bandwidth and dispersion for
arbitrary spectra
The following is a proof of the general relation
$\displaystyle\tau^{2}$ $\displaystyle=\tau_{0}^{2}+\gamma^{2}\delta^{2}$ (10)
between the duration $\tau$, the Fourier-limited duration $\tau_{0}$, the
bandwidth $\delta$ and the group-delay dispersion (GDD) $\gamma$ for a pulse
with an arbitrary spectrum and a constant GDD; $\tau$, $\tau_{0}$ and $\delta$
are taken as standard deviations of their respective distributions.
Let us first define spectral and temporal profiles as
$\displaystyle\tilde{f}(\omega)$
$\displaystyle=\tilde{f}_{0}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}$
(11a) $\displaystyle f(t)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{f}(\omega)\mathrm{e}^{\mathrm{i}\omega
t}\mathrm{d}\omega=\mathcal{F}^{-1}[\tilde{f}(\omega)](t).$ (11b)
We assume, without lack of generality, that $\tilde{f}(\omega)$ and $f(t)$ are
centered around $\omega=0$ and $t=0$, respectively.
The duration $\tau$ is defined as the standard deviation of $f(t)$, which is
the square-root of the variance
$\displaystyle\tau^{2}$
$\displaystyle=\int_{-\infty}^{\infty}t^{2}|f(t)|^{2}\mathrm{d}t=\int_{-\infty}^{\infty}\left|-\mathrm{i}tf(t)\right|^{2}\mathrm{d}t$
(12)
$\displaystyle=\int_{-\infty}^{\infty}\left|\mathcal{F}^{-1}[\tilde{f}^{\prime}(\omega)](t)\right|^{2}\mathrm{d}t.$
In the following derivation, the prime symbol (“′”) denotes differentiation
with respect to the argument and the pulse is normalized according to
$\int_{-\infty}^{\infty}|f(t)|^{2}\mathrm{d}t=\int_{-\infty}^{\infty}|f(\omega)|^{2}\mathrm{d}\omega=1$.
Assuming $\gamma$ is frequency-independent, then (11a) implies
$\tilde{f}^{\prime}(\omega)=\tilde{f}_{0}^{\prime}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}+\mathrm{i}\gamma\tilde{f}(\omega)$.
Inserting this expression for $\tilde{f}^{\prime}(\omega)$ into the rightmost-
hand-side of (12), we obtain
$\displaystyle\tau^{2}$
$\displaystyle=\int_{-\infty}^{\infty}|I(t;\gamma)|^{2}\mathrm{d}t+\gamma^{2}\int_{-\infty}^{\infty}|f^{\prime}(t)|^{2}\mathrm{d}t$
(13a)
$\displaystyle+\gamma\int_{-\infty}^{\infty}\big{(}I^{*}(t;\gamma)f^{\prime}(t)+I(t;\gamma){f^{\prime}}^{*}(t)\big{)}\mathrm{d}t,$
$\displaystyle I(t;\gamma)$
$\displaystyle=\mathcal{F}^{-1}[\tilde{f}^{\prime}_{0}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}](t).$
(13b)
In analogy to (12), the bandwidth-limited duration is given by
$\displaystyle\tau_{0}^{2}$
$\displaystyle=\int_{-\infty}^{\infty}\left|\mathcal{F}^{-1}[\tilde{f}_{0}^{\prime}(\omega)](t)\right|^{2}\mathrm{d}t.$
(14)
Now, $I(t;\gamma)$ and
$\tilde{f}^{\prime}_{0}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}$
are Fourier transforms of each other. Thus, from Parseval’s theorem, we have
$\displaystyle\int_{-\infty}^{\infty}|I(t;\gamma)|^{2}\mathrm{d}t$
$\displaystyle=\int_{-\infty}^{\infty}\left|\tilde{f}^{\prime}_{0}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}\right|^{2}\mathrm{d}\omega$
(15)
$\displaystyle=\int_{-\infty}^{\infty}\left|\tilde{f}^{\prime}_{0}(\omega)\right|^{2}\mathrm{d}\omega=\tau_{0}^{2},$
where (14) in combination with Parseval’s theorem was used for the last
equation on the RHS of (15).
The bandwidth $\delta$ is given, also in analogy to (12), as
$\displaystyle\delta^{2}$
$\displaystyle=\int_{-\infty}^{\infty}\omega^{2}|\tilde{f}(\omega)|^{2}\mathrm{d}\omega=\int_{-\infty}^{\infty}\left|\mathrm{i}\omega\tilde{f}(\omega)\right|^{2}\mathrm{d}\omega.$
(16)
Since $\mathrm{i}\omega\tilde{f}(\omega)$ and $f^{\prime}(t)$ are Fourier
transforms of each other, then from Parseval’s theorem,
$\displaystyle\delta^{2}$
$\displaystyle=\int_{-\infty}^{\infty}\left|\mathrm{i}\omega\tilde{f}(\omega)\right|^{2}\mathrm{d}\omega=\int_{-\infty}^{\infty}|f^{\prime}(t)|^{2}\mathrm{d}t.$
(17)
Using (15) and (17), we may now represent the duration $\tau$ as
$\displaystyle\tau^{2}$
$\displaystyle=\tau_{0}^{2}+\gamma^{2}\delta^{2}+2\gamma\int_{-\infty}^{\infty}\mathfrak{R}[I^{*}(t;\gamma)f^{\prime}(t)]\mathrm{d}t$
(18a)
$\displaystyle=\tau_{0}^{2}+\gamma^{2}\delta^{2}+2\gamma\int_{-\infty}^{\infty}\mathfrak{I}[\omega\tilde{f}^{\prime}_{0}(\omega)\tilde{f}^{*}_{0}(\omega)]\mathrm{d}\omega,$
(18b)
where we have used
$I(t;\gamma)=\mathcal{F}^{-1}[\tilde{f}^{\prime}_{0}(\omega)\mathrm{e}^{\frac{\mathrm{i}}{2}\gamma\omega^{2}}](t)$
and $f^{\prime}(t)=\mathcal{F}^{-1}[\mathrm{i}\omega\tilde{f}(\omega)](t)$ to
obtain (18b). Now, if the pulse’s GDD is constant over its spectrum,
$\tilde{f}_{0}(\omega)$ is a strictly _real_ quantity, and therefore the last
term on the RHS of (18b) is equal to zero, yielding (10). $\square$
Relation (10) is the reason why the ACE procedure can be applied for arbitrary
XUV spectra (of course, provided that the attosecond pulse is short compared
to the half-period of the streaking field).
## References
* (1) M Hentschel et al. Attosecond metrology. Nature, 414(6863):509–513, 2001.
* (2) A Baltus̆ka et al. Attosecond control of electronic processes by intense light fields. Nature, 421(6923):611–615, 2003.
* (3) F Krausz and M Y Ivanov. Attosecond physics. Rev. Mod. Phys., 81(1):163–234, 2009.
* (4) G Sansone et al. Isolated single-cycle attosecond pulses. Science, 314:443–446, 2006.
* (5) E Goulielmakis et al. Single-cycle nonlinear optics. Science, 320(5883):1614–1617, 2008.
* (6) J Itatani et al. Attosecond streak camera. Phys. Rev. Lett., 88:173903, 2002.
* (7) R Kienberger et al. Atomic transient recorder. Nature, 427(6977):817–821, 2004.
* (8) Y Mairesse and F Quéré. Frequency-resolved optical gating for complete reconstruction of attosecond bursts. Phys. Rev. A, 71:011401, 2005.
* (9) Yucheng Ge. Quantum enhancement in laser-assisted photoionization and a method for the measurement of an attosecond xuv pulse. Phys. Rev. A, 77(3):033851, 2008.
* (10) M Chini et al. Characterizing ultrabroadband attosecond lasers. Opt. Express, 18(12):13006–13016, 2010.
* (11) R Trebino et al. Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating. Rev. Sci. Instrum., 68:3277–3295, 1997.
* (12) M Schultze et al. Delay in photoemission. Science, 328(5986):1658–1662, 2010.
* (13) J Gagnon and V S Yakovlev. The robustness of attosecond streaking measurements. Opt. Express, 17(20):17678–17693, 2009.
* (14) J Gagnon et al. The accurate frog characterization of attosecond pulses from streaking measurements. Appl. Phys. B, Lasers and optics, 92(1):25–32, 2008.
* (15) V S Yakovlev et al. Attosecond streaking enables the measurement of quantum phase. Phys. Rev. Lett., 105(7):073001, 2010.
* (16) M Kitzler et al. Quantum theory of attosecond xuv pulse measurement by laser dressed photoionization. Phys. Rev. Lett., 88:173904, 2002.
|
arxiv-papers
| 2010-09-28T12:08:34 |
2024-09-04T02:49:13.185385
|
{
"license": "Public Domain",
"authors": "Justin Gagnon, Vladislav S. Yakovlev",
"submitter": "Justin Gagnon",
"url": "https://arxiv.org/abs/1009.5546"
}
|
1009.5646
|
# Stanley conjecture on intersections of four monomial prime ideals
Dorin Popescu Dorin Popescu, Institute of Mathematics ”Simion Stoilow”,
University of Bucharest, P.O.Box 1-764, Bucharest 014700, Romania
dorin.popescu@imar.ro
###### Abstract.
We show that the Stanley’s Conjecture holds for an intersection of four
monomial prime ideals of a polynomial algebra $S$ over a field and for an
arbitrary intersection of monomial prime ideals $(P_{i})_{i\in[s]}$ of $S$
such that
$P_{i}\not\subset\Sigma_{1=j\not=i}^{s}P_{j}$ for all $i\in[s]$.
Key words : Monomial Ideals, Stanley decompositions, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The Support from the CNCSIS grant PN II-542/2009 of Romanian Ministry of
Education, Research and Inovation is gratefully acknowledged.
## Introduction
Let $S=K[x_{1},\ldots,x_{n}]$, $n\in{\bf N}$ be a polynomial ring over a field
$K$. Let $I\subset S$ be a monomial ideal of $S$, $u\in I$ a monomial and
$uK[Z]$, $Z\subset\\{x_{1},\ldots,x_{n}\\}$ the linear $K$-subspace of $I$ of
all elements $uf$, $f\in K[Z]$. A presentation of $I$ as a finite direct sum
of such spaces ${\mathcal{D}}:\ I=\bigoplus_{i=1}^{r}u_{i}K[Z_{i}]$ is called
a Stanley decomposition of $I$. Set
$\operatorname{sdepth}(\mathcal{D})=\operatorname{min}\\{|Z_{i}|:i=1,\ldots,r\\}$
and
$\operatorname{sdepth}\ I:=\operatorname{max}\\{\operatorname{sdepth}\
({\mathcal{D}}):\;{\mathcal{D}}\;\text{is a Stanley decomposition of}\;I\\}.$
The Stanley’s Conjecture [10] says that $\operatorname{sdepth}\
I\geq\operatorname{depth}\ I$. This would be a nice connection between a
combinatorial invariant and a homological one. The Stanley’s Conjecture holds
for arbitrary squarefree monomial ideals if $n\leq 5$ by [7] (see especially
the arXiv version), or for intersections of three monomial prime ideals by
[6]. In the non squarefree monomial ideals a useful inequality is
$\operatorname{sdepth}I\leq\operatorname{sdepth}\sqrt{I}$ (see [3, Theorem
2.1]). In this paper we study only the case of squarefree monomial ideals $I$.
We extend the so called ”special Stanley decompositions” of [6] (see Theorem
1.5). This tool is very important because it gives lower bounds of
$\operatorname{sdepth}_{S}I$ in terms of $\operatorname{sdepth}$ of some
ideals in less variables for which we may apply induction hypothesis. More
precisely, we use it to find ”good” lower bounds of $\operatorname{sdepth}(I)$
when either $I$ is an intersection of four monomial prime ideals of $S$, or if
$I$ is an intersection of monomial prime ideals $(P_{i})_{i\in[s]}$ of $S$
such that $P_{i}\not\subset\Sigma_{1=j\not=i}^{s}P_{j}$ for all $i\in[s]$,
which shows that the Stanley’s Conjecture holds in these cases (see Theorems
2.3, 4.2). We introduce the so called the big size $t(I)$ of $I$ (usually
bigger then the size of $I$ given in [5]) and use it to find depth formulas.
If $t(I)=1$ then $\operatorname{depth}I=2$ and the Stanley’s Conjecture holds
(see Corollary 1.6). If $t(I)=2$ then we describe the possible values of
$\operatorname{depth}I$ (see Lemmas 3.2, 3.3) but we cannot show always that
the Stanley’s Conjecture holds. The problem is hinted by Example 4.3 and the
Remark 4.4.
## 1\. Big size one
Let $I=\cap_{i=1}^{s}P_{i}$, $s\geq 2$ be a reduced intersection of monomial
prime ideals of $S$. We assume that
$\Sigma_{i=1}^{s}P_{i}=m=(x_{1},\ldots,x_{n})$.
###### Definition 1.1.
Let $e$ be the minimal number such that there exists $e$-prime ideals among
$(P_{i})$ whose sum is $m$. After [5] the size of $I$ is $e-1$. We call the
big size of $I$ the minimal number $t=t(I)<s$ such that the sum of all
possible $(t+1)$-prime ideals of $\\{P_{1},\ldots,P_{s}\\}$ is $m$. In
particular, there exist $1\leq i_{1}<\ldots<i_{t}\leq s$ such that
$\Sigma_{k=1}^{t}P_{i_{k}}\not=m$ and for all
$j\in[s]\setminus\\{i_{1},\ldots,i_{t}\\}$ we have
$P_{j}+\Sigma_{k=1}^{t}P_{i_{k}}=m$. By convention we set $t(m)=0$. Clearly
the big size of $I$ is bigger than the size of $I$. We may define big size of
$I$ even when $\Sigma_{i=1}^{s}P_{i}\not=m$ adding to $t(I)$ the dimension of
$\Sigma_{i=1}^{s}P_{i}$ in $S$ (similarly to size).
###### Lemma 1.2.
Suppose that there exists $1\leq c<s$ such that $P_{i}+P_{j}=m$ for each
$c<j\leq s$ and $1\leq i\leq c$. Then $\operatorname{depth}I=2$. In
particular, if the big size of $I$ is $1$ then $\operatorname{depth}I=2$.
###### Proof.
Using the following exact sequence
$0\rightarrow S/I\rightarrow S/\cap_{i=1}^{c}P_{i}\oplus
S/\cap_{j>c}^{s}P_{j}\rightarrow
S/\cap_{i=1}^{c}\cap_{j>c}^{s}(P_{i}+P_{j})=S/m\rightarrow 0$
we get $\operatorname{depth}S/I=1$ by Depth Lemma.
###### Remark 1.3.
By [5, Proposition 2] $\operatorname{depth}_{S}S/I$ is always greater than the
size of $I$ and so if the size of $I$ is $1$ then necessary
$\operatorname{depth}_{S}I\geq 2$, the equality follows when also the big size
of $I$ is $1$.
###### Example 1.4.
Let $n=5$, $s=4$, $P_{1}=(x_{1},x_{5})$, $P_{2}=(x_{2},x_{5})$,
$P_{3}=(x_{3},x_{5})$, $P_{4}=(x_{1},x_{2},x_{3},x_{4})$. Since
$P_{1}+P_{2}+P_{3}\not=m$ the big size of $I=\cap_{i=1}^{4}P_{i}$ is $3$ but
the above lemma gives $\operatorname{depth}_{S}S/I=1$ because $P_{i}+P_{4}=m$
for all $1\leq i\leq 3$. Note that here the size of $I$ is $1$. In fact the
above lemma gives examples when $\operatorname{depth}_{S}S/I=1$ and $t(I)\geq
c$ for all positive integer $c$.
Next we want to extend [6, Proposition 2.3]. Let $r<n$ be a positive integer
and $S^{\prime}=K[x_{r+1},\ldots,x_{n}]$,
$S^{\prime\prime}=K[x_{1},\ldots,x_{r}]$. We suppose that one prime ideal
$P_{i}$ is generated in some of the first $r$ variables. If
$P_{i}=(x_{1},\ldots,x_{r})$ we say that $P_{i}$ is a main prime. For a subset
$\tau\subset[s]$ we set
$S_{\tau}=K[\\{x_{i}:1\leq i\leq r,x_{i}\not\in\Sigma_{i\in\tau}P_{i}\\}]$
and let ${\mathcal{F}}$ be the set of all nonempty subsets $\tau\subset[s]$
such that
$L_{\tau}=(\cap_{i\in\tau}P_{i})\cap S^{\prime}\not=(0),\ \
J_{\tau}=(\cap_{i\in[s]\setminus\tau}\ P_{i})\cap S_{\tau}\not=(0).$
For a $\tau\in{\mathcal{F}}$ consider the ideals $I_{0}=(I\cap
K[x_{1},\ldots,x_{r}])S$, and
$I_{\tau}=J_{\tau}S_{\tau}[x_{r+1},\ldots,x_{n}]\cap
L_{\tau}S_{\tau}[x_{r+1},\ldots,x_{n}].$
Define the integers
$A_{\tau}=\operatorname{sdepth}_{S_{\tau}[x_{r+1},\ldots,x_{n}]}I_{\tau}\geq\operatorname{sdepth}_{S_{\tau}}J_{\tau}+\operatorname{sdepth}_{S^{\prime}}L_{\tau}$
(see [8, Theorem 3.1], [6, Lemma 1.2]) and
$A_{0}=\operatorname{sdepth}_{S}I_{0}$ if $I_{0}\not=(0)$, otherwise take
$A_{0}=n$. Then
###### Theorem 1.5.
$\operatorname{sdepth}_{S}I\geq\operatorname{min}\\{A_{0},\\{A_{\tau}\\}_{\tau\in{\mathcal{F}}}\\}.$
###### Proof.
(after [6]) First we see that
$I=I_{0}\oplus(\oplus_{\tau\in{\mathcal{F}}}I_{\tau}),$
where the direct sum is of linear $K$-spaces. Let $a\in I\setminus I_{0}$ be a
monomial. We have $a=uv$, where $u\in S^{\prime\prime}$ and $v\in S^{\prime}$.
Set $\nu=\\{i\in[s]:u\not\in P_{i}\\}$. Clearly, $\nu\not=\emptyset$ because
$a\not\in I_{0}$. As $a\in I\subset P_{i}$, we get $v\in P_{i}$ for all
$i\in\nu$, that is $a\in I_{\nu}$. Since $u\not\in P_{i}$ for all $i\in\nu$ we
get $(x_{1},\ldots,x_{r})\not\subset\cap_{i\in\nu}\ P_{i}$. By the definition
of $\nu$ we have $u\in(\cap_{i\in[s]\setminus\nu}\ P_{i})\cap S_{\nu}\not=(0)$
and so $\nu\in{\mathcal{F}}$. The sum is direct because given a monomial $a\in
I\setminus I_{0}$ there exists just one $\nu\in{\mathcal{F}}$ such that $a\in
I_{\nu}$.
Now choose ”good” Stanley decompositions ${\mathcal{D}}_{0}$,
${\mathcal{D}}_{\tau}$ for $I_{0}$, respectively $I_{\tau}$ such that
$\operatorname{sdepth}_{S}{\mathcal{D}}_{0}=\operatorname{sdepth}_{S}I_{0}$,
$\operatorname{sdepth}_{S_{\tau}[x_{r+1},\ldots,x_{n}]}{\mathcal{D}}_{\tau}=\operatorname{sdepth}_{S_{\tau}[x_{r+1},\ldots,x_{n}]}I_{\tau}.$
They will induce a Stanley decomposition ${\mathcal{D}}$ of $I$ such that
$\operatorname{sdepth}_{S}I\geq\operatorname{sdepth}_{S}{\mathcal{D}}=\operatorname{min}\\{\operatorname{sdepth}_{S}I_{0},\\{\operatorname{sdepth}_{S_{\tau}[x_{r+1},\ldots,x_{n}]}I_{\tau}\\}_{\tau\in{\mathcal{F}}}\\}.$
###### Corollary 1.6.
If the big size of $I$ is $1$ then the Stanley’s Conjecture holds for $I$.
###### Proof.
It is easy to see that the corollary holds for $n\leq 2$. If $n\geq 3$ then
$\operatorname{sdepth}_{S}I\geq 2=\operatorname{depth}I$ by [1, Theorem 3.4],
which is enough as shows our Lemma 1.2. For the sake of the completeness we
give below another proof applying the above proposition.
Use induction on $s\geq 1$, the case $s=1$ being easy (see [8, Lemma 1.2]). We
may assume that $P_{1}=(x_{1},\ldots,x_{r})$ for some $r<n$. By Theorem 1.5 we
have
$\operatorname{sdepth}_{S}I\geq\operatorname{min}\\{A_{0},\\{A_{\tau_{i}}\\}_{\tau_{i}\in{\mathcal{F}}}\\},$
where $\tau_{i}=\\{i\\}$ for some $1<i\leq s$. Indeed, we have
${\mathcal{F}}\subset\\{\tau_{i}\\}_{1<i\leq s}$ because $P_{j}+P_{i}=m$ for
all $j\not=i$. The inclusion is in fact an equality because if
$P_{j}\cap K[\\{x_{e}:1\leq e\leq r,x_{e}\not\in P_{i}\\}]=(0)$
for some $1<j\not=i$ then $P_{j}\cap S^{\prime\prime}\subset P_{i}$ and so
$P_{1}\subset P_{j}$ since $P_{j}+P_{i}=m$ (contradiction). If $I_{0}\not=(0)$
then
$A_{0}=\operatorname{sdepth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})+n-r\geq 1+\dim S/P_{1}\geq
1+\operatorname{depth}S/I=\operatorname{depth}I.$
On the other hand, we have
$A_{\tau_{i}}\geq\operatorname{sdepth}_{S_{\tau_{i}}}(\cap_{j\not=i}P_{j}\cap
S_{\tau_{i}})+\operatorname{sdepth}_{S^{\prime}}(P_{i}\cap S^{\prime})\geq$
$\operatorname{depth}_{S_{\tau_{i}}}(\cap_{j\not=i}P_{j}\cap
S_{\tau_{i}})+\operatorname{depth}_{S^{\prime}}((x_{r+1},\ldots,x_{n})S^{\prime})\geq
2$
by induction hypothesis and because $P_{j}+P_{i}=m$ for all $j\not=i$. As
$\operatorname{depth}I=2$ by Lemma 1.2 we are done.
## 2\. Some results on general big size
Let $I=\cap_{i=1}^{s}P_{i}$, $s\geq 2$ be a reduced intersection of monomial
prime ideals of $S$.
###### Lemma 2.1.
If $P_{s}\not\subset\Sigma_{i=1}^{s-1}P_{i}$ then
$\operatorname{depth}I=\operatorname{min}\\{\operatorname{depth}(\cap_{i=1}^{s-1}P_{i}),1+\operatorname{depth}(\cap_{i=1}^{s-1}(P_{i}+P_{s}))\\}.$
###### Proof.
We have the following exact sequence
$0\rightarrow S/I\rightarrow S/(\cap_{i=1}^{s-1}P_{i})\oplus
S/P_{s}\rightarrow S/(\cap_{i=1}^{s-1}(P_{i}+P_{s}))\rightarrow 0$
where clearly $\operatorname{depth}S/I\leq\operatorname{depth}S/P_{s}$.
Choosing a variable $x_{i}\in P_{s}\setminus\Sigma_{i=1}^{s-1}P_{i}$ we see
that $I:x_{i}=\cap_{i=1}^{s-1}P_{i}$ and so
$\operatorname{depth}S/I\leq\operatorname{depth}S/(I:x_{i})=\operatorname{depth}S/(\cap_{i=1}^{s-1}P_{i})$
by [9, Corollary 1.3]. It follows
$\operatorname{depth}S/I=\operatorname{min}\\{\operatorname{depth}S/(\cap_{i=1}^{s-1}P_{i}),1+\operatorname{depth}S/(\cap_{i=1}^{s-1}(P_{i}+P_{s}))\\}$
using Depth Lemma (see [11, Lemma 1.3.9]), which is enough.
Next theorem uses an easy lemma of Ishaq [4, Lemma 3.1].
###### Lemma 2.2.
(Ishaq) Let $J\subset S[y]$ be a monomial ideal, $y$ being a new variable.
Then $\operatorname{sdepth}_{S}(J\cap S)\geq\operatorname{sdepth}_{S[y]}J-1$.
The following theorem extends [6, Theorem 1.4].
###### Theorem 2.3.
Let $I=\cap_{i=1}^{s}P_{i}$ be a reduced intersection of monomial prime ideals
of $S$. Assume that $P_{i}\not\subset\Sigma_{1=j\not=i}^{s}P_{j}$ for all
$i\in[s]$. Then
$\operatorname{sdepth}_{S}I\geq s=\operatorname{depth}_{S}I,$
that is the Stanley’s Conjecture holds for $I$.
###### Proof.
By [2, Lemma 3.6] it is enough to consider the case when
$\Sigma_{j=1}^{s}P_{j}=m$. Apply induction on $s$. If $s=1$ the result is
trivial. Suppose that $s>1$. We may assume that $P_{1}=(x_{1},\ldots,x_{r})$
for some $r<n$ and set $S^{\prime\prime}=K[x_{1},\ldots,x_{r}]$,
$S^{\prime}=K[x_{r+1},\ldots,x_{n}]$. By Lemma 2.1 we get
$\operatorname{depth}_{S}I=\operatorname{min}\\{\operatorname{depth}_{S}(\cap_{i>1}^{s}P_{i}),1+\operatorname{depth}_{S}(\cap_{i>1}^{s}(P_{i}+P_{1}))\\}.$
Note that $P_{i}\not\subset\Sigma_{1<j\not=i}^{s}P_{j}$ for all $1<i\leq s$
because otherwise we contradict the hypothesis. Then the induction hypothesis
gives
$\operatorname{depth}_{S}(\cap_{j>1}^{s}P_{j})=s-1+\dim
S/(\Sigma_{i>1}^{s}P_{i})\geq s.$
As $\cap_{i>1}^{s}(P_{i}+P_{1})$ satisfies also our assumption, the induction
hypothesis gives
$\operatorname{depth}_{S}(\cap_{i>1}^{s}(P_{i}+P_{1}))=s-1$. Hence
$\operatorname{depth}_{S}I=s$.
Now we apply Theorem 1.5 for ${\mathcal{F}}$ containing as usual some
$\tau\subset[s]$, $1\not\in\tau$ and we get
$\operatorname{sdepth}_{S}I\geq\operatorname{min}\\{A_{0},\\{A_{\tau}\\}_{\tau\in{\mathcal{F}}}\\}$
for $A_{0}=\operatorname{sdepth}(I\cap S^{\prime\prime})S$ if $I\cap
S^{\prime\prime}\not=0$ or $A_{0}=n$ otherwise, and
$A_{\tau}\geq\operatorname{sdepth}_{S_{\tau}}((\cap_{i\not\in\tau}P_{i})\cap
S_{\tau})+\operatorname{sdepth}_{S^{\prime}}(\cap_{i\in\tau}P_{i}\cap
S^{\prime}),$
where $S_{\tau}=K[\\{x_{i}:1\leq i\leq
r,x_{i}\not\in\Sigma_{i\in\tau}P_{i}\\}]$. Clearly $\cap_{i\in\tau}P_{i}\cap
S^{\prime}$ satisfies our assumption and so the induction hypothesis gives
$\operatorname{sdepth}_{S^{\prime}}(\cap_{i\in\tau}P_{i}\cap
S^{\prime})\geq\operatorname{depth}_{S^{\prime}}(\cap_{i\in\tau}P_{i}\cap
S^{\prime})=1+\operatorname{depth}_{S^{\prime}}S^{\prime}/(\cap_{i\in\tau}P_{i}\cap
S^{\prime})=1+$
$\operatorname{depth}_{S}S/(\cap_{i\in\tau}(P_{i}+P_{1}))\geq\operatorname{depth}_{S}S/(P_{1}\cap(\cap_{i\in\tau}P_{i}))=|\tau|+\dim
S/(P_{1}+\Sigma_{i\in\tau}P_{i}),$
by Lemma 2.1. Let
${\tilde{S}}_{\tau}=S_{\tau}[\\{x_{j}:j>r,x_{j}\not\in\Sigma_{i\in\tau}P_{i})\\}].$
Note that $(\cap_{i\not\in\tau}P_{i})\cap{\tilde{S}}_{\tau}$ satisfies our
hypothesis even $(\cap_{i\not\in\tau}P_{i})\cap S_{\tau}$ may be not. Indeed,
if $P_{i}\cap{\tilde{S}}_{\tau}\subset\Sigma_{j\not\in\tau,j\not=i}P_{j}$ for
some $i\not\in\tau$ then $P_{i}\subset\Sigma_{1=j\not=i}^{s}P_{j}$ which is
false. By Lemma 2.2 we have
$\operatorname{sdepth}_{S_{\tau}}((\cap_{i\not\in\tau}P_{i})\cap
S_{\tau})\geq\operatorname{sdepth}_{{\tilde{S}}_{\tau}}((\cap_{i\not\in\tau}P_{i})\cap{\tilde{S}}_{\tau})-|\\{i>r:x_{i}\not\in\Sigma_{j\in\tau}P_{j}\\}|\geq$
$s-|\tau|-\dim S/(P_{1}+\Sigma_{i\in\tau}P_{i}),$
using the induction hypothesis. Thus $A_{\tau}\geq s$. Finally note that if
$I\cap S^{\prime}\not=(0)$ then
$A_{0}=\operatorname{sdepth}_{S^{\prime}}(I\cap S^{\prime})+n-r\geq 1+\dim
S/P_{1}\geq 1+\operatorname{depth}_{S}S/I=\operatorname{depth}_{S}I$
using [2, Lemma 3.6].
## 3\. Depth on big size two
Let $I=\cap_{i=1}^{s}P_{i}$, $s\geq 3$ be a reduced intersection of monomial
prime ideals of $S$. Assume that $\Sigma_{i=1}^{s}P_{i}=m$ and the big size of
$I$ is two. We may suppose that $P_{1}+P_{2}=(x_{1},\ldots,x_{r})$ for some
$r<n$. We set
$q=\operatorname{min}\\{\dim S/(P_{i}+P_{j}):j\not=i,P_{i}+P_{j}\not=m\\}.$
Thus $q\leq n-r.$ Set $S^{\prime\prime}=K[x_{1},\ldots,x_{r}]$,
$S^{\prime}=K[x_{r+1},\ldots,x_{n}]$.
###### Lemma 3.1.
$\operatorname{depth}_{S}S/I\leq 1+q$.
###### Proof.
Note that for $i>2$ it holds $P_{i}\not\subset P_{1}+P_{2}$ because otherwise
$P_{1}+P_{2}=P_{1}+P_{2}+P_{i}=m$ since $t(I)=2$, which is a contradiction.
Then we may find a monomial $u\in\cap_{i>2}^{s}P_{i}\setminus(P_{1}+P_{2})$
and we have $(I:u)=P_{1}\cap P_{2}$. Thus
$\operatorname{depth}_{S}S/I\leq\operatorname{depth}_{S}S/(I:u)=\operatorname{depth}_{S}S/(P_{1}\cap
P_{2})=1+\dim S/(P_{1}+P_{2})$
by [9, Corollary 1.3]. In this way we see that
$\operatorname{depth}_{S}S/I\leq 1+\operatorname{min}\\{\dim
S/(P_{i}+P_{j}):j\not=i,P_{i}+P_{j}\not=m\\}.$
###### Lemma 3.2.
If $P_{k}+P_{e}=m$ for all $k,e>2$, $k\not=e$ then
1. (1)
$\operatorname{depth}_{S}S/I\in\\{1,2,1+q\\}$,
2. (2)
$\operatorname{depth}_{S}S/I=1$ if and only if there exists $j>2$ such that
$P_{1}+P_{j}=m=P_{2}+P_{j}$,
3. (3)
$\operatorname{depth}_{S}S/I>2$ if and only if $q>1$ and each $j>2$ satisfies
either
$P_{1}+P_{j}\not=m=P_{2}+P_{j},\ \mbox{or}$ $P_{2}+P_{j}\not=m=P_{1}+P_{j},$
4. (4)
$\operatorname{depth}_{S}S/I=2$ if and only if the following conditions hold:
1. (a)
each $j>2$ satisfies either $P_{1}+P_{j}\not=m$ or $P_{2}+P_{j}\not=m,$
2. (b)
$q=1$ or there exists a $k>2$ such that
$P_{1}+P_{k}\not=m\not=P_{2}+P_{k},$
5. (5)
$\operatorname{sdepth}_{S}I\geq\operatorname{depth}_{S}I$.
###### Proof.
Apply induction on $s+n$, $s\geq 3$. If $s=3$ then we may apply [6]. Suppose
that $s>3$. We have the following exact sequence
$0\rightarrow S/I\rightarrow S/(P_{1}\cap P_{2}\cap P_{3})\oplus S/(P_{1}\cap
P_{2}\cap P_{4}\cap\ldots\cap P_{s})\rightarrow S/(P_{1}\cap P_{2})\rightarrow
0$
because of our hypothesis. Then by Lemma 3.1 any module from the above exact
sequence has depth $\leq\operatorname{depth}_{S}S/(P_{1}\cap P_{2})$. Thus
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\operatorname{depth}_{S}S/(P_{1}\cap
P_{2}\cap P_{3}),\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap
P_{4}\cap\ldots\cap P_{s})\\}$
by Depth Lemma [11]. Using the induction hypothesis we get
$\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap P_{3})$,
$\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap P_{4}\cap\ldots\cap
P_{s})\in\\{1,2,1+q\\}$ because any three prime ideals of $(P_{i})$ have the
sum $m$. Hence (1) holds. Note that $\operatorname{depth}_{S}S/I=1$ if and
only if either $\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap P_{3})=1$, or
$\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap P_{4}\cap\ldots\cap P_{s})=1$
and (2) holds because of the induction hypothesis (see also Lemma 1.2).
Similarly, (3), (4) holds by induction hypothesis relying in fact on the case
$s=3$ stated in [6].
Now we apply Theorem 1.5 for ${\mathcal{F}}$ containing some
$\tau_{i}=\\{i\\}$, $2<i\leq s$ (note that $P_{i}+P_{j}=m$ for all $2<i<j\leq
s$ and so $\mathcal{F}$ does not contain $\tau=\\{i,j\\}$) and we get
$\operatorname{sdepth}_{S}I\geq\operatorname{min}\\{A_{0},\\{A_{\tau_{i}}\\}_{\tau_{i}\in{\mathcal{F}}}\\}$
for $A_{0}=\operatorname{sdepth}_{S}(I\cap S^{\prime\prime})S$ if $I\cap
S^{\prime\prime}\not=0$ or $A_{0}=n$ otherwise, and
$A_{\tau_{i}}\geq\operatorname{sdepth}_{S_{\tau_{i}}}((\cap_{j=1,j\not=i}^{s}P_{j})\cap
S_{\tau_{i}})+\operatorname{sdepth}_{S^{\prime}}(P_{i}\cap S^{\prime}),$
where $S_{\tau_{i}}=K[\\{x_{j}:x_{j}\in S^{\prime\prime},x_{j}\not\in
P_{i}\\}]$. Note that the big size of $J_{i}=(\cap_{j=1,j\not=i}^{s}P_{j})\cap
S_{\tau_{i}}$ is 1 or zero, because if $(P_{k}+P_{e})\cap S_{\tau_{i}}$ is not
the maximal ideal of $S_{\tau_{i}}$ for some two different $k,e$ which are not
$i$, then $P_{k}+P_{e}+P_{i}\not=m$ contradicting $t(I)=2$. By Corollary 1.6
we get
$\operatorname{sdepth}_{S_{\tau_{i}}}J_{i}\geq\operatorname{depth}_{S_{\tau_{i}}}J_{i}=1+\operatorname{depth}_{S_{\tau_{i}}}S_{\tau_{i}}/J_{i}=1+\operatorname{depth}_{S}S/(J_{i}S+P_{i}).$
Then $A_{\tau_{i}}\geq 2+\operatorname{depth}_{S}S/(J_{i}S+P_{i})$. By our
hypothesis $J_{i}S+P_{i}=(P_{1}+P_{i})\cap(P_{2}+P_{i})$. If
$P_{1}+P_{i}=m\not=P_{2}+P_{i}$ then
$\operatorname{depth}_{S}S/(J_{i}S+P_{i})=\dim S/(P_{2}+P_{i})\geq q$. Hence
$A_{\tau_{i}}\geq\operatorname{depth}_{S}I$ using (1). If
$P_{1}+P_{i}\not=m\not=P_{2}+P_{i}$ then we get $A_{\tau_{i}}\geq
3=\operatorname{depth}_{S}I$ using (4).
Suppose that $I\cap S^{\prime\prime}\not=0$. When $t(I\cap
S^{\prime\prime})=1$ we have $\operatorname{sdepth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})=2$ by Corollary 1.6 and so $A_{0}\geq 2+n-r\geq
2+q\geq\operatorname{depth}_{S}I$. If $t(I\cap S^{\prime\prime})=2$ then by
induction hypothesis (less variables) we have
$A_{0}\geq\operatorname{depth}_{S}(I\cap
S^{\prime\prime})S=n-r+\operatorname{depth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})\geq q+2\geq\operatorname{depth}_{S}I$
($I\cap S^{\prime\prime}$ cannot be the maximal ideal in $S^{\prime\prime}$).
Next we will consider the case when $t(I)=2$, but with the property:
(*) whenever there exist $i\not=j$ in $[s]$ such that $P_{i}+P_{j}\not=m$
there exist also $k\not=e$ in $[s]\setminus\\{i,j\\}$ such that
$P_{k}+P_{e}\not=m$,
that is the complementary case to the one solved by the above lemma. As above
we may suppose that $P_{1}+P_{2}\not=m$ and by (*) let suppose
$P_{s}+P_{s-1}\not=m$.
###### Lemma 3.3.
If $I$ satisfies (*) then
1. (1)
$\operatorname{depth}_{S}S/I\in\\{1,2,1+q\\}$.
2. (2)
$\operatorname{depth}_{S}S/I=1$ if and only if after a renumbering of
$(P_{i})$ there exists $1\leq c<s$ such that $P_{i}+P_{j}=m$ for each $c<j\leq
s$ and $1\leq i\leq c$.
###### Proof.
We use induction on $s\geq 3$, the case $s=3$ being done in [6]. Assume $s>3$.
Set $J=P_{1}\cap\ldots\cap P_{s-2}$. In the exact sequence:
$0\rightarrow S/I\rightarrow S/(J\cap P_{s-1})\oplus S/(J\cap
P_{s})\rightarrow S/(J\cap(P_{s-1}+P_{s}))\rightarrow 0$
we have $\operatorname{depth}_{S}S/(J\cap(P_{s-1}+P_{s}))=1$ by Lemma 1.2
because $P_{i}+P_{s-1}+P_{s}=m$ for all $i<s-1$ since $t(I)=2$. Note that
there exist no $i<s-1$ such that $P_{i}\subset P_{s-1}+P_{s}$ because
otherwise $P_{s-1}+P_{s}=P_{i}+P_{s-1}+P_{s}=m$, which is false. If
$\operatorname{depth}_{S}(S/(J\cap P_{s-1})\oplus S/(J\cap P_{s}))>1$
then $\operatorname{depth}_{S}S/I=2$. Otherwise, we may suppose that
$\operatorname{depth}_{S}(S/(J\cap P_{s-1}))=1$, where we may apply Lemma 3.2
or our induction hypothesis. Thus after a renumbering of $(P_{i})$ there
exists $1\leq k<s-1$ such that $P_{i}+P_{j}=m$ for each $k<j\leq s-1$ and
$1\leq i\leq k$. In the following exact sequence:
$0\rightarrow S/I\rightarrow S/(P_{1}\cap\ldots\cap P_{k}\cap P_{s})\oplus
S/(P_{k+1}\cap\ldots\cap P_{s})\rightarrow S/P_{s}\rightarrow 0$
all the modules have depth $\leq\operatorname{depth}_{S}S/P_{s}$. It follows
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\operatorname{depth}_{S}S/(P_{1}\cap\ldots\cap
P_{k}\cap P_{s}),\operatorname{depth}_{S}S/(P_{k+1}\cap\ldots\cap P_{s})\\}$
and applying Lemma 3.2 or our induction hypothesis we get (1). In (2) the
sufficiency follows from Lemma 1.2. If $\operatorname{depth}_{S}S/I=1$ then we
get let’ say
$\operatorname{depth}_{S}S/(P_{1}\cap\ldots\cap P_{k}\cap P_{s})=1$ and using
Lemma 3.2 and our induction hypothesis after a renumbering of $(P_{i})_{i<k}$
there exists $1\leq c\leq k$ such that $P_{i}+P_{j}=m$ for each $1\leq i\leq
c$ and $c<j\leq k$ or $j=s$. Thus using our assumptions on $k$ we get
$P_{i}+P_{j}=m$ for each $c<j\leq s$ and $1\leq i\leq c$.
## 4\. Intersections of four prime ideals
Let $I=\cap_{i=1}^{4}P_{i}$, be a reduced intersection of monomial prime
ideals of $S$. Assume that $\Sigma_{i=1}^{4}P_{i}=m$ and the big size of $I$
is two, let us say $P_{1}+P_{2}\not=m$ and $P_{1}=(x_{1},\ldots,x_{r})$,
$r<n$. Set
$q=\operatorname{min}\\{\dim S/(P_{i}+P_{j}):j\not=i,P_{i}+P_{j}\not=m\\},$
$S^{\prime\prime}=K[x_{1},\ldots,x_{r}]$,
$S^{\prime}=K[x_{r+1},\ldots,x_{n}].$
###### Proposition 4.1.
Then $\operatorname{sdepth}_{S}I\geq\operatorname{depth}_{S}I$.
###### Proof.
Using Lemma 3.2 we may suppose that $I$ satisfies (*) and in particular
$P_{3}+P_{4}\not=m$. The proof of Lemma 3.3 says that if
$\operatorname{depth}_{S}(S/(P_{1}\cap P_{2}\cap P_{3})\oplus S/(P_{1}\cap
P_{2}\cap P_{4}))>1$ then $\operatorname{depth}_{S}S/I=2$. Otherwise, let’ say
$\operatorname{depth}_{S}S/(P_{1}\cap P_{2}\cap P_{3})=1$ and it follows
$P_{1}+P_{3}=P_{2}+P_{3}=m$ by [6] since $P_{1}+P_{2}\not=m$. But (*) implies
also $P_{1}+P_{4}=P_{2}+P_{4}=m$, that is $\operatorname{depth}_{S}S/I=1$ by
Lemma 1.2.
We apply Theorem 1.5 for the main prime $P_{1}$ and ${\mathcal{F}}$ containing
only possible $\tau_{i}=\\{i\\}$, $i=2,3,4$ $\tau_{ij}=\\{i,j\\}$ for some
$1<i<j\leq 4$. We get
$\operatorname{sdepth}I\geq\operatorname{min}\\{A_{0},\\{A_{\tau}\\}_{\tau\in{\mathcal{F}}}\\}$
for $A_{0}=\operatorname{sdepth}(I\cap S^{\prime\prime})S$ if $I\cap
S^{\prime\prime}\not=0$ or $A_{0}=n$ otherwise, and
$A_{\tau_{i}}\geq\operatorname{sdepth}_{S_{\tau_{i}}}((\cap_{j=2,j\not=i}^{4}P_{j})\cap
S_{\tau_{i}})+\operatorname{sdepth}_{S^{\prime}}(P_{i}\cap S^{\prime}),$
for $i=3,4$ and
$A_{\tau_{ij}}\geq\operatorname{sdepth}_{S_{\tau_{ij}}}(P_{k}\cap
S_{\tau_{ij}})+\operatorname{sdepth}_{S^{\prime}}(P_{i}\cap P_{j}\cap
S^{\prime}),$
where $1<i<j\leq 4$, $k=[4]\setminus\\{1,i,j\\}$,
$S_{\tau_{ij}}=K[x_{j}:x_{j}\in S^{\prime\prime},x_{j}\not\in P_{i}+P_{j}]$
and $S_{\tau_{i}}=K[x_{j}:x_{j}\in S^{\prime\prime},x_{j}\not\in P_{i}]$ for
$i=2,3,4$. For example using [8, Lemma 4.3]
$A_{\tau_{4}}\geq\Sigma_{j=2}^{3}\lceil\frac{\dim
S^{\prime\prime}/((P_{j}+P_{4})\cap S^{\prime\prime})}{2}\rceil+1\geq
3\geq\operatorname{depth}_{S}I,$
if $(P_{2}+P_{4})\cap S^{\prime\prime}$ and $(P_{3}+P_{4})\cap
S^{\prime\prime}$ are not inclosed one in the other, where $\lceil a\rceil$,
$a\in{\bf Q}$ denotes the smallest integer $\geq a$. Otherwise, if $P_{2}\cap
S^{\prime\prime}\subset P_{3}+P_{4}$ then $P_{2}\cap S^{\prime}\not\subset
P_{4}$ since $P_{2}+P_{3}+P_{4}=m$ and $P_{3}+P_{4}\not=m$. Then
$A_{\tau_{4}}\geq\operatorname{sdepth}_{S_{\tau_{4}}}(P_{2}\cap
S_{\tau_{4}})+\operatorname{sdepth}_{S^{\prime}}(P_{4}\cap S^{\prime})\geq
2+\lceil\frac{\operatorname{height}(P_{4}\cap
S^{\prime})}{2}\rceil\geq\operatorname{depth}_{S}I.$
If $P_{3}\cap S^{\prime\prime}\subset P_{2}+P_{4}$ and $P_{3}\cap
S^{\prime}\not\subset P_{4}$ we proceed as above. If $P_{3}\cap
S^{\prime}\subset P_{4}$ then we get $P_{2}+P_{4}=m$ because
$P_{2}+P_{3}+P_{4}=m$. By (*) we get also $P_{1}+P_{3}=m$. It follows
$P_{3}\cap S_{\tau_{4}}$ is not maximal in $S_{\tau_{4}}$ because
$P_{3}+P_{4}\not=m$ and so
$A_{\tau_{4}}\geq\operatorname{sdepth}_{S_{\tau_{4}}}(P_{3}\cap
S_{\tau_{4}})+\operatorname{sdepth}_{S^{\prime}}(P_{4}\cap S^{\prime})\geq
2+\lceil\frac{\operatorname{height}(P_{3}\cap
S_{\tau_{4}})}{2}\rceil\geq\operatorname{depth}_{S}I.$
Now by [8, Lemma 4.3]
$A_{\tau_{34}}\geq\operatorname{sdepth}_{S_{\tau_{34}}}(P_{2}\cap
S_{\tau_{34}})+\operatorname{sdepth}_{S^{\prime}}(P_{3}\cap P_{4}\cap
S^{\prime})\geq$ $\lceil\frac{\operatorname{height}(P_{2}\cap
S_{\tau_{34}})}{2}\rceil+\lceil\frac{\dim S^{\prime}/(P_{3}\cap
S^{\prime})}{2}\rceil+\lceil\frac{\dim S^{\prime}/(P_{4}\cap
S^{\prime})}{2}\rceil\geq 3\geq\operatorname{depth}_{S}I$
if $P_{3}\cap S^{\prime}$ and $P_{4}\cap S^{\prime}$ are not inclosed one in
the other (note that $P_{2}+P_{3}+P_{4}=m$). Otherwise, if for example
$P_{3}\cap S^{\prime}\subset P_{4}$ we get $P_{1}+P_{4}=m$ because
$P_{1}+P_{3}+P_{4}=m$, and so $P_{2}+P_{3}=m$ by (*). If $P_{1}+P_{3}\not=m$
then $P_{3}\cap S^{\prime}$ is not the maximal ideal of $S^{\prime}$ and it
follows $\operatorname{sdepth}_{S^{\prime}}(P_{3}\cap S^{\prime})\geq
1+\lceil\frac{\operatorname{height}(P_{3}\cap S^{\prime})}{2}\rceil$, that is
$A_{34}\geq 3\geq\operatorname{depth}_{S}I$. If $P_{1}+P_{3}=m$ then
$P_{2}+P_{4}=m$ by (*) and so $A_{34}\geq 2\geq\operatorname{depth}_{S}I$ as
we know already. Similarly, if $\tau_{23}\in{\mathcal{F}}$ we get
$A_{\tau_{23}}\geq\operatorname{depth}I$ if $P_{2}\cap S^{\prime}\not\subset
P_{3}\cap S^{\prime}$, otherwise we see that $P_{2}\cap S^{\prime}$ is not the
maximal ideal in $S^{\prime}$ and so
$A_{\tau_{23}}\geq 2+\lceil\frac{\operatorname{height}(P_{2}\cap
S^{\prime})}{2}\rceil\geq\operatorname{depth}I.$
Since $A_{0}\geq\operatorname{depth}I$ as in Lemma 3.2 we are done.
###### Theorem 4.2.
Let $I=\cap_{i=1}^{4}P_{i}$ be a reduced intersection of four monomial prime
ideals of $S$. Then the Stanley’s Conjecture holds for $I$.
###### Proof.
By [2, Lemma 3.6] it is enough to consider the case when
$\cap_{j=1}^{4}P_{j}=m$. If $t(I)\leq 2$ then the result follows by Corollary
1.6 and Proposition 4.1. Otherwise, there exists $i\in[s]$ such that
$P_{i}\not\subset\Sigma_{1=j\not=i}^{4}P_{j}$, let us say
$P_{4}\not\subset\Sigma_{1=1}^{3}P_{j}$. Apply induction on $n$, the case
$n\leq 5$ being done in [7]. Assume
$\Sigma_{1=1}^{3}P_{j}=(x_{1},\ldots,x_{r})$ for some $r<n$. Apply Theorem 1.5
as above with $\mathcal{F}$ containing just $\tau=\\{4\\}$. We have
$A_{\tau}\geq\operatorname{sdepth}_{S_{\tau}}((\cap_{j=1}^{3}P_{j})\cap
S_{\tau})+\operatorname{sdepth}_{S^{\prime}}(P_{4}\cap
S^{\prime})\geq\operatorname{depth}_{S_{\tau}}((\cap_{j=1}^{3}P_{j})\cap
S_{\tau})+1$
by [6] and so
$A_{\tau}\geq\operatorname{depth}_{S_{\tau}}S_{\tau}/((\cap_{j=1}^{3}P_{j})\cap
S_{\tau})+2=2+\operatorname{depth}_{S}S/((\cap_{j=1}^{3}(P_{j}+P_{4}))=$
$1+\operatorname{depth}_{S}((\cap_{j=1}^{3}(P_{j}+P_{4}))\geq\operatorname{depth}_{S}I$
by Lemma 2.1. Suppose $I\cap S^{\prime\prime}\not=0$. Then $A_{0}\geq
n-r+\operatorname{sdepth}_{S^{\prime\prime}}(I\cap S^{\prime\prime})$ by [2,
Lemma 3.6]. If $t(I\cap S^{\prime\prime})\leq 2$ we get
$\operatorname{sdepth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})\geq\operatorname{depth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})$ as above. Otherwise there exists $i\in[4]$ such that
$(P_{i}\cap S^{\prime\prime})\not\subset\Sigma_{1=j\not=i}^{4}(P_{j}\cap
S^{\prime})$ and we get the same thing using the induction hypothesis (less
variables). Thus $A_{0}\geq n-r+\operatorname{depth}_{S^{\prime\prime}}(I\cap
S^{\prime\prime})\geq\operatorname{depth}_{S}I$ by [9, Proposition 1.2].
###### Example 4.3.
Let $n=10$, $P_{1}=(x_{1},\ldots,x_{7})$, $P_{2}=(x_{3},\ldots,x_{8})$,
$P_{3}=(x_{1},\ldots,x_{4},x_{8},\ldots,x_{10})$,
$P_{4}=(x_{1},x_{2},x_{5},x_{8},x_{9},x_{10})$, $P_{5}=(x_{5},\ldots,x_{10})$.
We have
$P_{1}+P_{3}=P_{2}+P_{3}=P_{1}+P_{4}=P_{2}+P_{4}=P_{3}+P_{5}=P_{1}+P_{5}=m$,
$P_{2}+P_{5}=m\setminus\\{x_{1},x_{2}\\}$,
$P_{3}+P_{4}=m\setminus\\{x_{6},x_{7}\\}$,
$P_{4}+P_{5}=m\setminus\\{x_{3},x_{4}\\}$,
$P_{1}+P_{2}=m\setminus\\{x_{9},x_{10}\\}$. We have $t(I)=2$. Applying Lemma
3.3 in fact the first subcase of Case ii) we get
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\operatorname{depth}_{S}S/(P_{1}\cap
P_{2}),\operatorname{depth}_{S}S/(P_{2}\cap\ldots\cap P_{5})\\}.$
We have $\operatorname{depth}_{S}S/(P_{1}\cap P_{2})=3$ and for
$a=\operatorname{depth}_{S}S/(P_{2}\cap\ldots\cap P_{5})$ we apply (3) of
Lemma 3.2, with $P_{4}+P_{5}\not=m$ and $P_{2}+P_{3}=m$. As for $j=2$ we have
$P_{2}+P_{4}=m\not=P_{2}+P_{5}$ and for $j=3$ we have
$P_{3}+P_{4}\not=m=P_{3}+P_{5}$ it follows that $a=1+\dim S/(P_{1}+P_{2})=3$
and so $\operatorname{depth}_{S}I=4$. Applying Theorem 1.5 for $P_{1}$ as main
prime we see that $A^{(1)}_{3,4}\geq 3$, that is $A_{\tau}\geq 3$ for
$\tau=\\{3,4\\}$. Indeed,
$A^{(1)}_{3,4}\geq\operatorname{sdepth}_{K[x_{6},x_{7}]}(x_{6},x_{7})K[x_{6},x_{7}]+\operatorname{sdepth}_{K[x_{8},x_{9},x_{10}]}(x_{8},x_{9},x_{10})K[x_{8},x_{9},x_{10}]=3.$
Similarly choosing $P_{2}$ as a main prime we get $A^{(2)}_{3,4}\geq 3$ and
taking $P_{3}$,$P_{4}$ as main primes we get $A^{(3)}_{2,5}\geq 3$,
respectively $A^{(4)}_{2,5}\geq 3$. Thus from these we cannot conclude that
$\operatorname{sdepth}_{S}I\geq\operatorname{depth}_{S}I$. Fortunately,
choosing $P_{5}$ as a main prime you can see that all $A_{\tau}\geq 4$, which
is enough ($\\{2\\}\not\in{\mathcal{F}}^{(5)}$ that is $A_{2}^{(5)}$ is not
considered because $P_{1}\cap K[\\{x_{i}:5\leq i\leq 10,x_{i}\not\in
P_{2}\\}]=(0)$). Note that $\dim S/P_{5}=4$ is maximum possible among $\dim
S/P_{i}$, but we have also $\dim S/P_{2}=\dim S/P_{4}=4$.
###### Remark 4.4.
The above example shows that it is not clear how one can use the special
Stanley decompositions from [6] in general, because it is not clear that we
may find always a ”good” main prime $P_{i}$ and if it really exists then it is
not clear how we could pick it, the maximum dimension of $S/P_{i}$ seems to be
not enough. On the other hand, if we apply Theorem 1.5 for $r=8$, that is to
the case $P_{1}+P_{2}=(x_{1},\ldots,x_{8})$, then
$A_{5}^{(12)}\geq\operatorname{sdepth}((x_{3},x_{4})\cap(x_{1},x_{2})\cap
K[x_{1},\ldots,x_{4}])+\operatorname{sdepth}(x_{9},x_{10})\cap
K[x_{9},x_{10}]=4,$ $\operatorname{depth}((x_{3},x_{4})\cap(x_{1},x_{2})\cap
K[x_{1},\ldots,x_{4}])+\operatorname{depth}(x_{9},x_{10})\cap
K[x_{9},,x_{10}]=3<\operatorname{depth}_{S}I$
and so the idea to apply here the Stanley’s Conjecture for less variables does
not work.
## References
* [1] G. Fløystad, J.Herzog, Gröbner basis of syzygies and Stanley depth, to appear in J. Algebra, arXiv:AC/1003.4495v1.
* [2] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [3] M. Ishaq, Upper bounds for the Stanley depth, to appear in Comm. Algebra, arXiv:AC/1003.3471.
* [4] M. Ishaq, Values and bounds of the Stanley depth, arXiv:AC/1010.4692.
* [5] G. Lyubeznik, On the arithmetic rank of monomial ideals. J. Algebra 112, 86 89 (1988).
* [6] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
* [7] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
* [8] D. Popescu, I. Qureshi, Computing the Stanley depth, J. Algebra, 323 (2010), 2943-2959.
* [9] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784.
* [10] R.P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
* [11] R. H. Villarreal, Monomial Algebras, Marcel Dekker Inc., New York, 2001.
|
arxiv-papers
| 2010-09-28T17:49:48 |
2024-09-04T02:49:13.194383
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1009.5646"
}
|
1009.5966
|
# Path Integral Methods for Stochastic Differential Equations
Carson C. Chow and Michael A. Buice Laboratory of Biological Modeling, NIDDK,
NIH, Bethesda, MD 20892
###### Abstract
We give a pedagogical review of the application of field theoretic and path
integral methods to obtaining perturbative solutions of stochastic
differential equations.
## I Introduction
There are many applications of stochastic differential equations (SDE) for
mathematical modeling. In the realm of neuroscience, SDEs are utilized to
model stochastic phenomena that range in scale from molecular transport in
neurons, to neuronal firing, to networks of coupled neurons, to even cognitive
phenomena such as decision problems Tuckwell (1989). In many applications,
what is often desired is the ability to obtain closed form solutions or
approximations of quantities such as the moments of stochastic processes.
However, generally these SDEs are nonlinear and difficult to solve. For
example one often encounters equations of the form
$\displaystyle\frac{dx}{dt}=f(x)+g(x)\eta(t)$
where $\eta(t)$ represents some noise process, in the simplest case a white
noise process where $\langle\eta(t)\rangle=0$ and
$\langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-t^{\prime})$. Often of
interest are the moments of $x(t)$ or the probability density function
$p(x,t)$. Traditional methods use Langevin or Fokker-Planck approaches to
compute these quantities, which can still be difficult and unwieldy to apply
perturbation theory Risken (1996); Gardiner (2004); Kampen (2007). Here, we
will show how methods developed in nonequilibrium statistical mechanics using
path or functional integrals Doi (1976a, b); Peliti (1985); Janssen and Tauber
(2005); Cardy (a, b); Kleinert (2004); Tauber can be applied to solve SDEs.
While these methods have been recently applied at the level of networks, the
methods are applicable to more general stochastic processes Buice and Cowan
(2007, 2009); Buice et al. (2010); Bressloff (2009); Hildebrand et al. (2006);
Buice and Chow (2007) . Path integral methods provide a convenient tool to
compute quantities such as moments and transition probabilities
perturbatively. They also make renormalization group methods available when
perturbation theory breaks down.
Although Wiener introduced path integrals to study stochastic processes, these
methods are not commonly used nor familiar to much of the neuroscience or
applied mathematics community. There are many textbooks on path integrals but
most are geared towards quantum field theory or statistical mechanics Zinn-
Justin (2002); Kardar (2007); Chaichian and Demichev (2001). Here we give a
pedagogical review of these methods specifically applied to SDEs. In
particular, we review the response function method Martin et al. (1973);
Tauber et al. (2005), which is particularly convenient to compute desired
quantities such as moments.
The goal of this review is to present methods to compute actual quantities.
Thus, mathematical rigor will be dispensed for convenience. This review will
be elementary. In Section II, we cover moment generating functionals, which
expand the definition of generating functions to cover distributions of
functions, such as the trajectory $x(t)$ of a stochastic process. We continue
in Section III by constructing functional integrals appropriate for the study
of SDEs, using the Ornstein-Uhlenbeck process as an example. Section IV
introduces the concept of Feynman diagrams as a tool for carrying our
perturbative expansions and introduces the “loop expansion”, a tool for
constructing semiclassical approximations. The following section V provides
the connection between SDEs and equations for the density $p(x,t)$ such as
Fokker-Planck equations. Finally, we end the paper by pointing the reader
towards important entries to the literature.
## II Moment generating functionals
The strategy of path integral methods is to derive a generating function or
functional for the moments and response functions for SDEs. The generating
functional will be an infinite dimensional generalization for the familiar
generating function for a single random variable. In this section we review
moment generating functions and show how they can be generalized to functional
distributions.
Consider a probability density function (PDF) $P(x)$ for a single real
variable $x$. The moments of the PDF are given by
$\displaystyle\langle x^{n}\rangle=\int x^{n}P(x)dx$
and can be obtained directly by taking derivatives of the generating function
$\displaystyle Z(\lambda)=\langle e^{\lambda x}\rangle=\int e^{\lambda
x}P(x)dx$
with
$\displaystyle\langle
x^{n}\rangle=\frac{1}{Z[0]}\left.\frac{d^{n}}{d\lambda^{n}}Z(\lambda)\right|_{\lambda=0}$
Note that in explicitly including $Z[0]$ we are allowing for the possibility
that $P(x)$ is not normalized. This freedom will be convenient especially when
we apply perturbation theory.
For example, the generating function for a Gaussian PDF, $P(x)\propto
e^{-\frac{(x-a)^{2}}{2\sigma^{2}}}$, is
$\displaystyle
Z(\lambda)=\int_{-\infty}^{\infty}e^{-\frac{(x-a)^{2}}{2\sigma^{2}}+\lambda
x}dx$ (1)
The integral can be computed by completing the square so that the exponent of
the integrand can be written as a perfect square
$\displaystyle-\frac{(x-a)^{2}}{2\sigma^{2}}+\lambda x=-A(x-x_{c})^{2}+B$
This is equivalent to shifting $x$ by $x_{c}$, which is the critical or
stationary point of the exponent
$\displaystyle\frac{d}{dx}\left(-\frac{(x-a)^{2}}{2\sigma^{2}}+\lambda
x\right)=0$
yielding $x_{c}=\lambda\sigma^{2}+a$. The constants are then
$\displaystyle A=\frac{1}{2\sigma^{2}}$
and
$\displaystyle
B=\frac{x_{c}^{2}}{2\sigma^{2}}-\frac{a^{2}}{2\sigma^{2}}=\frac{\lambda^{2}\sigma^{2}}{2}+\lambda
a$
The integral in (1) can then be computed to obtain
$\displaystyle
Z(\lambda)=\int_{-\infty}^{\infty}e^{-\frac{(x-\lambda\sigma^{2}-a)^{2}}{2\sigma^{2}}+\lambda
a+\frac{\lambda^{2}\sigma^{2}}{2}}dx=Z(0)e^{\lambda
a+\frac{\lambda^{2}\sigma^{2}}{2}}$
where
$\displaystyle
Z(0)=\int_{-\infty}^{\infty}e^{-\frac{x^{2}}{2\sigma^{2}}}dx=\sqrt{2\pi}\sigma$
is a normalization factor. The mean of $x$ is then given by
$\displaystyle\langle x\rangle=\frac{d}{d\lambda}\left.e^{\lambda
a+\frac{\lambda^{2}\sigma^{2}}{2}}\right|_{\lambda=0}=a$
The cumulant generating function is defined as
$\displaystyle W(\lambda)=\ln Z(\lambda)$
so that the cumulants are
$\displaystyle\langle
x^{n}\rangle_{C}=\left.\frac{d^{n}}{d\lambda^{n}}W(\lambda)\right|_{\lambda=0}$
In the Gaussian case
$\displaystyle W(\lambda)=\lambda a+\frac{1}{2}\lambda^{2}\sigma^{2}+\ln Z(0)$
yielding $\langle x\rangle_{C}=\langle x\rangle=a$ and $\langle
x^{2}\rangle_{C}\equiv{\rm var}(x)=\langle x^{2}\rangle-\langle
x\rangle^{2}=\sigma^{2}$, and $\langle x^{n}\rangle_{C}=0$, $n>2$.
The generating function can be generalized for an n-dimensional vector
$x=\\{x_{1},x_{2},\cdots,x_{n}\\}$ to become a generating functional that maps
the $n$-dimensional vector
$\lambda=\\{\lambda_{1},\lambda_{2},\dots,\lambda_{n}\\}$ to a real number
with the form
$\displaystyle
Z[\lambda]=\int\prod_{i=1}^{n}dx_{i}e^{-\frac{1}{2}\sum_{j,k}x_{j}K^{-1}_{jk}x_{k}+\sum_{j}\lambda_{j}x_{j}}$
where $K^{-1}_{jk}\equiv(K^{-1})_{jk}$ and we use square brackets to denote a
functional. This integral can be solved by transforming to orthonormal
coordinates, which is always possible if $K^{-1}_{ij}$ is symmetric, as it can
be assumed to be. Hence, let $\omega_{\alpha}$ and $v^{\alpha}$ be the
$\alpha$th eigenvalues and orthonormal eigenvectors of $K^{-1}$ respectively,
i.e.
$\displaystyle\sum_{j}K^{-1}_{ij}v_{j}^{\alpha}=\omega_{\alpha}v_{i}^{\alpha}$
and
$\displaystyle\sum_{j}v_{j}^{\alpha}v_{j}^{\beta}=\delta_{\alpha\beta}$
Now, expand $x$ and $\lambda$ in terms of the eigenvectors with
$\displaystyle x_{k}=\sum_{\alpha}c_{\alpha}v^{\alpha}_{k}$
$\displaystyle\lambda_{k}=\sum_{\alpha}d_{\alpha}v^{\alpha}_{k}$
Hence
$\displaystyle\sum_{j,k}x_{j}K^{-1}_{jk}x_{k}=\sum_{j}\sum_{\alpha,\beta}c_{\alpha}\omega_{\beta}c_{\beta}v_{j}^{\alpha}v_{j}^{\beta}=\sum_{\alpha,\beta}c_{\alpha}\omega_{\beta}c_{\beta}\delta_{\alpha\beta}=\sum_{\alpha}\omega_{\alpha}c^{2}_{\alpha}$
Since, the Jacobian is $1$ for an orthonormal transformation the generating
functional is
$\displaystyle Z[\lambda]$ $\displaystyle=$
$\displaystyle\int\prod_{\alpha}dc_{\alpha}e^{\sum_{\alpha}(-\frac{1}{2}\omega_{\alpha}c^{2}_{\alpha}+d_{\alpha}c_{\alpha})}$
$\displaystyle=$
$\displaystyle\prod_{\alpha}\int_{-\infty}^{\infty}dc_{\alpha}e^{-\frac{1}{2}\omega_{\alpha}c^{2}_{\alpha}+d_{\alpha}c_{\alpha}}$
$\displaystyle=$ $\displaystyle
Z[0]\prod_{\alpha}e^{\frac{1}{2}\omega_{\alpha}^{-1}d^{2}_{\alpha}}$
$\displaystyle=$ $\displaystyle
Z[0]e^{\sum_{jk}\frac{1}{2}\lambda_{j}K_{jk}\lambda_{k}}$
where
$\displaystyle Z[0]=(2\pi\det{K})^{n/2}$
The cumulant generating functional is
$\displaystyle W[\lambda]=\ln Z[\lambda]$
Moments are given by
$\displaystyle\left\langle\prod_{i=1}^{s}x_{i}\right\rangle=\left.\frac{1}{Z[0]}\prod_{i=1}^{s}\frac{\partial}{\partial\lambda_{i}}Z[\lambda]\right|_{\lambda_{i}=0}$
However, since the exponent is quadratic in the components $\lambda_{l}$, only
even powered moments are non-zero. From this we can deduce that
$\displaystyle\left\langle\prod_{i=1}^{2s}x_{i}\right\rangle=\sum_{\rm all\
possible\ pairings}K_{i_{1},i_{2}}\cdots K_{i_{2s-1}i_{2s}}$
which is known as Wick’s theorem. Any Gaussian moment about the mean can be
obtained by taking the sum of all the possible ways of “contracting” two of
the variables. For example
$\displaystyle\langle
x_{a}x_{b}x_{c}x_{d}\rangle=K_{ab}K_{cd}+K_{ad}K_{bc}+K_{ac}K_{bd}$
In the continuum limit, a generating functional for a function $x(t)$ on the
real domain $t\in[0,T]$ is obtained by taking a limit of the generating
functional for the vector $x_{i}$. Let the interval $[0,T]$ be divided into
$n$ segments of length $h$ so that $T=nh$ and $x(t/h)=x_{i}$ for $t\in[0,T]$.
We then take the limit of $n\rightarrow\infty$ and $h\rightarrow 0$ preserving
$T=nh$. We similarly identify $\lambda_{i}\rightarrow\lambda(t)$ and
$K_{ij}\rightarrow K(s,t)$ and obtain
$\displaystyle Z[\lambda]=\int{\cal D}x(t)e^{-\frac{1}{2}\int
x(s)K^{-1}(s,t)x(t)dsdt+\int\lambda(t)x(t)dt}$
$\displaystyle=Z[0]e^{\int\frac{1}{2}\lambda(s)K(s,t)\lambda(t)dsdt}$ (2)
where the the measure for integration
$\displaystyle{\cal
D}x(t)\equiv\lim_{n\rightarrow\infty}\prod_{i=0}^{n}dx_{i}$
is over functions. Although $Z[0]=\lim_{n\rightarrow\infty}(2\pi\det K)^{n/2}$
is formally infinite, the moments of the distributional are well defined. The
integral is called a path integral or a functional integral. Note that
$Z[\lambda]$ refers to a functional that maps different “forms” of the
function $\lambda(t)$ over the time domain to a real number. Defining the
functional derivative to obey all the rules of the ordinary derivative with
$\displaystyle\frac{\delta\lambda(s)}{\delta\lambda(t)}=\delta(s-t)$
the moments again obey
$\displaystyle\left\langle\prod_{i}x(t_{i})\right\rangle=\frac{1}{Z[0]}\prod_{i}\frac{\delta}{\delta\lambda(t_{i})}Z[\lambda]$
$\displaystyle=\sum_{\rm all\ possible\ pairings}K(t_{i_{1}},t_{i_{2}})\cdots
K(t_{i_{2s-1}},t_{t_{i_{2s}}})$
For example
$\displaystyle\langle
x(t_{1})x(t_{2})\rangle=\frac{1}{Z[0]}\frac{\delta}{\delta\lambda(t_{1})}\frac{\delta}{\delta\lambda(t_{2})}Z[\lambda]=K(t_{1},t_{2})$
We can further generalize the generating functional to describe the
probability distribution of a function $\varphi(\vec{x})$ of a real vector
$\vec{x}$, instead of a single variable $t$ with
$\displaystyle Z[\lambda]$ $\displaystyle=$ $\displaystyle\int{\cal D}\varphi
e^{-\frac{1}{2}\int\varphi(\vec{y})K^{-1}(\vec{y},\vec{x})\varphi(\vec{x})d^{d}yd^{d}x+\int\lambda(\vec{x})\varphi(\vec{x})d^{d}x}$
$\displaystyle=$ $\displaystyle
Z[0]e^{\int\frac{1}{2}\lambda(\vec{y})K(\vec{y},\vec{x})\lambda(\vec{x})d^{d}yd^{d}x}$
Historically, computing moments and averages of a probability density
functional of a function of more than one variable is called field theory. In
general, the probability density functional is usually written in exponential
form
$\displaystyle P[\varphi]=e^{-S[\varphi(\vec{t})]}$
where $S[\varphi]$ is called the action and the generating functional is often
written as
$\displaystyle Z[J]=\int{\cal D}\varphi e^{-S[\phi]+J\cdot\varphi}$
where
$\displaystyle J\cdot\varphi=\int J(\vec{t})\varphi(\vec{t})d^{d}t$
For example, the action given by
$\displaystyle
S[\varphi]=\int\varphi(\vec{t})K^{-1}(\vec{t},\vec{t^{\prime}})\varphi(\vec{t^{\prime}})d^{d}td^{d}t^{\prime}+g\int\varphi^{4}(\vec{t})d^{d}t$
is called $\varphi^{4}$ (“$\varphi$-4”) theory.
The analogy between stochastic systems and quantum theory, where path
integrals are commonly used, is seen by transforming the time coordinates in
the path integrals via $t\rightarrow it$ (where $i^{2}=-1$). When the field
$\phi$ is a function of a single variable $t$, then this would be analogous to
single particle quantum mechanics where the quantum amplitude can be expressed
in terms of a path integral over a configuration variable $\phi(t)$. When the
field is a function of two or more variables $\phi(\vec{r},t)$, then this is
analogous to quantum field theory, where the quantum amplitude is expressed as
a path integral over the quantum field $\phi(\vec{r},t)$.
## III Application to SDE
Building on the previous section, here we derive a generating functional for
SDEs. Consider a Langevin equation
$\displaystyle\frac{dx}{dt}=f(x,t)+g(x,t)\eta(t)$
with initial condition $x(t_{0})=y$, on the domain $t\in[0,T]$. Equation (III)
is to be interpreted as the Ito stochastic differential equation
$\displaystyle dx=f(x,t)dt+g(x,t)dB_{t}$ (3)
where $dB_{t}$ is a Brownian stochastic process. We will show how to
generalize to other stochastic processes later. According to the convention
for an Ito stochastic process, $g(x,t)$ is _non-anticipating_ , which means
that in evaluating the integrals over time and $B_{t}$, $g(x,t)$ is
independent of $B_{\tau}$ for $\tau>t$. The choice between Ito and
Stratonovich conventions amounts to a choice of the measure for the path
integrals, which will be manifested in a condition on the linear response or
“propagator” that we introduce below.
The goal is to derive a probability density functional (PDF) and moment
generating functional for the stochastic variable $x(t)$. For the path
integral formulation, it is more convenient to take $x(t_{0})=0$ in (III) and
enforce the initial condition with a source term so that
$\displaystyle\frac{dx}{dt}=f(x,t)+g(x,t)\eta(t)+y\delta(t-t_{0})$ (4)
where $\delta(\cdot)$ is the point mass or Dirac delta functional. The
discretized form of (4) with the Ito interpretation for small time step $h$ is
given by
$\displaystyle
x_{i+1}-x_{i}=f_{i}(x_{i})h+g_{i}(x_{i})w_{i}\sqrt{h}+y\delta_{i,o}$ (5)
$i\in\\{0,1,\dots,N\\}$, $T=Nh$, $\delta_{i,j}$ is the Kronecker delta,
$x_{0}=0$, and $w_{i}$ is a discrete random variable with $\langle
w_{i}\rangle=0$ and $\langle w_{i}w_{j}\rangle=\delta_{i,j}$. Hence, the
discretized stochastic variable vector $x_{i}$ depends on the discretized
white noise process $w_{i}$ and the initial condition $x_{0}$. We use $x$ and
$w$ without indices to denote the vectors $x=(x_{1},\dots,x_{N})$ and
$w=(w_{0},w_{1},\dots,w_{N-1})$. Formally, the joint PDF for the vector $x$
can be written as
$\displaystyle
P[x|w;y]=\prod_{i=0}^{N}\delta[x_{i+1}-x_{i}-f_{i}(x_{i})h-g_{i}(x_{i})w_{i}\sqrt{h}-y\delta_{i,0}]$
i.e. the probability density function is given by the point mass (Dirac delta)
constrained at the solution of the SDE.
Inserting the Fourier representation of the Dirac delta
$\displaystyle\delta(z_{i})=\frac{1}{2\pi}\int e^{-ik_{i}z_{i}}dk_{i}$
gives
$\displaystyle
P[x|w;y]=\int\prod_{j=0}^{N}\frac{dk_{j}}{2\pi}e^{-i\sum_{j}k_{j}(x_{j+1}-x_{j}-f_{j}(x_{j})h-g_{j}(x_{j})w_{j}\sqrt{h}-y\delta_{j,0})}$
The PDF is now expressed in exponential form.
For Gaussian white noise the PDF of $w_{i}$ is given by
$\displaystyle P(w_{i})=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}w_{i}^{2}}$
Hence
$\displaystyle P[x|y]$ $\displaystyle=$ $\displaystyle\int
P[x|w;y]\prod_{j=0}^{N}P(w_{j})dw_{j}$ $\displaystyle=$
$\displaystyle\int\prod_{j=0}^{N}\frac{dk_{j}}{2\pi}e^{-i\sum_{j}k_{j}(x_{j+1}-x_{j}-f_{j}(x_{j})h-y\delta_{j,0})}\int\prod_{j=0}^{N}\frac{dw_{j}}{\sqrt{2\pi}}e^{ik_{j}g_{j}(x_{j})w_{j}\sqrt{h}}e^{-\frac{1}{2}w_{j}^{2}}$
can be integrated by completing the square as demonstrated in the previous
section to obtain
$\displaystyle
P[x|y]=\int\prod_{j=0}^{N}\frac{dk_{j}}{2\pi}e^{-\sum_{j}(ik_{j})\left(\frac{x_{j+1}-x_{j}}{h}-f_{j}(x_{j})-y\frac{\delta_{j,0}}{h}\right)h+\sum_{j}\frac{1}{2}g_{j}^{2}(x_{j})(ik_{j})^{2}h}$
Taking the continuum limit $h\rightarrow 0$, $N\rightarrow\infty$ such that
$T=Nh$ gives
$\displaystyle P[x(t)|y,t_{0}]=\int{\cal
D}\tilde{x}(t)e^{-\int\left[\tilde{x}(t)(\dot{x}(t)-f(x(t),t)-y\delta(t-t_{0}))-\frac{1}{2}\tilde{x}^{2}(t)g^{2}(x(t),t)\right]dt}$
with a newly defined complex variable $ik_{i}\rightarrow\tilde{x}(t)$. We
include the argument of $x(t)$ as a reminder that this is a functional of $x$
conditioned on two scalars $y$ and $t_{0}$. The moment generating functional
for $x(t)$ and $\tilde{x}(t)$ is then given by
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x(t){\cal
D}\tilde{x}(t)e^{-S[x,\tilde{x}]+\int\tilde{J}(t)x(t)dt+\int
J(t)\tilde{x}(t)dt}$
with action
$\displaystyle
S[x,\tilde{x}]=\int\left[\tilde{x}(t)(\dot{x}(t)-f(x(t),t)-y\delta(t-t_{0}))-\frac{1}{2}\tilde{x}^{2}(t)g^{2}(x(t),t)\right]dt$
(6)
The probability density functional can be derived directly from the SDE (III)
by considering the infinite dimensional Dirac delta functional and taking the
path integral:
$\displaystyle P[x(t)|y,t_{0}]$ $\displaystyle=$ $\displaystyle\int{\cal
D}\eta(t)\delta[\dot{x}(t)-f(x,t)-g(x,t)\eta(t)-y\delta(t-t_{0})]e^{-\int\eta^{2}(t)dt}$
$\displaystyle=$ $\displaystyle\int{\cal D}\eta(t){\cal
D}\tilde{x}(t)e^{-\int\tilde{x}(t)(\dot{x}(t)-f(x,t)-y\delta(t-t_{0}))+\tilde{x}(t)g(x,t)\eta(t)-\eta^{2}(t)dt}$
$\displaystyle=$ $\displaystyle\int{\cal
D}\tilde{x}(t)e^{-\int\tilde{x}(t)(\dot{x}(t)-f(x,t)-y\delta(t-t_{0}))+\frac{1}{2}\tilde{x}^{2}(t)g^{2}(x,t)dt}$
yielding the action (6) 111This derivation is, strictly speaking, incorrect
because the delta functional fixes the value of $\dot{x}(t)$, not $x(t)$. It
works because the Jacobian under a change of variables from $\dot{x}(t)$ to
$x(t)$ is $1$.. Owing to the definition $ik_{i}\rightarrow\tilde{x}(t)$ the
integrals over $\tilde{x}(t)$ are along the _imaginary_ axis, which is why no
explicit $i$ appears in the action above.
In a similar manner, we can define the path integral for more general
processes than the Brownian motion processes that we are using. Let $\eta(t)$
instead be a process with cumulant generating functional $W[\lambda(t)]$ so
that the cumulants of $\eta(t)$ (which may depend upon $x(t)$) are given by
functional derivatives with respect to $\lambda(t)$. This process will have
its own action $S[\eta(t)]$ and the path integral can be written as
$\displaystyle P[x(t)|y,t_{0}]$ $\displaystyle=$ $\displaystyle\int{\cal
D}\eta(t)\delta[\dot{x}(t)-f(x,t)-\eta(t)-y\delta(t-t_{0})]e^{-S[\eta(t)]}$
$\displaystyle=$ $\displaystyle\int{\cal D}\eta(t){\cal
D}\tilde{x}(t)e^{-\int\tilde{x}(t)(\dot{x}(t)-f(x,t)-y\delta(t-t_{0}))+\tilde{x}(t)\eta(t)dt-S[\eta(t)]}$
Noting that
$\displaystyle\int{\cal
D}\eta(t)e^{\int\tilde{x}(t)\eta(t)dt-S[\eta(t)]}=e^{W[\tilde{x}(t)]}$
is the definition of the cumulant generating functional for $\eta(t)$, we have
that the path integral can be written as
$\displaystyle P[x(t)|y,t_{0}]$ $\displaystyle=$ $\displaystyle\int{\cal
D}\eta(t){\cal
D}\tilde{x}(t)e^{-\int\tilde{x}(t)(\dot{x}(t)-f(x,t)-y\delta(t-t_{0}))dt+W[\tilde{x}(t)]}$
In the cases where the input $\eta(t)$ is delta-correlated in time, we obtain
$\displaystyle W[\tilde{x}(t)]=\sum_{n=1}^{\infty}\int
g_{n}(x(t))\tilde{x}(t)^{n}dt=\sum_{n=1,m=0}^{\infty}\frac{v_{nm}}{n!}\int\tilde{x}^{n}(t)x^{m}(t)dt$
where we have Taylor expanded the functions $g_{n}(x)$. For example, the Ito
process above gives
$\displaystyle W[\tilde{x}(t)]=\frac{D}{2}\int\tilde{x}(t)^{2}dt$
i.e. $v_{20}=D$ and all other $v_{nm}=0$.
### III.1 Ornstein-Uhlenbeck Process
Consider the Ornstein-Uhlenbeck process
$\displaystyle\dot{x}(t)+ax(t)-\sqrt{D}\eta(t)=0$
with initial condition $x(0)=y$. The action is
$\displaystyle
S[x,\tilde{x}]=\int\left[\tilde{x}(t)\left(\dot{x}(t)+ax(t)-y\delta(t-t_{0})\right)-\frac{D}{2}\tilde{x}^{2}(t)\right]dt$
Defining an inverse propagator
$\displaystyle
G^{-1}(t-t^{\prime})=\left(\frac{d}{dt}+a\right)\delta(t-t^{\prime})$
the action is
$\displaystyle
S[x,\tilde{x}]=\int\tilde{x}(t)G^{-1}(t-t^{\prime})x(t^{\prime})dtdt^{\prime}-\int
y\tilde{x}(t)\delta(t-t_{0})dt-\int\frac{D}{2}\tilde{x}(t)^{2}dt$
and the generating functional is
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x(t){\cal
D}\tilde{x}(t)e^{-S[x,\tilde{x}]+\int\tilde{J}(t)x(t)dt+\int
J(t)\tilde{x}(t)dt}$
This path integral can be evaluated directly as a Gaussian integral since the
action is quadratic. In fact integrating by $\tilde{x}$(t) gives the Onsager-
Machlup path integral Risken (1996); Chaichian and Demichev (2001), which will
have a Jacobian factor depending upon whether we use Ito, Stratonovich, or
some other convention for our SDE. With the Ito convention, this Jacobian is
$1$. However, the generating functional can also be evaluated by expanding the
exponent around the “free” action given by
$S_{F}[x(t),\tilde{x}(t)]=\int\tilde{x}(t)G^{-1}(t-t^{\prime})x(t^{\prime})dtdt^{\prime}$.
We will demonstrate this method since it forms the basis for perturbation
theory for non-quadratic actions. Expand the integrand of the generating
functional as
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x(t){\cal D}\tilde{x}(t)e^{-\int
dtdt^{\prime}\tilde{x}(t)G^{-1}(t-t^{\prime})x(t^{\prime})}\left(1+\mu+\frac{1}{2!}\mu^{2}+\frac{1}{3!}\mu^{3}+\cdots\right)$
(7)
where
$\displaystyle\mu=y\int\tilde{x}(t)\delta(t-t_{0})dt+\int\frac{D}{2}\tilde{x}^{2}(t)dt+\int\tilde{J}(t)x(t)dt+\int
J(t)\tilde{x}(t)dt$
The generating functional is now expressed as a sum of moments of the free
action, which are calculated from the free generating functional
$\displaystyle Z_{F}[J,\tilde{J}]=\int{\cal D}x(t){\cal D}\tilde{x}(t)e^{-\int
dtdt^{\prime}\tilde{x}(t)G^{-1}(t-t^{\prime})x(t^{\prime})+\int\tilde{x}(t)J(t)dt+\int
x(t)\tilde{J}(t)dt}$
From equation (2), we can write this as
$\displaystyle
Z_{F}[J,\tilde{J}]=e^{\int\tilde{J}(t)G(t,t^{\prime}){J}(t^{\prime})}$ (8)
where $G(t,t^{\prime})$ is the operator inverse of $G^{-1}(t,t^{\prime})$,
i.e.
$\displaystyle\int
dt^{\prime\prime}G^{-1}(t,t^{\prime\prime})G(t^{\prime\prime},t^{\prime})=\left(\frac{d}{dt}+a\right)G(t,t^{\prime})=\delta(t-t^{\prime})$
Therefore
$\displaystyle G(t,t^{\prime})=H(t-t^{\prime})e^{-a(t-t^{\prime})}$
where $H(t)$ is the left continuous Heaviside step function (i.e. $H(0)=0$,
$\lim_{t\rightarrow 0^{+}}H(t)=1$ and thus $\lim_{t_{1}\rightarrow
t_{2}^{+}}G(t_{1},t_{2})=1$, $G(t,t)=0$). The choice of $H(0)=0$ is consistent
with the Ito condition for the SDE and insures that the configuration variable
$x(t)$ is uncorrelated with future values of the stochastic driving term.
Other choices for $H(0)$ represent other forms of stochastic calculus (e.g.
$H(0)=1/2$ is the choice consistent with Stratonovich calculus) 222This is
also a manifestation of the normal-ordering convention chosen for the theory.
Zinn-JustinZinn-Justin (2002) refers to this as the “$\epsilon(0)$ problem”..
The free moments are given by
$\displaystyle\left\langle\prod_{ij}x(t_{i})\tilde{x}(t_{j})\right\rangle_{F}=\prod_{ij}\left.\frac{\delta}{\delta\tilde{J}(t_{i})}\frac{\delta}{\delta
J(t_{j})}e^{\int\tilde{J}(t)G(t,t^{\prime})J(t^{\prime})dtdt^{\prime}}\right|_{J=\tilde{J}=0}$
since $Z_{F}[0,0]=1$. We use a subscript $F$ to denote expectation values with
respect to the free action. From the action of (8), it is clear the nonzero
free moments must have equal numbers of $x(t)$ and $\tilde{x}(t)$ due to
Wick’s theorem, which applies here for contractions between $x(t)$ and
$\tilde{x}(t)$. For example, one of the fourth moments is given by
$\displaystyle\langle
x(t_{1})x(t_{2})\tilde{x}(t_{3})\tilde{x}(t_{4})\rangle_{F}=G(t_{1},t_{3})G(t_{2},t_{4})+G(t_{1},t_{4})G(t_{2},t_{3})$
Now the generating functional for the OU process (7) can be evaluated. The
only surviving terms in the expansion will have equal numbers of $x(t)$ and
$\tilde{x}(t)$. Thus only terms with factors of
$\int\tilde{x}(t_{0})\tilde{J}(t_{1})x(t_{1})dt_{1}$,
$(D/2)\int\tilde{x}^{2}(t_{1})\tilde{J}^{2}(t_{2})x^{2}(t_{2})dt_{1}dt_{2}$
and $\int\tilde{J}(t_{1})x(t_{1})J(t_{2})\tilde{x}(t_{2})dt_{1}dt_{2}$ (and
combinations of the three) will survive. For the OU process, the entire series
is summable. First consider the case where $D=0$. Because there must be equal
numbers of $\tilde{x}(t)$ and $x(t)$ factors in any non-zero moment due to
Wick’s theorem, in this case the generating functional has the form
$\displaystyle Z$ $\displaystyle=$ $\displaystyle
1+\sum_{m=1}\frac{1}{m!m!}\int\left\langle\prod_{i,j=1}^{m}\tilde{J}(t_{i})x(t_{i})\tilde{x}(t_{j})[y\delta(t_{j}-t_{0})+J(t_{j})]\right\rangle_{F}\,\prod_{i,j=1}^{m}dt_{i}dt_{j}$
(9)
From Wick’s theorem, the free expectation value in (9) will be a sum over all
possible contractions between $x(t)$ and $\tilde{x}(t)$ leading to $m!$
combinations. Thus (9) is
$\displaystyle
Z=\sum_{m=1}\frac{1}{m!}\left(y\int\tilde{J}(t_{1})G(t_{1},t_{0})dt_{1}+\int\tilde{J}(t^{\prime})J(t^{\prime\prime})G(t^{\prime},t^{\prime\prime})dt^{\prime}dt^{\prime\prime}\right)^{m}$
which means the series is an exponential function. The other term in the
exponent of (10) can be similarly calculated resulting in
$\displaystyle
Z[J(t),\tilde{J}(t)]=\exp\left(y\int\tilde{J}(t_{1})G(t_{1},t_{0})dt_{1}+\int\tilde{J}(t_{1})J(t_{2})G(t_{1},t_{2})dt_{1}dt_{2}\right.$
$\displaystyle\left.+\frac{D}{2}\int\tilde{J}(t_{1})\tilde{J}(t_{2})G(t_{1},t^{\prime\prime})G(t_{2},t^{\prime\prime})dt^{\prime\prime}dt_{1}dt_{2}\right)$
(10)
The cumulant generating functional is
$\displaystyle
W[J(t),\tilde{J}(t)]=y\int\tilde{J}(t)G(t,t_{0})dt+\int\tilde{J}(t^{\prime})J(t^{\prime\prime})G(t^{\prime},t^{\prime\prime})dt^{\prime}dt^{\prime\prime}$
$\displaystyle+\frac{D}{2}\int\tilde{J}(t^{\prime})\tilde{J}(t^{\prime\prime})G(t^{\prime},t)G(t^{\prime\prime},t)dtdt^{\prime}dt^{\prime\prime}$
(11)
The only nonzero cumulants are the mean
$\displaystyle\langle x(t)\rangle=yG(t,t_{0})$
the response function
$\displaystyle\left\langle
x(t_{1})\tilde{x}(t_{2})\right\rangle_{C}=\frac{\delta}{\delta\tilde{J}(t_{1})}\frac{\delta}{\delta
J(t_{2})}W[J,\tilde{J}]_{J=\tilde{J}=0}=G(t_{1},t_{2})$
and covariance
$\displaystyle\langle x(t_{1})x(t_{2})\rangle_{C}$ $\displaystyle\equiv$
$\displaystyle\langle x(t_{1})x(t_{2})\rangle-\langle x(t_{1})\rangle\langle
x(t_{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{\delta}{\delta\tilde{J}(t_{1})}\frac{\delta}{\delta\tilde{J}(t_{2})}W[J,\tilde{J}]_{J=\tilde{J}=0}$
$\displaystyle=$ $\displaystyle D\int G(t_{1},t)G(t_{2},t)dt$
Closed form expressions for the cumulants are obtained by using the solution
for the propagator $G$. Hence, the mean is
$\displaystyle\left\langle x(t)\right\rangle=ye^{-a(t-t_{0})}H(t-t_{0})$ (12)
the response function is
$\displaystyle\left\langle
x(t_{1})\tilde{x}(t_{2})\right\rangle=e^{-a(t_{1}-t_{2})}H(t_{1}-t_{2})$
and the covariance is
$\displaystyle\langle
x(t_{1})x(t_{2})\rangle_{C}=D\int_{t_{0}}^{t_{2}}e^{-a(t_{1}-t^{\prime})}e^{-a(t_{2}-t^{\prime})}H(t_{1}-t^{\prime})H(t_{2}-t^{\prime})dt^{\prime}$
For $t_{2}\geq t_{1}\geq t_{0}$
$\displaystyle\langle
x(t_{1})x(t_{2})\rangle_{C}=D\frac{e^{2a(t_{1}-t_{2})}-e^{-a(t_{1}+t_{2}-2t_{0})}}{2a}$
For $t_{1}=t_{2}=t$
$\displaystyle\langle x(t)^{2}\rangle_{C}=\frac{D}{2a}(1-e^{-2at})$ (13)
## IV Perturbative methods and Feynman diagrams
If the SDE is nonlinear, then the generating functional cannot be computed
exactly as in the linear case. However, propagators and moments can be
computed perturbatively. The method we use is an infinite dimensional
generalization of Laplace’s method for finite dimensional integrals Bender and
Orszag (1999). In fact, the method was used to compute the generating
functional for the Ornstein-Uhlenbeck process. The only difference is that for
nonlinear SDEs the resulting asymptotic series is not generally summable.
The strategy is again to split the action $S[x,\tilde{x}]=S_{F}+S_{I}$, where
$S_{F}$ is called the “free” action and $S_{I}$ is called the “interacting”
action. The generating functional is
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x{\cal
D}\tilde{x}e^{-S[x,\tilde{x}]+\int\tilde{J}xdt+\int J\tilde{x}dt}$ (14)
The moments satisfy
$\displaystyle\left\langle\prod_{i}^{m}\prod_{j}^{n}x(t_{i})\tilde{x}(t_{j})\right\rangle=\frac{1}{Z[0,0]}\prod_{i}^{m}\prod_{j}^{n}\left.\frac{\delta}{\delta\tilde{J}(t_{i})}\frac{\delta}{\delta
J(t_{j})}Z\right|_{J=\tilde{J}=0}$ (15)
and the cumulants satisfy
$\displaystyle\left\langle\prod_{i}^{m}\prod_{j}^{n}x(t_{i})\tilde{x}(t_{j})\right\rangle_{C}=\prod_{i}^{m}\prod_{j}^{n}\left.\frac{\delta}{\delta\tilde{J}(t_{i})}\frac{\delta}{\delta
J(t_{j})}\ln Z\right|_{J=\tilde{J}=0}$ (16)
The generating functional is computed perturbatively by expanding the
integrand of (14) around the free action
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x{\cal
D}\tilde{x}e^{-S_{F}[x,\tilde{x}]}\left(1+S_{I}+\int\tilde{J}xdt+\int
J\tilde{x}dt\right.$
$\displaystyle\left.+\frac{1}{2!}\left(S_{I}+\int\tilde{J}xdt+\int
J\tilde{x}dt\right)^{2}+\frac{1}{3!}S_{I}^{3}+\cdots\right)$
Hence, the generating functional can be expressed in terms of a series of free
moments.
Now we apply this idea to the example nonlinear SDE
$\displaystyle\dot{x}=-ax+bx^{2}+y\delta(t-t_{0})+\sqrt{D}x^{\frac{n}{2}}\eta(t)$
for some $n\geq 0$. For example, $n=0$ corresponds to standard additive noise
(as in the OU process), while $n=1$ gives multiplicative noise with variance
proportional to $x$. The action for this equation is
$\displaystyle S[x,\tilde{x}]$ $\displaystyle=$ $\displaystyle\int
dt\tilde{x}(\dot{x}+ax-bx^{2}-y\delta(t-t_{0}))-\tilde{x}^{2}x^{n}\frac{D}{2}$
(17) $\displaystyle\equiv$ $\displaystyle
S_{F}[x,\tilde{x}]-y\tilde{x}(t_{0})-b\int dt\tilde{x}(t)x^{2}(t)-\int
dt\tilde{x}^{2}x^{n}\frac{D}{2}$
where we have implicitly defined the “free” action $S_{F}[x,\tilde{x}]=\int
dt\tilde{x}(\dot{x}+ax)$. Expectations with respect to this free action are
$\displaystyle\langle x(t)\tilde{x}(t^{\prime})\rangle_{F}=G(t,t^{\prime})$
where the propagator obeys
$\displaystyle\left(\frac{d}{dt}+a\right)G(t,t^{\prime})=\delta(t-t^{\prime})$
and all other moments are zero. The generating functional is
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x{\cal
D}\tilde{x}e^{-S_{F}[x,\tilde{x}]+\int\tilde{x}bx^{2}\,dt+\int\tilde{x}y\delta(t-t_{0})\,dt+\int\tilde{x}^{2}x^{n}\frac{D}{2}\,dt+\int\tilde{J}xdt+\int
J\tilde{x}dt}$
The Taylor expansion of the exponential around the free action gives
$\displaystyle Z[J,\tilde{J}]$ $\displaystyle=$ $\displaystyle\int{\cal
D}x{\cal
D}\tilde{x}e^{-S_{F}[x,\tilde{x}]}\left(1+b\int\tilde{x}x^{2}\,dt+\tilde{x}(t_{0})y+\frac{D}{2}\int\tilde{x}^{2}x^{n}\,dt+\int\tilde{J}xdt+\int
J\tilde{x}dt\right.$ $\displaystyle+$
$\displaystyle\left.\frac{1}{2!}\left(b\int\tilde{x}x^{2}\,dt+\tilde{x}(t_{0})y+\frac{D}{2}\int\tilde{x}^{2}x^{n}\,dt+\int\tilde{J}xdt+\int
J\tilde{x}dt\right)^{2}+\cdots\right)$
Because the free action $S_{F}$ is bilinear in $\tilde{x},x$, the only
surviving terms in the expansion are those with equal numbers of $x$ and
$\tilde{x}$ factors. Also, because of the Ito condition, $H(0)=0$, these
pairings must come from _different_ terms in the expansion, e.g. the only term
surviving from the first line is the very first term, regardless of the value
of $n$. All other terms come from the quadratic and higher terms in the
expansion. For simplicity in the remainder of this example we limit ourselves
to $n=0$. Hence, the expansion includes terms of the form
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x{\cal
D}\tilde{x}e^{-S_{F}[x,\tilde{x}]}\left(1\right.$ $\displaystyle+$
$\displaystyle\frac{1}{2!}2\left(b\int\tilde{x}x^{2}\tilde{x}(t_{0})y\,dt+b\int\tilde{x}x^{2}\,dt\int
J\tilde{x}dt+\int\tilde{J}x\tilde{x}(t_{0})y\,dt+\int\tilde{J}xdt\int
J\tilde{x}dt\right)$ $\displaystyle+$
$\displaystyle\frac{1}{3!}\frac{3!}{2!}b^{2}\frac{D}{2}\int\tilde{x}x^{2}dt\int\tilde{x}x^{2}dt\int\tilde{x}^{2}dt$
$\displaystyle+$
$\displaystyle\frac{1}{3!}\frac{3!}{2!}\frac{D}{2}\int\tilde{x}^{2}\,dt\int\tilde{J}xdt\int\tilde{J}xdt+\frac{1}{3!}3!b\frac{D}{2}\int\tilde{x}x^{2}\,dt\int\tilde{x}^{2}dt\int\tilde{J}xdt$
$\displaystyle+$
$\displaystyle\frac{1}{4!}\frac{4!}{2!}b\int\tilde{x}x^{2}\,dt(\tilde{x}(t_{0})y)^{2}\int\tilde{J}xdt+\frac{1}{4!}\frac{4!}{2!2!}(\tilde{x}(t_{0})y)^{2}\int\tilde{J}xdt\int\tilde{J}xdt$
$\displaystyle+$
$\displaystyle\left.\frac{1}{5!}5!b\frac{D}{2}\int\tilde{x}x^{2}\,dt\int\tilde{x}^{2}dt\tilde{x}(t_{0})y\int\tilde{J}xdt\int\tilde{J}xdt+\cdots\right)$
Note that this not an exhaustive list of terms up to fifth order. Many of
these terms will vanish because of $G(t,t^{\prime})\propto H(t-t^{\prime})$
and $H(0)=0$. The combinatorial factors arise from the multiple ways of
combining terms in the expansion. There are $n!$ ways of combining terms at
order $n$ and terms with $m$ repeats are divided by a factor of $m!$.
Completing the Gaussian integrals using Wick’s theorem then yields
$\displaystyle Z[J,\tilde{J}]=Z_{F}[0,0](1$ (18) $\displaystyle+$
$\displaystyle y\int
G(t_{1},t_{0})\tilde{J}(t_{1})\,dt_{1}+\int\tilde{J}(t_{1})G(t_{1},t_{2})J(t_{2})dt_{1}dt_{2}$
$\displaystyle+$ $\displaystyle D\int
G(t_{2},t_{1})G(t_{3},t_{1})\tilde{J}(t_{2})\tilde{J}(t_{3})\,dt_{1}dt_{2}dt_{3}$
$\displaystyle+$ $\displaystyle bD\int
G(t_{1},t_{2})^{2}G(t_{3},t_{1})\tilde{J}(t_{3})dt_{1}dt_{2}dt_{3}$
$\displaystyle+$ $\displaystyle by^{2}\int
G(t_{1},t_{0})^{2}G(t_{2},t_{1})\tilde{J}(t_{2})dt_{1}dt_{2}$ $\displaystyle+$
$\displaystyle y^{2}\int G(t_{1},t_{0})\tilde{J}(t_{1})\,dt_{1}\int
G(t_{2},t_{0})\tilde{J}(t_{2})\,dt_{2}$ $\displaystyle+$ $\displaystyle
2bDy\int
G(t_{1},t_{2})G(t_{1},t_{0})G(t_{3},t_{1})G(t_{4},t_{2})\tilde{J}(t_{3})\tilde{J}(t_{4})\,dt_{1}dt_{2}dt_{3}dt_{4}+\cdots)$
As above, we have $Z_{F}[0,0]=1$.
The moments and cumulants are obtained from (15) and (16) respectively. For
example, the mean is given by
$\displaystyle\langle
x(t)\rangle=\frac{1}{Z[0,0]}\frac{\delta}{\delta\tilde{J}(t)}Z[J(t),\tilde{J}(t)]_{J=0,\tilde{J}=0}$
(19) $\displaystyle=$ $\displaystyle yG(t,t_{0})+bD\int
G(t,t_{1})G(t_{1},t_{2})^{2}\,dt_{1}dt_{2}+by^{2}\int
G(t,t_{1})G(t_{1},t_{0})^{2}\,dt_{1}+\cdots$
The covariance is
$\displaystyle\langle
x(s)x(t)\rangle=\frac{\delta}{\delta\tilde{J}(s)}\frac{\delta}{\delta\tilde{J}(t)}Z[J(t),\tilde{J}(t)]_{J=0,\tilde{J}=0}$
$\displaystyle=$ $\displaystyle D\int
G(s,t_{1})G(t,t_{1})\,dt_{1}+y^{2}G(s,t_{0})G(t,t_{0})$ $\displaystyle+$
$\displaystyle 2bDy\int
G(t_{1},t_{2})G(t_{1},t_{0})G(s,t_{1})G(t,t_{2})\,dt_{1}dt_{2}$
$\displaystyle+$ $\displaystyle 2bDy\int
G(t_{1},t_{2})G(t_{1},t_{0})G(t,t_{1})G(s,t_{2})\,dt_{1}dt_{2}\cdots$
The first cumulant is the same as the mean but the second cumulant or
covariance is
$\displaystyle\langle
x(s)x(t)\rangle_{C}=\frac{\delta}{\delta\tilde{J}(s)}\frac{\delta}{\delta\tilde{J}(t)}\ln
Z[J(t),\tilde{J}(t)]_{J=0,\tilde{J}=0}$ (20) $\displaystyle=$
$\displaystyle\frac{1}{Z}\left.\frac{\delta}{\delta\tilde{J}(s)}\frac{\delta}{\delta\tilde{J}(t)}Z\right|_{J=0,\tilde{J}=0}-\left.\frac{\delta}{\delta\tilde{J}(s)}Z\frac{\delta}{\delta\tilde{J}(t)}Z\right|_{J=0,\tilde{J}=0}$
$\displaystyle=$ $\displaystyle D\int G(s,t_{1})G(t,t_{1})\,dt_{1}$
$\displaystyle+$ $\displaystyle 2bDy\int
G(t_{1},t_{2})G(t_{1},t_{0})G(s,t_{1})G(t,t_{2})\,dt_{1}dt_{2}$
$\displaystyle+$ $\displaystyle 2bDy\int
G(t_{1},t_{2})G(t_{1},t_{0})G(t,t_{1})G(s,t_{2})\,dt_{1}dt_{2}\cdots$
As can be seen in this example, the terms in the perturbation series become
rapidly unwieldy. However, a convenient means to keep track of the terms is to
use Feynman diagrams, which are graphs with edges connected by vertices that
represents each term in the expansion of a moment. The edges and vertices
represent terms (i.e. interactions) in the action and hence SDE, which are
combined according to a set of rules that reproduces the perturbation
expansion shown above. These are directed graphs (unlike the Feynman diagrams
usually used for equilibrium statistical mechanics or particle physics). The
flow of each graph, which represents the flow of time, is directed from right
to left, points to the left being considered to be at times after points to
the right. The vertices represent points in time and separate into two groups:
_endpoint_ vertices and _interior_ vertices. The moment
$\left\langle\prod_{i=1}^{N}x(t_{i})\prod_{j=1}^{M}\tilde{x}(t_{j})\right\rangle$
is represented by diagrams with $N$ _final_ endpoint vertices which represent
the times $t_{i}$ and $M$ _initial_ endpoint vertices which represent the
times $t_{j}$. Interior vertices are determined from terms in the action.
Consider the interacting action expressed as the power series
$\displaystyle S_{I}=\sum_{n\geq 2,m\geq 0}V_{nm}=\sum_{n\geq 2,m\geq
0}\frac{v^{nm}}{n!}\int_{t_{0}}^{\infty}dt\tilde{x}^{n}x^{m}$ (21)
where $n$ and $m$ cannot both be $\leq 1$ (those terms are part of the free
action). (Nonpolynomial functions in the action are expanded in a Taylor
series to obtain this form.) There is a vertex type associated with each
$V_{nm}$. The moment
$\left\langle\prod_{i=1}^{N}x(t_{i})\prod_{j=1}^{M}\tilde{x}(t_{j})\right\rangle$
is given by a perturbative expansion of free action moments that are
proportional to
$\left\langle\prod_{i=1}^{N}x(t_{i})\prod_{j=1}^{M}\tilde{x}(t_{j})V(N_{v})\right\rangle_{F}$
where $V(N_{v})$ represents a product of $N_{v}$ vertices. Each term in this
expansion corresponds to a graph with $N_{v}$ interior vertices. We label the
$k$th vertex with time $t_{k}$. As indicated in equation (21), there is an
integration over each such interior time point, over the interval
$(t_{0},\infty)$. The interaction $V_{nm}$ produces vertices with $n$ edges to
the left of the vertex (towards increasing time) and $m$ edges to the right of
the vertex (towards decreasing times). Edges between vertices are due to
Wick’s theorem, which tells us that every $\tilde{x}(t^{\prime})$ must be
joined by a factor of $x(t)$ _in the future_ , i.e. $t>t^{\prime}$, because
$G(t,t^{\prime})\propto H(t-t^{\prime})$. Also, by the Ito condition $H(0)=0$,
each edge must connect two _different_ vertices. All edges must be connected,
a vertex for the interaction $V_{nm}$ must connect to $n$ edges on the left
and $m$ edges on the right.
Hence, terms at the $N_{v}$th order of the expansion for the moment
$\left\langle\prod_{i=1}^{N}x(t_{i})\prod_{j=1}^{M}\tilde{x}(t_{j})\right\rangle$
are given by directed Feynman graphs with $N$ final endpoint vertices, $M$
initial endpoint vertices, and $N_{v}$ interior vertices with edges joining
all vertices in all possible ways. The sum of the terms associated with these
graphs is the value of the moment to $N_{v}$th order. Figure 1 shows the
vertices applicable to action (17) with $n=0$. Arrows indicate the flow of
time, from right to left. These components are combined into diagrams for the
respective moments. Figure 2 shows three diagrams in the sum for the mean and
second moment of $x(t)$. The entire expansion for any given moment can be
expressed by constructing the Feynman diagrams for each term.
The application of Feynman diagrams for computing the diagrams corresponding
to terms in a perturbative expansion is encapsulated in the Feynman rules:
A) For each vertex type, $V_{nm}$, which appears $k$ times in the diagram,
there is a factor of $\frac{1}{k!}$.
B) For $n$ distinct ways of connecting edges to vertices that yields the same
diagram, i.e. the same topology, there is an overall factor of $n$. This is
the combinatoric factor from the number of different Wick contractions that
yield the same diagram.
C) Each vertex interaction $V_{nm}$ adds a factor $-\frac{v_{nm}}{n!}$. The
minus sign enters because the action appears in the path integral with a minus
sign.
D) For each edge between times $t$ and $t^{\prime}$, there is a factor of
$G(t,t^{\prime})$.
E) There is an integration over the times $t$ of each interior vertex over the
domain $(t_{0},\infty)$.
Figure 1: Feynman diagram components for a) an edge, the propagator
$G(t,t^{\prime})$, and vertices b) $\int b\tilde{x}x^{2}dt$, c) $\int
y\tilde{x}\delta(t-t_{0})dt$, and d) $\int\frac{D}{2}\tilde{x}^{2}dt$. Figure
2: Feynman diagrams for a) the mean and b) second moment.
Comparing these rules with the diagrams in Figure 2, one can see the terms in
the expansions in equations (19) and (20), with the exception of the middle
diagram in Figure 2b. An examination of Figure 2a shows that this middle
diagram is two copies of the first diagram of the mean. Topologically, the
diagrams have two forms. There are connected graphs and disconnected graphs.
The disconnected graphs represent terms that can be completely factored into a
product of moments of lower order (cf. the middle diagram in Figure 2b).
Cumulants consist only of connected graphs since the products of lower ordered
moments are subtracted from the moment by definition. Thus, moments and
cumulants can be computed directly from the diagrams that represent them. The
connected diagrams in Figure 2 lead to the expressions (19) and (20). In the
expansion (18), the terms that do not include the source factors $J$ and
$\tilde{J}$ only contribute to the normalization $Z[0,0]$ and do not affect
moments because of (15). Borrowing terminology from quantum theory, these
terms are called vacuum graphs and consist of closed graphs, i.e. they have no
initial or trailing edges. In the cases we consider, all of these terms are
$0$, which implies $Z[0,0]=1$.
The diagrammatic expansion is particularly useful if the series can be
truncated so that only a few diagrams need to be computed. There are two types
of expansions depending on whether the nonlinearity is small or the noise
source is small. In quantum theory, the small nonlinearity expansion is called
a weak coupling expansion and the small fluctuation expansion is called a
semiclassical or loop expansion. The weak coupling expansion is
straightforward. Suppose one or more of the vertices is associated with a
small parameter $\alpha$. These vertices define the interacting action $S_{I}$
as demonstrated above. Each appearance of that particular vertex diagram
contributes a factor of $\alpha$ and the expansion can be continued to any
order in $\alpha$.
For the loop expansion, let us introduce the factor $h$ into the generating
functional:
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x(t){\cal
D}\tilde{x}(t)e^{-\frac{1}{h}\left(S[x(t),\tilde{x}(t)]-\int\tilde{J}(t)x(t)dt-\int
J(t)\tilde{x}(t)dt\right)}$ (22)
According to the Feynman rules described above, with this change each diagram
gains a factor of $h$ for each edge (internal or external) and a factor of
$1/h$ for each vertex. Let $E$ be the number of external edges, $I$ the number
of internal edges, and $V$ the number of vertices. Then each connected graph
now has a factor $h^{I+E-V}$. It can be shown via induction that the number of
closed loops $L$ in a given connected graph must satisfy $L=I-V+1$ Zinn-Justin
(2002). To see this note that for diagrams without loops any two vertices must
be connected by at most one internal edge. Since the diagrams are connected we
must have $V=I+1$ when $L=0$. Adding an internal edge between any two vertices
increases the number of loops by precisely one. Thus we see that the total
factor for each diagram may be written $h^{E+L-1}$. We can organize the
diagrammatic expansion in terms of the number of loops in the graphs. This is
called the loop expansion. For the mean which has one external edge there are
no factors of $h$ at lowest order. Higher cumulants (which are determined by
connected graphs) gain additional factors of $h$, e.g. the variance goes as
$h$ at lowest order in the loop expansion.
Loop diagrams arise because of nonlinearities in the SDE that couple to
moments of the driving noise source. For example, the middle graph in Figure
2a describes the coupling of the variance to the mean through the nonlinear
$x^{2}$ term. This produces a single loop diagram which is of order $h$,
compared to the order $1$ “tree” level mean graph. Compare this factor of $h$
to that from the tree level diagram for the variance, which is order $h$. This
same construction holds for higher nonlinearities and higher moments for
general theories. The loop expansion is thus a series organized around the
magnitude of the coupling of higher moments to lower moments.
As an example, consider the action
$\displaystyle
S[x,\tilde{x}]=\int\tilde{x}(\dot{x}-f(x(t),t))-\sigma^{2}\frac{1}{2}\tilde{x}^{2}g^{2}(x(t),t)\,dt$
where $\sigma$ is a small parameter and $f$ and $g$ are of order one. Rescale
the action with the transformation $\tilde{x}\rightarrow\tilde{x}/\sigma^{2}$
and $\tilde{J}\rightarrow\tilde{J}/\sigma^{2}$. The rescaled action now has
the form
$\displaystyle
S[x,\tilde{x}]=\frac{1}{\sigma^{2}}\int\tilde{x}(\dot{x}-f(x(t),t))-\frac{1}{2}\tilde{x}^{2}g^{2}(x(t),t)\,dt$
The generating functional is
$\displaystyle Z[J,\tilde{J}]=\int{\cal D}x{\cal
D}\tilde{x}e^{-\frac{1}{\sigma^{2}}\left(\int\tilde{x}(\dot{x}-f(x(t),t))-\frac{1}{2}\tilde{x}^{2}g^{2}(x(t),t)\,dt+\int\tilde{J}xdt+\int
J\tilde{x}dt\right)}$
The loop expansion in this construction is explicitly a small noise expansion
because $\sigma^{2}$ plays the role of $h$ in the loop expansion.
Figure 3: Vertex for multiplicative noise with $n=1$ in the action (17). This
vertex replaces the one in Figure 1d.
Now consider the one loop correction to the linear response, $\langle
x(t)\tilde{x}(t^{\prime})\rangle$, when $n=1$ in action (17). For simplicity,
we will assume the initial condition $y=0$. In this case, the vertex in Figure
1d now appears as in Figure 3. The linear response $\langle
x(t)\tilde{x}(t^{\prime})\rangle$ will be given by the sum of all diagrams
with one entering edge and one exiting edge. At tree level, there is only one
such graph, equal to $G(t,t^{\prime})$. At one loop order, we can combine the
vertices in Figures 1b and 1d to get the second graph shown in Figure 4 to
obtain
$\displaystyle\langle x(t)\tilde{x}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle G(t,t^{\prime})+bD\int
dt_{1}dt_{2}G(t,t_{2})G(t_{2},t_{1})^{2}G(t_{2},t^{\prime})$ $\displaystyle=$
$\displaystyle
e^{-a(t-t^{\prime})}H(t-t^{\prime})\left[1+\frac{t-t^{\prime}}{a}+\frac{1}{a^{2}}\left(e^{-a(t-t^{\prime})}-1\right)\right]$
Figure 4: Feynam diagrams for the linear response, $\langle
x(t)\tilde{x}(t^{\prime})\rangle$, to one loop order.
This loop correction arises because of two types of vertices. There are
vertices that we call “branching” (as in Figure 3), which have more exiting
edges then entering edges. The opposite case occurs for those vertices which
we call “aggregating”. Noise terms in the SDE produce vertices with more than
one exiting edge. As can be seen from the structure of the Feynman diagrams,
all moments can be computed exactly when the deterministic part of the SDE is
linear because it only involves convolving the propagator (i.e. Green’s
function) of the deterministic part of the SDE with the driving noise term, as
in the case of the OU process above. On the other hand, nonlinearities give
rise to vertices with more than one entering edge.
If there are no branching vertices in the action (i.e. terms quadratic or
higher in $\tilde{x}$), we do not even have an SDE at all, but just an
ordinary differential equation. Consider the expansion of the mean for action
(17) for the case where $D=0$ (so that there is no noise term). From equation
(19), we have
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$ $\displaystyle
yG(t,t_{0})+by^{2}\int G(t,t_{1})G(t_{1},t_{0})^{2}\,dt_{1}+\cdots$
The expansion for $D=0$ will be the sum of all tree level diagrams. It is easy
to see that in general this expansion will be the perturbative expansion for
the solution of the ordinary differential equation obtained by discarding the
stochastic driving term. In other words, the sum of the tree level diagrams
for the mean satisfies
$\displaystyle\frac{d}{dt}\langle x(t)\rangle_{\rm tree}=-a\langle
x(t)\rangle_{\rm tree}+b\langle x(t)\rangle_{\rm tree}^{2}$ (23)
along with the initial condition $\langle x(t_{0})\rangle_{\rm tree}=y$.
Similarly, the sum of the tree level diagrams for the linear response,
$\langle x(t)\tilde{x}(t^{\prime})\rangle_{\rm tree}=G_{\rm
tree}(t,t^{\prime})$, is the solution of the linearization of (23) with a
Dirac delta functional initial condition, i.e. the propagator
$\displaystyle\frac{d}{dt}G_{\rm tree}(t,t^{\prime})=-aG_{\rm
tree}(t,t^{\prime})+2b\langle x(t)\rangle_{\rm tree}G_{\rm
tree}(t,t^{\prime})+\delta(t-t^{\prime})$
The semiclassical approximation amounts to a small noise perturbation around
the solution to this equation. We can represent the sum of the tree level
diagrams graphically by using bold edges, which we call “classical” edges, as
in Figure 5. We can then use the classical edges within the loop expansion to
compute semiclassical approximations to the moments of the solution to the
SDE. The one loop semiclassical approximation of the mean for the case $n=0$
is given by the sum of the first two graphs in Figure 2a with the thin edges
replaced by bold edges. For the covariance, the first graph in Figure 2b
suffices, again with thin edges replaced by bold edges. These graphs are
equivalent to the equations:
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$ $\displaystyle\langle
x(t)\rangle_{\rm
tree}+bD\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}G_{\rm
tree}(t,t_{2})G_{\rm tree}(t_{2},t_{1})^{2}$ (24)
and
$\displaystyle\langle x(t)x(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle D\int_{t_{0}}^{{\rm min}(t,t^{\prime})}dt_{1}G_{\rm
tree}(t,t_{1})G_{\rm tree}(t^{\prime},t_{1})$ (25)
Using equation (25) in (24) gives
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$ $\displaystyle\langle
x(t)\rangle_{\rm tree}+bD\int_{t_{0}}^{t}dt_{1}G_{\rm tree}(t,t_{2})\langle
x(t_{2})x(t_{2})\rangle$
This approximation is first order in the dummy loop parameter $h$ for the mean
(one loop) and covariance (tree level). For the case $n=1$, equations (24) and
(25) are
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$ $\displaystyle\langle
x(t)\rangle_{\rm
tree}+bD\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}G_{\rm
tree}(t,t_{2})G_{\rm tree}(t_{2},t_{1})^{2}\langle x(t)\rangle_{\rm tree}$
and
$\displaystyle\langle x(t)x(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle D\int_{t_{0}}^{{\rm min}(t,t^{\prime})}G_{\rm
tree}(t,t_{1})G_{\rm tree}(t^{\prime},t_{1})\langle x(t)\rangle_{\rm tree}$
Using the definition of $G_{\rm tree}(t,t^{\prime})$, the self-consistent
semiclassical approximation for $\langle x(t)\rangle$ to one-loop order is
$\displaystyle\frac{d}{dt}\langle x(t)\rangle+a\langle x(t)\rangle-b\langle
x(t)\rangle^{2}$ $\displaystyle=$ $\displaystyle
bD\int_{t_{0}}^{t}dt_{1}G_{\rm tree}(t,t_{1})^{2}\langle x(t)\rangle$
or
$\displaystyle\frac{d}{dt}\langle x(t)\rangle+a\langle x(t)\rangle-b\langle
x(t)\rangle^{2}$ $\displaystyle=$ $\displaystyle
b\int_{t_{0}}^{t}dt_{1}\langle x(t_{1})x(t_{1})\rangle_{C}$
The semiclassical approximation known as the “linear noise” approximation
takes the tree level computation for the mean and covariance. The formal way
of deriving these self-consistent equations is via the _effective action_ ,
which is beyond the scope of this review. We refer the interested reader to
Zinn-Justin (2002).
Figure 5: Bold edges represent the sum of all tree level diagrams contributing
to that moment. Top) the mean $\langle x(t)\rangle_{\rm tree}$. Bottom) linear
response $G_{\rm tree}(t,t^{\prime})$.
## V Connection to Fokker-Planck equation
In stochastic systems, one is often interested in the PDF $p(x,t)$, which
gives the probability density of position $x$ at time $t$. This is in contrast
with the probability density functional $P[x(t)]$ which is the probability
density of all possible functions or paths $x(t)$. Previous sections have been
devoted to computing the moments of $P[x(t)]$, which provide the moments of
$p(x,t)$ as well. In this section we leverage knowledge of the moments of
$p(x,t)$ to determine an equation it must satisfy. In simple cases, this
equation is a Fokker-Planck equation for $p(x,t)$.
The PDF $p(x,t)$ can be formally obtained from $P[x(t)]$ by marginalizing over
the interior points of the function $x(t)$. Consider the transition
probability $U(x_{1},t_{1}|x_{0},t_{0})$ between two points $x_{0},t_{0}$ and
$x_{1},t_{1}$. This is equal to $p(x,t)$ given the initial condition
$p(x,t_{0})=\delta(x-x_{0})$. In terms of path integrals this can be expressed
as
$\displaystyle
U(x_{1},t_{1}|x_{0},t_{0})=\int^{(x(t_{1})=x_{1})}_{(x(t_{0})=x_{0})}{\cal
D}x(t)\,P[x(t)]$
where the upper limit in the integral is fixed at $x(t_{1})=x_{1}$ and the
lower at $x(t_{0})=x_{0}$. The lower limit appears as the initial condition
term in the action and can thus be considered part of $P[x(t)]$. The upper
limit on the path integral can be imposed with a functional Dirac delta via
$\displaystyle U(x_{1},t_{1}|x_{0},t_{0})=\int{\cal
D}x(t)\,\delta(x(t_{1})-x_{1})P[x(t)]$
which in the Fourier representation is given by
$\displaystyle U(x_{1},t_{1}|x_{0},t_{0})=\frac{1}{2\pi i}\int
d\lambda\int{\cal D}x(t)\,e^{\lambda(x(t_{1})-x_{1})}P[x(t)]$
where the contour for the $\lambda$ integral runs along the imaginary axis.
This can be rewritten as
$\displaystyle U(x_{1},t_{1}|x_{0},t_{0})=\frac{1}{2\pi i}\int
d\lambda\,e^{-\lambda(x_{1}-x_{0})}Z_{\rm CM}(\lambda)$ (26)
in terms of an initial condition centered moment generating function
$\displaystyle Z_{\rm CM}(\lambda)=\int{\cal
D}x\,e^{\lambda(x(t_{1})-x_{0})}P[x(t)]$
where the measure ${\cal D}x(t)$ is defined such that $Z_{\rm CM}(0)=1$. Note
that this generating function $Z_{\rm CM}(\lambda)$ is different from the
generating functionals we presented in previous sections. $Z_{\rm
CM}(\lambda)$ generates moments of the deviations of $x(t)$ from the initial
value $x_{0}$ at a specific point in time $t$. Taylor expanding the
exponential gives
$\displaystyle Z_{\rm
CM}(\lambda)=1+\sum_{n=1}^{\infty}\frac{1}{n!}\lambda^{n}\langle(x(t_{1})-x_{0})^{n}\rangle_{x(t_{0})=x_{0}}$
where
$\displaystyle\langle(x(t_{1})-x_{0})^{n}\rangle_{x(t_{0})=x_{0}}=\int{\cal
D}x\,(x(t_{1})-x_{0})^{n}P[x(t)]$
Inserting into (26) gives
$\displaystyle U(x_{1},t_{1}|x_{0},t_{0})=\frac{1}{2\pi i}\int
d\lambda\,e^{-\lambda(x_{1}-x_{0})}\left(1+\sum_{n=1}^{\infty}\frac{1}{n!}\lambda^{n}\langle(x(t_{1})-x_{0})^{n}\rangle\right)$
Using the identity
$\displaystyle\frac{1}{2\pi i}\int
d\lambda\,e^{-\lambda(x_{1}-x_{0})}\lambda^{n}=\left(-\frac{\partial}{\partial
x_{1}}\right)^{n}\delta(x_{1}-x_{0})$
results in
$\displaystyle
U(x_{1},t_{1}|x_{0},t_{0})=\left(1+\sum_{n=1}^{\infty}\frac{1}{n!}\left(-\frac{\partial}{\partial
x_{1}}\right)^{n}\langle(x(t_{1})-x_{0})^{n}\rangle_{x(t_{0})=x_{0}}\right)\delta(x_{1}-x_{0})$
(27)
The probability density function $p(y,t)$ obeys
$\displaystyle p(y,t+\Delta t)=\int U(x,t+\Delta
t|y^{\prime},t)p(y^{\prime},t)dy^{\prime}$ (28)
Inserting (27) gives
$\displaystyle p(y,t+\Delta
t)=\left(1+\sum_{n=1}^{\infty}\frac{1}{n!}\left(-\frac{\partial}{\partial
y}\right)^{n}\langle(x(t+\Delta t)-y)^{n}\rangle_{x(t)=y}\right)p(y,t)$
Expanding $p(y,t+\Delta t)$ and the moments in a Taylor series in $\Delta t$
gives
$\displaystyle\frac{\partial p(y,t)}{\partial t}\Delta
t=\sum_{n=1}^{\infty}\left(-\frac{\partial}{\partial
y}\right)^{n}\frac{1}{n!}\langle(x(t+\Delta
t)-y)^{n}\rangle_{x(t)=y}p(y,t)+O(\Delta t^{2})$
since $x(t)=y$. In the limit $\Delta t\rightarrow 0$ we obtain the Kramers-
Moyal expansion
$\displaystyle\frac{\partial p(y,t)}{\partial
t}=\sum_{n=1}^{\infty}\frac{1}{n!}\left(-\frac{\partial}{\partial
y}\right)^{n}D_{n}(y,t)p(y,t)+O(\Delta t^{2})$
where the _jump moments_ are defined by
$\displaystyle D_{n}(y,t)=\lim_{\Delta t\rightarrow
0}\left.\frac{\langle\left(x(t+\Delta t)-y\right)^{n}\rangle}{\Delta
t}\right|_{x(t)=y}$ (29)
As long as these limits are convergent, then it is relatively easy to see that
only connected Feynman graphs will contribute to the jump moments. In
addition, we can define $z=x-y$, $\tilde{z}=\tilde{x}$ and use the action
$S[z(t)+y,\tilde{z}(t)]$. This shift in $x$ removes the initial condition
term. This means we can calculate the $n$th jump moment by using this shifted
action to compute the sum of all graphs with no initial edges and $n$ final
edges (as in Figure 1d for $n=2$).
As an example, consider the Ito SDE (III). From the discretization (5), where
$h=\Delta t$, it is found that
$\displaystyle\lim_{\Delta t\rightarrow 0}\left.\frac{\langle\left(x(t+\Delta
t)-y\right)^{n}\rangle}{\Delta t}\right|_{x(t)=y}=\lim_{\Delta t\rightarrow
0}\frac{\left\langle\left(f_{i}(y)\Delta t-g_{i}(y)w_{i}\sqrt{\Delta
t}\right)^{n}\right\rangle}{\Delta t}$ (30)
Which yields $D_{1}(y,t)=f(y,t)$, $D_{2}=g(y,t)^{2}$ and $D_{n}=0$ for $n>2$.
Thus for the Ito SDE (III), the Kramers-Moyal expansion becomes the the
Fokker-Planck equation
$\displaystyle\frac{\partial p(y,t)}{\partial
t}=\left(-\frac{\partial}{\partial
y}D_{1}(y,t)+\frac{1}{2}\frac{\partial^{2}}{\partial^{2}y}D_{2}(y,t)\right)p(y,t)$
We have $D_{n}=0$ for $n>2$ even though there are non-zero contributions from
connected graphs to these moments for $n>2$ in general. However, all of these
moments require the repeated use of the vertex with two exiting edges; this
will cause $D_{n}\propto\Delta t^{m}$ for some $m>1$ and thus the jump moment
will be zero in the limit.
We can envision actions for more general stochastic processes by considering
vertices which have more than two exiting edges, i.e. we can add a term to the
action of the form
$\displaystyle S_{V}[x,\tilde{x}]=\frac{1}{n!}\int dt\tilde{x}^{n}h(x)$
for some $n$ and function $h(x)$. This will produce a non-zero $D_{n}$. The
PDF for this kind of process will not in general be describable by a Fokker-
Planck equation, but will need the full Kramers-Moyal expansion. If we wished
to provide an initial distribution for $x(t_{0})$ instead of specifying a
single point, we could likewise add similar terms to the action. In fact, the
completely general initial condition term is given by
$\displaystyle S_{\rm initial}[\tilde{x}(t_{0})]=-\ln Z_{y}[\tilde{x}(t_{0})]$
where $Z_{y}$ is the generating functional for the initial distribution. In
other words, the initial state terms in the action are the cumulants of the
initial distribution multiplied by the corresponding powers of
$\tilde{x}(t_{0})$.
Returning to the Ito process (III), the solution to the Fokker-Planck equation
can be obtained directly from the path integral formula for the transition
probability (26). Let $\ln Z[\lambda]$ be the cumulant generating function for
the moments of $x(t)$ at time $t$. It can be expanded as
$\displaystyle Z_{\rm
CM}[\lambda]=\exp\left[\sum_{n=1}\frac{1}{n!}\lambda^{n}\langle
x(t)^{n}\rangle_{C}\right]$
yielding
$\displaystyle p(x,t)=\frac{1}{2\pi i}\int d\lambda\,e^{-\lambda
x}\exp\left[\sum_{n=1}\frac{1}{n!}\lambda^{n}\langle
x(t)^{n}\rangle_{C}\right]$
For the Ornstein-Uhlenbeck process the first two cumulents are given in (12)
and (13) yielding (assuming initial condition $x(t_{0})=y$)
$\displaystyle p(x,t)=\sqrt{\frac{a}{\pi
D(1-e^{-2a(t-t_{0})})}}\exp\left(\frac{-a(x-ye^{-a(t-t_{0})})^{2}}{D(1-e^{-2a(t-t_{0})}}\right)$
(31)
## VI Further reading
The reader interested in this approach is encouraged to explore the extensive
literature on path integrals and field theory. The reader should be aware that
most of the references listed will concentrate on applications and
formulations appropriate for equilibrium statistical mechanics and particle
physics, which means that they will not explicitly discuss the response
function approach we have demonstrated here. For application driven
examinations of path integration there is SchulmanSchulman (2005) and
KleinertKleinert (2004). More mathematically rigorous treatments can be found
in SimonSimon (2005) and Glimm and Jaffe Glimm and Jaffe (1981). For the
reader seeking more familiarity with concepts of stochastic calculus such as
Ito or Stratonovich integration there are applied approaches Gardiner (2004)
and rigorous treatments Karatzas and Shreve (1991) as well. Zinn-Justin Zinn-
Justin (2002) covers a wide array of topics of interest in quantum field
theory from statistical mechanics to particle physics. Despite the
exceptionally terse and dense presentation, the elementary material in this
volume is recommended to those new to the concept of path integrals. Note that
Zinn-Justin covers SDEs in a somewhat different manner than that presented
here (the Onsager-Machlup integral is derived; although see chapters 16 and
17), as does Kleinert. We should also point out the parallel between the form
of the action for exponential decay (i.e. $D=0$ in the OU process) and the
holomorphic representation of the harmonic oscillator presented in Zinn-Justin
(2002). The response function formalism was introduced by Martin, Siggia, Rose
Martin et al. (1973). Closely related path integral formalisms have been
introduced via the work of Doi Doi (1976a, b) and Peliti Peliti (1985) which
have been used in the analysis of reaction-diffusion system Cardy (a, b);
Janssen and Tauber (2005); Tauber et al. (2005). Uses of path integrals in
neuroscience have appeared in Buice and Cowan (2007, 2009); Buice et al.
(2010); Bressloff (2009).
## References
* Tuckwell (1989) H. C. Tuckwell, _Stochastic processes in the neurosciences_ , vol. 56 (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1989), ISBN 0898712327.
* Risken (1996) H. Risken, _The Fokker-Planck equation: methods of solution and applications_ , vol. v. 18 (Springer-Verlag, New York, 1996), 2nd ed., ISBN 354061530X (alk. paper), URL http://www.loc.gov/catdir/enhancements/fy0817/96033182-d.html%.
* Gardiner (2004) C. W. Gardiner, _Handbook of stochastic methods: for physics, chemistry, and the natural sciences_ , Springer series in synergetics (Springer, Berlin, 2004), 3rd ed., ISBN 3540208828 (acid-free paper).
* Kampen (2007) N. G. v. Kampen, _Stochastic processes in physics and chemistry_ , North-Holland personal library (Elsevier, Amsterdam, 2007), 3rd ed., ISBN 9780444529657 (pbk.).
* Doi (1976a) M. Doi, Journal of Physics A: Mathematical and General 9, 1465 (1976a).
* Doi (1976b) M. Doi, Journal of Physics A: Mathematical and General 9, 1479 (1976b).
* Peliti (1985) L. Peliti, Journal de Physique 46, 1469 (1985).
* Janssen and Tauber (2005) H.-K. Janssen and U. C. Tauber, Annals of Physics 315, 147 (2005), ISSN 0003-4916, special Issue, URL http://www.sciencedirect.com/science/article/B6WB1-4F29SKJ-3/%2/336bbc9919ba2102ceee9b65c89c620a.
* Cardy (a) J. Cardy, _Renormalization group approach to reaction-diffusion problems_ , review article, cond-mat/9607163.
* Cardy (b) J. Cardy, _Field theory and nonequilibrium statistical mechanics_ , review article, Année acad’emique 1998-99, semestre d’été.
* Kleinert (2004) H. Kleinert, _Path integrals in quantum mechanics, statistics polymer physics, and financial markets_ (Singapore: World Scientific Publishing, 2004).
* (12) U. Tauber, _Critical Dynamics_ (????), in progress, http://www.phys.vt.edu/ tauber/.
* Buice and Cowan (2007) M. A. Buice and J. D. Cowan, Physical Review E 75, 051919 (2007).
* Buice and Cowan (2009) M. A. Buice and J. D. Cowan, Progress in Biophysics and Molecular Biology 99 (2009).
* Buice et al. (2010) M. A. Buice, J. D. Cowan, and C. C. Chow, Neural Computation 22 (2010).
* Bressloff (2009) P. C. Bressloff, SIAM Journal of Applied Math 70 (2009).
* Hildebrand et al. (2006) E. J. Hildebrand, M. A. Buice, and C. C. Chow, arXiv nlin.CD (2006), eprint nlin/0612029v1, URL http://arxiv.org/abs/nlin/0612029v1.
* Buice and Chow (2007) M. A. Buice and C. C. Chow, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 76, 031118 (pages 25) (2007), URL http://link.aps.org/abstract/PRE/v76/e031118.
* Zinn-Justin (2002) J. Zinn-Justin, _Quantum Field Theory and Critical Phenomena_ (Oxford Science Publications, 2002), 4th ed.
* Kardar (2007) M. Kardar, _Statistical physics of fields_ (Cambridge University Press, Cambridge, 2007), ISBN 9780521873413 (hbk.), URL http://www.loc.gov/catdir/enhancements/fy0803/2007279232-t.ht%%****␣pathintegralreview10.bbl␣Line␣150␣****ml.
* Chaichian and Demichev (2001) M. Chaichian and A. P. Demichev, _Path integrals in physics_ (Institute of Physics, Bristol, 2001), ISBN 0750307137.
* Martin et al. (1973) P. C. Martin, E. D. Siggia, and H. A. Rose, Physical Review A 8, 423 (1973).
* Tauber et al. (2005) U. C. Tauber, M. Howard, and B. P. Vollmayr-Lee, Journal of Physics A: Mathematical and General 38, R79 (2005), URL http://stacks.iop.org/0305-4470/38/R79.
* Bender and Orszag (1999) C. M. Bender and S. A. Orszag, _Advanced mathematical methods for scientists and engineers_ (Springer, New York, 1999), ISBN 0387989315 (hc. : alk. paper), URL http://www.loc.gov/catdir/enhancements/fy0816/99044783-d.html%.
* Schulman (2005) L. Schulman, _Techniques and Applications of Path Integration_ (Dover, New York, 2005).
* Simon (2005) B. Simon, _Functional Integration and Quantum Physics_ (AMS Chelsea, 2005).
* Glimm and Jaffe (1981) J. Glimm and A. Jaffe, _Quantum Physics: A Functional Integral Point of View_ (Springer-Verlag, New York, 1981).
* Karatzas and Shreve (1991) I. Karatzas and S. Shreve, _Brownian Motion and Stochastic Calculus_ (Springer-Verlag New York, 1991).
|
arxiv-papers
| 2010-09-29T18:16:41 |
2024-09-04T02:49:13.209514
|
{
"license": "Public Domain",
"authors": "Carson C. Chow and Michael A. Buice",
"submitter": "Carson C. Chow",
"url": "https://arxiv.org/abs/1009.5966"
}
|
1009.5979
|
# Performance Analysis of the Matrix Pair Beamformer with Matrix Mismatch
Jianshu Chen, , Jian Wang, Xiu-Ming Shan,
Ning Ge, and Xiang-Gen Xia,
EDICS: SAM-PERF This work was supported by the National Natural Science
Foundation of China under contract No. 60972019 and No. 60928001. Jianshu Chen
was with the Department of Electronic Engineering, Tsinghua University,
Beijing, P. R. China, 100084. He is with the Department of Electrical
Engineering, University of California, Los Angeles, CA 90095, USA. (e-mail:
jshchen@ee.ucla.edu) Jian Wang, Xiu-Ming Shan and Ning Ge are with the
Department of Electronic Engineering, Tsinghua University, Beijing, P. R.
China, 100084. (e-mail: {jian-wang, shanxm, gening} @tsinghua.edu.cn) Xiang-
Gen Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA. (e-mail: xxia@ee.udel.edu). His
work was suppported in part by the Air Force Office of Scientific Research
(AFOSR) under Grant No. FA9550-08-1-0219, the National Science Foundation
(NSF) under Grant CCF-0964500, and the World Class University (WCU) Program
2008-000-20014-0, National Research Foundation, Korea.
###### Abstract
Matrix pair beamformer (MPB) is a blind beamformer. It exploits the temporal
structure of the signal of interest (SOI) and applies generalized eigen-
decomposition to a covariance matrix pair. Unlike other blind algorithms, it
only uses the second order statistics. A key assumption in the previous work
is that the two matrices have the same interference statistics. However, this
assumption may be invalid in the presence of multipath propagations or certain
“smart” jammers, and we call it as matrix mismatch. This paper analyzes the
performance of MPB with matrix mismatch. First, we propose a general framework
that covers the existing schemes. Then, we derive its normalized output SINR.
It reveals that the matrix mismatch leads to a threshold effect caused by
“steering vector competition”. Second, using matrix perturbation theory, we
find that, if there are generalized eigenvalues that are infinite, the
threshold will increase unboundedly with the interference power. This is
highly probable when there are multiple periodical interferers. Finally, we
present simulation results to verify our analysis.
###### Index Terms:
Adaptive beamforming, Matrix pair beamformer, Generalized eigen-decomposition,
Matrix mismatch
## I Introduction
Beamforming is a spatial filter, which combines the outputs of multiple sensor
elements by an appropriately designed weight vector so as to pass a signal of
interest (SOI) while rejecting interfering signals. Since the pioneering work
of Howells[1], Applebaum[2] and Widrow[3], it has been intensively studied
during the past decades and widely applied in radar, sonar and wireless
communications [3, 4, 5, 6, 7, 8, 9, 10, 11] etc. For a comprehensive review,
we refer to [12, 13, 14, 15] and the references therein.
There are various forms of implementations for a beamformer. Some use the
direction of arrival (DOA) of the SOI and directly calculate the weight vector
by sample matrix inversion (SMI) [16]. Some employ a reference signal (e.g.
training signal[17] and decision feedback signal[6]) to iteratively calculate
the weight vector. And there are also blind beamformers which do not require
the DOA or the reference signal[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31].
To achieve blind beamforming, we need to exploit the special properties of the
SOI, such as constant modulus, non-Gaussianness, high order statistics etc.
One important property is the temporal structure of the SOI. Such kind of
blind beamformers are extensively studied for CDMA systems, where the inherent
structure of the spreading codes can be exploited[23, 24, 25, 26, 28, 29, 30,
31]. The advantage of this scheme is that it only relies on the second order
statistics of the covariance matrix pair. Although the implementation details
differ, the main idea of these approaches is to exploit a pair of array
covariance matrix and hence will be referred to as matrix pair beamformer
(MPB) in this paper. The MPB projects the discrete sequence in each antenna
onto the space spanned by the SOI’s signature vector (for a CDMA system, it is
the spreading code vector) and another carefully designed base vector. Then
two sets of array snapshots (i.e. signal snapshot and interference snapshot)
are acquired to calculate a pair of covariance matrices. With the processing
gain, the desired signal power in the signal snapshot is generally greater
than that in the interference snapshot. _A key and common assumption is that
the interference statistics in the two snapshots are identical._ These two
features enable the separation of the signal steering vector and the
interference covariance matrix from the two matices. And the weight vector
derived from the dominant eigenvector of the matrix pair will maximize the
output signal to interference plus noise ratio (SINR).
However, this key assumption that the two matrices share the same interference
statistics is not valid in many cases, which we refer to as _matrix mismatch_.
Matrix mismatch may occur when there are interferers with certain periodical
structure, such as multiple access interference (MAI) in CDMA systems, tones
and some other “smart jammers” in radar systems and it may lead to the failure
of a system. In practical applications, to avoid such an unexpected failure,
it is necessary to understand the detailed performance of a scheme when the
assumption/condition is not satisfied. To our best knowledge, little effort
has been devoted to analyzing the effect of matrix mismatch on the performance
of MPB. This paper aims to analyze the MPB’s performance under matrix
mismatch, and our contributions are:
* •
proposing a general framework to model various existing MPB schemes;
* •
deriving analytical expressions for the normalized output SINR as the
performance measure;
* •
discovering a threshold effect for MPB, i.e., due to matrix mismatch, the
performance of MPB degrades rapidly when the input signal to noise ratio (SNR)
is below a predicted threshold, and the main beam points to the directions of
the interferers;
* •
explaining how MPB works “blindly” by “steering vector competition”;
* •
discussing various factors that have impact on the threshold, and showing that
when there is an generalized eigenvalue that is infinite which is called _the
noise-free covariance matrix pair_ , the threshold SNR increases unboundedly
with the interference power;
* •
discussing several typical scenarios and showing that MPB is very vulnerable
to multiple periodical interferers.
The rest of the paper is organized as follows. Sec. II presents a general
framework of MPB to cover and reinterpret the basic ideas in [23, 24, 25, 26,
27, 28, 29, 30, 31], followed by a formulation of the matrix mismatch problem.
In Sec. III, we present the expressions for MPB’s weight vector and its
normalized output SINR, which uncovers the inherent threshold effect caused by
matrix mismatch. The discussion relies heavily on the approximation of the
generalized eigenvalue, which is derived in Appendix A using Gerschgorin
theorem[32, 33]. Sec. IV applies the Weyl-Lidskii type theorem in matrix
perturbation theory[32] to analyze MPB’s threshold SNR, and discusses it in
two typical scenarios. Finally, Sec. V presents simulation results to verify
our theoretical analysis and Sec. VI concludes the whole paper.
## II Problem Formulation
### II-A Signal Model
Figure 1: Block diagram of the matrix pair beamformer in a CDMA system. Data
corresponding to the $k$ symbol are blocked and projected onto two spaces to
produce signal channel snapshot $\mathbf{x}_{\mathcal{S}}(k)$ and interference
channel snapshot $\mathbf{X}_{\mathcal{I}}(k)$, respectively.
Consider an antenna array of $L$ isotropic elements that receives $D+1$
signals from a far field. After preprocessing (e.g. mixing, filtering, etc.)
and sampling, the output $L\times 1$ array vector can be written as
$\mathbf{x}(n)=\sum_{i=0}^{D}\sqrt{P_{i}}s_{i}(n)\mathbf{a}_{i}+\mathbf{v}(n),$
(1)
where $s_{i}(n)$ is the discrete sequence of the $i$th signal with normalized
power; $P_{i}$ is its power; $\mathbf{a}_{i}$ is the steering vector, which
depends on the DOA and the array geometry; $\mathbf{v}(n)$ is the additive
white Gaussian noise (AWGN) vector with zero mean and covariance matrix
$\sigma^{2}\mathbf{I}$. Assume $s_{0}(n)$ is the SOI, and
$s_{1}(n),s_{2}(n),\ldots,s_{D}(n)$ model all the possible interferers like
MAI and jammer etc. This paper considers the case when $s_{0}(n)$ has some
inherent temporal structure expressed as
$s_{0}(n)=\sum_{k=-\infty}^{+\infty}b_{0}(k)c_{0}[n-\tau(k)],$ (2)
where $c_{0}(n)$ is known and supported on $0\leq n\leq N-1$. Thus, $s_{0}(n)$
can be viewed as a train of pulses, with each of them being a delayed and
scaled replica of $c_{0}(n)$. And $\tau(k)$ and $b_{0}(k)$ are the
corresponding delay and amplitude of the $k$th pulse. This model is common in
modern communication systems. For instance, in CDMA system, $c_{0}(n)$ is the
spreading code of the desired user, $b_{0}(k)$ is its data bits and $\tau(k)$
is the delay of the $k$th symbol. Without loss of generality, let us consider
$b_{0}(k)\in\\{\pm 1\\}$, $c_{0}(n)\in\\{\pm 1\\}$ and $\tau(k)=kN+n_{0}$.
Then $N$ is the processing gain and $n_{0}$ is the propagation delay.
### II-B A General Framework of the Matrix Pair Beamformer
We first propose a framework called _matrix pair beamformer_ (MPB) to cover
the common ideas in [23, 24, 25, 26, 27, 28, 29, 30, 31]. Our strategy is to
use orthogonal projection operation to model their ways of estimating the
covariance matrix pair. By this framework, we will find a common threshold
effect in these methods.
The steering vector $\mathbf{a}_{i}$ in (1) is a spatial signature of the
$i$th signal, which is different from other $\mathbf{a}_{j}$ so long as they
arrive from different directions. Beamformer is a spatial filter that exploits
such difference to pass the desired signal $s_{0}(n)$ while suppressing
$s_{1}(n)\ldots s_{D}(n)$ and $\mathbf{v}(n)$. A statistically optimum
beamformer [12, 15] generally requires at least, either explicitly or
implicitly, the information about the steering vector $\mathbf{a}_{0}$ and the
interference covariance matrix. The latter one may be replaced by the data
covariance matrix, so the remaining problem is how to acquire
$\mathbf{a}_{0}$. In DOA-based beamformer, it is calculated by the DOA and
array manifold information. As for training-based method or decision directed
method, it is inherent in the correlation vector between the reference signal
and the data vector.
To work “blindly”, i.e. without any explicit information of DOA, the methods
in [23, 24, 25, 26, 27, 28, 29, 30, 31] exploit the SOI’s temporal signature
$c_{0}(n)$ to acquire these spatial statistical information. Specifically, it
is implemented by two orthogonal projections and a generalized eigen-
decomposition of a covariance matrix pair. Hence, we refer to them as matrix
pair beamformer (MPB) in this literature. In Fig. 1, we summarize the common
structure of MPB. With the data segmentation, the array outputs corresponding
to the $k$th symbol of the desired user can be expressed in the following
matrix form:
$\displaystyle\mathbf{X}(k)\\!$
$\displaystyle\triangleq\\!\begin{bmatrix}\mathbf{x}(kN+n_{0})&\cdots&\mathbf{x}(kN+n_{0}+N-1)\end{bmatrix}=\\!\left[\sqrt{P_{0}}b_{0}(k)\right]\\!\mathbf{a}_{0}\mathbf{c}_{0}^{T}\\!\\!+\\!\\!\underbrace{\mathbf{A}_{I}\boldsymbol{\Theta}_{I}^{\frac{1}{2}}\mathbf{S}_{I}^{T}(k)\\!\\!+\\!\\!\mathbf{V}(k)}_{\mathbf{Z}(k)}$
(3)
where
$\displaystyle\mathbf{A}_{I}$
$\displaystyle\triangleq\big{[}\;\mathbf{a}_{1}\;\mathbf{a}_{2}\;\cdots\;\mathbf{a}_{D}\;\big{]}$
$\displaystyle\mathbf{c}_{0}$
$\displaystyle\triangleq\big{[}\;c_{0}(0)\;c_{0}(1)\;\cdots\;c_{0}(N-1)\;\big{]}^{T}$
$\displaystyle\mathbf{S}_{I}(k)$
$\displaystyle\triangleq\big{[}\;\mathbf{s}_{1}(k)\;\mathbf{s}_{2}(k)\;\cdots\;\mathbf{s}_{D}(k)\;\big{]}$
$\displaystyle\mathbf{s}_{i}(k)$
$\displaystyle\triangleq\big{[}\;s_{i}(kN+n_{0})\;\cdots\;s_{i}(kN+n_{0}+N-1)\;\big{]}^{T}$
$\displaystyle\mathbf{V}(k)$
$\displaystyle\triangleq\big{[}\;\mathbf{v}(kN+n_{0})\;\cdots\;\mathbf{v}(kN+n_{0}+N-1)\;\big{]}.$
$\displaystyle\boldsymbol{\Theta}_{I}$
$\displaystyle\triangleq\mathrm{diag}\\{P_{1},P_{2},\cdots,P_{D}\\}.$
Then, the $k$th data block in each antenna is projected onto two subspaces:
signal space $\mathcal{S}$ spanned by the SOI’s temporal signature vector
$\mathbf{h}_{\mathcal{S}}\\!=\\!\mathbf{c}_{0}/\sqrt{N}$, and a specifically
designed interference space
$\mathcal{I}\\!=\\!\mathcal{R}\\{\mathbf{H}_{\mathcal{I}}\\}$, respectively.
Without loss of generality, assume the columns of
$\mathbf{H}_{\mathcal{I}}\\!\\!\in\\!\\!\mathbb{C}^{N\times r_{\mathcal{I}}}$
are orthonormal. Then, the projections produce the signal snapshot
$\mathbf{x}_{\mathcal{S}}(k)\\!\\!=\\!\\!\mathbf{X}(k)\mathbf{h}_{\mathcal{S}}^{\ast}$
and the interference snapshot
$\mathbf{X}_{\mathcal{I}}(k)\\!\\!=\\!\\!\mathbf{X}(k)\mathbf{H}_{\mathcal{I}}^{\ast}$.
Define $\sigma_{\mathcal{S}_{0}}^{2}\\!\\!\triangleq\\!\\!NP_{0}$ and
$\sigma_{\mathcal{I}_{0}}^{2}\\!\\!\triangleq\\!\\!\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}^{H}\mathbf{c}_{0}\|^{2}$.
Then, the covariance matrices of $\mathbf{x}_{\mathcal{S}}(k)$ and
$\mathbf{X}_{\mathcal{I}}(k)$ are
$\displaystyle\mathbf{R}_{\mathcal{S}}$
$\displaystyle\triangleq\textsf{E}\\{\mathbf{x}_{\mathcal{S}}(k)\mathbf{x}_{\mathcal{S}}^{H}(k)\\}=\sigma_{\mathcal{S}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}^{H}_{0}+\mathbf{Q}_{\mathcal{S}}$
(4) $\displaystyle\mathbf{R}_{\mathcal{I}}$
$\displaystyle\triangleq\frac{1}{r_{\mathcal{I}}}\textsf{E}\\{\mathbf{X}_{\mathcal{I}}(k)\mathbf{X}_{\mathcal{I}}^{H}(k)\\}=\sigma_{\mathcal{I}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}^{H}_{0}+\mathbf{Q}_{\mathcal{I}}$
(5)
where $\sigma_{\mathcal{S}_{0}}^{2}$ and $\sigma_{\mathcal{I}_{0}}^{2}$ are
the SOI’s powers in the _signal channel_ and _interference channel_ (c.f. Fig.
1), respectively. $\mathbf{Q}_{\mathcal{S}}$ and $\mathbf{Q}_{\mathcal{I}}$
are the covariance matrices of the interference-plus-noise in them, defined by
$\displaystyle\mathbf{Q}_{\mathcal{S}}$
$\displaystyle=\textsf{E}\\{\mathbf{Z}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{Z}^{H}(k)\\}=\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}+\sigma^{2}\mathbf{I},$
(6) $\displaystyle\mathbf{Q}_{\mathcal{I}}$
$\displaystyle=\textsf{E}\\{\mathbf{Z}(k)\mathbf{H}_{\mathcal{S}}^{\ast}\mathbf{H}_{\mathcal{S}}^{T}\mathbf{Z}^{H}(k)\\}=\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}}\mathbf{A}_{I}^{H}+\sigma^{2}\mathbf{I},$
(7)
where
$\bm{\Phi}_{\mathcal{S}}\\!=\\!\sigma^{2}\textsf{INR}\cdot\bm{\Phi}_{\mathcal{S}_{0}}$,
$\bm{\Phi}_{\mathcal{I}}\\!=\\!\sigma^{2}\textsf{INR}\cdot\bm{\Phi}_{\mathcal{I}_{0}}$,
$\textsf{INR}\\!=\\!P_{1}/\sigma^{2}$, and
$\displaystyle\bm{\Phi}_{\mathcal{S}_{0}}=\bm{\Omega}_{I}^{\frac{1}{2}}\textsf{E}\left\\{\mathbf{S}_{I}^{T}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{S}_{I}^{\ast}(k)\right\\}\bm{\Omega}_{I}^{\frac{1}{2}},\qquad\bm{\Phi}_{\mathcal{I}_{0}}=\frac{1}{r_{\mathcal{I}}}\bm{\Omega}_{I}^{\frac{1}{2}}\textsf{E}\left\\{\mathbf{S}_{I}^{T}(k)\mathbf{H}_{\mathcal{I}}^{\ast}\mathbf{H}_{\mathcal{I}}^{T}\mathbf{S}_{I}^{\ast}(k)\right\\}\bm{\Omega}_{I}^{\frac{1}{2}},$
(8)
with $\bm{\Omega}_{I}\\!=\\!(\bm{\Theta}_{I}/\sigma^{2})/\textsf{INR}$. We can
see that $\bm{\Omega}_{I}$ is independent of INR and only depends on the
relative strength of the interferers. So are $\bm{\Phi}_{\mathcal{S}_{0}}$ and
$\bm{\Phi}_{\mathcal{I}_{0}}$, and we will use this conclusion in Sec. IV.
The MPB uses the eigenvector corresponding to the largest generalized
eigenvalue of the matrix pair
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$ as the weight vector
$\mathbf{w}$, which is the solution to the following equation
$\displaystyle\mathbf{R}_{\mathcal{S}}\mathbf{w}=\lambda_{\mathrm{max}}\mathbf{R}_{\mathcal{I}}\mathbf{w},$
(9)
where $\lambda_{\mathrm{max}}$ is the largest generalized eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$. $\mathbf{w}$ is applied
to $\mathbf{x}_{\mathcal{S}}(k)$ to yield the output
$\displaystyle
y_{o}(k)\\!\\!=\\!\\!\mathbf{w}^{H}\mathbf{x}_{\mathcal{S}}(k)\\!\\!=\\!\\!\underbrace{\left[\sigma_{\mathcal{S}_{0}}b_{0}(k)\right]\mathbf{w}^{H}\mathbf{a}_{0}}_{y_{S}(k)}\\!\\!+\\!\\!\underbrace{\mathbf{w}^{H}\mathbf{Z}(k)\mathbf{h}_{\mathcal{S}}^{\ast}}_{y_{I}(k)}$
(10)
where $y_{S}(k)$ and $y_{I}(k)$ are the output signal and interference-plus-
noise. Eq. (9) is also equivalent to
$\left(\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}\right)\mathbf{w}=(\lambda_{\mathrm{max}}-1)\mathbf{R}_{\mathcal{I}}\mathbf{w}.$
(11)
Using (4) and (5), we have
$\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}=(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}+\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}}.$
(12)
Since the columns of $\mathbf{H}_{\mathcal{I}}$ are orthonormal, i.e.
$\mathbf{H}_{\mathcal{I}}^{H}\mathbf{H}_{\mathcal{I}}=\mathbf{I}$, its
spectral norm is one and we can have
$\displaystyle\sigma_{\mathcal{I}_{0}}^{2}\triangleq\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}^{H}\mathbf{c}_{0}\|^{2}\leq\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}\|^{2}\|\mathbf{c}_{0}\|^{2}=\frac{NP_{0}}{r_{\mathcal{I}}}\leq\sigma_{\mathcal{S}_{0}}^{2}.$
The above inequality holds strictly when choosing
$\mathbf{H}_{\mathcal{I}}\neq\mathbf{h}_{\mathcal{S}}$ (either
$r_{\mathcal{I}}\\!>\\!1$ or
$\mathbf{H}_{\mathcal{I}}\\!\neq\\!\mathbf{h}_{\mathcal{S}}$ for
$r_{\mathcal{I}}\\!=\\!1$). Furthermore, it is commonly assumed [23, 24, 25,
26, 27, 28, 29, 30, 31] that
$\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$. Then,
$\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}}\\!=\\!\mathbf{O}$ in
(12), and the dominant eigenvector of
$(\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}},\mathbf{R}_{\mathcal{I}})=\left((\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H},\mathbf{R}_{\mathcal{I}}\right)$
is
$\displaystyle\mathbf{w}_{\mathrm{opt}}=\mu\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}=\mu^{\prime}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0},$
(13)
where $\mu$ and $\mu^{\prime}$ are scalars. Then $\mathbf{w}_{\mathrm{opt}}$
will maximize the output interference plus noise ratio (SINR) [15], and the
optimal SINR is
$\displaystyle\textsf{SINR}_{\textsf{opt}}=\left.\frac{\textsf{E}\\{|y_{S}(k)|^{2}\\}}{\textsf{E}\\{|y_{I}(k)|^{2}\\}}\right|_{\mathbf{w}_{\mathrm{opt}}}=\sigma_{\mathcal{S}_{0}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}$
(14)
All the methods in[23, 24, 25, 26, 27, 28, 29, 30, 31] share the structures
described above. They only differ in the dominant eigenvector searching
algorithm and the interference space $\mathcal{I}$ (i.e.
$\mathbf{H}_{\mathcal{I}}$). In most existing approaches, $\mathcal{I}$ is a
one dimension space ($r_{\mathcal{I}}=1$). The pre- and post-correlation
(PAPC) scheme[23, 24, 25, 26, 27] uses
$\mathbf{R}_{\mathcal{I}}\\!=\\!\textsf{E}\\{\mathbf{x}(n)\mathbf{x}^{H}(n)\\}$,
thus it is equivalent to choosing $\mathbf{H}_{\mathcal{I}}$ as
$\mathbf{H}_{\mathcal{I}}=\big{[}\;0\;\;\cdots\;\;0\;\;1\;\;0\;\;\cdots\;\;0\;\big{]}^{T}.$
(15)
where only one component in $\mathbf{H}_{\mathcal{I}}$ is nonzero. The Maximin
scheme in [28] and [29] employs a _monitor filter_ to isolate the
interference, which can be interpreted as
$\mathbf{H}_{\mathcal{I}}=\mathbf{c}_{0}\odot\left[\;1\;\;e^{j2\pi
f_{\mathrm{MF}}}\;\;\cdots\;\;e^{j2\pi f_{\mathrm{MF}}(N-1)}\;\right]^{T},$
(16)
where $f_{\mathrm{MF}}\in(0,1]$ is the normalized center frequency of monitor
filter, and $\odot$ is the Hadamard product.
### II-C Matrix Mismatch
We see that the MPB relies heavily on the key assumption that
$\mathbf{Q}_{\mathcal{S}}=\mathbf{Q}_{\mathcal{I}}$, namely the interferers
have the same second order statistics in the two channels. By (6)–(8), we know
it is valid when each interferer is random enough in the temporal domain, say
directional white noise (c.f. Sec. IV-B1). However, it is generally not
satisfied, especially when there are multiple deterministic periodical
interferers, like tones and MAI (c.f. Sec. IV-B2). This is because when the
interferers are deterministic and periodical, the expectations in (8) can be
eliminated. Then $\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$
requires
$\mathbf{S}_{I}^{T}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\\!=\\!\mathbf{S}_{I}^{T}(k)\mathbf{H}_{\mathcal{I}}^{\ast}$,
which is highly improbable when
$\mathbf{h}_{\mathcal{S}}\\!\neq\\!\mathbf{H}_{\mathcal{I}}$. We term this as
_“matrix mismatch”_. To our best knowledge, very little effort has been
devoted to analyze this problem. Therefore, we will investigate the
performance of MPB in this more general case. Before we proceed, we define the
normalized output SINR to measure performance degradation with respect to that
of no matrix mismatch.
###### Definition 1
The _normalized output SINR_ is defined as the actual output SINR of MPB
normalized by the optimal value, i.e.
$\displaystyle\textsf{G}\triangleq\left.\frac{\textsf{E}\\{|y_{S}(k)|^{2}\\}}{\textsf{E}\\{|y_{I}(k)|^{2}\\}}\right|_{\mathbf{w}}\cdot\frac{1}{\textsf{SINR}_{\textsf{{opt}}}}$
(17)
where $\textsf{SINR}_{\textsf{opt}}$ is given by (14), and $\mathbf{w}$ is the
solution to (11) without the assumption of
$\mathbf{Q}_{\mathcal{S}}=\mathbf{Q}_{\mathcal{I}}$.
G generally depends on the input SNR and the interference powers. So a
reasonable way to characterize the performance is to plot G against the input
$\textsf{SNR}\triangleq\sigma_{\mathcal{S}_{0}}^{2}/\sigma^{2}$, when fixing
the interference powers. In the following sections, we will base our analysis
on $\textsf{G}(\textsf{SNR})$, which we will refer to as _operating curve_.
Moreover, we assume infinite sample size so that the finite sample effect is
ignored.
## III Performance Analysis of the Matrix Pair Beamformer with Matrix
Mismatch
In this section, we will derive the operating curve of MPB, and discuss how it
works blindly.
### III-A Operating Curve of Matrix Pair Beamformer
We base our discussions on the following assumptions, and summarize the main
result in theorem 1.
###### Assumption 1
The spacing of DOA between any two signals is large enough (greater than a
mainlobe), so that $\\{\mathbf{a}_{i}\\}_{i=0}^{D}$ are linearly independent
and the projection of $\mathbf{a}_{0}$ onto
$\mathrm{span}\\{\\!\mathbf{a}_{i}\\}_{i=1}^{D}$ is much less than
$\|\mathbf{a}_{0}\|$.
###### Assumption 2
The steering vectors of all signals are normalized so that
$\|\mathbf{a}_{i}\|^{2}=L$, ($i=0,1,\ldots,D$).
###### Theorem 1 (operating curve)
The normalized output SINR of MPB with matrix mismatch is
$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{\displaystyle
P_{I}\\!+\\!1}{\displaystyle
P_{I}/\left[1\\!-\\!{\textsf{SNR}_{\textsf{T}0}}/{\textsf{SNR}}\right]^{2}\\!+\\!1}G_{U},&\textsf{SNR}>\textsf{SNR}_{\textsf{T2}}\\\
{\displaystyle\left[\frac{1\\!\\!+\\!\\!K_{0}}{1\\!\\!-\\!\\!{\textsf{SNR}}/{\textsf{SNR}_{\textsf{T}0}}\\!\\!+\\!\\!K_{0}\left({L\beta}\textsf{SNR}/{N}\\!\\!+\\!\\!1\right)}\right]^{2}G_{L},}&\textsf{SNR}<\textsf{SNR}_{\textsf{T1}}\end{array}\right.$
(20)
where $G_{U}$ and $G_{L}$ are the normalized output SINR when
$\textsf{SNR}\\!\\!=\\!\\!+\infty$ and $\textsf{SNR}\\!\\!=\\!\\!0$
($-\infty$dB), respectively.
$\beta\\!\\!\triangleq\\!\\!N\sigma_{\mathcal{I}_{0}}^{2}/\sigma_{\mathcal{S}_{0}}^{2}$
is the normalized power leakage ratio (PLR) in interference channel.
$P_{I}\\!\\!\triangleq\\!\\!1/G_{U}\\!\\!-\\!\\!1$ is the output interference
to noise ratio, and $\textsf{SNR}_{\textsf{T}0}$ is the empirical threshold
SNR. Their expressions are
$\displaystyle
G_{U}=\frac{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}}\cdot\frac{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{Q}_{\mathcal{S}}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}$
(21)
$\displaystyle\textsf{SNR}_{\textsf{T0}}=\frac{N}{L}\cdot\frac{1}{[{(N-\beta)}/{(\gamma_{1})^{+}}-\beta]^{+}}$
(22) $\displaystyle
K_{0}=\frac{\beta+\frac{N}{L}/\textsf{SNR}_{\textsf{T}0}}{N-\beta}\left(\gamma_{1}-\frac{N-\beta}{\beta}\right)^{+}$
(23)
where $(\cdot)^{+}=\max\\{\cdot,0\\}$, and $\gamma_{1}$ is the largest nonzero
generalized eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.111If
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
has less than $D$ nonzero eigenvalues, pad them with zeros up to $D$ and order
them decreasingly. The SNR at which $\textsf{G}(\textsf{SNR})$ is close to
$G_{L}$ and $G_{U}$ (within $3$dB), which are given by
$\displaystyle\textsf{SNR}_{\textsf{T}1}=\left(1-\sqrt{\frac{1}{2}}\right)\textsf{SNR}_{\textsf{T}0},\qquad\textsf{SNR}_{\textsf{T}2}=\left(1-\sqrt{\frac{P_{I}}{2P_{I}+1}}\right)^{-1}\\!\\!\\!\textsf{SNR}_{\textsf{T}0},$
(24)
respectively.
Though we will give expression for $G_{L}$ in Sec. III-D, its specific values
are of no interest to us. Fig. 2 shows a typical curve of
$\textsf{G}(\textsf{SNR})$. We plot the curve in failure area and operating
area given in (20), and connect their ends by a dashed line. We can see that
the performance of beamformer degrades rapidly when the input SNR is below
$\textsf{SNR}_{\textsf{T2}}$. And it fails completely after reaching
$\textsf{SNR}_{\textsf{T1}}$. Therefore, matrix mismatch causes a threshold
effect in MPB, and $\textsf{SNR}_{\textsf{T2}}$ is a critical parameter to be
optimized.
There are two special cases of $\textsf{G}(\textsf{SNR})$. The first one is
$\textsf{SNR}_{\textsf{T1}}\\!\\!=\\!\\!\textsf{SNR}_{\textsf{T2}}\\!\\!=\\!\\!0$
($-\infty$dB). This happens when there is no matrix mismatch, i.e.
$\mathbf{Q}_{\mathcal{S}}\\!\\!=\\!\\!\mathbf{Q}_{\mathcal{I}}$, so that
$\gamma_{1}\\!\\!=\\!\\!0$ and $\textsf{SNR}_{\textsf{T}i}$ given by (22) and
(24) are zero. Then, (20) implies the operating curve is a horizontal line as
shown in Fig. 2. Another interesting case happens in PAPC schemes mentioned in
II-B, whose PLR is $\beta\\!\\!=\\!\\!1\\!\\!>\\!\\!0$. If it further
satisfies $(N-\beta)/\gamma_{1}\\!\\!<\\!\\!\beta$, then by (22) and (24),
$\textsf{SNR}_{\textsf{T}1}\\!\\!=\\!\\!\textsf{SNR}_{\textsf{T}2}=+\infty$
and $\textsf{G}(\textsf{SNR})$ only has the failure area, which decreases in
the order of $\mathcal{O}(\textsf{SNR}^{-2})$ (c.f. Fig. 2). In the rest of
the section, we will derive $\textsf{G}(\textsf{SNR})$ and reveal how MPB
works blindly under matrix mismatch. The discussion of
$\textsf{SNR}_{\textsf{T0}}$ is left to Sec. IV.
\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Regular}
\put(36.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}$}}
\put(61.0,25.0){\small{$\textsf{SNR}_{\textsf{T2}}$}} \end{overpic}
\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Operating}
\put(20.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}=\textsf{SNR}_{\textsf{T2}}=-\infty$
dB}} \end{overpic}
\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Failure}
\put(20.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}=\textsf{SNR}_{\textsf{T2}}=+\infty$
dB}} \put(66.0,57.0){\small{$\propto-20\log\textsf{SNR}$}} \end{overpic}
Figure 2: (a) A typical curve of $\textsf{G}(\textsf{SNR})$ has failure area,
threshold area and operating area, separated by $\textsf{SNR}_{\textsf{T1}}$
and $\textsf{SNR}_{\textsf{T2}}$. (b) Without matrix mismatch, the curve has
only operating area. (c) A curve has only failure area for some cases in PAPC.
### III-B Derivation of the Weight Vector for MPB
As a step to prove Theorem 1, we first derive the expression of the weight
vector for MPB. By the arguments in Sec. II-B, it is the solution to (11).
Define
$\bm{\Phi}_{\Delta}\triangleq\bm{\Phi}_{\mathcal{S}}-\bm{\Phi}_{\mathcal{I}}$.
Then, by (6), (7) and (12), the problem becomes solving the following
generalized eigenequation
$\displaystyle\left[(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}+\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\right]\mathbf{w}=(\lambda_{\mathrm{max}}-1)\mathbf{R}_{\mathcal{I}}\mathbf{w},$
(25)
where $\lambda_{\mathrm{max}}$ is the largest eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$. Sec. II-B already gave
the result for $\mathbf{w}$ when there is no matrix mismatch, i.e.
$\bm{\Phi}_{\Delta}\\!\\!=\\!\\!\mathbf{O}$ (or
$\mathbf{Q}_{\mathcal{S}}\\!\\!=\\!\\!\mathbf{Q}_{\mathcal{I}}$). We concluded
that it is optimal in the sense of maximizing the output SINR. However, in the
presence of matrix mismatch, we have
$\bm{\Phi}_{\Delta}\\!\\!\neq\\!\\!\mathbf{O}$, which is the key challenge for
solving (25). To deal with this problem, we first have the following two
observations:
* •
By left-multiplying
$\frac{1}{\lambda_{\mathrm{max}}-1}\mathbf{R}_{\mathcal{I}}^{-1}$ to both
sides of (25), we can see that $\mathbf{w}$ can be expressed as
$\displaystyle\mathbf{w}=\eta_{0}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\eta_{i}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{i}$
(26)
i.e. it is a linear combination of
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{1},\ldots,\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{D}$,
where
$\eta_{0}=\frac{\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2}}{\lambda_{\mathrm{max}}-1}\mathbf{a}_{0}^{H}\mathbf{w}$,
$\mathbf{a}_{i}$ is the $i$th column of $\mathbf{A}_{I}$, and $\eta_{i}$ is
the $i$th component of
$\bm{\eta}=\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\mathbf{w}$. To determine
$\eta_{0},\eta_{1},\ldots,\eta_{D}$, we only need to substitute (26) back into
(25) and solve a linear equation. However, we will immediately discover that
the solution is intractable for further analysis because
$\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I}$ and
$\bm{\Phi}_{\Delta}$ are not diagonal. To overcome this, we need the next
observation.
* •
Suppose we can factorize $\bm{\Phi}_{\Delta}$ into diagonal form such that
$\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}=\mathbf{A}_{\epsilon}\bm{\Gamma}\mathbf{A}_{\epsilon}$
and
$\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}=\mathbf{I}$,
where $\bm{\Gamma}$ is a diagonal matrix. Then, by repeating the above
procedure except for replacing $\mathbf{a}_{1},\ldots,\mathbf{a}_{D}$ by
$\mathbf{a}_{\epsilon_{1}},\ldots,\mathbf{a}_{\epsilon_{D}}$, with
$\mathbf{a}_{\epsilon_{i}}$ being the $i$th column of $\mathbf{A}_{\epsilon}$,
we can solve $\mathbf{w}$ as
$\displaystyle\mathbf{w}=\eta_{0}\left[\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\frac{\gamma_{i}\tilde{\psi}_{T_{i}}}{\lambda_{\mathrm{max}}-1-\gamma_{i}}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{i}}\right],$
(27)
where $\gamma_{i}$ is the $i$th diagonal component of $\bm{\Gamma}$,
$\eta_{0}$ is an arbitrary constant, and
$\tilde{\psi}_{T_{i}}=\mathbf{a}_{\epsilon_{i}}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$.
So far, the only thing left is to figure out a way to factorize
$\bm{\Phi}_{\Delta}$ so that it meets the above requirement. Consider the
generalized eigen-decomposition of the matrix pair
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
By the simultaneous diagonalization theorem (c.f. [34], pp.133), there exists
a nonsingular matrix $\mathbf{T}$ such that
$\displaystyle\mathbf{T}^{H}\bm{\Phi}_{\Delta}\mathbf{T}=\bm{\Gamma},\qquad\mathbf{T}^{H}(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\mathbf{T}=\mathbf{I},$
(28)
where $\bm{\Gamma}=\mathrm{diag}(\gamma_{1},\gamma_{2},\ldots,\gamma_{D})$ is
a diagonal matrix whose diagonal terms are the generalized eigenvalues of
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
Define $\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$. Then, by
(28), we can verify that
$\displaystyle\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}\bm{\Gamma}\mathbf{T}^{-1}\mathbf{A}_{I}^{H}=\mathbf{A}_{\epsilon}\bm{\Gamma}\mathbf{A}_{\epsilon}^{H}$
$\displaystyle\mathbf{A}_{\epsilon}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}=\mathbf{T}^{-1}\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}=\mathbf{I}.$
Therefore, (28) is the exact decomposition of $\bm{\Phi}_{\Delta}$ we are
looking for. Furthermore, by
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$, we notice that
each column of $\mathbf{A}_{\epsilon}$ is in fact a linear combination of
$\mathbf{a}_{1},\ldots,\mathbf{a}_{D}$. This, together with (27), implies that
$\mathbf{w}$ is still a a linear combination of
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{1},\ldots,\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{D}$,
just as that in (26).
### III-C How MPB Works Blindly
To fully understand the behavior of MPB given by (27) and how it works
blindly, we still need the expressions of $\lambda_{\mathrm{max}}$ and
$\gamma_{i}$. However, $\lambda_{\mathrm{max}}$ and $\gamma_{i}$ are the
solutions to
$\displaystyle\mathrm{det}\left\\{(\lambda-1)\mathbf{R}_{\mathcal{I}}-(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}-\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\right\\}=0$
(29)
$\displaystyle\mathrm{det}\left\\{\lambda(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}-\bm{\Phi}_{\Delta}\right\\}=0$
(30)
which are polynomial equations of degree $D+1$ and $D$, respectively. It is
known that there are no general closed-form solutions if their degrees are
higher than four. Thus, our approach here is to derive approximate expressions
for $\lambda_{\mathrm{max}}$ and $\gamma_{i}$. The main idea is to transform
(29) and (30) into eigenvalue problems of a diagonal matrix perturbed by a
small term. Then we can approximate the eigenvalues by the diagonal entries,
and bound the error using matrix perturbation theory.
We first discuss $\lambda_{\mathrm{max}}$. Before this, let’s introduce the
following identity which will be used repeatedly in the analysis and can be
derived from Properties 16 and 17 in [34, pp.5].
$\displaystyle\det(\lambda\mathbf{I}-\mathbf{X}\mathbf{Y})=\lambda^{m-n}\cdot\det(\lambda\mathbf{I}-\mathbf{Y}\mathbf{X})$
(31)
where $\mathbf{X}\in\mathbb{C}^{m\times n}$ and
$\mathbf{Y}\in\mathbb{C}^{n\times m}$. Substituting the factorization of
$\bm{\Phi}_{\Delta}$ in (28) into (29) and using (31) as well as
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$, we can have the
equivalent form of (29) as below
$\displaystyle(\lambda-1)^{L-D-1}\cdot\det\mathbf{R}_{\mathcal{I}}\cdot\det\left\\{(\lambda-1)\mathbf{I}-\begin{bmatrix}\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2}&\mathbf{0}^{T}\\\
\mathbf{0}&\bm{\Gamma}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}&\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}\\\
\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}&\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}\end{bmatrix}\right\\}=0.$
(32)
Then, (32) implies that the solution of (29) is the same as the eigenvalues of
the following matrix
$\displaystyle\mathbf{M}=\begin{bmatrix}\gamma_{0}+1&\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\bm{\psi}_{T}^{H}\\\
\frac{\sqrt{\frac{\sigma^{2}}{L}}}{\frac{L\beta}{N}\textsf{SNR}+1}\bm{\Gamma}\bm{\psi}_{T}&\bm{\Gamma}+\mathbf{I}\end{bmatrix}$
(33)
except for the multiplicity of ones, where we have used the second equality of
(28). And $\gamma_{0}$ and $\bm{\psi}_{T}$ are
$\displaystyle\gamma_{0}\triangleq(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\qquad\bm{\psi}_{T}\triangleq\left(\frac{L\beta}{N}\textsf{SNR}+1\right)\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
(34)
Before proceeding on, we cite the following two lemmas. The first one is
summarized from Lemma 1 and Lemma 2 in [35], and the second one can be found
in [33, pp.344].
###### Lemma 1
The quantities $\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
and $\bm{\psi}_{T}$ have the following results
$\displaystyle\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\approx\frac{L}{\sigma^{2}}\frac{N}{{L\beta}\textsf{SNR}+N},\qquad\|\boldsymbol{\psi}_{T}\|\approx\sqrt{\frac{L}{\sigma^{2}}\kappa_{0}}\ll\sqrt{\frac{L}{\sigma^{2}}}.$
where $\beta$ is the PLR defined in Theorem 1, and $\kappa_{0}$ is a very
small positive number independent of SNR.
###### Lemma 2 (Gerschgorin)
Let matrix $\mathbf{A}\in\mathbb{C}^{n\times n}$ and $a_{ij}$ be its $i,j$th
element. Define
$\displaystyle R_{i}(\mathbf{A})\triangleq\sum_{j=1,j\neq
i}^{n}|a_{i,j}|,\quad 1\leq i\leq n$
as the _deleted absolute row sums_ of $\mathbf{A}$ . Then all the eigenvalues
of $\mathbf{A}$ are in the union of $n$ disks, i.e.
$\displaystyle\lambda(\mathbf{A})\in\bigcup_{i=1}^{n}G_{i}(\mathbf{A}),$
where
$G_{i}(\mathbf{A})\\!\\!\triangleq\\!\\!\\{z\\!\\!\in\\!\\!\mathbb{C}\\!\\!:\\!\\!|z\\!\\!-\\!\\!a_{ii}|\\!\\!\leq\\!\\!R_{i}(\mathbf{A})\\}$
is the $i$th Gerschgorin disk. Moreover, if a union of $k$ disks forms a
connected region disjoint from all the remaining disks, then there are $k$
eigenvalues of $\mathbf{A}$ in this region.
By Lemma 1, the off-diagonal terms of each row in (33) are much smaller than
the corresponding diagonal terms. Then, according to lemma 2, its eigenvalues
are approximately $\gamma_{0}\\!+\\!1,\ldots,\gamma_{D}\\!+\\!1$. We will give
a bound for the approximation error in the next subsection. Now we directly
use this approximation to discuss the behavior of MPB. Without loss of
generality, assume $\gamma_{1}>\ldots>\gamma_{D}$, then
$\displaystyle\lambda_{\mathrm{max}}\approx\max\\{\gamma_{0}+1,\gamma_{1}+1,\ldots,\gamma_{D}+1,1\\}=\max\\{\gamma_{0}+1,\gamma_{1}+1\\}.$
(35)
Substituting the expression of
$\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$ in Lemma 1
into its definition in (34), we can have
$\displaystyle\gamma_{0}=(N-\beta)\frac{L\textsf{SNR}}{L\beta\textsf{SNR}+N}$
(36)
which is a monotonically increasing function of SNR. Furthermore, we will show
later that $\gamma_{1}\\!\\!+\\!\\!1$ is approximate to one of the generalized
eigenvalue of $(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ and is
almost independent of SNR. Hence, there is a threshold
$\textsf{SNR}_{\textsf{T0}}$ such that $\lambda_{\mathrm{max}}$ switches from
$\gamma_{1}+1$ to $\gamma_{0}+1$ when SNR exceeds
$\textsf{SNR}_{\textsf{T0}}$. Its expression is in (22) and can be derived by
setting (36) to $\gamma_{1}$ and solving SNR. As a result,
* •
When $\textsf{SNR}>\textsf{SNR}_{\textsf{T0}}$,
$\lambda_{\mathrm{max}}\approx\gamma_{0}+1$. If $\gamma_{0}+1$ can be much
greater than $\gamma_{1}+1$ as SNR increases, then (27) implies
$\mathbf{w}\approx\mu_{1}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$. Hence,
$\mathbf{a}_{0}$ is the dominant steering vector, and the beamformer can
operate properly by steering the mainlobe to the direction of the desired
signal.
* •
When $\textsf{SNR}<\textsf{SNR}_{\textsf{T0}}$,
$\lambda_{\mathrm{max}}\approx\gamma_{1}+1$. Thus
$\mathbf{w}\approx\mu_{1}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{1}}$,
and $\mathbf{a}_{\epsilon_{1}}$ is the dominant steering vector in (27). By
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$,
$\mathbf{a}_{\epsilon_{1}}$ is a linear combination of all the interference
steering vectors. Therefore, the beamformer fails for its mainlobe pointing to
the directions of the interferers. If $\beta\\!\\!\neq\\!\\!0$, then there
would be SOI in $\mathbf{R}_{\mathcal{I}}$, which makes MPB treat it as
interference and null it out. This explains the reason why
$\textsf{G}(\textsf{SNR})$ decreases in the order of
$\mathcal{O}(\textsf{SNR}^{-2})$ in Fig. 2.
Now, we further analyze how this threshold effect happens. By the
factorizations (28), we can rewrite
$\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}$ on the left hand side of
(25) as
$\displaystyle\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}=\gamma_{0}\cdot\mathbf{a}_{S_{0}}\mathbf{a}_{S_{0}}^{H}+\sum_{i=1}^{D}\gamma_{i}\cdot\mathbf{a}_{\epsilon_{i}}\mathbf{a}_{\epsilon_{i}}^{H},$
(37)
where
$\mathbf{a}_{S_{0}}\triangleq\mathbf{a}_{0}/[\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}]^{\frac{1}{2}}$.
It is obvious that $\gamma_{0}$ is the generalized eigenvalue of the matrix
pair
$\left((\sigma_{\mathcal{S}_{0}}^{2}\\!\\!-\\!\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
with $\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{S_{0}}$ being its “normalized”
eigenvector. By applying (31) to (30), we can also verify that
$\gamma_{1},\gamma_{2},\ldots,\gamma_{D}$ are the generalized eigenvalues of
$\left(\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
with the eigenvectors being
$\\{\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{i}}\\}_{i=1}^{D}$.
Comparing (37) with the left hand side of (25), we can find that $\gamma_{0}$
actually measures the mismatch of the desired signal between the signal
channel and the interference channel in Fig. 1, and
$\\{\gamma_{i}\\}_{i=1}^{D}$ measures that of the interferers. From the
previous discussion, we know that
$\lambda_{\mathrm{max}}=\max\\{\gamma_{0}+1,\gamma_{1}+1\\}$, and $\mathbf{w}$
takes $\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$ or
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{1}}$ depending on which
one of $\gamma_{0}+1$ and $\gamma_{1}+1$ is larger. Therefore, MPB _blindly_
chooses the one with the largest mismatch metrix in (37) as its steering
vector. We term this as “ _steering vector competition_ ”. The threshold
happens when the SNR is large enough so that $\gamma_{0}$ exceeds
$\gamma_{1}$. If there is no mismatch, i.e. $\bm{\Phi}_{\Delta}=\mathbf{O}$ so
that $\gamma_{i}=0$, then the desired signal always wins out in the
competition ($\gamma_{0}>\gamma_{1}$), and MPB can work properly for all SNR.
### III-D Proof of Theorem 1
Let us return to our original problem of operating curve for MPB and prove
Theorem 1. We first use the expression of $\mathbf{w}$ in (27) to derive
$\textsf{G}(\textsf{SNR})$ in term of $\lambda_{\mathrm{max}}$. The result is
given by the following lemma, with its proof in Appendix B.
###### Lemma 3
The normalized output SINR in (17) can be expressed as the following function
of $\lambda_{\mathrm{max}}$
$\displaystyle\textsf{G}(\textsf{SNR})=\frac{\displaystyle\left[1+\frac{N}{L\beta\textsf{SNR}+N}\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}}{\displaystyle\left[\psi_{I}(\lambda_{\mathrm{max}})-\frac{L\beta\textsf{SNR}}{L\beta\textsf{SNR}+N}\psi_{S}^{2}(\lambda_{\mathrm{max}})\right]+\left[1+\frac{N}{L\beta\textsf{SNR}+N}\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}}.$
(38)
where $\psi_{S}(\lambda_{\mathrm{max}})$ and
$\psi_{I}(\lambda_{\mathrm{max}})$ are in the following forms with
$\psi_{T_{i}}$ being the $i$th component of $\bm{\psi}_{T}$.
$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})\\!\\!\triangleq\\!\\!\frac{\sigma^{2}}{L}\sum_{i=1}^{D}\\!\\!\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}|\psi_{T_{i}}|^{2},\qquad\psi_{I}(\lambda_{\mathrm{max}})\\!\\!\triangleq\\!\\!\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}\\!(\gamma_{i}\\!\\!+\\!\\!1)\\!\left[\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}\right]^{2}\\!\\!\\!|\psi_{T_{i}}|^{2}.$
(39)
To further simplify (38), we need the expression of $\lambda_{\mathrm{max}}$.
In the previous subsection, we used the approximation (35) to analyze MPB. The
following lemma quantifies how precise it is by bounding the approximation
error. Besides checking the validity of (35), this is also critical in
deriving $\textsf{G}(\textsf{SNR})$.
###### Lemma 4
The largest generalized eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$ defined in (4) and (5)
satisfies
$\displaystyle|\lambda_{\mathrm{max}}-(\lambda_{a}+1)|\leq\lambda_{a}\cdot
f\left(\frac{\lambda_{b}}{\lambda_{a}}\right)$ (40)
where $\lambda_{a}\triangleq\max\\{\gamma_{0},\gamma_{1}\\}$ and
$\lambda_{b}\triangleq\min\\{\gamma_{0},\gamma_{1}\\}$. $f(x)$ and $\delta$
are defined as
$\displaystyle f(x)$
$\displaystyle\triangleq\frac{1}{2}\left[1\\!\\!-\\!\\!x\\!\\!-\\!\\!\sqrt{(1\\!\\!-\\!\\!x)^{2}\\!\\!-\\!\\!4\delta|x|}\right],\quad\delta\triangleq\frac{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}{\frac{L\beta}{N}\textsf{SNR}+1}\ll
1,\quad
x\in(-\infty,1\\!-\\!2\gamma_{-}]\\!\cup\\![1\\!+\\!2\gamma_{+},+\infty)$
with
$\gamma_{\pm}\\!\triangleq\\!\sqrt{\delta^{2}\\!+\\!\delta}\\!\pm\\!\delta$.
Furthermore, $0\\!\leq\\!f(x)\\!\leq\\!\max\\{\delta,\gamma_{-}\\}\\!\ll\\!1$
when $x\in(-\infty,1\\!-\\!2\gamma_{-}]$.
###### Proof:
The key idea here is to design the appropriate similarity transform for
$\mathbf{M}$ in (33) and apply Lemma 2. Our detailed proof can be found in
Appendix A. ∎
Now we are ready to derive the expression of $\textsf{G}(\textsf{SNR})$. We
first consider the trivial case of
$-1\\!\\!\leq\\!\\!\gamma_{1}\\!\\!\leq\\!\\!0$.222Since $\gamma_{i}$ is the
eigenvalue of
$\left(\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
$\gamma_{i}\\!+\\!1$ is for
$\left(\sigma_{\mathcal{I}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}_{0}^{H}\\!\\!+\\!\\!\mathbf{Q}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}}\right)$,
a positive definite pair. Hence $\gamma_{i}\\!+\\!1\\!\\!>\\!\\!0$. Since
$\gamma_{D}\\!\\!<\\!\\!\ldots\\!\\!<\\!\\!\gamma_{1}$, we can immediately
have from (39) that
$0\\!\\!<\\!\\!\psi_{S}(\lambda_{\mathrm{max}})\\!\\!<\\!\\!\frac{\sigma^{2}}{L}\|\bm{\psi}_{T}\|^{2}\\!\\!\ll\\!\\!1$
and
$0\\!\\!<\\!\\!\psi_{I}(\lambda_{\mathrm{max}})\\!\\!<\\!\\!\frac{\sigma^{2}}{L}\|\bm{\psi}_{T}\|^{2}\\!\\!\ll\\!\\!1$.
Therefore, (38) becomes $\textsf{G}(\textsf{SNR})\\!\\!\approx\\!\\!1$ for all
SNR.
Next, we are going to discuss the case of $\gamma_{1}\\!\\!>\\!\\!0$ in two
separate cases.
#### III-D1 $\gamma_{0}>\gamma_{1}$
Then $\lambda_{a}\\!=\\!\gamma_{0}$, $\lambda_{b}\\!=\\!\gamma_{1}$ and
$\lambda_{\mathrm{max}}\\!\approx\\!\gamma_{0}\\!+\\!1$. So long as SNR is
slightly larger than $\textsf{SNR}_{\textsf{T}0}$ such that
$\gamma_{0}\\!\\!>\\!\\!\gamma_{1}/(1\\!\\!-\\!\\!\sqrt{{\sigma^{2}}/{L}}\|\bm{\psi}_{T}\|)\\!\\!\approx\\!\\!\gamma_{1}$,
we have the following approximation for $\psi_{S}$:
$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})\approx\frac{\sigma^{2}}{L}\sum_{i=1}^{D}\\!\\!\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{i}}|\psi_{T_{i}}|^{2}<\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{1}}\frac{\sigma^{2}}{L}\sum_{i=1}^{D}|\psi_{T_{i}}|^{2}<\sqrt{\frac{\sigma^{2}}{L}}\|\bm{\psi}_{T}\|\ll
1.$
Moreover, we approximate $\psi_{I}$ by the following bound, which is
asymptotically tight with respect to $\frac{\gamma_{0}}{\gamma_{1}}$.333In
fact, this approximation is precise enough when $\gamma_{0}$ is reasonably
larger than $\gamma_{1}$, say, $\gamma_{0}>2\gamma_{1}$.
$\displaystyle\psi_{I}(\lambda_{\mathrm{max}})\approx\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}\\!(\gamma_{i}\\!\\!+\\!\\!1)\\!\left(\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{i}}\right)^{2}\\!\\!\\!|\psi_{T_{i}}|^{2}\leq\left(\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{1}}\right)^{2}\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}(\gamma_{i}\\!\\!+\\!\\!1)|\psi_{T_{i}}|^{2}.$
Substituting the above two approximations together with (36) and (22) into
(38), we can finally have
$\displaystyle\textsf{G}(\textsf{SNR})=\frac{P_{I}+1}{\displaystyle
P_{I}/\left[1-{\textsf{SNR}_{\textsf{T}0}}/{\textsf{SNR}}\right]^{2}+1}G_{U}$
where $G_{U}=1/(P_{I}+1)$ and $P_{I}$ is defined as
$\displaystyle
P_{I}\triangleq\left[\frac{(N-\beta)/\beta}{(N-\beta)/\beta-\gamma_{1}}\right]^{2}\frac{\sigma^{2}}{L}\sum_{i=1}^{D}(\gamma_{i}+1)|\psi_{T_{i}}|^{2}$
However, this expression is difficult to evaluate. Instead, we can compute
$G_{U}$ first and have $P_{I}\\!\\!=\\!\\!1/G_{U}\\!\\!-\\!\\!1$. Noticing
that $G_{U}$ is the output SINR as $\textsf{SNR}\\!\\!=\\!\\!\infty$, we can
ignore the term $\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}$ on the
left hand side of (25). Hence,
$\mathbf{w}_{\infty}\\!\\!=\\!\\!\mu\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\\!\\!=\\!\\!\mu^{\prime}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$,
and by (17) and (14), we have its expression in (21).
#### III-D2 $\gamma_{0}<\gamma_{1}$
Then $\lambda_{a}\\!=\\!\gamma_{1}$, $\lambda_{b}\\!=\\!\gamma_{0}$ and
$\lambda_{\mathrm{max}}\\!\approx\\!\gamma_{1}\\!+\\!1$. As a result, the term
corresponding to $\gamma_{1}$ would dominate $\psi_{S}$ and $\psi_{I}$ in
(39). To further evaluate these two terms, we need the bound in (40) to
measure how close $\lambda_{\mathrm{max}}$ is to $\gamma_{1}+1$, namely, we
use it to evaluate $\lambda_{\mathrm{max}}-(\gamma_{1}+1)$ in $\psi_{S}$ and
$\psi_{I}$:
$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})$
$\displaystyle\\!\approx\\!\frac{\sigma^{2}}{L}\frac{\gamma_{1}}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}|\psi_{T_{1}}|^{2}\\!\approx\\!\frac{\sigma^{2}}{L}\frac{1}{f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)}|\psi_{T_{1}}|^{2}\\!=\\!\left(\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1\right)\frac{\gamma_{1}}{\gamma_{0}}\cdot
g\left(\frac{\gamma_{0}}{\gamma_{1}}\right)$
$\displaystyle\psi_{I}(\lambda_{\mathrm{max}})$
$\displaystyle\\!\approx\\!\frac{\sigma^{2}}{L}\frac{(\gamma_{1}\\!\\!+\\!\\!1)\gamma_{1}^{2}}{[\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{1}\\!\\!+\\!\\!1)]^{2}}|\psi_{T_{1}}|^{2}\\!\approx\\!\frac{\sigma^{2}}{L}\frac{\gamma_{1}+1}{\left[f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)\right]^{2}}|\psi_{T_{1}}|^{2}\\!=\\!\frac{\gamma_{1}+1}{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}\left[\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}$
where
$g(x)=\frac{1}{2}[1\\!\\!-\\!\\!x\\!\\!+\\!\\!\sqrt{(1\\!\\!-\\!\\!x)^{2}\\!\\!-\\!\\!4\delta
x}]$. In fact, we can have the approximation that $g(x)\approx 1-x$, because
$\displaystyle\left|(1-x)-g(x)\right|=|f(x)|\ll 1$
Substituting the above expressions of $\psi_{S}$ and $\psi_{I}$ as well as the
approximation of $g(x)$ into (3), we get
$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle\\!\\!=\\!\\!\left[\frac{N}{L\beta\textsf{SNR}\\!\\!+\\!\\!N}\right]^{2}\\!\\!\left[\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\right]^{2}\\!\\!\left[\frac{\gamma_{1}\\!\\!+\\!\\!1}{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}\\!\\!-\\!\\!\frac{\frac{L\beta}{N}\textsf{SNR}}{\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1}\\!\\!+\\!\\!\left(\\!\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\\!\right)^{2}\right]^{-1}\\!\\!\\!\\!\\!\\!\\!\approx\\!\\!\left[\frac{N}{L\beta\textsf{SNR}\\!\\!+\\!\\!N}\right]^{2}\\!\\!\left[\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\right]^{2}\\!\\!G_{L},$
where
$G_{L}\triangleq\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}/(\gamma_{1}\\!\\!+\\!\\!1)$
is the output SINR when $\textsf{SNR}\\!\\!=\\!\\!0$ ($-\infty$dB). The last
approximation holds when
$\gamma_{0}\\!\\!<\\!\\!(1\\!\\!-\\!\\!\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|)\gamma_{1}$,
i.e. $\gamma_{0}$ is slightly smaller than $\gamma_{1}$, which implies
$\gamma_{1}^{2}/(\gamma_{1}\\!\\!-\\!\\!\gamma_{0})^{2}\\!\\!<\\!\\!1/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}<(\gamma_{1}+1)/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}$.
Furthermore, because $\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}\\!\\!\ll\\!\\!1$,
${\frac{L\beta}{N}\textsf{SNR}}/{(\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1)}\\!\\!<\\!\\!1\\!\\!\ll\\!\\!(\gamma_{1}+1)/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}$.
There is, however, no easier way to evaluate $G_{L}$ than its definition.
Fortunately, we are not interested in its specific values but the threshold
$\textsf{SNR}_{\textsf{T}1}$ that G reaches this value, as discussed in Sec.
III-A. Noticing that $\textsf{SNR}_{\textsf{T}0}=\infty$ when
$(N-\beta)/\beta<\gamma_{1}$ (c.f. (22)), we have following fact regarding
$\gamma_{1}$ and $\textsf{SNR}_{\textsf{T}0}$,
$\displaystyle\gamma_{1}=\frac{L(N-\beta)\textsf{SNR}_{\textsf{T}0}}{L\beta\textsf{SNR}_{\textsf{T}0}+N}+\left(\gamma_{1}-\frac{N-\beta}{\beta}\right)^{+}.$
Combining the above expression and (36), $\textsf{G}(\textsf{SNR})$ can be
reduced to
$\displaystyle\textsf{G}(\textsf{SNR})=\left[\frac{1+K_{0}}{1-{\textsf{SNR}}/{\textsf{SNR}_{\textsf{T}0}}+K_{0}\left(\frac{L\beta}{N}\textsf{SNR}+1\right)}\right]^{2}G_{L}$
(41)
with $K_{0}$ defined in (23). Finally, the following lemma provides an easier
way to compute $\textsf{SNR}_{\textsf{T}0}$ from $\gamma_{1}$. The proof can
also be found in Appendix A.
###### Lemma 5
$\gamma_{1},\ldots,\gamma_{D}$ are approximate to all the nonzero generalized
eigenvalues of
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
(padded up to $D$ with zeros if not enough), and is almost independent of SNR.
## IV Discussion of the Threshold
From Theorem 1 in Sec. III-A, we know that the empirical threshold
$\textsf{SNR}_{\textsf{T}0}$ is a key parameter for MPB. By the discussion in
Sec. III-C, it is the intersection of the curves $\gamma_{0}\\!\\!+\\!\\!1$
and $\gamma_{1}\\!\\!+\\!\\!1$, as shown in Fig. 3(a). Therefore, we need to
investigate these two parameters to gain deeper insight of
$\textsf{SNR}_{\textsf{T}0}$.
### IV-A General Results of $\gamma_{0}$ and $\gamma_{1}$
$\gamma_{0}$ has a simple expression of (36), from which we know that the
parameter $\beta$ is critical. Fig. 3(b) shows the curve of
$\gamma_{0}\\!+\\!1$ with different $\beta$. We see that, if $\beta\\!=\\!0$,
i.e. $\mathbf{H}_{\mathcal{I}}$ and $\mathbf{h}_{\mathcal{S}}$ are orthogonal,
then $\gamma_{0}\\!\\!=\\!\\!L\textsf{SNR}$ is unbounded as SNR goes to
infinity. Otherwise, there would be a limiting value of
$(N\\!-\\!\beta)/\beta$ so that $\gamma_{0}$ might never exceed $\gamma_{1}$
and $\textsf{SNR}_{\textsf{T}0}\\!\\!=\\!\\!+\infty$. Therefore,
$\beta\\!\\!=\\!\\!0$ is the best choice for $\gamma_{0}$.
(a) The curves of $\gamma_{0}\\!+\\!1$, $\gamma_{1}\\!\\!+\\!\\!1$ and
$\lambda_{\mathrm{max}}$ vs. SNR. ($\beta=0$)
(b) $\gamma_{0}\\!+\\!1$ against SNR with different $\beta$.
Figure 3: The curve of $\gamma_{0}+1$ and $\gamma_{1}+1$ against SNR. The
intersection of them is the empirical threshold $\textsf{SNR}_{\textsf{T}0}$.
$\gamma_{1}$ is another critical parameter that determines
$\textsf{SNR}_{\textsf{T}0}$. Fig. 3 shows that, as $\gamma_{1}$ increases,
its intersection with $\gamma_{0}$ moves rightward and
$\textsf{SNR}_{\textsf{T}0}$ increases. Therefore, knowing how to control
$\gamma_{1}$ is important in designing MPB. By Lemma 5, $\gamma_{1}$ is the
largest nonzero generalized eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.
(It is zero if there are less than $D$ nonzero eigenvalues and all of them are
negative.) To solve it directly, we need the roots of a polynomial eigen-
equation of order $L$, which has no general formula when $L\\!\\!>\\!\\!4$.
Instead, we resort to matrix perturbation theory again to derive a bound for
it. We first notice that the eigen-decomposition of
$(\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
is equivalent to $(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ and
their eigenvalues only differ by one. By (6) and (7), the eigenvalue of
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ is further equivalent to
that of
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$,
where
$\displaystyle\mathbf{Y}_{\mathcal{S}}=\frac{1}{\sigma^{2}}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}=\textsf{INR}\cdot\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}_{0}}\mathbf{A}_{I}^{H},\qquad\mathbf{Y}_{\mathcal{I}}=\frac{1}{\sigma^{2}}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}}\mathbf{A}_{I}^{H}=\textsf{INR}\cdot\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}_{0}}\mathbf{A}_{I}^{H}.$
(42)
We term $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ as _noise free
pair_ , since it can be viewed as the covariance matrix pair of MPB without
noise. Our strategy here is to view
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$
as a perturbed version of
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ and apply the results in
matrix perturbation theory to derive a bound for $\gamma_{1}\\!\\!+\\!\\!1$.
Before we proceed on, we cite a more general definition of the generalized
eigenvalue of a matrix pair[32].
###### Definition 2 (Generalized eigenvalue)
The generalized eigenvalue of a matrix pair
$\left(\mathbf{A},\mathbf{B}\right)$ is a one dimensional space, denoted as
$\langle\nu,\mu\rangle\triangleq\left\\{\tau\cdot[\nu\;\mu]:\tau\in\mathbb{C}\right\\}$,
where $[\nu\;\mu]$ is a $1\times 2$ vector satisfying
$\displaystyle\mu\cdot\mathbf{A}\mathbf{x}=\nu\cdot\mathbf{B}\mathbf{x},$
with $\mathbf{x}$ being its eigenvector. If $\mathbf{B}$ is nonsingular, then
$\lambda\\!\\!=\\!\\!\nu/\mu$ becomes the conventional definition.
Comparing to the conventional definition of generalized eigenvalues, this one
includes the special case of $\mathbf{B}$ being singular so that
$\mu\\!\\!=\\!\\!0$ and $\nu\\!\\!\neq\\!\\!0$, namely,
$\lambda\\!\\!=\\!\\!+\infty$. We will see its importance later. Besides, we
also need the following definition and lemma from [32, pp.315–316].
###### Definition 3 (Definite pair)
If a matrix pair $(\mathbf{A},\mathbf{B})$ consists of two Hermitian matrices
and
$\displaystyle
C(\mathbf{A},\mathbf{B})\triangleq\min_{\|\mathbf{x}\|=1}\sqrt{\left(\mathbf{x}^{H}\mathbf{A}\mathbf{x}\right)^{2}+\left(\mathbf{x}^{H}\mathbf{B}\mathbf{x}\right)^{2}}>0,$
then $(\mathbf{A},\mathbf{B})$ is called definite pair, and
$C(\mathbf{A},\mathbf{B})$ is its Crawford number.
###### Theorem 2 (Weyl-Lidskii type)
Assume $\left(\mathbf{A},\mathbf{B}\right)$ is a definite pair and
$\left(\mathbf{A}\\!+\\!\mathbf{E},\mathbf{B}\\!+\\!\mathbf{F}\right)$ is its
perturbed version. Let $\langle\alpha_{i},\beta_{i}\rangle$ and
$\langle\widetilde{\alpha}_{i},\widetilde{\beta}_{i}\rangle$,
($i\\!\\!=\\!\\!1,\ldots,n$), be their ordered generalized eigenvalues,
respectively. If
$\displaystyle{\sqrt{\|\mathbf{E}\|_{S}^{2}+\|\mathbf{F}\|_{S}^{2}}}<{C(\mathbf{A},\mathbf{B})},$
(43)
where $\|\cdot\|_{S}$ denotes spectral norm of a matrix, then
$(\widetilde{\mathbf{A}},\widetilde{\mathbf{B}})$ is a definite pair, and
$\displaystyle\frac{|\alpha_{1}\widetilde{\beta}_{2}-\beta_{1}\widetilde{\alpha}_{2}|}{\sqrt{|\alpha_{1}|^{2}+|\beta_{1}|^{2}}\sqrt{|\widetilde{\alpha}_{2}|^{2}+|\widetilde{\beta}_{2}|^{2}}}\leq\frac{\sqrt{\|\mathbf{E}\|_{S}^{2}+\|\mathbf{F}\|_{S}^{2}}}{C(\mathbf{A},\mathbf{B})},$
We want to apply Theorem 2 to
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$
and $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ to derive a bound
for its eigenvalues. However,
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ is not a definite pair,
for the null spaces of $\mathbf{Y}_{\mathcal{S}}$ and
$\mathbf{Y}_{\mathcal{I}}$ may have nontrivial interset (larger than
$\\{0\\}$). Therefore, we need to transform
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ into a definite pair.
Let $\mathcal{N}_{\mathcal{S}}$ and $\mathcal{N}_{\mathcal{I}}$ be the null
spaces of $\mathbf{Y}_{\mathcal{S}}$ and $\mathbf{Y}_{\mathcal{I}}$,
respectively, and
$\mathcal{N}_{0}\\!\triangleq\\!\mathcal{N}_{\mathcal{S}}\bigcap\mathcal{N}_{\mathcal{I}}$.
Let $\mathbf{E}_{0}$ be a matrix whose columns are the orthonormal basis of
$\mathcal{N}_{0}^{\perp}$, with $\mathbf{E}_{0}^{H}\mathbf{E}_{0}=\mathbf{I}$.
Then, by the determinant identity in (31),
$\left(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I}\right)$
has the same generalized eigenvalues as
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$,
except for the multiplicity of ones. Similarly,
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
has the same eigenvalue as
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ does except for the
multiplicity of zero. Therefore, instead, we can apply Theorem 2 to
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
to derive the bound. The Crawford number of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
is
$\displaystyle C_{Y}$
$\displaystyle=\min_{\begin{subarray}{c}\mathbf{w}\in\mathcal{N}_{0}^{\perp},\;\|\mathbf{w}\|=1\end{subarray}}\sqrt{\left(\mathbf{w}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{w}\right)^{2}\\!\\!+\\!\\!\left(\mathbf{w}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{w}\right)^{2}}>0,$
(44)
which means
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
is a positive pair. Before applying Theorem 2, we first analyze the dependency
of $C_{Y}$ on the interference power. Substituting (42) into (44), we have
$C_{Y}\\!=\\!\textsf{INR}\cdot C_{Y_{0}}$, where
$\displaystyle C_{Y_{0}}$
$\displaystyle=\min_{{\mathbf{w}\in\mathcal{N}_{0}^{\perp},\;\|\mathbf{w}\|=1}}\sqrt{\left(\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}_{0}}\mathbf{A}_{I}^{H}\mathbf{w}\right)^{2}\\!\\!+\\!\\!\left(\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}_{0}}\mathbf{A}_{I}^{H}\mathbf{w}\right)^{2}}.$
Since $\bm{\Phi}_{\mathcal{S}_{0}}$ and $\bm{\Phi}_{\mathcal{I}_{0}}$ are
independent of INR (c.f. Sec. II-B), $C_{Y_{0}}$ is also independent of INR.
Therefore, $C_{Y}$ is proportional to INR. Now, we are ready to use Theorem 2,
and we only consider the case of INR being large. Let
$\langle\nu_{0},\mu_{0}\rangle$ and $\langle\nu,\mu\rangle$ be the
corresponding generalized eigenvalues of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$,
respectively. Then, for large INR, (43) is satisfied and
$\displaystyle\frac{|\nu\mu_{0}-\mu\nu_{0}|}{\sqrt{|\nu|^{2}\\!\\!+\\!\\!|\mu|^{2}}\\!\sqrt{|\nu_{0}|^{2}\\!\\!+\\!\\!|\mu_{0}|^{2}}}\\!\\!<\\!\\!\frac{\sqrt{2}}{C_{Y_{0}}}\cdot\frac{1}{\textsf{INR}}.$
(45)
We now derive the bound for $\gamma_{1}\\!+\\!1$ in two separate cases.
#### IV-A1 $\mathcal{N}_{\mathcal{I}}\nsubseteq\mathcal{N}_{\mathcal{S}}$
There is a nonzero $\mathbf{w}_{0}$ such that
$\mathbf{Y}_{\mathcal{S}}\mathbf{w}_{0}\\!\neq\\!\mathbf{0}$ and
$\mathbf{Y}_{\mathcal{I}}\mathbf{w}_{0}\\!=\\!\mathbf{0}$. By Definition 2,
$\langle\nu_{0},0\rangle$ ($\forall\nu_{0}\\!\neq\\!0$) is a generalized
eigenvalue of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$.
This means the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ has an infinite
generalized eigenvalue. As a result, (45) becomes
$\displaystyle\frac{|\mu|}{\sqrt{|\nu|^{2}\\!\\!+\\!\\!|\mu|^{2}}}\\!\\!<\\!\\!\frac{\sqrt{2}}{C_{Y_{0}}}\cdot\frac{1}{\textsf{INR}}\quad\Leftrightarrow\quad\lambda\triangleq\frac{\nu}{\mu}>\sqrt{\frac{C_{Y_{0}}^{2}}{2}\textsf{INR}^{2}\\!\\!-\\!\\!1}\approx\frac{C_{Y_{0}}}{\sqrt{2}}\textsf{INR}.$
This means
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
would always have an eigenvalue that satisfies the above inequality. Since
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ has the same eigenvalue
except for the multiplicity of ones, $\gamma_{1}\\!+\\!1$ satisfies
$\displaystyle\gamma_{1}+1>\frac{C_{Y_{0}}}{\sqrt{2}}\textsf{INR},$ (46)
which gives a lower bound for $\gamma_{1}\\!+\\!1$. We can see that it goes up
unboundedly as INR increases. Hence, by (22) and (24), the threshold SNR also
increases unboundedly with INR.
#### IV-A2 $\mathcal{N}_{\mathcal{I}}\subseteq\mathcal{N}_{\mathcal{S}}$
Then, $\mathcal{N}_{0}\\!=\\!\mathcal{N}_{\mathcal{I}}$ and
$\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}$ is nonsingular. As
a result, all eigenvalues of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are bounded. When
INR is large, (45) becomes
$\displaystyle\frac{|\lambda-\lambda_{0}|}{\sqrt{1+\lambda^{2}}\sqrt{1+\lambda_{0}^{2}}}<\frac{C_{Y_{0}}}{\sqrt{2}}\cdot\frac{1}{\textsf{INR}}\quad\Longrightarrow\quad|\lambda-\lambda_{0}|<\frac{C_{Y_{0}}}{\sqrt{2}}(1+\lambda_{0}^{2})\cdot\frac{1}{\textsf{INR}}$
where $\lambda_{0}\\!=\\!\nu_{0}/\mu_{0}$ is the largest eigenvalue of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0})$
and we used $\lambda\\!\approx\lambda_{0}$ in the second inequality. Since
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ has the same eigenvalue
as
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
except for the multiplicity of ones, $\gamma_{1}+1$ is bounded around the
largest eigenvalue of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ or
one. Furthermore, from (42), we know that $\lambda_{0}$ is independent of INR.
Therefore, as all eigenvalues of noise free pair is finite,
$\gamma_{1}\\!+\\!1$ is bounded and independent of INR.
### IV-B Two Typical Scenarios of $\gamma_{1}$
From the previous part, we know that whether the threshold of MPB is unbounded
is determined by the existence of infinite eigenvalue of the noise free pair.
Now, we discuss two typical classes of interferences. In the first case, all
eigenvalues are finite, and in the second case, there may be infinite
eigenvalues.
#### IV-B1 Directional White Noise
By uncorrelated directional white noise, we mean an interference that arrives
from a specific direction, with its samples in time domain being uncorrelated.
This means the entries of $\mathbf{S}_{I}(k)$ are independent identically
distributed random variables with zero mean and unit variance. (In section
II-A, we have already normalized the interference power in
$\mathbf{s}_{i}(k)$.) Then, by (8), we can immediately have
$\bm{\Phi}_{\mathcal{S}_{0}}\\!=\\!\bm{\Phi}_{\mathcal{I}_{0}}$, and all the
eigenvalue of the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are ones. Thus, in this
case, $\gamma_{1}\\!+\\!1$ is bounded and is independent of INR. In fact, we
can further have $\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$
according to (6) and (7). Therefore, $\gamma_{1}\\!=\\!0$,
$\textsf{SNR}_{\textsf{T}0}\\!=\\!0$ and $\textsf{G}(\textsf{SNR})\\!=\\!1$.
#### IV-B2 Directional Periodical Interference
If an interference has periodical structure in time domain and arrives from
certain direction, then we term it as directional periodical interference. By
periodical, we mean the interfer is periodic with respect to the projection
basis, i.e.
$\displaystyle\mathbf{H}_{\mathcal{I}}^{H}\mathbf{S}_{I}(k)=e^{j\phi_{k}}\mathbf{H}_{\mathcal{I}}^{H}\mathbf{S}_{I},\qquad\mathbf{h}_{\mathcal{S}}^{H}\mathbf{S}_{I}(k)=e^{j\phi_{k}^{\prime}}\mathbf{h}_{\mathcal{S}}^{H}\mathbf{S}_{I}.$
(47)
A stronger condition would be $\mathbf{S}_{I}(k)\\!=\\!\mathbf{S}_{I}$.
However, (47) is good enough for our analysis. Now, we will discuss the
existence of infinite generalized eigenvalue in the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$. To do this, we need to
consider the relationship between their null spaces.
In practice, $\mathbf{R}_{\mathcal{S}}$ and $\mathbf{R}_{\mathcal{I}}$ are
estimated from sample average. Therefore, replacing the expectation in (8) by
sample average and using (47), we can get
$\displaystyle\bm{\Phi}_{\mathcal{S}_{0}}=\bm{\Omega}_{I}^{1/2}\mathbf{S}_{I}^{T}\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{S}_{I}^{\ast}\bm{\Omega}_{I}^{1/2},\qquad\bm{\Phi}_{\mathcal{I}_{0}}=\frac{1}{r_{\mathcal{I}}}\bm{\Omega}_{I}^{1/2}\mathbf{S}_{I}^{T}\mathbf{H}_{\mathcal{I}}^{\ast}\mathbf{H}_{\mathcal{I}}^{T}\mathbf{S}_{I}^{\ast}\bm{\Omega}_{I}^{1/2}.$
Let
$\bm{\Pi}\\!\triangleq\\!\mathbf{S}_{I}\bm{\Omega}_{I}^{1/2}\mathbf{A}_{I}^{T}$.
Then, by (42), $\mathbf{Y}_{\mathcal{S}}$ and $\mathbf{Y}_{\mathcal{I}}$ in
this case can be expressed as
$\displaystyle\mathbf{Y}_{\mathcal{S}}=\textsf{INR}\cdot\left(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}\right)^{\ast},\qquad\mathbf{Y}_{\mathcal{I}}=\textsf{INR}\cdot\frac{1}{r_{\mathcal{I}}}\left(\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}\right)^{\ast}.$
(48)
Therefore, the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ has the same eigenvalue
as
$(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi},\frac{1}{r_{\mathcal{I}}}\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi})$
does, and we only have to check the null spaces of later matrix pair. Let
$\mathcal{V}_{I}\\!\triangleq\\!\mathcal{R}(\bm{\Pi})$ be the range space of
$\bm{\Pi}$. By definition, it is also the range space of $\mathbf{S}_{I}$,
namely the space spanned by all interference waveforms in one period. Then, we
can express the null spaces of
$\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}$ and
$\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}$ as
$\displaystyle\mathcal{N}_{\mathcal{S}}^{\prime}\triangleq\mathcal{N}\left(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}\right)=\left\\{\bm{\Pi}^{\dagger}\mathbf{x}:\;\mathbf{x}\in\mathcal{S}^{\perp}\cap\mathcal{V}_{I}\right\\}\oplus\mathcal{N}(\bm{\Pi})$
$\displaystyle\mathcal{N}_{\mathcal{I}}^{\prime}\triangleq\mathcal{N}\left(\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}\right)=\left\\{\bm{\Pi}^{\dagger}\mathbf{x}:\;\mathbf{x}\in\mathcal{I}^{\perp}\cap\mathcal{V}_{I}\right\\}\oplus\mathcal{N}(\bm{\Pi}),$
where $\mathcal{S}\\!=\\!\mathcal{R}(\mathbf{h}_{\mathcal{S}})$,
$\mathcal{I}\\!=\\!\mathcal{R}(\mathbf{H}_{\mathcal{I}})$ as defined in Sec.
II-B, and $\dagger$ denotes Moore-Penrose pseudoinverse [34]. To see the
relationship between $\mathcal{N}_{\mathcal{S}}^{\prime}$ and
$\mathcal{N}_{\mathcal{I}}^{\prime}$, we only need to check the relationship
between $\mathcal{S}^{\perp}\cap\mathcal{V}_{I}$ and
$\mathcal{I}^{\perp}\cap\mathcal{V}_{I}$ which can be expressed as
$\displaystyle\mathcal{S}^{\perp}\cap\mathcal{V}_{I}=\\{\mathbf{x}:\;\mathbf{B}_{\mathcal{S}}^{H}\mathbf{x}=\mathbf{0}\\},\qquad\mathcal{I}^{\perp}\cap\mathcal{V}_{I}=\\{\mathbf{x}:\;\mathbf{B}_{\mathcal{I}}^{H}\mathbf{x}=\mathbf{0}\\},$
(49)
where
$\displaystyle\mathbf{B}_{\mathcal{I}}\triangleq\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{I}^{\perp}}&\mathbf{H}_{\mathcal{I}}\\\
\end{bmatrix},\qquad\mathbf{B}_{\mathcal{S}}\triangleq\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{I}^{\perp}}&\mathbf{h}_{\mathcal{S}}\\\
\end{bmatrix}.$
with $\mathbf{H}_{\mathcal{V}_{I}^{\perp}}$ being the matrix whose columns are
the orthonormal basis of $\mathcal{V}_{I}^{\perp}$. Then, by (49), whether the
eigenvalues of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are
finite is equivalent to the validity of
$\mathcal{R}\\{\mathbf{B}_{\mathcal{I}}\\}\\!\\!\supseteq\\!\\!\mathcal{R}\\{\mathbf{B}_{\mathcal{S}}\\}$.
Define
$\displaystyle\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}\triangleq\begin{bmatrix}\mathbf{I}&-\mathbf{H}_{\mathcal{V}_{I}^{\perp}}^{H}\mathbf{H}_{\mathcal{I}}\\\
0&\mathbf{I}\end{bmatrix},\qquad\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}\triangleq\begin{bmatrix}\mathbf{I}&-\mathbf{H}_{\mathcal{V}_{I}^{\perp}}^{H}\mathbf{h}_{\mathcal{S}}\\\
0&\mathbf{I}\end{bmatrix}.$
Right-multiplying $\mathbf{B}_{\mathcal{I}}$ and $\mathbf{B}_{\mathcal{S}}$ by
$\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}$ and
$\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}$, respectively, we can have
$\displaystyle\mathbf{B}_{\mathcal{I}}\cdot\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}=\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{c}^{\perp}}&\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\end{bmatrix},\qquad\mathbf{B}_{\mathcal{S}}\cdot\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}=\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{c}^{\perp}}&\mathbf{P}_{\mathcal{V}_{I}}\mathbf{h}_{\mathcal{S}}\end{bmatrix},$
where
$\mathbf{P}_{\mathcal{V}_{I}}\\!\triangleq\\!\mathbf{H}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{V}_{I}}^{H}$
is the projection matrix onto $\mathcal{V}_{I}$. Since
$\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}$ and
$\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}$ are nonsingular, the above two
matrices have the same range spaces as $\mathbf{B}_{\mathcal{S}}$ and
$\mathbf{B}_{\mathcal{I}}$ do, namely we can have
$\displaystyle\mathcal{R}\\{\mathbf{B}_{\mathcal{I}}\\}=\mathcal{V}_{I}^{\perp}\oplus\mathcal{R}\\{\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\\},\qquad\mathcal{R}\\{\mathbf{B}_{\mathcal{S}}\\}=\mathcal{V}_{I}^{\perp}\oplus\mathcal{R}\\{\mathbf{P}_{\mathcal{V}_{I}}\mathbf{h}_{\mathcal{S}}\\}$
Therefore, whether the generalized eigenvalues of the noise free pair are
bounded is finally equivalent to whether
$\displaystyle\mathcal{R}\big{\\{}\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\big{\\}}\\!\\!\supseteq\\!\\!\mathcal{R}\big{\\{}\mathbf{P}_{\mathcal{V}_{c}}\mathbf{h}_{\mathcal{S}}\big{\\}}.$
(50)
Figure 4: Geometrical interpretation of (50).
$\mathbf{h}_{\mathcal{I}}^{(1)}$ and $\mathbf{h}_{\mathcal{I}}^{(2)}$ denote
the columns of $\mathbf{H}_{\mathcal{I}}$. For one interference channel,
$\mathbf{H}_{\mathcal{I}}\\!=\\!\mathbf{h}_{\mathcal{I}}$.
Fig. 4 shows the geometrical interpretation of (50). To see if $\gamma_{1}$ is
bounded, we can project $\mathbf{h}_{\mathcal{S}}$ and all the columns of
$\mathbf{H}_{\mathcal{I}}$ onto $\mathcal{V}_{I}$. If the former projection
lies in the space spanned by the later ones, then $\gamma_{1}$ is bounded.
Otherwise, there is an infinite eigenvalue in the noise free pair and
$\gamma_{1}$ will increase unboundedly with INR. Since
$\mathbf{H}_{\mathcal{I}}$ of the Maximin scheme and PAPC scheme has only one
column, this means the projections of $\mathbf{h}_{\mathcal{S}}$ and
$\mathbf{H}_{\mathcal{I}}$ should be on the same line, which is hardly valid.
Thus, these two approaches are sensitive to directional periodical
interferences, as we will see in later simulation results. Furthermore, Fig. 4
also shows the case of multiple interference channels, i.e. the dimension of
$\mathcal{I}$ is larger than one, then $\gamma_{1}$ is more likely to be
bounded. This issue is out of the scope of this paper and will be discussed in
in [36].
## V Simulation Results
In this section, we simulate various scenarios and compare them to our
theoretical results. The matrix pair beamformers implemented include the
Maximin algorithm and PAPC algorithm, which have many kinds of adaptive
algorithms. However, we are only interested in their steady state performance.
The reason for this is that, if the methods suffer from poor steady state
performance, then it is meaningless to investigate their adaptive algorithms.
Therefore, we directly calculate their weight vectors by performing
generalized eigen-decompositions on the estimated matrix pairs. The
interference signals encountered in the simulation include BPSK signal, tones,
periodical noise and multiuser interference. The first one is the directional
white noise and the others are directional periodical interferences.
In all cases, we consider a uniform linear array (ULA) with eight isotropic
antennas ($L=8$) spaced half a wavelength apart. For each user, a $100$ kbps
DPSK signal is randomly generated and spreaded by a distinct 31-chip Gold
sequence in each simulation trial. Then it is modulated onto a $1$ GHz carrier
to form a RF signal with bandwidth $3.1$ MHz. In all the simulations, we
assume the SOI arrives from $0^{\circ}$ and the interferers have equal power.
Fig. 5(a)–Fig. 5(d) show the operating curve $\textsf{G}(\textsf{SNR})$ of the
Maximin and PAPC beamformers in the four scenarios. The simulated values are
obtained by using the simulated data and the theoretical ones are computed by
the piecewise function $\textsf{G}(\textsf{SNR})$ in (20). To eliminate the
randomness caused by finite sample effects, $K=10^{6}$ symbols are simulated
for each SNR and INR in every experiment.
(a) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under BPSK jammers.
(b) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under periodical noise.
(c) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under tone jammers.
(d) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under multipath MAI.
Figure 5: Simulated and theoretical operating curve $\textsf{G}(\textsf{SNR})$
under different kinds of interference for Maximin and PAPC.
In Fig. 5(a), the interferers are three uncorrelated BPSK modulated signals,
i.e. random sequences of $\pm 1$. The rates of the interferers are all
3.1Mbps, which is the same as the chip rate of the SOI and covers its whole
bandwidth. Therefore, they belong to the type of directional white noise. We
can see that $\textsf{G}(\textsf{SNR})\approx 1$ and there is no threshold
effect, which is consistent with the analysis in Section IV-B1.
Fig. 5(b)–Fig. 5(d) show the results for three types of directional periodical
interferences: periodical noise, tones and multiple access interference (MAI)
with multipath. In Fig. 5(b), we consider two periodical noises, which arrives
from $30^{\circ}$ and $-40^{\circ}$, respectively, and each of them is
generated by repeating a segment of Gaussian white noise over times, with the
repeating frequency being $100$ kHz (same as that of SOI’s symbol rate). The
frequency offsets of the tones with respect to the carrier in Fig. 5(c) are
$100$kHz, $-300$kHz, $0$Hz, $400$kHz and $-100$kHz, respectively, and their
DOAs are $30^{\circ}$, $-50^{\circ}$, $-20^{\circ}$, $19^{\circ}$ and
$45^{\circ}$. In Fig. 5(d), there is one incident MAI signal with three-ray
multipath delays of $3$ chips, $5$ chips and $4$ chips arriving from
$30^{\circ}$, $-20^{\circ}$ and $-50^{\circ}$, respectively. We can see that
the theoretical values of $\textsf{G}(\textsf{SNR})$ match with the simulated
ones very well.
To verify the validity of the approximation given by Lemma 4, in Fig. 6(a) and
Fig. 6(b), we also show the curves of $\gamma_{0}\\!+\\!1$,
$\gamma_{1}\\!+\\!1$, and $\lambda_{\mathrm{max}}$. We can see that
$\max\\{\gamma_{0}+1,\gamma_{1}+1\\}$ can be an excellent approximation for
$\lambda_{\mathrm{max}}$. In Fig. 7(a) and Fig. 7(b), we show the array
patterns that are below and above the threshold. The results in these four
figures are simulated under two periodical noises, with the same parameter as
Fig. 5(b). The case for the tone jammers and the multiple access interference
cases are quite similar, and is thus omitted.
For the Maximin algorithm, all the curves of $\textsf{G}(\textsf{SNR})$ have
failure area, threshold area and operating area, which are consistent with the
typical curve in Fig. 2. We have also marked the predicted thresholds of
$\textsf{SNR}_{\textsf{T}1}$ and $\textsf{SNR}_{\textsf{T}2}$, computed by
(24), in the figures as well, which confirm our theoretical calculations.
Furthermore, they also show that the threshold SNR would increase with the
interference INR. This is because $\gamma_{1}+1$ (c.f. Fig. 6(a)) would moving
upward as INR increases, making the intersecting point of $\gamma_{0}+1$ and
$\gamma_{1}+1$ move rightwards. This is consistent with our claim in Section
IV-B2 that $\gamma_{1}$ would increase unboundedly with the INR. Fig. 7(a) and
Fig. 7(b) show its array pattern below and above threshold, respectively. We
can see that the mainlobe of the beamformer would mistakenly point to the
interferers once the SNR is below the threshold, as we have predicted in
Section III-C.
For the PAPC algorithm, Fig. 5(b)–Fig. 5(d) show that there are only failure
areas. This is because, no matter how large SNR is, $\gamma_{0}\\!\\!+\\!\\!1$
will never exceeds $\gamma_{1}\\!\\!+\\!\\!1$, as shown by Fig. 6(b).
Therefore, the threshold of PAPC in this case is infinity. The array pattern
shown in Fig. 7(a) also indicates that its mainlobe has pointed to the
interferers. We have stated in section III-C that the presence of SOI in
$\mathbf{R}_{\mathcal{I}}$ will make the beamformer mistakenly null the SOI.
The curves of PAPC in Fig. 5(b)–Fig. 5(d) confirms this observation, and Fig.
7(b) shows that PAPC beamformer has a deep null in the direction of the SOI.
(a) Maximin scheme
(b) PAPC scheme
Figure 6: The largest and second largest generalized eigenvalues of the
Maximin scheme vs SNR under two periodical noises.
(a) Case I: below threshold ($\textsf{SNR}\\!\\!=\\!\\!-10.1$dB,
$\textrm{INR}\\!\\!=\\!\\!30$dB)
(b) Case II: above threshold ($\textsf{SNR}\\!\\!=\\!\\!40.9$dB,
$\textrm{INR}\\!\\!=\\!\\!30$dB)
Figure 7: The array patterns corresponding to the Maximin and PAPC under two
periodical noises.
In summary, the conventional MPB like Maximin and PAPC work well in the
presence of directional white noise, even when the INR is large. However, they
are very vulnerable to multiple directional periodical interferers with
repeating structures in the time domain and arrive from different directions.
## VI Conclusions
Matrix pair beamformer (MPB) is a general framework we proposed to model a
class of blind beamformers that exploit the temporal signature of the signal
of interest (SOI). It has the advantages only relying on the second order
statistics to achieve blind processing. In this paper, we have analyzed the
mechanism of MPB with matrix mismatch, and showed how it worked “blindly”. We
have discovered that there is a threshold effect in MPB, i.e. the beamformer
would fail completely if the SOI’s input SNR is below that threshold.
Meanwhile, its normalized output SINR has been derived as the performance
measure, and the threshold SNR has also been predicted. We have also observed
that the existence of infinite generalized eigenvalue in what is called _noise
free pair_ makes the threshold increase unboundedly with the interference
power. This is highly probable when there are multiple periodical interferers.
All our theoretical analysis matches with the simulation results very well.
Our analysis indicates that the conventional MPB is very vulnerable to
multiple periodical interferers. Moreover, it also implies the importance of
choosing the appropriate projection space for the interference channel. And we
will address this issue in another paper[36].
## Appendix A Approximation of $\lambda_{\mathrm{max}}$ and $\gamma_{i}$
### A-A Approximation of $\lambda_{\mathrm{max}}$
Since similarity transform does not change the eigenvalues, we can apply it to
$\mathbf{M}$ before using Lemma 2. As we will see later, the bounds derived in
this way can be surprisingly tight. Though any transform matrix can be used,
we prefer the following diagonal matrix that can preserve the diagonal terms
of $\mathbf{M}$
$\displaystyle\mathbf{F}=\mathrm{diag}\\{1,f_{1},f_{2},\ldots,f_{D}\\},\quad(f_{i}>0,\;i=1,2,\ldots,D)$
Applying it to $\mathbf{M}$ in (33) and using Lemma 2, we can have the
following Gerschgorin disks for $\mathbf{F}\mathbf{M}\mathbf{F}^{-1}$:
$\displaystyle G_{i}$
$\displaystyle=\\{\lambda:\;|\lambda-(\gamma_{i}+1)|<R_{i}(\mathbf{F})\\},\qquad
i=0,1,2,\ldots,D,$
where the radii of the disks are
$\displaystyle R_{0}(\mathbf{F})$
$\displaystyle=\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\sum_{i=1}^{D}|\psi_{T_{i}}|\frac{1}{f_{i}}$
(51) $\displaystyle R_{i}(\mathbf{F})$
$\displaystyle=\gamma_{i}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{i}}|\cdot\mathrm{sgn}(\gamma_{i})}{\frac{L\beta}{N}\textsf{SNR}+1}f_{i},\quad
1\leq i\leq D,$ (52)
with $\psi_{T_{i}}$ being the $i$th component of $\bm{\psi}_{T}$. Now, we are
going to optimize $f_{1},\ldots,f_{D}$. To derive an effective bound for
$\lambda_{\mathrm{max}}$, we should ensure the rightmost disk is separated
from the others (c.f. Lemma 2). Therefore, our criterion for finding the
optimal $\mathbf{F}$ is _minimizing the radius of the rightmost Gerschgorin
disk subject to the constraint that it is separated from all the remaining
ones_. Depending on $\gamma_{0}+1>\gamma_{1}+1$ or
$\gamma_{0}+1<\gamma_{1}+1$, the rightmost disk might be $G_{0}$ or $G_{1}$,
and we now discuss them separately.
If $\gamma_{0}+1>\gamma_{1}+1$, then $G_{0}$ is the rightmost disk, and we can
formulate the optimization of $\mathbf{F}$ as
$\displaystyle\min~{}$
$\displaystyle~{}R_{0}(\mathbf{F})=\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\sum_{i=1}^{D}|\psi_{T_{i}}|\frac{1}{f_{i}}$
(53) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{i}+1+R_{i}(F)\leq\gamma_{0}+1-R_{0}(\mathbf{F}),\quad
1\leq i\leq D.$ (54)
To minimize $R_{0}(\mathbf{F})$, we want $f_{1},\ldots,f_{D}$ to be as large
as possible. But this will increase the radius of $G_{1},\ldots,G_{D}$, and to
avoid connecting $G_{0}$, they cannot be too large. In fact,
$f_{1},\ldots,f_{D}$ have different importance in this tradeoff. Since
$\gamma_{0}>\ldots>\gamma_{D}$, $G_{2},\ldots,G_{D}$ are farther away from
$G_{0}$ than $G_{1}$. Therefore, $f_{2},\ldots,f_{D}$ can be reasonably larger
than $f_{1}$ when keeping separate from $G_{0}$. Thus, to simplify analysis,
we can ignore all the terms except $|\psi_{T_{1}}|/f_{1}$ in
$R_{0}(\mathbf{F})$, i.e.
$R_{0}(\mathbf{F})\approx\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\frac{|\psi_{T_{1}}|}{f_{1}}$
in (53) and (54), and only the constraint for $i=1$ is effective in (54).
Then, the problem can be reduced to
$\displaystyle\min~{}$
$\displaystyle~{}R_{0}(\mathbf{F})\approx\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\frac{1}{f_{1}}$
$\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\cdot\mathrm{sgn}(\gamma_{1})}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}^{2}-(\gamma_{0}-\gamma_{1})f_{1}+\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\leq
0.$
This is a simple convex optimization problem, feasible when
$\frac{\gamma_{1}}{\gamma_{0}}\in(-\infty,1+2\delta-\sqrt{(1+2\delta)^{2}-1}]$,
where
$\delta\triangleq\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}/(\frac{L\beta}{N}\textsf{SNR}+1)\ll
1$. Thus, we can easily solve the above optimization and get
$\displaystyle|\lambda_{\mathrm{max}}-(\gamma_{0}+1)|<\gamma_{0}\cdot
f\left(\frac{\gamma_{1}}{\gamma_{0}}\right)$
where
$f(x)\triangleq\frac{1}{2}[1\\!-\\!x\\!-\\!\sqrt{(1\\!-\\!x)^{2}\\!-\\!4\delta|x|}]$
with
$x\in(-\infty,1\\!-\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta]\\!\cup\\![1\\!+\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta,+\infty)$.
Furthermore, by taking the derivative of $f(x)$, we can easily get
$0\\!\\!\leq\\!\\!f(x)\\!\\!\leq\\!\\!\max\\{\delta,\sqrt{\delta^{2}\\!\\!+\\!\\!\delta}\\!\\!-\\!\\!\delta\\}\\!\\!\ll\\!\\!1$
when $x\in(-\infty,1\\!-\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta]$.
If $\gamma_{0}+1<\gamma_{1}+1$, then $G_{1}$ is the rightmost disk and
$\gamma_{1}>0$. The optimization problem is
$\displaystyle\min~{}$
$\displaystyle~{}R_{1}(\mathbf{F})=\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}$
(55) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{0}+1+R_{0}(F)\leq\gamma_{1}+1-R_{1}(\mathbf{F}),$
(56)
$\displaystyle~{}\gamma_{i}+1+R_{i}(F)\leq\gamma_{1}+1-R_{1}(\mathbf{F}),\quad
2\leq i\leq D.$ (57)
According to (51) and (52), $f_{1}$ should be as small as possible to minimize
$R_{1}(\mathbf{F})$ in (55), (56) and (57). However, this will increase
$R_{0}(\mathbf{F})$ in (56), making $G_{0}$ connect with $G_{1}$. Therefore,
$f_{1}$ cannot be arbitrarily small. On the other hand, $f_{2},\ldots,f_{D}$
should be as large as possible to reduce $R_{0}(\mathbf{F})$ in (56) while
keeping (57) valid. As a result, $|\psi_{T_{1}}|/f_{1}$ is still the dominant
term in $R_{0}(\mathbf{F})$ (c.f. (51)) and the key point here remains the
tradeoff between $R_{1}(\mathbf{F})$ and $R_{0}(\mathbf{F})$. In other words,
the optimization becomes
$\displaystyle\min~{}$
$\displaystyle~{}R_{1}(\mathbf{F})=\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}$
$\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}^{2}-(\gamma_{1}-\gamma_{0})f_{1}+\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\leq
0.$
The feasible region for this convex optimization is
$\frac{\gamma_{1}}{\gamma_{0}}\in[1+2\delta+\sqrt{(1+2\delta)^{2}-1},+\infty)$,
with $\delta$ defined in the previous case. By solving it, we can finally get
the bound for $\lambda_{\mathrm{max}}$ as
$\displaystyle|\lambda_{\mathrm{max}}-(\gamma_{1}+1)|\leq\gamma_{1}\cdot
f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)$
where $f(x)$ is the same as in the previous case.
### A-B Approximation of $\gamma_{i}$
Then, we discuss the approximation of $\gamma_{i}$. By substituting (5) into
(30) and using matrix inversion lemma, we can have the equivalent equation of
(30) as
$\displaystyle\det(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\cdot\det\left\\{\lambda\mathbf{I}-\left[\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}-\frac{\sigma_{\mathcal{I}}^{2}\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\cdot\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}}{1+\sigma_{\mathcal{I}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}\right]\bm{\Phi}_{\Delta}\right\\}=0$
(58)
To further reduce the above expression, we need factorizations of
$\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}$ and
$\bm{\Phi}_{\Delta}$ like (28). Let $\sigma_{\mathcal{I}_{0}}^{2}=0$ in (28).
Then, $\mathbf{R}_{\mathcal{I}}=\mathbf{Q}_{\mathcal{I}}$, and (28) become
$\displaystyle\mathbf{T}^{H}_{0}\bm{\Phi}_{\Delta}\mathbf{T}_{0}=\bm{\Gamma}_{0},\qquad\mathbf{T}^{H}_{0}(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\mathbf{T}_{0}=\mathbf{I}.$
where $\mathbf{T}_{0}$ and $\bm{\Gamma}_{0}$ are the counterparts of
$\mathbf{T}$ and $\bm{\Gamma}$, respectively. Substituting them into (58), we
have
$\displaystyle\det(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\cdot\det\left\\{\lambda\mathbf{I}-\left[\mathbf{I}-\frac{\sigma_{\mathcal{I}}^{2}\tilde{\bm{\psi}}_{T_{0}}\tilde{\bm{\psi}}_{T_{0}}^{H}}{1+\sigma_{\mathcal{I}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}\right]\bm{\Gamma}_{0}\right\\}=0$
where
$\tilde{\bm{\psi}}_{T_{0}}=\mathbf{T}_{0}^{-1}\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
is the counterpart of $\tilde{\bm{\psi}}_{T}$ when
$\sigma_{\mathcal{I}_{0}}^{2}=0$. By letting $\textsf{SNR}=0$ in Lemma 1,
$\mathbf{R}_{\mathcal{I}}$ becomes $\mathbf{Q}_{\mathcal{I}}$ and we can have
$\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\approx
L/\sigma^{2}$ and $\|\bm{\psi}_{T_{0}}\|^{2}=L\kappa_{0}/\sigma^{2}\ll
L/\sigma^{2}$. Therefore, by the similar argument of $\lambda_{\mathrm{max}}$
using Gerschgorin theorem, we can have $\gamma_{i}\approx\lambda_{i,0}$, where
$\lambda_{i,0}$ is the $i$th diagonal term of $\bm{\Gamma}_{0}$, namely, the
$i$th eigenvalue of
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
In fact, there is a correspondence between $\lambda_{i,0}$ and the generalized
eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.
By (31),
$\displaystyle\det\left\\{\lambda(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}-\bm{\Phi}_{\Delta}\right\\}$
$\displaystyle=\lambda^{D-L}\cdot\det(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\det\mathbf{Q}_{\mathcal{I}}^{-1}\cdot\det\\{\lambda\mathbf{Q}_{\mathcal{I}}-\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\\}.$
Since
$\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}}=\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}$,
the above expression implies that
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$
has the same eigenvalues as
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
except for multiplicity of zeros. Thus, we can estimate $\gamma_{i}$ like
this: 1) take out all nonzero eigenvalues of
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$,
2) pad them up to $D$ eigenvalues with zeros, 3) order them decreasingly to
get $\lambda_{1,0},\ldots,\lambda_{D,0}$, and 4) let
$\gamma_{i}\approx\lambda_{i,0}$.
## Appendix B Proof of Lemma 3
Substituting the definitions of $y_{S}(k)$ and $y_{I}(k)$ (c.f. (10)), (6) and
(14) into (17) , we can have
$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle=\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}\mathbf{w}+\sigma^{2}\|\mathbf{w}\|^{2}}\frac{1}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}}\approx\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}\mathbf{w}+\sigma^{2}\|\mathbf{w}\|^{2}}\big{/}\frac{L}{\sigma^{2}},$
(59)
where we used
$\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}\\!\\!\approx\\!\\!{L}/{\sigma^{2}}$,
derived by replacing $\mathbf{R}_{\mathcal{I}}$ with
$\mathbf{R}_{\mathcal{S}}$ and letting
$\sigma_{\mathcal{S}_{0}}^{2}\\!\\!=\\!\\!0$ in Lemma 1.
To further derive G, we first need the expression for $\|\mathbf{w}\|^{2}$.
The key trick is to recognize that $\mathbf{w}\in\mathcal{R}(\mathbf{A})$ so
that its projection onto $\mathcal{R}(\mathbf{A})$ equals itself, where
$\mathcal{R}(\cdot)$ is the range space of a matrix and
$\mathbf{A}\triangleq[\mathbf{a}_{0}\;\mathbf{A}_{I}]$. This can be proved by
substituting (7) into (5), applying matrix inversion lemma and pluging it into
(27). Furthermore, $\mathcal{R}(\mathbf{A}_{I})$ is a subspace of
$\mathcal{R}(\mathbf{A})$ with the dimension lower by one. Thus,
$\mathcal{R}(\mathbf{A})\\!\\!=\\!\\!\mathcal{R}(\mathbf{A})\\!\oplus\\!\mathcal{R}(\hat{\mathbf{b}}_{0})$,
where $\oplus$ denotes the direct sum and $\hat{\mathbf{b}}_{0}$ is the unit
vector in $\mathcal{R}(\mathbf{A})$ that is orthogonal to
$\mathcal{R}(\mathbf{A}_{I})$. Then, the projection matrix of
$\mathcal{R}(\mathbf{A})$ can be written as
$\displaystyle\mathbf{P}_{A}=\mathbf{A}_{I}(\mathbf{A}_{I}^{H}\mathbf{A}_{I})^{-1}\mathbf{A}_{I}^{H}+\mathbf{\hat{b}}_{0}\mathbf{\hat{b}}_{0}^{H}.$
This together with $\mathbf{P}_{A}\mathbf{w}=\mathbf{w}$ and
$\mathbf{P}_{A}^{2}=\mathbf{P}_{A}$ yields
$\displaystyle\|\mathbf{w}\|^{2}=\mathbf{w}^{H}\mathbf{P}_{A}\mathbf{w}=\mathbf{w}^{H}\mathbf{A}_{I}(\mathbf{A}_{I}^{H}\mathbf{A}_{I})^{-1}\mathbf{A}_{I}^{H}\mathbf{w}+|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}$
(60)
Assumption 1 in Sec. III-A implies that $\mathbf{a}_{0}$ is almost orthogonal
to $\mathcal{R}(\mathbf{A}_{I})$. Thus, it is nearly aligned with
$\hat{\mathbf{b}}_{0}$, and intuitively,
$|\mathbf{w}^{H}\hat{\mathbf{b}}_{0}|\\!\\!\approx\\!\\!|\mathbf{w}^{H}\hat{\mathbf{a}}_{0}|$,
where
$\hat{\mathbf{a}}_{0}\\!\\!\triangleq\\!\\!\mathbf{a}_{0}/\|\mathbf{a}_{0}\|$
is the unit vector of $\mathbf{a}_{0}$. To prove it, let
$\mathcal{R}_{0}\\!\\!\triangleq\\!\\!\mathrm{span}\\{\hat{\mathbf{a}}_{0},\hat{\mathbf{b}}_{0}\\}$
and $\\{\hat{\mathbf{b}}_{0},\hat{\mathbf{b}}_{1}\\}$ be the orthonormal basis
of $\mathcal{R}_{0}$. Its projection matrix becomes
$\mathbf{P}_{\mathcal{R}_{0}}\\!\\!=\\!\\!\hat{\mathbf{b}}_{0}\hat{\mathbf{b}}_{0}^{H}\\!\\!+\\!\\!\hat{\mathbf{b}}_{1}\hat{\mathbf{b}}_{1}^{H}$,
which satisfies
$\mathbf{P}_{\mathcal{R}_{0}}\hat{\mathbf{a}}_{0}\\!\\!=\\!\\!\hat{\mathbf{a}}_{0}$
and
$\mathbf{P}_{\mathcal{R}_{0}}\hat{\mathbf{b}}_{0}\\!\\!=\\!\\!\hat{\mathbf{b}}_{0}$.
Define
$\mathbf{w}_{0}\\!\\!=\\!\\!\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}/\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|$.
Then,
$\displaystyle|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\left[|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\right]=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\left[|\mathbf{w}^{H}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}^{H}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{\hat{a}}_{0}|^{2}\right]$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}\left[|\mathbf{w}_{0}^{H}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}_{0}^{H}\hat{\mathbf{b}}_{0}\cdot\hat{\mathbf{b}}_{0}^{H}\mathbf{\hat{a}}_{0}\\!\\!+\\!\\!\mathbf{w}_{0}^{H}\hat{\mathbf{b}}_{1}\cdot\hat{\mathbf{b}}_{1}^{H}\mathbf{\hat{a}}_{0}|^{2}\right]$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}\cdot\cos\phi_{I}\cdot\cos(\phi_{1}+2\phi_{I})$
(61)
where $\phi_{I}$ is the angle between $\hat{\mathbf{a}}_{0}$ and
$\mathcal{R}(\mathbf{A}_{I})$, and $\phi_{1}$ is the angle between
$\hat{\mathbf{a}}_{0}$ and $\hat{\mathbf{b}}_{0}$. And the derivation of the
last step in (61) involves some simple trigonometry identities like product-
to-sum formula. Let
$\boldsymbol{\Psi}_{I}\\!\\!\triangleq\\!\\!\mathbf{A}_{I}^{H}\mathbf{A}_{I}/L$.
Combining (60), (61),
$\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|\\!\\!\leq\\!\\!\|\mathbf{w}\|$ and
$|\cos\phi_{I}|\\!\\!\ll\\!\\!1$ (c.f. Assumption 1), we can have
$\displaystyle\|\mathbf{w}\|^{2}$
$\displaystyle=\frac{\frac{1}{L}\left[|\mathbf{a}_{0}^{H}\mathbf{w}|^{2}+\mathbf{w}^{H}\mathbf{A}_{I}\boldsymbol{\Psi}_{I}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}\right]}{1-\frac{\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}}{\|\mathbf{w}\|^{2}}\cdot\cos\phi_{I}\cdot\cos(\phi_{1}+2\phi_{I})}\approx\frac{1}{L}\left[|\mathbf{a}_{0}^{H}\mathbf{w}|^{2}+\mathbf{w}^{H}\mathbf{A}_{I}\boldsymbol{\Psi}_{I}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}\right],$
where we also used $\|\mathbf{a}_{0}\|^{2}\\!\\!=\\!\\!L$ in Assumption 2.
Substitute the above expression back to (59), and we get
$\displaystyle\textsf{G}(\textsf{SNR})\approx\
\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\\!\left[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\right]\\!\mathbf{A}_{I}^{H}\mathbf{w}\\!\\!+\\!\\!|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}.$
(62)
Next, we are going to evaluate $\mathbf{w}^{H}\mathbf{a}_{0}$ and
$\mathbf{w}^{H}\mathbf{A}_{I}[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}]\mathbf{A}_{I}^{H}\mathbf{w}$.
By the expression of $\mathbf{w}$ in (27),
$\displaystyle\mathbf{a}_{0}^{H}\mathbf{w}$
$\displaystyle=\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\frac{\gamma_{i}}{\lambda_{\mathrm{max}}\\!-\\!(\gamma_{i}\\!+\\!1)}|\widetilde{\psi}_{T_{i}}|^{2}=\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\\!\\!-\\!\\!\|\widetilde{\boldsymbol{\psi}}_{T}\|^{2}\\!\\!+\\!\\!\sum_{i=1}^{D}\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!-\\!(\gamma_{i}+1)}|\widetilde{\psi}_{T_{i}}|^{2}$
$\displaystyle\approx\frac{L}{\sigma^{2}}\left(\frac{L\beta}{N}\textsf{SNR}+1\right)\left[1+\left(\frac{L\beta}{N}\textsf{SNR}+1\right)^{-1}\psi_{S}(\lambda_{\mathrm{max}})\right],$
(63)
where we used Lemma 1 in the approximation and
$\psi_{S}(\lambda_{\mathrm{max}})$ is defined as (39) in Lemma 3. Before the
derivation of
$\mathbf{w}^{H}\mathbf{A}_{I}[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}]\mathbf{A}_{I}^{H}\mathbf{w}$,
we first cite the following identity from Lemma 2 in [35].
$\displaystyle\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{I}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\\!\\!=\\!\\!\frac{L}{\sigma^{2}}(\mathbf{T}^{-1})^{H}\\!\\!\left[\mathbf{I}\\!-\\!\frac{{L\beta}\textsf{SNR}\cdot\frac{\sigma^{2}}{L}\boldsymbol{\psi}_{T}\boldsymbol{\psi}_{T}^{H}}{{L\beta}(1\\!\\!-\\!\\!\rho_{0})\textsf{SNR}\\!\\!+\\!\\!N}\right]\\!\\!\mathbf{T}^{-1}.$
Then, combining the above expression together with
$\bm{\Phi}_{\Delta}\\!\\!=\\!\\!\bm{\Phi}_{\mathcal{S}}\\!\\!-\\!\\!\bm{\Phi}_{\mathcal{I}}$,
(28) and
$\mathbf{A}_{\epsilon}\\!\\!\triangleq\\!\\!\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$
we can have
$\displaystyle\mathbf{w}^{H}\mathbf{A}_{I}\left[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\right]\mathbf{A}_{I}^{H}\mathbf{w}$
$\displaystyle=\frac{L}{\sigma^{2}}\mathbf{w}^{H}\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}\\!\\!\left[\mathbf{I}\\!+\\!\bm{\Gamma}\\!-\\!\frac{{L\beta}\textsf{SNR}\cdot\frac{\sigma^{2}}{L}\boldsymbol{\psi}_{T}\boldsymbol{\psi}_{T}^{H}}{{L\beta}(1\\!\\!-\\!\\!\rho_{0})\textsf{SNR}\\!\\!+\\!\\!N}\right]\\!\\!\mathbf{T}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}$
$\displaystyle\approx\left[\frac{N{L}/{\sigma^{2}}}{{L\beta}\textsf{SNR}+N}\right]^{2}\left[\psi_{I}(\lambda_{\mathrm{max}})\\!-\\!\frac{{L\beta}\textsf{SNR}}{{L\beta}\textsf{SNR}+N}\psi_{S}^{2}(\lambda_{\mathrm{max}})\right],$
(64)
where $\psi_{I}(\lambda_{\mathrm{max}})$ is also given by (39) in Lemma 3, and
$\rho_{0}\\!\\!\ll\\!\\!1$ is a small number independent of SNR.
Finally, substituting (63) and (64) into (62), we can derive
$\textsf{G}(\textsf{SNR})$ in (38) immediately.
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|
arxiv-papers
| 2010-09-29T19:27:42 |
2024-09-04T02:49:13.219291
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianshu Chen, Jian Wang, Xiu-Ming Shan, Ning Ge, Xiang-Gen Xia",
"submitter": "Jianshu Chen",
"url": "https://arxiv.org/abs/1009.5979"
}
|
1009.6112
|
# Higher-order derivatives of the QR and of the real symmetric eigenvalue
decomposition in forward and reverse mode algorithmic differentiation
Sebastian F. Walter sebastian.walter@gmail.com Department of Mathematics,
Faculty of Mathematics and Natural Sciences II,
Humboldt-Universität zu Berlin, Rudower Chaussee 25, Adlershof, 12489 Berlin,
mail address: Unter den Linden 6, 10099 Berlin, Germany
Lutz Lehmann Department of Mathematics, Faculty of Mathematics and Natural
Sciences II,
Humboldt-Universität zu Berlin, Rudower Chaussee 25, Adlershof, 12489 Berlin,
mail address: Unter den Linden 6, 10099 Berlin, Germany
René Lamour Department of Mathematics, Faculty of Mathematics and Natural
Sciences II,
Humboldt-Universität zu Berlin, Rudower Chaussee 25, Adlershof, 12489 Berlin,
mail address: Unter den Linden 6, 10099 Berlin, Germany
###### Abstract
We address the task of higher-order derivative evaluation of computer programs
that contain QR decompositions and real symmetric eigenvalue decompositions.
The approach is a combination of univariate Taylor polynomial arithmetic and
matrix calculus in the (combined) forward/reverse mode of Algorithmic
Differentiation (AD). Explicit algorithms are derived and presented in an
accessible form. The approach is illustrated via examples.
Keywords: univariate Taylor polynomial arithmetic; higher-order derivatives;
QR decomposition; real symmetric eigenvalue decomposition; algorithmic
differentiation; automatic differentiation
Classification: 65D25; 12E05; 58C15; 65F15; 65F25
## 1 Introduction and related work
This paper is concerned with the efficient evaluation of higher-order
derivatives of functions $F:{\mathbb{R}}^{N}\rightarrow{\mathbb{R}}^{M}$ which
are implemented as computer programs that contain numerical linear algebra
functions like the QR or the real symmetric eigenvalue decomposition.
Traditionally, Algorithmic Differentiation (AD) tools like ADOL-C [GJM+99] or
CppAD [Bel10] regard the functions defined in the C header file math.h as
elementary functions. In the forward mode of AD, their approach to compute
higher-order derivatives is to generalize from real arithmetic to univariate
Taylor polynomial (UTP) arithmetic [GJM+99, GUW00, GW08]. For the reverse mode
of AD, the program evaluation is traced and stored in a computational graph or
on a sequential tape. During the so-called reverse sweep the stored
intermediate values are retrieved and used to compute derivatives (c.f.
Section 4).
As explained in Section 3, the functions in math.h suffice since all
computable functions are a concatenation of these functions. However, working
only at the expression level has also disadvantages since no global knowledge
of the function’s structure can be used. A particularly important class of
algorithms in science and engineering are numerical linear algebra (NLA)
functions. Though NLA functions are typically locally smooth, their
implementations often contain non-differentiable operations and program
branches. If no special care is taken, this may result in incorrect
computations of derivatives. Also, many NLA functions on
${\mathbb{R}}^{N\times N}$ matrices require $\mathcal{O}(N^{3})$ arithmetic
operations. Since during the reverse mode intermediate results are required,
this would yield an $\mathcal{O}(N^{3})$ memory requirement. Though it may be
possible to adapt codes to yield reduced memory requirements, as for instance
reported for the LU decomposition [Gri03], in practice it can be a cumbersome
and error-prone process. Also one would like to reuse existing, high-
performance implementations of NLA algorithms. Adding the NLA functions to the
list of elementary functions circumvents this problem. This has been realized
before [Büc02, BH96] and also UTP algorithms for some NLA functions (e.g. the
solution of linear equations) have been implemented in software [Eri03].
The contribution of this paper is to provide explicit algorithms for UTP
arithmetic applied to the QR decomposition and the real symmetric eigenvalue
decomposition. Note that our approach to the real symmetric eigenvalue
decomposition is similar to [AT98, vdAMM07] but our algorithmic result
differs. In addition, we also treat the reverse mode of AD.
The document is structured as follows: In Section 2 we give two application
examples for the algorithms presented in this document, followed by a brief
review of the underlying computational model in Section 3. We shortly describe
the basics of AD in Section 4 where we make use of the results from 3. In
Section 5 we describe the general approach of NLA functions. After that, we
apply the results from Section 4 to find extended functions for the QR and
eigenvalue decomposition in Section 6 and 7 and also provide pullback
algorithms that are necessary in in the reverse mode of AD. Finally, we
present some numerical results in Section 8.
## 2 Application examples for the proposed algorithms
The purpose of this section is to show two application examples where higher-
order derivatives of computer programs that contain the QR and the real
symmetric eigenvalue decomposition are necessary.
### 2.1 Optimum experimental design
The goal in optimum experimental design (OED) is to minimize some cost
function representing the size of the confidence region of parameters of
interest. We consider here a popular formulation where the objective function
$\Phi(C)\in{\mathbb{R}}$ depends on the covariance matrix
$C\in{\mathbb{R}}^{{N_{p}}\times{N_{p}}}$ of a constrained parameter
estimation problem, where the covariance matrix is computed by
$\displaystyle C$ $\displaystyle=$
$\displaystyle(1\\!\\!\\!\mathrm{I},0)\begin{pmatrix}J_{1}^{T}J_{1}&J_{2}^{T}\\\
J_{2}&0\\\ \end{pmatrix}^{-1}\begin{pmatrix}1\\!\\!\\!\mathrm{I}\\\
0\end{pmatrix}\;,$ (2.1)
and we assume that $J_{1}\equiv
J_{1}(q)\in{\mathbb{R}}^{{N_{m}}\times{N_{p}}}$, $J_{2}\equiv
J_{2}(q)\in{\mathbb{R}}^{{N_{\mathrm{r}}}\times{N_{p}}}$,
$1\\!\\!\\!\mathrm{I}\in{\mathbb{R}}^{N_{p}\times N_{p}}$ and
$q\in{\mathbb{R}}^{{N_{q}}}$. The notation is motivated as follows:
$p\in{\mathbb{R}}^{N_{p}}$ are model (pseudo-)parameters,
$q\in{\mathbb{R}}^{{N_{q}}}$ are control variables and $J_{1}$ and $J_{2}$ are
Jacobians of the residuals resp. of the constraint function with respect to
the parameters $p$. Typical choices for cost function $\Phi$ are the trace,
the determinant or the maximum eigenvalue of the covariance matrix $C$. Though
Eqn. (2.1) correctly describes the covariance matrix $C$, the actual
algorithmic implementation is often a code like
$\displaystyle C$ $\displaystyle=$ $\displaystyle
Q_{2}^{T}\left(Q_{2}J_{1}^{T}J_{1}Q_{2}^{T}\right)^{-1}Q_{2}\;,$ (2.2)
where $Q_{2}$ results from a QR-like decomposition of $J_{2}$, i.e.
$J_{2}^{T}=(Q_{1}^{T},Q_{2}^{T})(L,0)^{T}$. The matrices $J_{1}$ and $J_{2}$
are assumed to satisfy the constraint qualification
${\mathrm{rank}\,}(J_{2}))={N_{\mathrm{r}}}$ and the condition
${\mathrm{rank}\,}(J)={N_{p}}$, where $J=(J_{1}^{T},J_{2}^{T})^{T}$. The
matrix $Q_{2}$ spans the nullspace of $J_{2}$. For a detailed discussion we
refer to Körkel [Kör02] and to Bock and Kostina [KKSB07].
Newton-type optimization algorithms require at least the gradient
$\nabla_{q}\Phi(q)$ of the objective function $\Phi$. To obtain good
convergence near the local minimizer, it is often advantageous if exact
second-order derivatives are available. Since the number of controls ${N_{q}}$
can be large, one would like to have the possibility to compute these
derivatives in the reverse mode of AD. Robust objective functions are often
formulated in a way that require third and even higher-order derivatives, so
it is necessary to have algorithms that scale easily to arbitrary order.
Thus, this example requires the differentiation of the nullspace of a matrix,
the matrix product, matrix inversion, the QR decomposition and the objective
function evaluation, e.g. the eigenvalue decomposition.
### 2.2 Index determination of differential algebraic equations
Many dynamical problems in chemical engineering, rigid body mechanics, circuit
simulation and control theory are described by Differential Algebraic
Equations (DAEs) of the form
$0=f\left(\tfrac{{\rm d}}{{\rm d}x}d(y,x),y,x\right),\;x\in
I=[a,b]\subset{\mathbb{R}},$ (2.3)
where $y:{\mathbb{R}}\to{\mathbb{R}}^{m}$ lives in suitable function space,
$f:{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}\times I\to{\mathbb{R}}^{m}$,
$d:{\mathbb{R}}^{m}\times I\to{\mathbb{R}}^{n}$ are sufficiently smooth and
typically $n$ is smaller than $m$.
Using higher–order derivatives of the functions in the DAE one can, in
general, transform the DAE system into an ODE system of order one. The
_differentiation index_ is the highest derivative order required in this
process, that is, derivatives of up to this order of the original equations
are part of any solution of the DAE. The knowledge of the index allows to
estimate the difficulty to solve the DAE.
There are many different index definitions. Here we consider the tractability
index. To compute it, the DAE is linearized along a given function
$\bar{y}(x)$ as
$\underbrace{\tfrac{\partial}{\partial
z}f(\bar{w}(x),\bar{y}(x),x)}_{=A(x)\in{\mathbb{R}}^{m\times n}}\tfrac{{\rm
d}}{{\rm d}x}\Bigl{(}\underbrace{\tfrac{\partial}{\partial
y}d(\bar{y}(x),x)}_{=D(x)\in{\mathbb{R}}^{n\times
m}}z(x)\Bigr{)}+\underbrace{\tfrac{\partial}{\partial
y}f(\bar{w}(x),\bar{y}(x),x)}_{=B(x)\in{\mathbb{R}}^{m\times
m}}z(x)=\underbrace{-f\left(\bar{w}(x),\bar{y}(x),x\right)}_{q(x)}$
with $\bar{w}(x)=\frac{{\rm d}}{{\rm d}x}d(\bar{y}(x),x)$. The coefficient
functions $A=A(x),\;D=D(x)$ and $B=B(x)$ give rise to a matrix sequence
$\displaystyle G_{0}$ $\displaystyle=AD,$ $\displaystyle B_{0}$
$\displaystyle=B,$ $\displaystyle G_{i+1}$ $\displaystyle=G_{i}+B_{i}Q_{i},$
$\displaystyle B_{i+1}$ $\displaystyle=B_{i}P_{i}-G_{i+1}D^{-}\tfrac{{\rm
d}}{{\rm d}x}(DP_{0}\cdots P_{i+1}D^{-})DP_{0}\cdots P_{i},$ (2.4)
where $Q_{i}$ describes a projector onto ${\rm ker}\,G_{i}$, $P_{i}=I-Q_{i}$
and $D^{-}$ is a generalized reflexive inverse of $D$. Now, the tractability
index is the smallest number $\mu\in{\mathbb{N}}$ where $G_{\mu}$ is
nonsingular. The projectors $Q_{i}$ can be determined mainly by use of a QR
decomposition.
A QR decomposition of the potentially singular matrix
$G\in{\mathbb{R}}^{M\times M}$ with ${\mathrm{rank}\,}G=r$ results in
$G\mathnormal{\Pi}=Q\begin{pmatrix}R_{1}&R_{2}\\\ 0&0\end{pmatrix},$
where $\mathnormal{\Pi}$ describes a column permutation matrix, $Q$ an
orthogonal matrix and $R_{1}\in{\mathbb{R}}^{r\times r}$ an upper triangular
matrix. Then a nullspace projector $Q_{G}$ onto ${\rm ker}\,G$ is given by
$Q_{G}=\mathnormal{\Pi}\begin{pmatrix}0&-R^{-1}_{1}R_{2}\\\
0&I\end{pmatrix}\mathnormal{\Pi}^{T}.$
The computation of $B_{i+1}$ via (2.4) needs the differentiation of
$DP_{0}\cdots P_{i+1}D^{-}$ with respect to $x$. Thus, higher–order
derivatives of a function that contains the QR decomposition are necessary.
For a in-depth discussion of index definition of DAEs see März [Mär02, Mär03].
## 3 Computational model
We consider functions
$\displaystyle F:{\mathbb{R}}^{N}$ $\displaystyle\rightarrow$
$\displaystyle{\mathbb{R}}^{M}$ $\displaystyle x$ $\displaystyle\mapsto$
$\displaystyle y=F(x)\;,$
that can be described by the _three-part form_
$\displaystyle v_{n-N}$ $\displaystyle=$ $\displaystyle x_{n}\quad\quad\quad
n=1,\dots,N$ $\displaystyle v_{l}$ $\displaystyle=$
$\displaystyle\phi_{l}(v_{j\prec l})\quad l=1,\dots,L$ $\displaystyle y_{M-m}$
$\displaystyle=$ $\displaystyle v_{L-m}\quad\quad m=M-1,\dots,0\;,$
where $\phi_{l}\in\\{+,-,\cdot,/,\sin,\exp,\dots\\}$ are called _elementary
functions_ , $v_{l}$ are intermediate values and $v_{i\prec l}$ denote the
tuples of input arguments of $\phi_{l}$. For instance the function
$F:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$, $x\mapsto
y=F(x)=\sin(x_{1}+\cos(x_{2})*x_{1})$ is described by
independent | $v_{-1}$ | = | $x_{1}$ | = | $3$
---|---|---|---|---|---
independent | $v_{0}$ | = | $x_{2}$ | = | $7$
| $v_{1}$ | = | $\phi_{1}(v_{0})$ | = | $\cos(v_{0})$
| $v_{2}$ | = | $\phi_{2}(v_{1},v_{-1})$ | = | $v_{1}v_{-1}$
| $v_{3}$ | = | $\phi_{3}(v_{-1},v_{2})$ | = | $v_{-1}+v_{2}$
| $v_{4}$ | = | $\phi_{4}(v_{3})$ | = | $\sin(v_{3})$
dependent | $y$ | = | $v_{4}$ | |
It shows a sequential representation of the computation. Alternatively, one
can describe the function evaluation as composite function
$\displaystyle F(x)$ $\displaystyle=$ $\displaystyle
P_{y}\circ\Phi_{L}\circ\Phi_{L-1}\circ\dots\circ\Phi_{1}\circ P_{x}^{T}(x)\;,$
(3.1)
where $\Phi_{l}:\mathcal{H}\rightarrow\mathcal{H}$, $s^{(l-1)}\mapsto
s^{(l)}=\Phi_{l}(s^{(l-1)})$ are called _elementary transitions_ that operate
the _state space_ $\mathcal{H}$. Each elementary transition can be written as
$\Phi_{l}=P_{l}\circ\phi_{l}\circ
Q_{l}+(1\\!\\!\\!\mathrm{I}-(1-\sigma_{l})P_{l}\circ P_{l}^{T})\;.$ (3.2)
where the functions
$\phi_{l}:\mathcal{D}_{l}\subseteq\mathcal{H}_{l}\rightarrow\mathcal{H}_{l}\in\\{+,-,*,/,\sin,\exp,\dots\\}$
are the elementary functions. The
$Q_{l}:\mathcal{H}\rightarrow\mathcal{H}_{l}$ map to the domains of the
elementary functions and the $P_{l}:\mathcal{H}_{l}\rightarrow\mathcal{H}$ map
back to the overall state space. The functions $P_{x}^{T}$ and $P_{y}$ are
used to map the independent variables $x$ to the state $s^{(0)}$ and $s^{(L)}$
to $y$. The case $\sigma_{l}=1$ corresponds to an augmented assignment
$s_{l}=s_{l}+\phi_{l}(s_{l})$ and $\sigma_{l}=0$ to the usual assignment
$s_{l}=\phi_{l}(s_{l})$. For our purposes it suffices to consider a real
vector space as state space, i.e., the mappings $P_{l}$ and $Q_{l}$ can be
written as matrices. For a more detailed discussion see Griewank [Gri03].
## 4 Algorithmic differentiation
In this section we briefly review some key results from the theory of AD that
will be necessary in Section 6 and 7. For a detailed discussion we refer to
the standard reference “Evaluating Derivatives” by Griewank and Walther
[GW08].
### 4.1 The forward mode
One can use univariate Taylor series expansions to compute higher-order
(directional) derivatives. The basic observation is that given a smooth curve
$x(t)=x_{0}+x_{1}t$ with $t\in(-\epsilon,\epsilon)$, $\epsilon>0$, and a
smooth function $F$ one obtains a smooth curve $y(t)=F(x(t))$ with the Taylor
series expansion
$y(t)=\sum_{d=0}^{D-1}y_{d}t^{d}+\mathcal{O}(t^{D})=\sum_{d=0}^{D-1}\left.\frac{1}{d!}\frac{{\rm
d}^{d}}{{\rm d}t^{d}}F(x(t))\right|_{t=0}t^{d}+\mathcal{O}(t^{D})\;.$ (4.1)
By application of the chain rule one can interpet the terms of the expansion.
The zeroth derivative is the normal function evaluation $y_{0}=F(x_{0})$ and
the first coefficient $y_{1}=\left.\frac{{\rm d}}{{\rm
d}t}F(x(t))\right|_{t=0}=\frac{{\rm d}}{{\rm d}x}F(x_{0})\cdot x_{1}$ is a
directional derivative.
In the context of AD it is advantageous to generalize the notion of Taylor
series expansions to a purely algebraic task. In other words, for arithmetic
with univariate Taylor polynomials (UTP) one extends functions
$F:{\mathbb{R}}^{N}\rightarrow{\mathbb{R}}^{N}$ to functions
$E_{D}(F):{\mathbb{R}}^{N}[T]/(T^{D})\rightarrow{\mathbb{R}}^{M}[T]/(T^{D})$.
We denote representing elements of the polynomial factor ring
${\mathbb{R}}^{N}[T]/(T^{D})$ as
$[x]_{D}:=[x_{1},\dots,x_{D-1}]:=\sum_{d=0}^{D-1}x_{d}T^{d}\;,$ (4.2)
where $x_{d}\in{\mathbb{R}}^{N}$ is called _Taylor coefficient_. The quantity
$T$ is an indeterminate, i.e., a formal variable. The _extended function_
$E_{D}(F)$ is defined by its action
$\displaystyle[y]_{D}$ $\displaystyle=$ $\displaystyle
E_{D}(F)([x]_{D})=\sum_{d=0}^{D-1}y_{d}T^{d}=\left.\sum_{d=0}^{D-1}\frac{1}{d!}\frac{{\rm
d}^{d}}{{\rm d}t^{d}}F(\sum_{d=0}^{D-1}x_{d}t^{d})\right|_{t=0}T^{d}\;.$ (4.3)
The notation $[x]_{D}\equiv[x]_{:D-1}$ and
$[x]_{d+1:D-1}\equiv[x]_{d+1:}\equiv[x_{d+1},\dots,x_{D-1}]$ will be useful
later on. One can show that this definition is compatible with the usual
polynomial addition and multiplication. Furthermore, any composite function
$F(x)=(H\circ G)(x)=H(G(x))$ satisfies
$\displaystyle E_{D}(F)$ $\displaystyle=$ $\displaystyle E_{D}(H)\circ
E_{D}(G)\;.$ (4.4)
I.e., the extension function $E_{D}$ is a homomorphism which preserves
function composition. An immediate consequence is that it is necessary to find
algorithms only for the very limited set of elementary functions
$\phi\in\\{+,-,*,/,\sin,\cos,\exp,\dots\\}$. Explicitly, one performs the
program transformation $E_{D}(F)=E_{D}(\Phi_{L})\circ\dots\circ
E_{D}(\Phi_{1})([x]_{D})$. We call the action of computing
$[y]_{D}=E_{D}(F)([x]_{D})$, i.e., the resolution of the symbolic dependence
to obtain the numerical value $[y]_{D}$, the _pushforward_ of the function
$E_{D}(F)$.
Many functions are implicitly defined by equations of the type
$0=F(x,y)\in{\mathbb{R}}^{M}\;$, where $x\in{\mathbb{R}}^{N}$ are the inputs
and $y\in{\mathbb{R}}^{M}$ the outputs. The idea is to demand that the
_defining equations of order $D$_
$0{\stackrel{{\scriptstyle D}}{{=}}}E_{D}(F)([x]_{D},[y]_{D})$ (4.5)
should be satisfied. By ${\stackrel{{\scriptstyle D}}{{=}}}$ it is meant that
$[x]{\stackrel{{\scriptstyle D}}{{=}}}[y]$ if $x_{d}=y_{d}$ for
$d=0,\dots,D-1$. This is also often written either as
$[x]=[y]+\mathcal{O}(T^{D})$ or $[x]=[y]\mod T^{D}$. The defining equations
lead directly to an algorithmic approach to compute $[y]_{D}$, the so-called
_Newton-Hensel lifting_. In the literature it is often also just called
Hensel-lifting or Newton’s method [GW08]. Assuming $[y]_{D}$ is already known
and satisfies $0{\stackrel{{\scriptstyle D}}{{=}}}E_{D}(F)([x],[y]_{D})$, one
can lift the computation to a higher degree. Explicitly, one tries to solve
$0{\stackrel{{\scriptstyle D+E}}{{=}}}E_{D+E}(F)([x],[y]_{D+E})$. Splitting
$[y]_{D+E}=[y]_{D}+[\Delta y]_{E}T^{D}$ and performing a first order Taylor
expansion of $F$ about $[y]_{D}$ yields after a short calculation
$\displaystyle[\Delta y]_{E}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle-[F_{y}]_{E}^{-1}[\Delta F]_{E}\;,$ (4.6)
where $E_{D+E}(F)([x],[y]_{D}){\stackrel{{\scriptstyle D+E}}{{=}}}[\Delta
F]_{E}T^{D}$ and $[F_{y}]_{E}:=E_{E}(\frac{{\rm d}F}{{\rm d}y})([x],[y]_{E})$.
Setting $E=D$ means that at each step the number of correct coefficients is
doubled. In this case we call it _Newton’s method_. In the case $E=1$ only the
next coefficient is computed. We call the special case $E=1$ _sequential
Hensel lifting_ which is also the formula that is often given as part of the
implicit function theorem. The difference is that Newton-Hensel lifting is a
purely algebraic task. For a discussion on how to obtain asymptotically fast
algorithms and for the nomenclature see e.g. Bernstein [Ber01, Ber08].
### 4.2 The reverse mode
The basic idea of the reverse mode of AD is to pullback linear forms $\alpha$
to obtain an explict mapping $\bar{y}\mapsto\bar{x}$. I.e., given
$F:{\mathbb{R}}^{N}\rightarrow{\mathbb{R}}^{M},\;y=F(x)$ one has
$\displaystyle\alpha(\bar{y},y)$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\bar{y}_{m}{\rm
d}y_{m}=\sum_{m=1}^{M}\bar{y}_{m}\sum_{n=1}^{N}\frac{\partial F_{m}}{\partial
x_{n}}{\rm d}x_{n}=\sum_{n=1}^{N}\bar{x}_{n}{\rm d}x_{n}=\alpha(\bar{x},x)\;,$
(4.7)
where $\bar{x}_{n}=\sum_{m=1}^{M}\bar{y}_{m}\frac{\partial F_{m}}{\partial
x_{n}}$. For notational reasons one uses $\sum_{n=1}^{N}\bar{x}_{n}{\rm
d}x_{n}\equiv\bar{x}^{T}{\rm d}x$. We call the action of going back one level
of the symbolic dependence the _pullback_ of the linear form
$\alpha(\bar{y},y)$. For a more detailed discussion on calculations with
differentials see Magnus and Neudecker [MN99].
It is also possible to compute higher-order derivatives by combining UTP
arithmetic and the reverse mode of AD. For that, the UTP algorithms are
interpreted as functions mapping $D$ coefficients $x_{d}$, $0\leq d<D$ to $D$
coefficients $y_{d}$, $0\leq d<D$, i.e., a mapping from ${\mathbb{R}}^{N\times
D}\rightarrow{\mathbb{R}}^{M\times D}$ with a special structure. One can
formally define a linear form by
$E_{D}(\alpha)([\bar{y}]_{D},[y]_{D}):=[y]_{D}^{T}{\rm d}[y]_{D}\;.$ (4.8)
Here, ${\rm d}[y]_{D}=\sum_{d=0}^{D-1}{\rm d}y_{d}T^{d}$ is a formal
polynomial where each coefficient is a differential and $[\bar{y}]_{D}^{T}{\rm
d}[y]_{D}=\sum_{m=1}^{M}[\bar{y}_{m}]_{D}{\rm d}[y_{m}]_{D}$ computes the
polynomial multiplication of formal polynomials. One can show that
$\displaystyle E_{D}(\alpha)([\bar{y}]_{D},[y]_{D})$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[\bar{y}]_{D}^{T}E_{D}(\frac{\partial F}{\partial
x})([x]_{D}){\rm d}[x]_{D}{\stackrel{{\scriptstyle
D}}{{=}}}[\bar{x}_{n}]_{D}^{T}{\rm
d}[x_{n}]_{D}=E_{D}(\alpha)([\bar{x}]_{D},[x]_{D})$ (4.9)
holds [Chr91]. One can interpret this result as follows: If
$[\bar{y}]_{D}=w\in{\mathbb{R}}^{M}$ then
$[\bar{x}]_{D}=E_{D}(w^{T}\frac{\partial F}{\partial x})([x]_{D})$. Setting
$w=e_{i}$ a Cartesian basis vector would yield the Taylor expansion of the
$i$’th row of the Jacobian. The interpretation of the Taylor coefficients as
derivatives yields higher-order derivatives. If $M=1$ and $w=1$ one obtains
the Taylor expansion of the gradient $[\bar{x}]_{D}=E_{D}(\nabla F)([x]_{D})$.
E.g., propagating the UTP $[x]_{2}=x_{0}+x_{1}T$ would yield
$[\bar{x}]_{2}=\bar{x}_{0}+\bar{x}_{1}T$ where $\bar{x}_{0}=\nabla_{x}F(x)$
and $\bar{x}_{1}=\nabla_{x}^{2}F(x)\cdot x_{1}$, i.e., a Hessian-vector
product.
## 5 Defining equations of numerical linear algebra functions
As briefly mentioned in the introduction, Numerical Linear Algebra (NLA)
functions can be viewed as algorithms representing a concatenation of
functions like $+,-,*,/,\sin,\cos,\dots$ and thus it is possible to apply the
AD techniques described in the previous section directly to the algorithm.
However, there is also the possibility to regard the problem from a more
abstract point of view. Many NLA functions are implicitly defined by a system
of equations.
For instance the QR decomposition is defined by the defining equations
$\displaystyle 0$ $\displaystyle=$ $\displaystyle QR-A$ (5.1) $\displaystyle
0$ $\displaystyle=$ $\displaystyle Q^{T}Q-1\\!\\!\\!\mathrm{I}$ (5.2)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle P_{L}\circ R\;,$ (5.3)
where $A,R\in{\mathbb{R}}^{M\times N}$ with $M\geq N$ and
$Q\in{\mathbb{R}}^{M\times M}$. The functional dependence of the defining
equations is denoted
$\displaystyle Q,R={\rm qr\,}(A)\;.$ (5.4)
Only the first $N$ rows $R_{:N,:}\in{\mathbb{R}}^{N\times N}$ of $R$ are
nonzero. For convenience reasons we use the slicing notation
$i:j=(i,i+1,\dots,j)$.
The defining equations of the symmetric eigenvalue decomposition are given by
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}AQ-\Lambda$ (5.5)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}Q-1\\!\\!\\!\mathrm{I}$
(5.6) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(P_{L}+P_{R})\circ\Lambda\;,$ (5.7)
where $A\in{\mathbb{R}}^{M\times M}$ is symmetric. The functional dependence
is denoted $\Lambda,Q={\rm eigh\,}(A)$. We call the matrices
$(P_{L})_{ij}=\delta_{j<i}$ and $(P_{R})_{ij}=\delta_{i<j}$ skeletal
projectors since their elementwise product with a matrix returns strictly
lower resp. strictly upper triangular matrices.
## 6 The QR decomposition
Before we derive algorithms based on the defining equations, we briefly
investigate what can go wrong if a typical implementation of the QR
decomposition using Householder reflections is evaluated in UTP arithmetic.
Consider Algorithm 5.1.1 from the book “Matrix Computations” by Golub and Van
Loan [GVL96] which we adapted to our notation in Algorithm 1. From the AD
point of view, the problematic part in the code is the check $\sigma=0$. Since
a paradigm of AD tools is that the control flow must remain unchanged, the
check $\sigma=0$ only considers the zeroth coefficient $x_{0}$ of a UTP.
Hence, if $[x]_{2}=e_{1}+x_{1}T$ is given as input and $x_{1}\neq 0$, the
algorithm will simply evaluate $\beta=0$ and return. As final result, one
obtains a matrix $[R]_{2}$ where $R_{1}$ is not upper triangular. The LAPACK
implementation (LAPACK-3.2.2) of DGEQRFP.f calls the subroutine DLARFGP.f
which contains a similar check. Hence, automatic augmentation based on AD
principles can go wrong in such cases.
As a side remark, note that additionally the function realized by this
algorithm has a pole at $\sigma=0$, producing numerical overflow for
$\sigma\approx 0$.
input : $x\in{\mathbb{R}}^{N}$
output : $v\in{\mathbb{R}}^{N}$ with $v_{1}=1$
output : $\beta\in{\mathbb{R}}$
$\sigma=x_{2:}^{T}x_{2:}$
$v=\begin{pmatrix}1\\\ x_{2:}\end{pmatrix}$
if _$\sigma=0$_ then $\beta=0$ else $\mu=\sqrt{x_{1}^{2}+\sigma}$
if _$x_{1}\leq 0$_ then $v_{1}=x_{1}-\mu$ else $v_{1}=-\sigma/(x_{1}+\mu)$ end
$\beta=2v_{1}^{2}/(\sigma+v_{1}^{2})$
$v=v/v_{1}$ end
Algorithm 1 Householder Vector. The reflector is $P_{v}=I-\beta vv^{\top}$,
with $v_{1}=1$.
### 6.1 Pushforward in Taylor arithmetic
We now come to the higher–level approach that is based on the defining
equations given in Section 5. To compute $[Q]_{D},[R]_{D}=E_{D}({\rm
qr\,})([A]_{D})$ one can apply Newton-Hensel lifting to solve
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}[R]_{D}-[A]_{D}$ (6.1) $\displaystyle 0$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I}$ (6.2) $\displaystyle 0$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$ $\displaystyle
P_{L}\circ[R]_{D}\;.$ (6.3)
The direct application of Eqn. (4.6) should be avoided since $F_{y}$ is sparse
and has a lot of structure. Rather, one assumes that one has already computed
$[Q]_{D}$ and $[R]_{D}$ and computes the next $1\leq E\leq D$ coefficients by
performing a first order Taylor expansion $[Q]_{D+E}=[Q]_{D}+[\Delta
Q]_{E}T^{D}$ and $[R]_{D+E}=[R]_{D}+[\Delta R]_{E}T^{D}$ and tries to solve
for the yet unknown $[\Delta R]_{E}$ and $[\Delta Q]_{E}$. As result one
obtains Proposition 1. For convenience, we use the convention that $R_{d;i,j}$
is the $i$’th row and $j$’th column of the $d$’th coefficient of $[R]_{D}$.
###### Proposition 1.
Let $[A]_{D+E}\in{\mathbb{R}}^{M\times N}[T]/(T^{D+E})$ with $M\geq N$ and
$1\leq E\leq D$, $[R]_{D}\in{\mathbb{R}}^{M\times N}[T]/(T^{D})$ where
$[R_{:N,:}]_{D}$ is upper triangular with nonsingular $R_{0;:N,:}$ and
$[Q]_{D}\in{\mathbb{R}}^{M\times M}[T]/(T^{D})$ orthogonal be given and
satisfy the defining equations of order $D$. Then $[\Delta
R_{:N,:}]_{E}\equiv[R_{:N,:}]_{D:D+E-1}$ and $[\Delta
Q]_{E}\equiv[Q]_{D:D+E-1}$ are given by
$\displaystyle\;[\Delta F]_{E}T^{D}$ $\displaystyle{\stackrel{{\scriptstyle
D+E}}{{=}}}$ $\displaystyle-[Q]_{D}[R]_{D}+[A]_{D+E}$ (6.4)
$\displaystyle\;[\Delta G]_{E}T^{D}$ $\displaystyle{\stackrel{{\scriptstyle
D+E}}{{=}}}$ $\displaystyle-[Q]^{T}_{D}[Q]_{D}+1\\!\\!\\!\mathrm{I}$ (6.5)
$\displaystyle\;[S]_{E}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle\frac{1}{2}[\Delta G]_{E}$ (6.6)
$\displaystyle\;P_{L}\circ([X_{:,:N}]_{E})$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle
P_{L}\circ\left([Q]^{T}_{E}[\Delta
F]_{E}[R_{:N,:}]_{E}^{-1}\right)-P_{L}\circ[S_{:,:N}]_{E}$ (6.7)
$\displaystyle\;[\Delta R]_{E}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle[Q]_{E}^{T}[\Delta F]_{E}-([S]_{E}+[X]_{E})[R]_{E}$
(6.8) $\displaystyle\;[\Delta Q]_{E}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle[Q]_{E}\left([S]_{E}+[X]_{E}\right)\;,$ (6.9)
where $P_{L}\in{\mathbb{R}}^{M\times N}$ with $(P_{L})_{ij}=\delta_{j<i}$.
###### Proof.
In the Appendix A.1.1. ∎
### 6.2 Pullback
###### Proposition 2.
Let $A,R,\bar{R}\in{\mathbb{R}}^{M\times N}$ resp.
$Q,\bar{Q}\in{\mathbb{R}}^{M\times M}$ be given and it holds $M\geq N$,
${\mathrm{rank}\,}(A)=N$, $Q,R={\rm qr\,}(A)$. Then
$\bar{A}\in{\mathbb{R}}^{M\times N}$ can be computed by
$\displaystyle\bar{A}$ $\displaystyle=$ $\displaystyle
Q\left(\bar{R}+\left(P_{L}\circ\left(R\bar{R}^{T}-\bar{R}R^{T}+Q^{T}\bar{Q}-\bar{Q}^{T}Q\right)\right)R^{+T}\right)\;.$
(6.10)
Here, $R^{+}$ denotes the Moore-Penrose pseudoinverse of $R$. That means it
satisfies $RR^{+}R=R$ and since $R$ has full column rank also
$R^{+}R=1\\!\\!\\!\mathrm{I}$.
###### Proof.
In Appendix A.1.2. ∎
### 6.3 Explicit algorithms
One can use Proposition 1 to derive an explicit algorithm as shown in
Algorithm 2, where at each step $E=1$ is used.
input : $[A]_{D}=[A_{0},\dots,A_{D-1}]$, where $A_{d}\in{\mathbb{R}}^{M\times
N}$, $d=0,\dots,D-1$ and ${\mathrm{rank}\,}(A_{0})=N$, $M\geq N$.
output : $[Q]_{D}=[Q_{0},\dots,Q_{D-1}]$ matrix with orthonormal column
vectors, where $Q_{d}\in{\mathbb{R}}^{M\times N}$ , $d=0,\dots,D-1$
output : $[R]_{D}=[R_{0},\dots,R_{D-1}]$ upper triangular, where
$R_{d}\in{\mathbb{R}}^{N\times N}$ , $d=0,\dots,D-1$
$Q_{0},R_{0}={\rm qr\,}(A_{0})$
for _$d=1$ to $D-1$_ do $\Delta F=A_{d}-\sum_{k=1}^{d-1}Q_{d-k}R_{k}$
$S=-\frac{1}{2}\sum_{k=1}^{d-1}Q^{T}_{d-k}Q_{k}$
$X_{:,:N}=P_{L}\circ(Q_{0}^{T}\Delta FR_{0;:N,:N}^{-1}-S_{:,:N})$
$X_{:,N+1:}=0$
$X=X-X^{T}$
$R_{d}=Q_{0}^{T}\Delta F-(S+X)R_{0}$
$Q_{d}=Q_{0}(S+X)$ end
Algorithm 2 Sequential Hensel lifting for the QR decomposition.
The pullback can be computed in Taylor arithmetic. In the global derivative
accumulation it is necessary to update the value of $[\bar{A}]_{D}$. This
happens if $[A]_{D}$ is input of more than one function. The algorithm for the
pullback takes this into consideration.
input : $[A]_{D}=[A_{0},\dots,A_{D-1}]$, where $A_{d}\in{\mathbb{R}}^{M\times
N}$, $d=0,\dots,D-1$, $M\geq N$.
input : $[Q]_{D}=[Q_{0},\dots,Q_{D-1}]$, where $Q_{d}\in{\mathbb{R}}^{M\times
M}$ , $d=0,\dots,D-1$
input : $[R]_{D}=[R_{0},\dots,R_{D-1}]$, where $R_{d}\in{\mathbb{R}}^{M\times
N}$ , $d=0,\dots,D-1$
input/output : $[\bar{A}]_{D}=[\bar{A}_{0},\dots,\bar{A}_{D-1}]$, where
$\bar{A}_{d}\in{\mathbb{R}}^{M\times N}$, $d=0,\dots,D-1$, $M\geq N$.
input : $[\bar{Q}]_{D}=[\bar{Q}_{0},\dots,\bar{Q}_{D-1}]$, where
$\bar{Q}_{d}\in{\mathbb{R}}^{M\times M}$ , $d=0,\dots,D-1$
input : $[\bar{R}]_{D}=[\bar{R}_{0},\dots,\bar{R}_{D-1}]$, where
$\bar{R}_{d}\in{\mathbb{R}}^{M\times N}$ , $d=0,\dots,D-1$
$\displaystyle[\bar{A}]_{D}$ $\displaystyle=$
$\displaystyle[\bar{A}]_{D}+[Q]_{D}\cdot$
$\displaystyle\cdot\left([\bar{R}]_{D}+\left(P_{L}\circ\left([R]_{D}[\bar{R}]_{D}^{T}-[\bar{R}]_{D}[R]_{D}^{T}+[Q]_{D}^{T}[\bar{Q}]_{D}-[\bar{Q}]_{D}^{T}[Q]_{D}\right)\right)[R]_{D}^{+T}\right)$
Algorithm 3 Pullback of the QR decomposition in Taylor arithmetic. The inputs
$[A]_{D},[Q]_{D},[R]_{D}$ must satisfy the defining equations.
## 7 The real symmetric eigenvalue decomposition
The problem of finding eigenvalues and eigenvectors arises in a wide variety
of practical applications. As for the QR decomposition, we want to have
algorithms that compute the real symmetric eigenvalue decomposition in UTP
arithmetic as well as pullback algorithms. The symmetric eigenvalue
decomposition is also important since the Singular Value Decomposition (SVD)
of real matrices is closely related to it. More explicitly, one can compute
the SVD of a matrix $A\in{\mathbb{R}}^{M\times N}$ of rank $r$., i.e.,
$A=U\Sigma V^{T}$, where $\Sigma=\mathop{\rm diag}(\Sigma_{1},0)$,
$U=(U_{1},U_{2})$, $U_{1}\in{\mathbb{R}}^{M\times r}$, $V=(V_{1},V_{2})$,
$V_{1}\in{\mathbb{R}}^{N\times r}$ as
$\displaystyle C=\begin{pmatrix}0&A\\\
A^{T}&0\end{pmatrix}=P^{T}\begin{pmatrix}\Sigma_{1}&0&0\\\ 0&-\Sigma_{1}&0\\\
0&0&0\\\ \end{pmatrix}P\;,$
where
$\displaystyle
P=\frac{1}{\sqrt{2}}\begin{pmatrix}U_{1}&U_{1}&\sqrt{2}U_{2}&0\\\
V_{1}&-V_{1}&0&\sqrt{2}V_{2}\end{pmatrix}^{T}$
is orthogonal [Bjö96].
### 7.1 Pushforward in Taylor arithmetic
Given the symmetric polynomial matrix $[A]_{D}\in{\mathbb{R}}^{N\times
N}[T]/(T^{D})$. The eigenvalue decomposition is the solution
$[\Lambda]_{D},[Q]_{D}\in{\mathbb{R}}^{N\times N}[T]/(T^{D})$ of the implicit
system
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[A]_{D}[Q]_{D}-[\Lambda]_{D}$ (7.1) $\displaystyle 0$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I}$ (7.2) $\displaystyle 0$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle(P_{L}+P_{R})\circ[\Lambda]_{D}\;,$ (7.3)
which is called the _defining equations of order $D$_. We also assume that the
eigenvalues are sorted as
$[\Lambda_{11}]_{D}\leq[\Lambda_{22}]_{D}\leq\dots\leq[\Lambda_{NN}]_{D}$. The
functional dependence is denoted
$\displaystyle[\Lambda]_{D},[Q]_{D}$ $\displaystyle={\rm eigh\,}([A]_{D})\;.$
(7.4)
Let $\Lambda,Q={\rm eigh\,}(A)$ be the usual symmetric eigenvalue
decomposition. We denote the diagonal of $[\Lambda]_{D}$ as
$[\lambda]_{D}=\mathop{\rm diag}([\Lambda]_{D})$. If eigenvalues are repeated,
i.e., multiple, the eigenvectors generalize to eigenspaces and the columns of
$Q$, that are associated to such a multiple eigenvalue, are not unique.
Rather, any orthonormal basis could be the result. This has consequences for
the Hensel-Newton lifting approach, because given $[Q]_{D}$ and $[R]_{D}$ that
satisfy the defining equations of order $D$ it is generally not possible to
find a $[\Delta Q]_{E}$ and $[\Delta R]_{E}$ such that
$[Q]_{D+E}=[Q]_{D}+[\Delta Q]_{E}T^{D}$ and $[R]_{D+E}=[R]_{D}+[\Delta
R]_{E}T^{D}$ satisfy the defining equations of order $D+E$. The higher-order
coefficients $[\Delta A]_{E}$ enforce additional conditions on the chosen
basis of the eigenspaces. A wrong choice of $[Q]_{D}$ means that
$0{\stackrel{{\scriptstyle D+E}}{{=}}}(P_{L}+P_{R})\circ[\Lambda]_{D+E}$
cannot be satisfied. However, $0{\stackrel{{\scriptstyle
D}}{{=}}}P_{b}^{D}\circ[\Lambda]_{D+E}$ can be satisfied. The matrix
$P_{b}^{D}$ is a skeletal projector with zero blocks on the main diagonal
whose size corresponds to the multiplicity of an eigenvalue $[\lambda]_{D}$
and all other entries are ones. The _multiplicity_ $m^{d}([\lambda_{j}]_{D})$
of an eigenvalue $[\lambda_{j}]_{D}$ of _level_ $d$ is defined to be the
number of $i\in{\mathbb{N}}$ s.t. $[\lambda_{j}]_{D}\stackrel{{\scriptstyle
d}}{{=}}[\lambda_{i}]_{D}$. I.e.,
$\displaystyle\mathop{\rm diag}([\Lambda]_{d})$ $\displaystyle=$
$\displaystyle(\underbrace{[\lambda_{1}]_{d},\dots,[\lambda_{1}]_{d}}_{m^{d}([\lambda_{1}]_{D})\mbox{
times}},\dots,\underbrace{[\lambda_{N_{\mathrm{b}}^{d}}]_{d},\dots,[\lambda_{N_{\mathrm{b}}^{d}}]_{d}}_{m^{d}([\lambda_{N_{\mathrm{b}}^{d}}]_{D})\mbox{
times}}),$
where $N_{\mathrm{b}}^{d}$ is the number of different eigenvalues at level
$d$. We define $b^{d}\in{\mathbb{N}}^{N_{\mathrm{b}}^{d}+1}$ to be a vector
satisfying
$m^{d}([\lambda_{{n_{\mathrm{b}}}}]_{D})=b_{{n_{\mathrm{b}}}+1}^{d}-b_{{n_{\mathrm{b}}}}^{d}$.
The symbol $b$ is used because it relates to blocks in the matrix. The
elements of $P_{b}^{d}$ satisfy
$(P^{d}_{b})_{ij}=1-\sum_{{n_{\mathrm{b}}}=1}^{N_{\mathrm{b}}^{d}+1}\delta_{b_{n_{\mathrm{b}}}^{d}\leq
i<b_{{n_{\mathrm{b}}}+1}^{d}}\delta_{b_{n_{\mathrm{b}}}^{d}\leq
j<b_{{n_{\mathrm{b}}}+1}^{d}}$. This notation is a little cumbersome but turns
out to be helpful. One defines $b^{0}=[0,N+1]$. The vector $b^{1}$ represents
the multiplicities of the usual symmetric eigenvalue decomposition. E.g., for
$N=3$ and $b^{d}=[1,3,4]$ one has
$\displaystyle P_{b}^{d}=\begin{pmatrix}0&0&1\\\ 0&0&1\\\
1&1&0\end{pmatrix}\;.$
We reformulate the overall problem as a sequence of subproblems. We call the
implicit system
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q^{d}]_{D}^{T}[A]_{D}[Q^{d}]_{D}-[\Lambda^{d}]_{D}$ (7.5)
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q^{d}]_{D}^{T}[Q^{d}]_{D}-1\\!\\!\\!\mathrm{I}$ (7.6)
$\displaystyle 0$ $\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle(P_{L}+P_{R})\circ[\Lambda]_{d}$ (7.7) $\displaystyle 0$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$ $\displaystyle
P_{b}^{d}\circ[\Lambda^{d}]_{D}\;,$ (7.8)
the _relaxed problem of level $d$ and order $D$_. I.e., it is assumed that up
to order $d$ the original problem is solved but only block diagonalized for
the higher order coefficients.
To give an illustrative example consider this relaxed problem of order $3$ and
level $2$. At this point of the algorithm, one has potentially obtained a
matrix polynomial $[\Lambda]_{3}=\sum_{d=0}\Lambda_{d}T^{d}$ with coefficients
of the form
$\displaystyle\Lambda_{0}=\begin{pmatrix}1\\\ &1\\\ &&1\\\ &&&2\\\ &&&&2\\\
&&&&&3\end{pmatrix},\quad\Lambda_{1}=\begin{pmatrix}2\\\ &2\\\ &&3\\\ &&&2\\\
&&&&2\\\ &&&&&2\end{pmatrix},\quad\Lambda_{2}=\begin{pmatrix}1&3\\\ 3&5\\\
&&7\\\ &&&1&2\\\ &&&2&3\\\ &&&&&7\end{pmatrix}\;.$
I.e., $\Lambda_{0}$ and $\Lambda_{1}$ are already diagonal. Since there are
two eigenvalues with multiplicity $m^{2}([\lambda]_{3})=2$ it follows that
$\Lambda_{2}$ is only block diagonal. Note that the eigenvalues are not
globally sorted by value in the higher coefficients but only in the subblocks
defined by the lower order coefficients. In this example, the repeated
eigenvalues in the first block split at the lift from $d=0$ to $d=1$. The
blocks are defined by $b^{1}=[1,4,6,7]$ and $b^{2}=[1,3,4,6,7]$. The blocks in
$\Lambda_{2}$ are defined by $b^{2}$.
The function that solves the relaxed problem of order $D$ and level $d$ is
denoted
$\displaystyle[\Lambda^{d}]_{D},[Q^{d}]_{D}$ $\displaystyle=$
$\displaystyle{\rm eigh\,}_{d}([A]_{D})\;.$ (7.9)
The idea is to implement an algorithm that successively increases $d$ by one.
For convenience we define $[Q^{0}]_{D}:=1\\!\\!\\!\mathrm{I}$ and
$[\Lambda^{0}]_{D}:=[A]_{D}$.
###### Theorem 3.
Let $[A]_{D}$ be given. Then the solution of
$\displaystyle[Q^{d+1}]_{D},[\Lambda^{d+1}]_{D}{\stackrel{{\scriptstyle
D}}{{=}}}{\rm eigh\,}_{d+1}([A]_{D})$ (7.10)
can be computed from the solution
$[Q^{d}]_{D},[\Lambda^{d}]_{D}{\stackrel{{\scriptstyle D}}{{=}}}{\rm
eigh\,}_{d}([A]_{D})$ by computing
$\displaystyle[\hat{\Lambda}_{s,s}]_{D-d},[\hat{Q}_{s,s}]_{D-d}$
$\displaystyle\stackrel{{\scriptstyle D-d}}{{=}}$ $\displaystyle{\rm
eigh\,}_{1}([\Lambda^{d}_{s,s}]_{d:})\;,$ (7.11)
where $s={b^{d}_{{n_{\mathrm{b}}}}:b^{d}_{{n_{\mathrm{b}}}+1}-1}$ are slice
indices and ${n_{\mathrm{b}}}=1,\dots,N_{\mathrm{b}}^{d}$. All other elements
of $[\hat{Q}]_{D-d}$ and $[\hat{\Lambda}]_{D-d}$ are zero. I.e.,
$[\hat{Q}]_{D-d}$ and $[\hat{\Lambda}]_{D-d}$ are block diagonal. It holds
that
$\displaystyle\;[\Lambda^{d+1}]_{D}$ $\displaystyle{\stackrel{{\scriptstyle
D}}{{=}}}$ $\displaystyle[\Lambda^{d}]_{d}+[\hat{\Lambda}]_{D-d}T^{d}$ (7.12)
$\displaystyle\;[Q^{d+1}]_{D}$ $\displaystyle{\stackrel{{\scriptstyle
D}}{{=}}}$ $\displaystyle[Q^{d}]_{D}[Q]_{D}\;,$ (7.13)
where $[Q]_{D}=[\hat{Q}]_{D-d}+[\Delta Q]_{d}T^{D-d}$ for some $[\Delta
Q]_{D-d}$ that satisfies
$\displaystyle 0{\stackrel{{\scriptstyle
D}}{{=}}}[Q]^{T}_{D}[Q]_{D}-1\\!\\!\\!\mathrm{I}\;.$ (7.14)
###### Proof.
We need to show that $[\Lambda^{d+1}]_{D}$, $[Q^{d+1}]_{D}$ is a solution to
the relaxed equations of level $d+1$ and order $D$. From the definition of
${\rm eigh\,}_{1}$ it follows that $0=(P_{L}+P_{R})\circ[\Lambda^{d+1}]_{d+1}$
and $0=P_{b}^{d+1}\circ[\Lambda^{d+1}]_{D}$ is satisfied. We also know that
$0{\stackrel{{\scriptstyle
D}}{{=}}}[Q^{d+1}]_{D}^{T}[Q^{d+1}]_{D}-1\\!\\!\\!\mathrm{I}{\stackrel{{\scriptstyle
D}}{{=}}}[Q]_{D}^{T}[Q^{d}]_{D}^{T}[Q^{d}]_{D}[Q]_{D}-1\\!\\!\\!\mathrm{I}$ is
fulfilled because $0{\stackrel{{\scriptstyle
D}}{{=}}}[Q^{d}]_{D}^{T}[Q^{d}]_{D}-1\\!\\!\\!\mathrm{I}$ and
$0{\stackrel{{\scriptstyle
D}}{{=}}}[Q]_{D}^{T}[Q]_{D}^{T}-1\\!\\!\\!\mathrm{I}$. Hence, it only remains
to show that the third defining equation is satisfied which is shown by the
following straight-forward calculation:
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[Q^{d}]_{D}^{T}[A]_{D}[Q^{d}]_{D}[Q]_{D}-[\Lambda^{d+1}]_{D}$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[\Lambda^{d}]_{D}[Q]_{D}-[\Lambda^{d+1}]_{D}$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}([\Lambda^{d}]_{d}+[\Lambda^{d}]_{d:}T^{d})[Q]_{D}-[\Lambda^{d}]_{d}-[\hat{\Lambda}]_{D-d}T^{d}$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[Q]_{D}^{T}[\Lambda^{d}]_{d}[Q]_{D}+[Q]_{D}^{T}[\Lambda^{d}]_{d:}[Q]_{D}T^{d}-[\Lambda^{d}]_{d}-[\hat{\Lambda}]_{D-d}T^{d}$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[\Lambda^{d}]_{d}[Q]_{D}^{T}[Q]_{D}+[\hat{Q}]_{D-d}^{T}[\Lambda^{d}]_{d:}[\hat{Q}]_{D-d}T^{d}-[\Lambda^{d}]_{d}-[\hat{\Lambda}]_{D-d}T^{d}$
$\displaystyle{\stackrel{{\scriptstyle D}}{{=}}}$
$\displaystyle[\hat{Q}]_{D-d}^{T}[\Lambda^{d}]_{d:}[\hat{Q}]_{D-d}T^{d}-[\hat{\Lambda}]_{D-d}T^{d}$
$\displaystyle\stackrel{{\scriptstyle D-d}}{{=}}$
$\displaystyle[\hat{Q}]_{D-d}^{T}[\Lambda^{d}]_{d:}[\hat{Q}]_{D-d}-[\hat{\Lambda}]_{D-d}\;.$
In the fifth line it has been used that the diagonalization has only to be
performed for block diagonal matrices. If the eigenvalues are already distinct
there is nothing to diagonalize and the step can be skipped. It also means
that one may interchange $[\Lambda^{d}]_{d}$ with $[Q]_{D}$. ∎
The following proposition gives us the means to diagonalize a matrix in the
zeroth degree and block diagonalize w.r.t. the blocks defined by the repeated
eigenvalues. I.e., it gives the justification that the solution of Eqn. (7.11)
can be found. In the case of distinct eigenvalues the application of this
algorithm already solves the original problem.
###### Proposition 4.
Let $[A]_{D+E}=[A]_{D}+[\Delta A]_{E}T^{D}$ be given and $[\Lambda^{d}]_{D}$,
$[Q^{d}]_{D}$ be a solution of the relaxed problem of level $d=1$ and order
$D$. Then it exist $[\Delta\Lambda^{d}]_{E}$ and $[\Delta Q^{d}]_{E}$ such
that $[\Lambda^{d}]_{D+E}=[\Lambda^{d}]_{D}+[\Delta\Lambda^{d}]_{E}T^{D}$ and
$[\Delta Q^{d}]_{D+E}=[\Delta Q^{d}]_{D}+[\Delta Q^{d}]_{E}T^{D}$ are a
solution of the relaxed problem of level $d=1$ and order $D+E$. A closed form
solution is
$\displaystyle\;[\Delta\Lambda^{d}]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle\bar{P}_{b}^{d}\circ[K]_{E}$ (7.15) $\displaystyle\;[\Delta
Q^{d}]_{E}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle[Q^{d}]_{E}\left([\Delta
G]_{E}+P_{b}^{d}\circ\left([K]_{E}/[E]_{E}\right)\right)$ (7.16)
where $[\Delta F]_{E}T^{D}{\stackrel{{\scriptstyle
D+E}}{{=}}}[Q^{d}]^{T}_{D}[A]_{D}[Q^{d}]_{D}-[\Lambda^{d}]_{D}$ and $[\Delta
G]_{E}T^{D}{\stackrel{{\scriptstyle
D+E}}{{=}}}-\frac{1}{2}\left([Q^{d}]^{T}_{D}[Q^{d}]_{D}-1\\!\\!\\!\mathrm{I}\right)$,
$[K]_{E}{\stackrel{{\scriptstyle E}}{{=}}}[\Delta F]_{E}+([\Lambda]_{E}[\Delta
G]_{E}+[\Delta G]_{E}[\Lambda]_{E})+[Q^{d}]^{T}_{E}[\Delta A]_{E}[Q^{d}]_{E}$
and $[E_{ij}]_{E}{\stackrel{{\scriptstyle
E}}{{=}}}[\Lambda^{d}_{jj}]_{E}-[\Lambda^{d}_{ii}]_{E}$. The expression
$[K]_{E}/[E]_{E}$ denotes an element-wise division. $P_{b}^{d}$ is a matrix
with only ones on the diagonal blocks defined by the multiplicity of
eigenvalues in $\Lambda_{0}$. $\bar{P}_{b}^{d}$ is defined s.t.
$\bar{P}_{b}^{d}+P_{b}^{d}$ is a matrix full of ones. One can see here that if
the eigenvalues are distinct, then $\bar{P}_{b}^{d}$ is the identity matrix
$1\\!\\!\\!\mathrm{I}$.
###### Proof.
We set $Q^{d}\equiv Q$ etc. for notational simplicity. Applying Newton-Hensel
lifting to the defining equations yields
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle([Q]_{D}+[\Delta Q]_{E}T^{D})^{T}([Q]_{D}+[\Delta
Q]_{E}T^{D})-1\\!\\!\\!\mathrm{I}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle-2[\Delta G]_{E}+[\Delta
Q]_{E}^{T}[Q]_{E}+[Q]_{E}^{T}[\Delta Q]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle-2[\Delta
G]_{E}+2[S]_{E}\;,$ $\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle
D+E}}{{=}}}$ $\displaystyle([Q]_{D}+[\Delta Q]_{E}T^{D})^{T}([A]_{D}+[\Delta
A]_{E}T^{D})([Q]_{D}+[\Delta
Q]_{E}T^{D})-([\Lambda]_{D}+[\Delta\Lambda]_{E}T^{D})$ (7.17)
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle[\Delta
F]_{E}+[Q]_{E}^{T}[\Delta A]_{E}[Q]_{E}+[\Delta
Q]_{E}^{T}[Q]_{E}[\Lambda]_{E}+[\Lambda]_{E}[Q]_{E}^{T}[\Delta
Q]_{E}-[\Delta\Lambda]_{E}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle[K]_{E}+[X]_{E}[\Lambda]_{E}-[\Lambda]_{E}[X]_{E}-[\Delta\Lambda]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle[K]_{E}+[E]_{E}\circ[X]_{E}-[\Delta\Lambda]_{E}\;.$
Thus $[\Delta\Lambda]_{E}{\stackrel{{\scriptstyle
E}}{{=}}}\bar{P}_{b}^{d}\circ[K]_{E}$ and $[X]_{E}^{T}{\stackrel{{\scriptstyle
E}}{{=}}}P_{b}^{d}\circ([K]_{E}/[E]_{E})$. Above, $[\Delta
Q]_{E}^{T}[Q]_{E}{\stackrel{{\scriptstyle E}}{{=}}}[S]_{E}+[X]_{E}$, $[S]_{E}$
symmetric and $[X]_{E}$ antisymmetric (Lemma 15) has been used. ∎
It remains to show that Eqn. (7.14) can be satisfied.
###### Lemma 5.
Let $[Q]_{D}$ be given and it satisfies the defining equation
$0{\stackrel{{\scriptstyle D}}{{=}}}[Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I}$.
Then the solution can be lifted to $D+E$ with $E\leq D$. I.e., it is possible
to find $[Q]_{D+E}:=[Q]_{D}+[\Delta Q]_{E}T^{D}$ s.t.
$0{\stackrel{{\scriptstyle
D+E}}{{=}}}[Q]_{D+E}^{T}[Q]_{D+E}-1\\!\\!\\!\mathrm{I}$. A closed form
solution for $[\Delta Q]_{E}$ is given by
$\displaystyle\;[\Delta Q]_{E}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle[Q]_{E}[S]_{E}\;,$ (7.18)
where $[S]_{E}T^{D}\stackrel{{\scriptstyle
D+E}}{{=}}-\frac{1}{2}\left([Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I}\right)$.
###### Proof.
In Appendix A.1.3. ∎
### 7.2 Pullback
The eigenvalue decomposition is non-differentiable at points where eigenvalues
are repeated and hence the defining equations do not define a _well behaved
implicit mapping_ as described by Christianson [Chr98]. However, the
eigenvalue decomposition is typically used within a global context where the
non-uniqueness and non-differentiability can be worked around. Here, we give
only the pullback algorithm that is correct for unique eigenvalues.
###### Proposition 6 (Pullback of the Symmetric Eigenvalue Decomposition with
Distinct Eigenvalues:).
Given $A,Q,\Lambda,\bar{Q},\bar{\Lambda}$, where all eigenvalues are distinct,
one can compute $\bar{A}$ by
$\displaystyle\;H_{ij}$ $\displaystyle=$
$\displaystyle(\lambda_{j}-\lambda_{i})^{-1}\quad\mbox{if}\quad i\neq j,\quad
0\quad\mbox{ else}$ (7.19) $\displaystyle\bar{A}$ $\displaystyle=$
$\displaystyle Q\left(\bar{\Lambda}+H\circ(Q^{T}\bar{Q})\right)Q^{T}$ (7.20)
###### Proof.
In Appendix A.1.4. ∎
### 7.3 Explicit algorithms
input : $[Q]_{d}=[Q_{0},\dots,Q_{d-1}]$ with $0\stackrel{{\scriptstyle
d}}{{=}}[Q]_{d}^{T}[Q]_{d}-1\\!\\!\\!\mathrm{I}$
input : $D\in{\mathbb{N}}$
output : $[Q]_{D}=[Q_{0},\dots,Q_{D-1}]$, where $0\stackrel{{\scriptstyle
D}}{{=}}[Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I}$
for _$k=d$ to $D-1$_ do
$Q_{k}=-\frac{1}{2}Q_{0}\sum_{i=1}^{k-1}Q_{i}^{T}Q_{k-i}$ end
Algorithm 4 This algorithm computes $[Q]_{D}=\mbox{qlift}([Q]_{d},D)$ as
described in Proposition 5 using sequential Hensel-lifting ($E=1$).
input : $[A]_{D}=[A_{0},\dots,A_{D-1}]$, where $A_{d}\in{\mathbb{R}}^{N\times
N}$ symmetric positive definite, $d=0,\dots,D-1$
output : $[\Lambda]_{D}=[\Lambda_{0},\dots,\Lambda_{D-1}]$, where
$\Lambda_{0}\in{\mathbb{R}}^{N\times N}$ diagonal and
$\Lambda_{d}\in{\mathbb{R}}^{N\times N}$ block diagonal $d=1,\dots,D-1$.
output : $[Q]_{D}=[Q_{0},\dots,Q_{D-1}]$ orthogonal, where
$Q_{d}\in{\mathbb{R}}^{N\times N}$
output : $b\in{\mathbb{N}}^{N_{b}+1}$, array of integers defining the blocks.
The integer $N_{b}$ is the number of blocks. Each block has the size of the
multiplicity of an eigenvalue $\lambda_{n_{b}}$ of $\Lambda_{0}$ s.t. for
$s=b_{n_{b}}:b_{n_{b}+1}-1$ one has
$(Q_{0;:,s})^{T}A_{0}Q_{0;:,s}=\lambda_{n_{b}}1\\!\\!\\!\mathrm{I}$.
$\Lambda_{0},Q_{0}={\rm eigh\,}(A_{0})$
compute $b\in{\mathbb{R}}^{N_{b}+1}$
$E_{ij}=\Lambda_{0;jj}-\Lambda_{0;ii}$
$H=P_{b}\circ(1/E)$
for _$d=1$ to $D-1$_ do $\Delta F=\sum_{|i|=d}Q_{i_{1}}^{T}A_{i_{2}}Q_{i_{3}}$
$S=-\frac{1}{2}\sum_{k=1}^{d-1}Q^{T}_{d-k}Q_{k}$
$K=\Delta F+Q_{0}^{T}A_{d}Q_{0}+S\Lambda_{0}+\Lambda_{0}S$
$Q_{d}=Q_{0}(S+H\circ K)$
$\Lambda_{d}=\bar{P}_{b}\circ K$ end
Algorithm 5 This algorithm computes $[\Lambda]_{D},[Q]_{D},b={\rm
eigh\,}_{1}([A]_{D})$ as specified by 4 using sequential Hensel-lifting
($E=1$). I.e., the zeroth coefficient is diagonalized and the higher order
coefficients are block diagonalized. The symbol $i\in{\mathbb{N}}_{0}^{3}$
denotes a multi-index, i.e., the summation $\sum_{|i|=d}$ goes over all
possible $i$ such that $|i|\equiv\sum_{k=1}^{3}i_{k}=d$.
input : $[A]_{D}=[A_{0},\dots,A_{D-1}]$ symmetric with
$A_{d}\in{\mathbb{R}}^{N\times N}$
output : $[\Lambda]_{D}=[\Lambda_{0},\dots,\Lambda_{D-1}]$, where
$\Lambda_{d}\in{\mathbb{R}}^{N\times N}$ diagonal for $d=0,\dots,D-1$.
output : $[Q]_{D}=[Q_{0},\dots,Q_{D-1}]$ orthogonal, where
$Q_{d}\in{\mathbb{R}}^{N\times N}$
$[\Lambda^{0}]_{D}=[A]_{D}$, $[Q^{0}]_{D}=1\\!\\!\\!\mathrm{I}$ and
$b^{0}=[1,N+1]$
for _$d=0$ to $D-1$_ do for _$n_{b}=1$ to $N_{b}^{d}$_ do
$s=b^{d}_{n_{b}}:b^{d}_{n_{b}+1}-1$ (slice index)
$[\hat{\Lambda}_{s,s}]_{D-d},[\hat{Q}_{s,s}]_{D-d},b^{d+1}={\rm
eigh\,}_{1}([\Lambda^{d}_{s,s}]_{d:})$
$[Q_{s,s}]_{D}=\mbox{qlift}([\hat{Q}_{s,s}]_{D-d},D)$ end
$[\Lambda^{d+1}]_{D}=[\Lambda^{d}]_{d}+[\hat{\Lambda}]_{D-d}T^{d}$
$[Q^{d+1}]_{D}=[Q^{d}]_{D}[Q]_{D}$ end
Algorithm 6 This algorithm computes $[\Lambda]_{D},[Q]_{D}={\rm
eigh\,}([A]_{D})$ as described in Theorem 3. The algorithm uses internally
Algorithm 5 and 4.
## 8 Numerical tests and examples
### 8.1 Taylor polynomial arithmetic on real symmetric eigenvalue problem
As an example to test the validity of the pushforward in UTP arithmetic we
consider the following system [AT98]:
$\displaystyle Q(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\begin{pmatrix}\cos(x(t))&1&\sin(x(t))&-1\\\
-\sin(x(t))&-1&\cos(x(t))&-1\\\ 1&-\sin(x(t))&1&\cos(x(t))\\\
-1&\cos(x(t))&1&\sin(x(t))\\\ \end{pmatrix}$ $\displaystyle\Lambda(t)$
$\displaystyle=$ $\displaystyle\mathop{\rm
diag}(x^{2}-x+\frac{1}{2},4x^{2}-3x,\delta(-\frac{1}{2}x^{3}+2x^{2}-\frac{3}{2}x+1)+(x^{3}+x^{2}-1),3x-1)\;,$
where $x\equiv x(t):=1+t$. The constant $\delta$ is some predefined constant.
In Taylor arithmetic one obtains
$\displaystyle\Lambda_{0}$ $\displaystyle=$ $\displaystyle\mathop{\rm
diag}(1/2,1,1+\delta,2)$ $\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\mathop{\rm diag}(1,5,5+\delta,3)$ $\displaystyle\Lambda_{2}$
$\displaystyle=$ $\displaystyle\mathop{\rm diag}(2,8,8+\delta,0)$
$\displaystyle\Lambda_{3}$ $\displaystyle=$ $\displaystyle\mathop{\rm
diag}(0,0,6-3\delta,0)$ $\displaystyle\Lambda_{d}$ $\displaystyle=$
$\displaystyle\mathop{\rm diag}(0,0,0,0),\quad\forall d\geq 4\;.$
One can see that in the case $\delta=0$ one obtains one repeated eigenvalue
that splits at $d=3$. We apply Algorithm 6 to reconstruct $[\Lambda]_{D}$. The
reconstructed values are denoted $[\tilde{\Lambda}]_{D}$. The numerical
results are shown in Fig. (8.1).
Figure 8.1: One left side one can see that error between the true
$\lambda_{1}$ and reconstructed $\tilde{\lambda}_{1}$ is close to machine
precision. On the right side one can see that the absolute error
$|\tilde{\lambda}_{2}-\lambda_{2}|$ has a jump at $\delta\approx 10^{-7}$.
This is due to the fact that that the algorithm treats eigenvalues
$|\lambda_{i}-\lambda_{j}|<10^{-7}$ as repeated eigenvalues. One can see that
when $\delta$ approaches $10^{-16}$ the error gets smaller. The eigenvalue
$\lambda_{4}$ shows the same qualitative behavior as $\lambda_{1}$ and
$\lambda_{3}$ the same as $\lambda_{2}$.
### 8.2 Covariance matrix computation
To test the validity of the covariance matrix computation of Eqn. (2.1) we
first check that the first directional derivatives of the covariance matrix
$C$ w.r.t. $J_{1}$ and $J_{2}$ coincide with the results from the complex-step
derivative approximation, abbreviated here for convenience as CSDA. The CSDA
computes directional derivatives of a real-valued function $y=f(x)$ as
$\dot{y}\approx\frac{\Im(f(x+i\epsilon\dot{x}))}{\epsilon}=\frac{f(x+i\epsilon\dot{x})-f(x-i\epsilon\dot{x})}{2i\epsilon}$,
i.e., $\Im$ extracts the imaginary part and $i\equiv\sqrt{-1}$. The number
$\epsilon\in{\mathbb{R}}$ can be made very small. For a detailed discussion
that also shows the relation to AD see Martins et al. [MSA01, MSA03]. Having
verified the first order derivatives by UTP arithmetic we can check if the UTP
arithmetic on Eqn. (2.2) yields the same result. Unfortunately, it is not
possible to use the CSDA in a straight-forward fashion since for complex
matrices the QR decomposition does not yield an orthogonal but a unitary $Q$.
For reproducibility we define $J_{1}$ and $J_{2}$ rather arbitrarily as
$\displaystyle J_{1}(x)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\sin(x_{1})x_{2}&\cos(x_{2})\\\
\exp(x_{1})&x_{1}x_{2}\\\ x_{1}\log(x_{2})&\log(1+\exp(\cos(x_{1})))\\\
x_{2}+x_{1}&x_{1}(x_{2}+\cos(x_{1})\\\
\end{pmatrix}\;,\quad\;J_{2}(x)^{T}=\begin{pmatrix}x_{1}\log(x_{2}+3\sin(x_{1}x_{2}))\\\
x_{2}\exp(\sin(x_{1})+\cos(x_{1}x_{2}))\end{pmatrix}\;.$
The numerical results are shown in Figure 8.2. Note that $x_{1}$ and $x_{2}$
are here elements of the vector $x$ and not coefficients.
Figure 8.2: This plot shows the maximum absolute differences of the
directional derivatives at $x=t(3,1)^{T}$, where $t\in[0.1,1]$ in direction
$\dot{x}=(5,7)^{T}$. The circles show the difference between the CSDA solution
and the first order UTP solution using Eqn. (2.1). The diamonds show the
difference between the UTP solution of Eqn. (2.1) and Eqn. (2.2). One can see
that the difference is close to machine precision of IEEE 754 float64, which
is approximately $10^{-16}$.
## 9 Summary and outlook
We have shown how computer codes containing real symmetric eigenvalue
decompositions and QR decompositions can be evaluated in univariate Taylor
polynomial arithmetic. Furthermore, the reverse mode of AD has been treated.
Explicit algorithms have been presented that can be used in combination with
existing AD software, e.g. general purpose AD tools like ADOL-C [GJM+99] or
CppAD [Bel10] but also differentiated DAE solvers like SolvIND [AK]. Numerical
tests have been used to check the algorithms.
Other algorithms that contain the the QR decomposition and the real symmetric
eigenvalue decompostion can be differentiated using the shown algorithms. In
particular, we think of the differentiation of the SVD or the computation of
pseudoinverses. We believe that these algorithms, in modified form, may also
be valuable for the eigenvalue optimization problem where eigenvalues are
repeated in the solution point.
## Acknowledgements
The authors wish to thank Bruce Christianson for his comments that helped to
gain a deeper understanding of the matter and also greatly helped to improve
the readability of this work.
This research was partially supported by the Bundesministerium für Bildung und
Forschung (BMBF) within the project NOVOEXP (Numerische Optimierungsverfahren
für die Parameterschätzung und den Entwurf optimaler Experimente unter
Berücksichtigung von Unsicherheiten für die Modellvalidierung
verfahrenstechnischer Prozesse der Chemie und Biotechnologie) (03GRPAL3),
Humboldt Universität zu Berlin.
## Appendix A Additional proofs
### A.1 Proofs of QR decomposition
#### A.1.1 Proof of Proposition 1
###### Proof.
We look at the first defining equation and try to separate the known from the
unknown quantities:
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle[Q]_{D+E}[R]_{D+E}-[A]_{D+E}$ (A.1)
$\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle([Q]_{D}+[\Delta Q]_{E}T^{D})([R]_{D}+[\Delta
R]_{E}T^{D})-[A]_{D+E}$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle[Q]_{D}[R]_{D}-[A]_{D+E}+([\Delta Q]_{E}[R]_{D}+[Q]_{D}[\Delta
R]_{E})T^{D}$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle-[\Delta F]_{E}T^{D}+([\Delta Q]_{E}[R]_{D}+[Q]_{D}[\Delta
R]_{E})T^{D}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-[\Delta F]_{E}+[\Delta Q]_{E}[R]_{E}+[Q]_{E}[\Delta R]_{E}\;.$
Similarly for the second defining equation
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle[Q]^{T}_{D+E}[Q]_{D+E}-1\\!\\!\\!\mathrm{I}$
$\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle[Q]^{T}_{D}[Q]_{D}-1\\!\\!\\!\mathrm{I}+([Q]^{T}_{D}[\Delta
Q]_{E}+[\Delta Q]^{T}_{E}[Q]_{D})T^{D}$ $\displaystyle\Rightarrow 0$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle-[\Delta
G]_{E}+[Q]^{T}_{E}[\Delta Q]_{E}+[\Delta Q]^{T}_{E}[Q]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle-[\Delta
G]_{E}+[S]_{E}+[X]_{E}+[S]_{E}-[X]_{E}$ $\displaystyle\Rightarrow\quad S$
$\displaystyle=$ $\displaystyle\frac{1}{2}[\Delta G]_{E}\;,$
where $[S]_{E}+[X]_{E}=[Q]_{E}^{T}[\Delta Q]_{E}$ and it has been used that
every matrix can be written as the sum of a symmetric and an antisymmetric
matrix. Now multiply (A.1) by $[Q]^{T}_{E}$ from the left to obtain
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-[Q]^{T}_{E}[\Delta F]_{E}+[Q]^{T}_{E}[\Delta
Q]_{E}[R]_{E}+[\Delta R]_{E}$ (A.2) $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle-[Q]^{T}_{E}[\Delta
F]_{E}+([S]_{E}+[X]_{E})[R]_{E}+[\Delta R]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-[Q]^{T}_{E}[\Delta F]_{E}+[S]_{E}[R]_{E}+[X]_{E}[R]_{E}+[\Delta
R]_{E}\;.$
Multiplication of $[R_{:N,:}]_{E}^{-1}$ from the right yields
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-[Q]^{T}_{E}[\Delta
F]_{E}[R_{:N,:}]_{E}^{-1}+[S]_{E}[R]_{E}[R_{:N,:}]_{E}^{-1}+[X]_{E}[R]_{E}[R_{:N,:}]_{E}^{-1}+[\Delta
R]_{E}[R_{:N,:}]_{E}^{-1}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-[Q]^{T}_{E}[\Delta
F]_{E}[R_{:N,:}]_{E}^{-1}+[S_{:,:N}]_{E}+[X_{:,:N}]_{E}+[\Delta
R]_{E}[R_{:N,:}]_{E}$ $\displaystyle\Rightarrow P_{L}\circ([X_{:,:N}]_{E})$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$ $\displaystyle
P_{L}\circ\left([Q]_{E}^{T}[\Delta
F]_{E}[R_{:N,:}]_{E}^{-1}-[S_{:,:N}]_{E}\right)\;.$
The coefficients of $X_{:,N+1:}$ are not specified and can for instance be set
to zero. Since $X$ is antisymmetric it is already defined by the above
equation. Since $[S]_{E}+[X]_{E}{\stackrel{{\scriptstyle
E}}{{=}}}[Q]_{E}^{T}[\Delta Q]_{E}$ one can obtain $[\Delta Q]_{E}$ as
$\displaystyle[\Delta Q]_{E}$ $\displaystyle=$
$\displaystyle[Q]_{E}([S]_{E}+[X]_{E})$
because for quadratic $Q$ one has the identity $QQ^{T}=1\\!\\!\\!\mathrm{I}$.
∎
#### A.1.2 Proof of Proposition 2
###### Proof.
We differentiate the implicit system
$\displaystyle 0$ $\displaystyle=$ $\displaystyle A-QR$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle Q^{T}Q-1\\!\\!\\!\mathrm{I}$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle P_{L}\circ R$
and obtain
$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\rm d}A-{\rm d}QR-Q{\rm
d}R\quad(*)$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle{\rm
d}Q^{T}Q+Q^{T}{\rm d}Q\quad(**)\;.$
We define the antisymmetric “matrix” $X:=Q^{T}{\rm d}Q$. Multiplication of
Eqn. (*) from the left with $Q^{T}$ yields
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}{\rm d}A-XR-{\rm d}R$
$\displaystyle\mbox{hence}\quad{\rm d}R$ $\displaystyle=$ $\displaystyle
Q^{T}{\rm d}A-XR\;.$
The multipication of this last equation from the right with the Moore-Penrose
pseudoinverse $R^{+}=(R_{:N,:}{}^{-1},0)$ yields the equivalent equation
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}{\rm
d}AR^{+}-XRR^{+}-{\rm d}RR^{+}$ $\displaystyle\mbox{ and thus }\;P_{L}\circ X$
$\displaystyle=$ $\displaystyle P_{L}\circ(Q^{T}{\rm d}AR^{+})\;,$
where we have chosen arbitrarily that $X_{:,N+1:}=0$. Since $X$ is
antisymmetric we have
$\displaystyle X$ $\displaystyle=$ $\displaystyle(P_{L}\circ X)-(P_{L}\circ
X)^{T}\;.$
We can use these results to compute the pullback:
$\displaystyle\mathop{\rm tr\,}(\bar{R}^{T}{\rm d}R)+\mathop{\rm
tr\,}(\bar{Q}^{T}{\rm d}Q)$ $\displaystyle=\mathop{\rm tr\,}(Q\bar{R}{\rm
d}A^{T})-\mathop{\rm tr\,}(R\bar{R}^{T}X)+\mathop{\rm
tr\,}(\bar{Q}^{T}QQ^{T}{\rm d}Q)$ $\displaystyle=\mathop{\rm
tr\,}(Q\bar{R}{\rm d}A^{T})+\mathop{\rm
tr\,}(\underbrace{(\bar{Q}^{T}Q-R\bar{R}^{T})}_{=:K}X)$
$\displaystyle=\mathop{\rm tr\,}(Q\bar{R}{\rm d}A^{T})+\mathop{\rm
tr\,}((K-K^{T})(P_{L}\circ X))$ $\displaystyle=\mathop{\rm tr\,}(Q\bar{R}{\rm
d}A^{T})+\mathop{\rm tr\,}(R^{+T}{\rm d}A^{T}Q(P_{L}\circ(K^{T}-K)))$
$\displaystyle=\mathop{\rm
tr\,}(Q[\bar{R}+\\{P_{L}\circ(Q^{T}\bar{Q}-\bar{Q}^{T}Q+R\bar{R}^{T}-\bar{R}R^{T})\\}R^{+T}]{\rm
d}A^{T})$ $\displaystyle=\mathop{\rm tr\,}(\bar{A}{\rm d}A^{T})\;.$
In the above derivation we have used Lemmas 7, 8 and 9.
∎
#### A.1.3 Proof of Lemma 5
###### Proof.
$\displaystyle 0$ $\displaystyle{\stackrel{{\scriptstyle D+E}}{{=}}}$
$\displaystyle([Q]_{D}+[\Delta Q]_{E}T^{D})^{T}([Q]_{D}+[\Delta
Q]_{E}T^{D})-1\\!\\!\\!\mathrm{I}$ $\displaystyle{\stackrel{{\scriptstyle
D+E}}{{=}}}$
$\displaystyle([Q]_{D}^{T}[Q]_{D}-1\\!\\!\\!\mathrm{I})+([Q]_{D}^{T}[\Delta
Q]_{E}+[\Delta Q]_{E}^{T}[Q]_{D})T^{D}$ $\displaystyle{\stackrel{{\scriptstyle
E}}{{=}}}$ $\displaystyle[\Delta G]_{E}+[Q]_{E}^{T}[\Delta Q]_{E}+[\Delta
Q]_{E}^{T}[Q]_{E}$ $\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle[\Delta G]_{E}+2[S]_{E}$ $\displaystyle\;[\Delta Q]_{E}$
$\displaystyle{\stackrel{{\scriptstyle E}}{{=}}}$
$\displaystyle-\frac{1}{2}[Q]_{E}[\Delta G]_{E}\;,$
where $[\Delta Q]_{E}^{T}[Q]_{E}=[S]_{E}+[X]_{E}$, $[S]_{E}$ symmetric and
$[X]_{E}$ antisymmetric and $[\Delta G_{E}]_{E}T^{D}\stackrel{{\scriptstyle
D+E}}{{=}}\left(Q^{T}Q-1\\!\\!\\!\mathrm{I}\right)$ . Since no condition
defines constraints on $[X]_{E}$ it has been set to zero. ∎
#### A.1.4 Proof of Proposition 6
###### Proof.
We want to compute $\mathop{\rm tr\,}(\bar{A}^{T}{\rm d}A)=\mathop{\rm
tr\,}(\bar{\Lambda}^{T}{\rm d}\Lambda)+\mathop{\rm tr\,}(\bar{Q}^{T}{\rm
d}Q)$. We differentiate the implicit system
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}AQ-\Lambda$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle Q^{T}Q-1\\!\\!\\!\mathrm{I}$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle(P_{L}+P_{R})\circ\Lambda$
and obtain
$\displaystyle{\rm d}\Lambda$ $\displaystyle=$ $\displaystyle Q^{T}{\rm
d}AQ+{\rm d}Q^{T}AQ+Q^{T}A{\rm d}Q$ $\displaystyle=$ $\displaystyle Q^{T}{\rm
d}AQ+{\rm d}Q^{T}Q\Lambda+\Lambda Q^{T}{\rm d}Q$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle{\rm d}Q^{T}Q+Q^{T}{\rm d}Q\;.$
A straight forward calculation shows:
$\displaystyle\mathop{\rm tr\,}(\bar{\Lambda}^{T}{\rm d}\Lambda)$
$\displaystyle=$ $\displaystyle\mathop{\rm tr\,}(Q\bar{\Lambda}Q^{T}{\rm
d}A)+\mathop{\rm tr\,}(\Lambda\bar{\Lambda}{\rm d}Q^{T}Q)+\mathop{\rm
tr\,}(\bar{\Lambda}\Lambda Q^{T}{\rm d}Q)$ $\displaystyle=$
$\displaystyle\mathop{\rm tr\,}(Q\bar{\Lambda}Q^{T}{\rm d}A)\;,$
$\displaystyle\mathop{\rm tr\,}(\bar{Q}^{T}{\rm d}Q)$ $\displaystyle=$
$\displaystyle\mathop{\rm tr\,}(\bar{Q}^{T}Q(H\circ(Q^{T}{\rm d}AQ)))$
$\displaystyle=$ $\displaystyle\mathop{\rm
tr\,}(Q(H^{T}\circ(\bar{Q}^{T}Q))Q^{T}{\rm d}A)\;,$ $\displaystyle\mathop{\rm
tr\,}(\bar{A}^{T}{\rm d}A)$ $\displaystyle=$ $\displaystyle\mathop{\rm
tr\,}\left((Q(\bar{\Lambda}+H^{T}\circ(\bar{Q}^{T}Q))Q^{T}){\rm d}A\right)$
where we have used
$\displaystyle{\rm d}\Lambda$ $\displaystyle=$ $\displaystyle Q^{T}{\rm
d}AQ-(Q^{T}{\rm d}Q)^{T}\Lambda+\Lambda Q^{T}{\rm d}Q$ $\displaystyle=$
$\displaystyle Q^{T}{\rm d}AQ-K\circ(Q^{T}{\rm d}Q)$ $\displaystyle\implies
Q^{T}{\rm d}Q$ $\displaystyle=$ $\displaystyle H\circ(Q^{T}{\rm d}AQ-{\rm
d}\Lambda)$ $\displaystyle=$ $\displaystyle H\circ(Q^{T}{\rm d}AQ)$
where we have defined $K_{ij}:=\Lambda_{jj}-\Lambda_{ii}$ and
$H_{ij}=(K_{ij})^{-1}$ for $i\neq j$ and $H_{ij}=0$ otherwise and used the
property $X\Lambda-\Lambda X=K\circ X$ for all $X\in{\mathbb{R}}^{N\times N}$
and diagonal $\Lambda\in{\mathbb{R}}^{N\times N}$. ∎
### A.2 Basic results used in the proofs
###### Lemma 7.
Let $X\in{\mathbb{R}}^{N\times N}$ be an antisymmetric matrix, i.e.,
$X^{T}=-X$ and $P_{L}$ defined as above. We then can write
$\displaystyle X$ $\displaystyle=$ $\displaystyle P_{L}\circ X-(P_{L}\circ
X)^{T}\;.$ (A.4)
###### Proof.
$X=P_{L}\circ X+P_{R}\circ X=P_{L}\circ X+(P_{L}\circ X^{T})^{T}=P_{L}\circ
X-(P_{L}\circ X)^{T}$ ∎
###### Lemma 8.
Let $A\in{\mathbb{R}}^{N\times N}$ and $P_{L}$ resp. $P_{R}$ defined as above.
Then
$\displaystyle(P_{L}\circ A)^{T}$ $\displaystyle=$ $\displaystyle P_{R}\circ
A^{T}\;.$ (A.5)
###### Proof.
$B_{ij}:=(P_{L}\circ A)_{ij}=A_{ij}(i>j)$ and
$B_{ij}^{T}=B_{ji}=A_{ji}(j>i)=A_{ij}^{T}P_{R}=P_{R}\circ A$ ∎
###### Lemma 9.
Let $A,B,C\in{\mathbb{R}}^{M\times N}$. We then have
$\displaystyle\mathop{\rm tr\,}\left(A^{T}(B\circ C)\right)$ $\displaystyle=$
$\displaystyle\mathop{\rm tr\,}\left(C^{T}(B\circ A)\right)$ (A.6)
###### Proof.
$\mathop{\rm tr\,}(A^{T}(B\circ
C))=\sum_{i=1}^{N}\sum_{j=1}^{M}A_{ij}B_{ij}C_{ij}=\mathop{\rm
tr\,}(C^{T}(B\circ A))$ ∎
###### Lemma 10.
Let $A,B$ be lower triangular matrices. Then the following expression holds:
$\displaystyle P_{D}\circ(AB)$ $\displaystyle=$ $\displaystyle(P_{D}\circ
A)(P_{D}\circ B)\;.$ (A.7)
###### Proof.
$AB$ is also lower trinangular and thus
$P_{D}\circ(AB)=diag(a_{ii}b_{ii})=diag(a_{ii})diag(b_{ii})=(P_{D}\circ
A)(P_{D}\circ B)$ ∎
###### Lemma 11.
The formula
$\displaystyle P_{D}\circ(A^{T})$ $\displaystyle=$ $\displaystyle P_{D}\circ
A$ (A.8)
holds for all quadratic matrices $A$.
###### Proof.
$(P_{D}\circ(A^{T}))_{ij}=\delta_{ij}A_{ji}=\delta_{ij}A_{ij}=(P_{d}\circ
A)_{ij}$ ∎
###### Lemma 12.
Let $A$ be a nonsingular quadratic lower triangular matrix. Then the formula
$\displaystyle P_{D}\circ(A^{-1})$ $\displaystyle=$ $\displaystyle(P_{D}\circ
A)^{-1}$ (A.9)
holds.
###### Proof.
Using Lemma 10 we obtain $(P_{D}\circ(A^{-1}))(P_{D}\circ A)=P_{D}\circ
1\\!\\!\\!\mathrm{I}=1\\!\\!\\!\mathrm{I}$. Since the quadratic matrices form
a group, the inverse is unique. Therefore, equality between
$(P_{D}\circ(A^{-1}))=(P_{D}\circ(A))^{-1}$ must hold. ∎
###### Lemma 13.
Let $A\in{\mathbb{R}}^{N\times N}$ be strictly lower triangular and
$B\in{\mathbb{R}}^{N\times N}$ lower triangular. Then their product $C=AB$ is
strictly lower triangular.
###### Proof.
$C$ is lower triangular and the diagonal entries are $C_{ii}=A_{ii}B_{ii}=0$
since $B$ has a zero diagonal. ∎
###### Corollary 14.
Let $A\in{\mathbb{R}}^{N\times N}$ be strictly lower triangular and
$D\in{\mathbb{R}}^{N\times N}$ diagonal. Then their product $C=AD$ is strictly
lower triagonal.
###### Lemma 15.
Every quadratic matrix $A$ can be written as the sum of a symmetric matrix
$S=\frac{1}{2}(A+A^{T})$ and an antisymmetric matrix $X=\frac{1}{2}(A-A^{T})$,
i.e.
$\displaystyle A$ $\displaystyle=$ $\displaystyle S+X$ (A.10)
###### Proof.
$A=\frac{1}{2}(A+A^{T}+A-A^{T})=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T})=S+X$.
∎
## References
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|
arxiv-papers
| 2010-09-30T12:15:49 |
2024-09-04T02:49:13.234543
|
{
"license": "Public Domain",
"authors": "Sebastian F. Walter and Lutz Lehmann and Ren\\'e Lamour",
"submitter": "Sebastian F. Walter",
"url": "https://arxiv.org/abs/1009.6112"
}
|
1009.6144
|
# Cover-Decomposition and Polychromatic Numbers
Béla Bollobás1 David Pritchard2 Thomas Rothvoß3 Alex Scott4 Béla Bollobás
(University of Memphis, University of Cambridge) David Pritchard (EPFL)
Thomas Rothvoß (MIT) Alex Scott (University of Oxford)
###### Abstract
A colouring of a hypergraph’s vertices is _polychromatic_ if every hyperedge
contains at least one vertex of each colour; the _polychromatic number_ is the
maximum number of colours in any polychromatic colouring. Its dual is the
_cover-decomposition number_ : the maximum number of disjoint hyperedge-
covers. In geometric settings, there is extensive work on lower-bounding these
numbers in terms of their trivial upper bounds (minimum hyperedge size and
minimum degree). Our goal is to get good lower bounds in natural hypergraph
families not arising from geometry. We obtain algorithms yielding near-tight
bounds for three hypergraph families: those with bounded hyperedge size, those
representing paths in trees, and those with bounded VC-dimension. One new
technique we use is a link between cover-decomposition and iterated relaxation
of linear programs.
## 1 Introduction
In a set system on vertex set $V$, a subsystem is a _set cover_ if each vertex
of $V$ appears in at least 1 set of the subsystem. Suppose each vertex appears
in at least $\delta$ sets of the set system, for some large $\delta$; does it
follow that we can partition the system into 2 subsystems, such that each
subsystem is a set cover?
Many natural families of set systems admit a universal constant $\delta$ for
which this question has an affirmative answer. Such families are typically
called _cover-decomposable_. But the family of _all_ set systems is not cover-
decomposable, as the following example shows. For any positive integer $k$,
consider the set system $\tbinom{[2k-1]}{k}^{*}$, which has $2k-1$ sets, and
where every subfamily of $k$ sets contain one mutually common vertex not
contained by the other $k-1$ sets. This system satisfies the hypothesis of the
question for $k=\delta$. But there cannot be a set cover of
$\tbinom{[2k-1]}{k}^{*}$ consisting of $\leq k-1$ sets, and it has only $2k-1$
sets in total; so no partition into two set covers is possible. This example
above shows that some sort of restriction on the family is necessary to ensure
cover-decomposability.
One positive example of cover-decomposition is where every set has size 2,
i.e. the setting of graphs. It is cover-decomposable with $\delta=3$: for a
graph with minimum degree 3, the edges can be partitioned into two edge
covers. More generally, Gupta [18] showed (see also [5, 1]) that we can
partition the edges of any multigraph into $\lfloor\frac{3\delta+1}{4}\rfloor$
edge covers (this bound is tight for all $\delta$, by considering a
parallelization of a triangle).
Set systems in many geometric settings have been studied with respect to
cover-decomposability; many positive and negative examples are known and there
is no easy way to distinguish one from the other. In the affirmative case, as
with Gupta’s theorem, the next natural problem is to find for each $t\geq 3$ a
small $\delta(t)$ such that when each vertex appears in at least $\delta(t)$
sets, a partition into $t$ set covers is possible. The goal of this paper is
to extend the study of cover-decomposition beyond geometric settings. Our
study gives (as far as we are aware) the first examples where linear
programming is used to obtain cover-decomposition theorems.
### 1.1 Terminology and Notation
A _hypergraph_ $H=(V,\mathcal{E})$ consists of a ground set $V$ of vertices,
together with a collection $\mathcal{E}$ of hyperedges, where each hyperedge
$E\in\mathcal{E}$ is a subset of $V$. Hypergraphs are the same as _set
systems_. We will sometimes call hyperedges just _edges_ or _sets_. We permit
$\mathcal{E}$ to contain multiple copies of the same hyperedge (e.g. to allow
us to define “duals” and “shrinking” later), and we also allow hyperedges of
cardinality 0 or 1. We only consider hypergraphs that are finite. (In many
geometric cases, infinite versions of the problem can be reduced to finite
ones, e.g. [30]; see also [13] for work on infinite versions of cover-
decomposability.)
A _polychromatic $k$-colouring_ of a hypergraph is a map from $V$ to a set of
$k$ colours so that for every edge, its image contains all colours.
Equivalently, the colour classes are a partition of $V$ into sets which each
meet every edge, so-called _vertex covers_ or _transversals_. The maximum
number of colours in a polychromatic colouring of $H$ is called its
_polychromatic number_ , which we denote by $\mathsf{p}(H)$.
A _cover $k$-decomposition_ of a hypergraph is a partition of $\mathcal{E}$
into $k$ subfamilies $\mathcal{E}=\biguplus_{i=1}^{k}\\{\mathcal{E}_{i}\\}$
such that each $\bigcup_{E\in\mathcal{E}_{i}}E=V$. In other words, each
$\mathcal{E}_{i}$ must be a set cover. The maximum $k$ for which the
hypergraph $H$ admits a cover $k$-decomposition is called its _cover-
decomposition number_ , which we denote by $\mathsf{p}^{\prime}(H)$.
The _dual_ $H^{*}$ of a hypergraph $H$ is another hypergraph such that the
vertex set of $H^{*}$ corresponds to the edge set of $H$, and vice-versa, with
incidences preserved. Thus the vertex-edge incidence matrices for $H$ and
$H^{*}$ are transposes of one another. From the definitions it is easy to see
that the polychromatic and cover-decomposition numbers are dual to one
another,
$\mathsf{p}^{\prime}(H)=\mathsf{p}(H^{*}).$
The _degree_ of a vertex $v$ in a hypergraph is the number of hyperedges
containing $v$; it is $d$-_regular_ if all vertices have degree $d$. We denote
the minimum degree by $\delta$, and the maximum degree by $\Delta$. We denote
the minimum size of any hyperedge by $r$, and the maximum size of any
hyperedge by $R$. Note that $\Delta(H)=R(H^{*})$ and $\delta(H)=r(H^{*})$. It
is trivial to see that $\mathsf{p}\leq r$ in any hypergraph and dually that
$\mathsf{p}^{\prime}\leq\delta$. So the cover-decomposability question asks if
there is a converse to this trivial bound: if $\delta$ is large enough, does
$\mathsf{p}^{\prime}$ also grow? To write this concisely, for a family
$\mathcal{F}$ of hypergraphs, let its extremal _cover-decomposition function_
$\overline{\mathsf{p}}^{\prime}(\mathcal{F},\delta)$ be
$\overline{\mathsf{p}}^{\prime}(\mathcal{F},\delta):=\min\\{\mathsf{p}^{\prime}(H)\mid
H\in\mathcal{F};~{}\forall v\in V(H):\textrm{degree}(v)\geq\delta\\},$
i.e. $\overline{\mathsf{p}}^{\prime}(\mathcal{F},\delta)$ is the best possible
lower bound for $\mathsf{p}^{\prime}$ among hypergraphs in $\mathcal{F}$ with
min-degree $\geq\delta$. So to say that $\mathcal{F}$ is cover-decomposable
means that $\overline{\mathsf{p}}^{\prime}(\mathcal{F},\delta)>1$ for some
constant $\delta$. We also dually define
$\overline{\mathsf{p}}(\mathcal{F},r):=\min\\{\mathsf{p}(H)\mid
H\in\mathcal{F};~{}\forall E\in\mathcal{E}(H):|E|\geq r\\}.$
In the rest of the paper we focus on computing these functions. When the
family $\mathcal{F}$ is clear from context, we write
$\overline{\mathsf{p}}^{\prime}(\delta)$ for
$\overline{\mathsf{p}}^{\prime}(\mathcal{F},\delta)$ and
$\overline{\mathsf{p}}(r)$ for $\overline{\mathsf{p}}(\mathcal{F},r)$.
### 1.2 Results
In Section 2 we generalize Gupta’s theorem to hypergraphs of bounded edge
size. Let ${\textsc{Hyp}}(R)$ denote the family of hypergraphs with all edges
of size at most $R$.
###### Theorem 1.
For all $R,\delta$ we have
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\geq\max\\{1,\delta/(\ln
R+O(\ln\ln R))\\}$.
In proving Theorem 1, we first give a simple proof which is weaker by a
constant factor, and then we refine the analysis. We use the Lovász Local
Lemma (LLL) as well as discrepancy-theoretic results which permit us to
partition a large hypergraph into two pieces with roughly-equal degrees. Next
we show that Theorem 1 is essentially tight:
###### Theorem 2.
(a) For a constant $C$ and all $R\geq 2,\delta\geq 1$ we have
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\leq\max\\{1,C\delta/\ln
R\\}.$ (b) As $R,\delta\to\infty$ with $\delta=\omega(\ln R)$ we have
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\leq(1+o(1))\delta/\ln(R).$
Here (a) uses an explicit construction while (b) uses the probabilistic
method.
By plugging Theorem 1 into an approach of [1], we deduce a good bound on the
cover-decomposition number of _sparse_ hypergraphs.
###### Corollary 3.
Suppose $H=(V,\mathcal{E})$ satisfies, for all $V^{\prime}\subseteq V$ and
$\mathcal{E}^{\prime}\subseteq\mathcal{E}$, that the number of incidences
between $V^{\prime}$ and $\mathcal{E}^{\prime}$ is at most
$\alpha|V^{\prime}|+\beta|\mathcal{E}^{\prime}|$. Then
$\mathsf{p}^{\prime}(H)\geq\frac{\delta(H)-\alpha}{\ln\beta+O(\ln\ln\beta)}$.
Duality yields a similar bound on the polychromatic number of sparse
hypergraphs.
In Section 3 we consider the following family of hypergraphs: the ground set
is the edge set of an undirected tree, and each hyperedge must correspond to
the edges lying in some path in the tree. We show that such systems are cover-
decomposable:
###### Theorem 4.
For hypergraphs defined by edges of paths in trees,
$\overline{\mathsf{p}}^{\prime}(\delta)\geq 1+\lfloor(\delta-1)/5\rfloor.$
To prove Theorem 4, we exploit the connection to discrepancy and iterated
rounding, using an extreme point structure theorem for paths in trees [21]. We
also determine the extremal polychromatic number for such systems:
###### Theorem 5.
For hypergraphs defined by edges of paths in trees,
$\overline{\mathsf{p}}(r)=\lceil r/2\rceil.$
This contrasts with a construction of Pach, Tardos and Tóth [28]: if we also
allow hyperedges consisting of sets of “siblings,” then
$\overline{\mathsf{p}}(r)=1$ for all $r$.
The _VC-dimension_ is a prominent measure of set system complexity used
frequently in geometry: it is the maximum cardinality of any $S\subseteq V$
such that $\\{S\cap E\mid E\in\mathcal{E}\\}=\mathbf{2}^{S}$. It is natural to
ask what role the VC-dimension plays in cover-decomposability. In Appendix B
(deferred for space reasons) we show the following:
###### Theorem 6.
For the family of hypergraphs with VC-dimension 1,
$\overline{\mathsf{p}}(r)=\lceil r/2\rceil$ and
$\overline{\mathsf{p}}^{\prime}(\delta)=\lceil\delta/2\rceil$.
By duality, the same holds for the family of hypergraphs whose duals have VC-
dimension 1. We find Theorem 6 is best possible in a strong sense:
###### Theorem 7.
For the family of hypergraphs $\\{H\mid\textrm{VC-dim}(H),\textrm{VC-
dim}(H^{*})\leq 2\\}$, we have $\overline{\mathsf{p}}(r)=1$ for all $r$ and
$\overline{\mathsf{p}}^{\prime}(\delta)=1$ for all $\delta$.
To prove this, we show the construction of [28] has primal and dual VC-
dimension at most 2.
All of our lower bounds on $\overline{\mathsf{p}}$ and
$\overline{\mathsf{p}}^{\prime}$ can be implemented as polynomial-time
algorithms. In the case of Theorem 1 this relies on the constructive LLL
framework of Moser-Tardos [24]. In the tree setting (Theorem 4) the tree
representing the hypergraph does not need to be explicitly given as input,
since the structural property used in each iteration (Lemma 15) is easy to
identify from the values of the extreme point LP solution. Note: since we also
have the trivial bounds $\overline{\mathsf{p}}\leq
r,\overline{\mathsf{p}}^{\prime}\leq\delta$ these give _approximation
algorithms_ for $\overline{\mathsf{p}}$ and $\overline{\mathsf{p}}^{\prime}$,
e.g. Theorem 1 gives a $(\ln R+O(\ln\ln R))$-approximation for
$\overline{\mathsf{p}}^{\prime}$.
### 1.3 Related Work
One practical motive to study cover-decomposition is that the hypergraph can
model a collection of sensors [9, 17], with each $E\in\mathcal{E}$
corresponding to a sensor which can monitor the set $E$ of vertices; then
monitoring all of $V$ takes a set cover, and $\overline{\mathsf{p}}^{\prime}$
is the maximum “coverage” of $V$ possible if each sensor can only be turned on
for a single time unit or monitor a single frequency. Another motive is that
if $\overline{\mathsf{p}}^{\prime}(\delta)=\Omega(\delta)$ holds for a family
closed under vertex deletion, then the size of a _dual $\epsilon$-net_ is
bounded by $O(1/\epsilon)$ [27].
A hypergraph is said to be _weakly $k$-colourable_ if we can $k$-colour its
vertex set so that no edge is monochromatic. Weak 2-colourability is also
known as _Property B_ , and these notions coincide with the property
$\mathsf{p}\geq 2$. However, weak $k$-colourability does not imply
$\mathsf{p}\geq k$ in general.
Given a plane graph, define a hypergraph whose vertices are the graph’s
vertices, and whose hyperedges are the faces. For this family of hypergraphs,
it was shown in [1] that
$\overline{\mathsf{p}}(\delta)\leq\lfloor(3\delta-5)/4\rfloor$ using Gupta’s
theorem and a sparsity argument. This is the same approach which we exploit to
prove Corollary 3.
For a graph $G=(V,E)$, the _(closed) neighbourhood hypergraph_
$\mathcal{N}(H)$ is defined to be a hypergraph on ground set $V$, with one
hyperedge $v\cup\\{u\mid\\{u,v\\}\in E\\}$ for each $v\in V$. Then
$\mathsf{p}(\mathcal{N}(G))$ equals the _domatic number_ of $G$, i.e. the
maximum number of disjoint dominating sets. The paper of Feige, Halldórsson,
Kortsarz & Srinivasan [15] obtains upper bounds for the domatic number and
their bounds are essentially the same as what we get by applying Theorem 1 to
the special case of neighbourhood hypergraphs; compared to our methods they
use the LLL but not discrepancy or iterated LP rounding. They give a hardness-
of-approximation result which implies that Theorem 1 is tight with respect to
the approximation factor, namely for all $\epsilon>0$, it is hard to
approximate $\overline{\mathsf{p}}^{\prime}$ within a factor better than
$(1-\epsilon)\ln R$, under reasonable complexity assumptions. A generalization
of results in [15] to packing polymatroid bases was given in [11]; this
implies a weak version of Theorem 1 where the $\log R$ term is replaced by
$\log|V|$.
A notable progenitor in geometric literature on cover-decomposition is the
following question of Pach [25]. Take a convex set $A\subset\mathbb{R}^{2}$.
Let $\mathbb{R}^{2}|\textsc{Translates}(A)$ denote the family of hypergraphs
where the ground set $V$ is a finite subset of $\mathbb{R}^{2}$, and each
hyperedge is the intersection of $V$ with some translate of $A$. Pach asked if
such systems are cover-decomposable, and this question is still open. A state-
of-the-art partial answer is due to Gibson & Varadarajan [17], who prove that
$\overline{\mathsf{p}}(\mathbb{R}^{2}|\textsc{Translates}(A),\delta)=\Omega(\delta)$
when $A$ is an open convex polygon.
The paper of Pach, Tardos and Tóth [28] obtains several negative results with
a combinatorial method. They define a family of non-cover-decomposable
hypergraphs based on trees and then they “embed” these hypergraphs into
geometric settings. By doing this, they prove that the following families are
not cover-decomposable: $\mathbb{R}^{2}|\textsc{Axis-Aligned-Rectangles}$;
$\mathbb{R}^{2}|\textsc{Translates}(A)$ when $A$ is a non-convex
quadrilateral; and $\mathbb{R}^{2}|\textsc{Strips}$ and its dual. In contrast
to the latter result, it is known that
$\overline{\mathsf{p}}(\mathbb{R}^{2}|\textsc{Axis-Aligned-
Strips},r)\geq\lceil r/2\rceil$ [3]. Recently it was shown [20] that
$\mathbb{R}^{3}|\textsc{Translates}(\mathbb{R}^{3}_{+})$ is cover-
decomposable, implying cover-decomposability of
$\mathbb{R}^{2}|\textsc{Bottomless-Axis-Aligned-Rectangles}$ in addition to
$\mathbb{R}^{2}|\textsc{Homothets}(T)$ for any triangle $T$; the latter
contrasts with the non-cover-decomposability of
$\mathbb{R}^{2}|\textsc{Homothets}(D)$ for $D$ the unit disc [28].
Pálvölgyi [30] poses a fundamental combinatorial question: is there a function
$f$ so that in hypergraph families closed under edge deletion and duplication,
$\overline{\mathsf{p}}^{\prime}(\delta_{0})\geq 2$ implies
$\overline{\mathsf{p}}^{\prime}(f(\delta_{0}))\geq 3$? This is open for all
$\delta_{0}\geq 2$ and no counterexamples are known to the conjecture
$f(\delta_{0})=O(\delta_{0})$.
We mention additional related work in Appendix A.
## 2 Hypergraphs of Bounded Edge Size
To get good upper bounds on
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)$, we will use the
Lovász Local Lemma (LLL):
###### Lemma 8 (LLL, [14]).
Consider a collection of “bad” events such that each one has probability at
most $p$, and such that each bad event is independent of the other bad events
except at most $D$ of them. (We call $D$ the _dependence degree_.) If
$p(D+1)\mathrm{e}\leq 1$ then with positive probability, no bad events occur.
Our first proposition extends a standard argument about Property B [2, Theorem
5.2.1].
###### Proposition 9.
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\geq\lfloor\delta/\ln(\mathrm{e}R\delta^{2})\rfloor.$
I.e. given any hypergraph $H=(V,\mathcal{E})$ where every edge has size at
most $R$ and such that each $v\in V$ is covered at least $\delta$ times, we
must show for $t=\lfloor\delta/\ln(\mathrm{e}R\delta^{2})\rfloor$ that
$\mathsf{p}^{\prime}(H)\geq t$, i.e. that $\mathcal{E}$ can be decomposed into
$t$ disjoint set covers. It will be helpful here and later to make the degree
of every vertex _exactly_ $\delta$, (this bounds the dependence degree).
Indeed $\mathrm{deg}(v)=\delta$ is without loss of generality since otherwise
we can repeatedly “shrink” — replace a hyperedge $E$ containing $v$ by
$E\backslash\\{v\\}$ until the degree of $v$ drops to $\delta$ — since a
vertex cover remains a vertex cover upon unshrinking.
###### Proof of Proposition 9.
Consider the following randomized experiment: for each hyperedge
$E\in\mathcal{E}$, assign a random colour between $1$ and $t$ to $E$. If we
can show that with positive probability, every vertex is incident with a
hyperedge of each colour, then we will be done. In order to get this approach
to go through,
For each vertex $v$ define the _bad event_ $\mathfrak{E}_{v}$ to be the event
that $v$ is not incident with a hyperedge of each colour. The probability of
$\mathfrak{E}_{v}$ is at most $t(1-\frac{1}{t})^{\delta},$ by using a union
bound. The event $\mathfrak{E}_{v}$ only depends on the colours of the
hyperedges containing $v$; therefore the events $\mathfrak{E}_{v}$ and
$\mathfrak{E}_{v^{\prime}}$ are independent unless $v,v^{\prime}$ are in a
common hyperedge. In particular the dependence degree is less than $R\delta$.
It follows by LLL that if
$R\delta t(1-\tfrac{1}{t})^{\delta}\leq 1/\mathrm{e},$
then with positive probability, no bad events happen and we are done. We can
verify that $t=\delta/\ln(\mathrm{e}R\delta^{2})$ satisfies this bound.∎
We will next show that the bound can be raised to $\Omega(\delta/\ln R)$.
Intuitively, our strategy is the following. We have that $\delta/\ln(R\delta)$
is already $\Omega(\delta/\ln R)$ unless $\delta$ is superpolynomial in $R$.
For hypergraphs where $\delta\gg R$ we will show that we can partition
$\mathcal{E}$ into $m$ parts $\mathcal{E}=\biguplus_{i=1}^{m}\mathcal{E}_{i}$
so that $\delta(V,\mathcal{E}_{i})$ is at least a constant of $\delta/m$, and
such that $\delta/m$ is polynomial in $R$. Thus by Proposition 9 we can
extract $\Omega((\delta/m)/\ln R)$ set covers from each $(V,\mathcal{E}_{i})$,
and their union proves $\overline{\mathsf{p}}^{\prime}\geq\Omega(\delta/\ln
R)$.
In fact, it will be enough to consider splitting $\mathcal{E}$ into two parts
at a time, recursively. Then ensuring
$\delta(V,\mathcal{E}_{i})\gtrsim\delta/2~{}(i=1,2)$ amounts to a discrepancy-
theoretic problem: given the incidence matrix $M_{H}$, we must 2-colour the
rows by $\pm 1$ so that for each vertex the sum of the incident weights is in
$[-d,d]$, with the _discrepancy_ $d$ as small as possible. To get a short
proof of a weaker version of Theorem 1, we can use an approach of Beck and
Fiala [8]; moreover it is important to review their proof since we will extend
it in Section 3.
###### Proposition 10 (Beck & Fiala [8]).
In a $\delta$-regular hypergraph $H=(V,\mathcal{E})$ with all edges of size at
most $R$, we can partition the edge set into
$\mathcal{E}=\mathcal{E}_{1}\uplus\mathcal{E}_{2}$ such that
$\delta(V,\mathcal{E}_{i})\geq\delta/2-R~{}(i=1,2)$.
###### Proof.
Define a linear program with nonnegative variables
$\\{x_{e},y_{e}\\}_{e\in\mathcal{E}}$ subject to $x_{e}+y_{e}=1$ and for all
$v$, degree constraints $\sum_{e:v\in e}x_{e}\geq\delta/2$ and $\sum_{e:v\in
e}y_{e}\geq\delta/2$. Note $x\equiv y\equiv\frac{1}{2}$ is a feasible
solution. Let us abuse notation and when $x$ or $y$ is 0-1, use them
interchangeably with the corresponding subsets of $\mathcal{E}$. So in the LP,
a feasible integral $x$ and $y$ would correspond to a discrepancy-zero
splitting of $\mathcal{E}$. We’ll show that such a solution can be _nearly_
found, allowing an additive $R$ violation in the degree constraints. We use
the following fact, which follows by double-counting. A constraint is _tight_
if it holds with equality.
###### Lemma 11.
In any extreme point solution to the linear program, there is some tight
degree constraint for whom at most $R$ of the variables it involves are
strictly between 0 and 1. This holds also if some variables are fixed at
integer values and some of the degree constraints have been removed.
Now we use the following iterated LP rounding algorithm. Each iteration starts
with solving the LP and getting an extreme point solution. Then perform two
steps: for each variable with an integral value in the solution, fix its value
forever; and discard the constraint whose existence is guaranteed by the
lemma. Eventually all variables are integral and we terminate.
For each degree constraint, either it was never discarded in which case the
final integral solution satisfies it, or else it was discarded in some
iteration. Now when the constraint was discarded it had at most $R$ fractional
variables, and was tight. So in the sum (say) $\sum_{e:v\in e}x_{e}=\delta$
there were at least $\delta-R$ variables fixed to 1 on the left-hand side.
They ensure $\sum_{e:v\in e}x_{e}\geq\delta-R$ at termination, proving what we
wanted. ∎
Here is how we use the Beck-Fiala theorem to get a roughly-optimal bound on
$\overline{\mathsf{p}}^{\prime}$.
###### Proposition 12.
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\geq\delta/O(\ln R).$
###### Proof.
If $\delta<R$ this already follows from Proposition 9. Otherwise apply
Proposition 10 to the initial hypergraph, and then use shrinking to make both
the resulting $(V,\mathcal{E}_{i})$’s into regular hypergraphs. Iterate this
process; stop splitting each hypergraphs once its degree falls in the range
$[R,4R)$, which is possible since $\delta\geq 4R\Rightarrow\delta/2-R\geq R$.
Let $M$ be the number of hypergraphs at the end.
Observe that in applying the splitting-and-shrinking operation to some
$(V,\mathcal{E})$ to get $(V,\mathcal{E}_{1})$ and $(V,\mathcal{E}_{2})$, the
sum of the degrees of $(V,\mathcal{E}_{1})$ and $(V,\mathcal{E}_{2})$ is at
least the degree of $(V,\mathcal{E})$, minus $2R$ “waste”. It follows that the
total waste is at most $2R(M-1)$, and we have that $4RM+2R(M-1)\geq\delta$.
Consequently $M\geq\delta/6R$. As sketched earlier, applying Proposition 9 to
the individual hypergraphs, and combining these vertex covers, shows that
$\mathsf{p}^{\prime}\geq M\lfloor R/\ln(\mathrm{e}R^{3})\rfloor$ which gives
the claimed bound. ∎
Now we get to the better bound with the correct multiplicative constant.
###### Proof of Theorem 1: $\forall R,\delta,$
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\geq\max\\{1,\delta/(\ln
R+O(\ln\ln R))\\}$.
Proposition 9 gives us the desired bound when $\delta$ is at most
polylogarithmic in $R$, so we assume otherwise. Due to the crude bound in
Proposition 12, we may assume $R$ is sufficiently large when needed. We will
need the following discrepancy bound which follows using Chernoff bounds; we
provide the standard proof in Appendix D.
###### Proposition 13.
For a constant $C_{1}$, in a $d$-regular hypergraph $H=(V,\mathcal{E})$ with
all edges of size at most $R$, we can partition the edge set into
$\mathcal{E}=\mathcal{E}_{1}\uplus\mathcal{E}_{2}$ such that
$\delta(V,\mathcal{E}_{i})\geq d/2-C_{1}\sqrt{d\ln(Rd)}~{}(i=1,2)$.
Let $d_{0}=\delta$ and $d_{i+1}=d_{i}/2-C_{1}\sqrt{d_{i}\ln(Rd_{i})}$. Thus
each hypergraph obtained after $i$ rounds of splitting has degree at least
$d_{i}$; evidently $d_{i}\leq\delta/2^{i}$. We stop splitting after $T$
rounds, where $T$ will be fixed later to make $d_{T}$ and $\delta/2^{T}$
polylogarithmic in $R$. The total degree loss due to splitting is
$\displaystyle\delta-2^{T}d_{T}=\sum_{i=0}^{T}2^{i}(d_{i}-2d_{i+1})\leq\sum_{i=0}^{T-1}2^{i}2C_{1}\sqrt{d_{i}\ln
Rd_{i}}$
$\displaystyle\leq\sum_{i=0}^{T-1}2^{i}2C_{1}\sqrt{\frac{\delta}{2^{i}}\ln\frac{R\delta}{2^{i}}}$
$\displaystyle=2C_{1}\sqrt{\delta}\sum_{i=0}^{T-1}\sqrt{2^{i}\ln\frac{R\delta}{2^{i}}}.$
This sum is an arithmetic-geometric series dominated by the last term, i.e.
$\delta-2^{T}d_{T}=O(\sqrt{\delta 2^{T}\ln(R\delta/2^{T})})$. Pick $T$ such
that $\delta/2^{T}$ is within a constant factor of $\ln^{3}R$, then we deduce
$d_{T}\geq\delta/2^{T}(1-O(\sqrt{2^{T}/\delta\ln(R\delta/2^{T})}))\geq\delta/2^{T}(1-O(\ln^{-1}(R))).$
Consequently with Proposition 9 we see that
$\mathsf{p}^{\prime}\geq 2^{T}d_{T}/(\ln R+O(\ln\ln
R))\geq\delta(1-O(\ln^{-1}(R)))/(\ln R+O(\ln\ln R))$
which gives the claimed bound. ∎
### 2.1 Sparse Hypergraphs: Proof of Corollary 3
The proof is analogous to that in [1]. The key claim is that it is possible to
shrink all hyperedges so that the minimum degree becomes $\delta(H)-\alpha$,
and the maximum edge size becomes at most $\beta$; then using Theorem 1, we
are clearly done.
Modelling this shrinking as a flow problem, we want to send flow from a
source, through $V$, past $E$, and to a sink such that $\delta-\alpha$ units
pass through each $v\in V$, at most $\beta$ units pass through each $e\in E$,
and each vertex can send up to 1 unit of flow to each incident hyperedge. Let
$I(V^{\prime},\mathcal{E}^{\prime})$ denote the number of incidences between
$V^{\prime}$ and $\mathcal{E}^{\prime}$. By the max-flow min-cut theorem the
only obstacle is a cut containing the source along with $V^{\prime}\subseteq
V$ and $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, with
$(\delta-\alpha)|V\backslash
V^{\prime}|+I(V^{\prime},\mathcal{E}\backslash\mathcal{E}^{\prime})+\beta|\mathcal{E}^{\prime}|<(\delta-\alpha)|V|$.
Using
$I(V^{\prime},\mathcal{E}\backslash\mathcal{E}^{\prime})\geq\delta|V^{\prime}|-I(V^{\prime},\mathcal{E}^{\prime})\geq\delta|V^{\prime}|-\alpha|V^{\prime}|-\beta|\mathcal{E}^{\prime}|$
we see no such obstacle exists.
### 2.2 Lower Bounds
Now we show that the bounds obtained previously are tight.
###### Proof of Theorem 2(a).
We want to show, for a constant $C$ and all $R\geq 2,\delta\geq 1$ we have
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\leq\max\\{1,C\delta/\ln
R\\}$. Consider the hypergraph $H=\tbinom{[2k-1]}{k}^{*}$ in the introduction.
It is $k$-regular, it has $\mathsf{p}^{\prime}(H)=1$, and
$R(H)=\tbinom{2k-2}{k-1}$.
Since $\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)$ is non-
increasing in $R$, we may reduce $R$ by a constant factor to assume that
either $R=2$, (in which case we are done by Gupta’s theorem) or
$R(H)=\tbinom{2k-2}{k-1}$ for some integer $k$. Note this gives $k=\Theta(\log
R)$. Moreover, if $\delta\leq k$ then $H$ proves the theorem, so assume
$\delta\geq k$. Again by monotonicity, we may increase $\delta$ by a constant
factor to make $\delta$ a multiple of $k$. Let $\mu=\delta/k$.
Consider the hypergraph $\mu H$ obtained by copying each of its edges $\mu$
times, for an integer $\mu\geq 1;$ note that it is $\delta$-regular. The
argument in the introduction shows that any set cover has size at least $k$
and therefore average degree at least
$k\tbinom{2k-2}{k-1}/\tbinom{2k-1}{k}=k^{2}/(2k-1)=\Theta(\ln R)$. Thus
$\overline{\mathsf{p}}^{\prime}(\mu H)=O(\delta/\ln R)$ which proves the
theorem. ∎
###### Proof of Theorem 2(b).
We want to show as $R,\delta\to\infty$ with $\delta=\omega(\ln R)$, we have
$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)\leq(1+o(1))\delta/\ln(R)$.
We assume an additional hypothesis, that $R\geq\delta$; this will be without
loss of generality as we can handle the case $\delta>R$ using the
$\mu$-replication trick from the proof of Theorem 2(a), since our argument is
again based on lower-bounding the minimum size of an set cover.
Let $\delta^{\prime}=\delta(1+o(1))$ and $R^{\prime}=R(1-o(1))$ be parameters
that will be specified shortly. We construct a random hypergraph with
$n=R^{\prime 2}\delta^{\prime}$ vertices and $m=R^{\prime}\delta^{\prime 2}$
edges, where for each vertex $v$ and each edge $E$, we have $v\in E$ with
independent probability $p=1/R^{\prime}\delta^{\prime}$. Thus each vertex has
expected degree $\delta^{\prime}$ and each edge has expected size
$R^{\prime}$. A standard Chernoff bound together with $np=\omega(\ln m)$ shows
the maximum edge size is $(1+o(1))R^{\prime}$ asymptotically almost surely
(_a.a.s._); pick $R^{\prime}$ such that this $(1+o(1))R^{\prime}$ equals $R$.
Likewise, since $mp=\omega(\ln n)$ a.a.s. the actual minimum degree is at
least $(1-o(1))\delta^{\prime}$ which we set equal to $\delta$.
We will show that this random hypergraph has
$\mathsf{p}^{\prime}\geq(1+o(1))\delta/\ln R$ a.a.s. via the following bound,
which is based off of an analogous bound for Erdős-Renyi random graphs in [15,
§2.5]:
###### Claim 14.
A.a.s. the minimum set cover size is at least $\frac{1}{p}\ln(pn)(1-o(1))$.
We give the proof in Appendix C. This claim finishes the proof since it
implies that the maximum number of disjoint set covers $\mathsf{p}^{\prime}$
is at most
$(1+o(1))mp/\ln(pn)=(1+o(1))\delta^{\prime}/\ln(R^{\prime})=(1+o(1))\delta/\ln(R).$
∎
Aside from the results above, not much else is known about specific values
of$\overline{\mathsf{p}}^{\prime}({\textsc{Hyp}}(R),\delta)$ for small
$R,\delta$. The Fano plane gives
$\overline{\mathsf{p}}({\textsc{Hyp}}(3),3)=1$: if its seven sets are
partitioned into two parts, one part has only three sets, and it is not hard
to verify the only covers consisting of three sets are pencils through a point
and therefore preclude the remaining sets from forming a cover. Moreover,
Thomassen [36] showed that every 4-regular, 4-uniform hypergraph has Property
B; together with monotonicity we deduce that
$\overline{\mathsf{p}}({\textsc{Hyp}}(3),4)\geq\overline{\mathsf{p}}({\textsc{Hyp}}(4),4)\geq
2$.
## 3 Paths in Trees
Let $\textsc{TreeEdges}|\textsc{Paths}$ denote the following family of
hypergraphs: the ground set is the edge set of an undirected tree, and each
hyperedge must correspond to the edges lying in some path in the tree. Such
systems are cover-decomposable:
###### Theorem 4.
$\overline{\mathsf{p}}^{\prime}(\textsc{TreeEdges}|\textsc{Paths},\delta)\geq
1+\lfloor(\delta-1)/5\rfloor$.
###### Proof.
In other words, given a family of paths covering each edge at least
$\delta=5k+1$ times, we can partition the family into $k+1$ covers. We use
induction on $k$; the case $k=0$ is evidently true.
We will use an iterated LP relaxation algorithm similar to the one used in
Proposition 10. However, it is more convenient to get rid of the $y$
variables; it is helpful to think of them implicitly as $y=1-x$. Thus our
linear program will have one variable $0\leq x_{P}\leq 1$ for every path $P$.
Fix integers $A,B$ such that $A+B=\delta$, and the LP will aim to make $x$ the
indicator vector of an $A$-fold cover, and $1-x$ the indicator vector of a
$B$-fold cover. So for each edge $e$ of the tree, we will have one _covering_
constraint $\sum_{P:e\in P}x_{e}\geq A$ and one _packing_ constraint
$\sum_{P:e\in P}x_{e}\leq|{P:e\in P}|-B$ (corresponding to coverage for $y$).
Note that the linear program has a feasible fractional solution $x\equiv
A/\delta$.
As before, the iterated LP relaxation algorithm repeatedly finds an extreme
point solution of the linear program, fixes the value of variables whenever
they have integral values, and discards certain constraints. We will use the
following analogue of Lemma 11, which is an easy adaptation of a similar
result for packing in [21]. (We explain in more detail how [21] implies Lemma
15 in Appendix D.)
###### Lemma 15.
Suppose some $x$ variables are fixed to 0 or 1, and some covering/packing
constraints are discarded. Any extreme point solution $x^{*}$ has the
following property: there is a tight covering or packing constraint involving
at most 3 variables which are fractional in $x^{*}$.
When such a constraint arises, we discard it. As before, any non-discarded
constraint is satisfied by the integral $x$ at termination. Additionally,
consider a discarded constraint, say a covering one $\sum_{P:e\in P}x_{e}\geq
A$ for some $P$. When it is discarded, it holds with equality, and the left-
hand side consists of 0’s, 1’s, and at most 3 fractional values. Since $A$ is
an integer, it follows that there are at least $A-2$ 1’s on the LHS. The final
$x$ still has these variables equal to 1; so overall, $x$ is the
characteristic vector of an $(A-2)$-fold cover, and likewise $1-x$ is the
characteristic vector of a $(B-2)$-fold cover.
Finally, fix $A=3$ and $B=\delta-3$. The final integral $x$ covers every edge
at least $3-2$ times — it is a cover. The final $1-x$ covers every edge at
least $\delta-5=5(k-1)+1$ times. Hence we can use induction to continue
splitting $1-x$, giving the theorem. ∎
We have little idea whether Theorem 4 is anywhere close to tight; the best
upper bound we know is
$\overline{\mathsf{p}}^{\prime}(\textsc{TreeEdges}|\textsc{Paths},\delta)\leq\lfloor(3\delta+1)/4\rfloor$.
Next, we give the optimal bound on the polychromatic number for systems of
paths in trees.
###### Theorem 5.
$\overline{\mathsf{p}}(\textsc{TreeEdges}|\textsc{Paths},r)=\lceil r/2\rceil.$
###### Proof.
The lower bound means that for every collection of paths of length at least
$2k-1$, we can $k$-colour the edges so that each path is polychromatic. To see
this, root the tree arbitrarily and give all edges at level $i$ the colour
$i\bmod k$. Every path uses at least $k$ consecutive levels, so we are done.
For the upper bound, it is enough to exhibit a tree and a family of
$(2k-2)$-edge paths such that no $k$-colouring of the tree’s edges is
polychromatic with respect to the paths. We consider a complete $t$-ary tree
of height $k-1$, for $t=2k^{k-1}$, with root $r$, and where every path of
length $2k-2$ (i.e. each leaf-leaf path passing through $r$) is a hyperedge.
Fix any colouring of the tree’s edges. By the pigeonhole principle, there is a
set $C_{r}$ of at least $t/k$ children of $r$, such that all edges $\\{rc\mid
c\in C_{r}\\}$ get the same colour. Likewise, each node in $C_{r}$ exhibits
the same phenomenon for some colour, so there is a subset
$C^{\prime}_{r}\subset C_{r}$ and node sets $\\{C_{c}\\}_{c\in
C^{\prime}_{r}}$ such that $C_{c}$ is a subset of $c$’s children of size at
least $t/k$, such that all edges between $C^{\prime}_{r}$ and
$\bigcup_{c}C_{c}$ have a common colour, and such that $|C^{\prime}_{r}|\geq
t/k^{2}$. Continuing in this way, since $t/k^{k-1}\geq 2$, we find leaves
$u,v$ such that the paths $ru$ and $rv$ contain the same ordered sequence of
colours. Since the path has at most $k-1$ colours, it is not polychromatic, as
needed. ∎
## 4 Future Work
In the _sensor cover_ problem (e.g. [17]) each hyperedge has a given duration;
we seek to schedule each hyperedge at an offset so that every item in the
ground set is covered for the contiguous time interval $[0,T]$ with the
_coverage_ $T$ as large as possible. Cover-decomposition is the special case
where all durations are unit. Clearly $T$ is at most the minimum of the
duration-weighted degrees, which we denote by $\overline{\delta}$. Is there
always a schedule with $T=\Omega(\overline{\delta}/\ln R)$ if all hyperedges
have size at most $R$? The LLL is viable but splitting does not work and new
ideas are needed. For $R=2$ (i.e. graphs) we have an 8-approximation algorithm
for maximum coverage (see Appendix E):
###### Theorem 16.
Every instance of sensor cover in graphs has a schedule of coverage at least
$\overline{\delta}/8$.
### Acknowledgments
We thank Jessica McDonald, Dömötör Pálvölgyi, and Oliver Schaud for helpful
discussions on these topics.
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## Appendix A Additional Related Work
We mention some terminology from the literature on hypergraphs (e.g. see [33])
relevant to the study of $\mathsf{p}$. For each hypergraph $H$ let $M_{H}$ be
its 0-1 incidence matrix, where rows correspond to edges and columns to
vertices. Then the following three terms are equivalent by definition: (i) for
every hypergraph obtained from $H$ by deleting or duplicating vertices
(columns), $\mathsf{p}=r$; (ii) the fractional vertex cover polytope $\\{x\mid
x\geq 0,M_{H}x\geq 1\\}$ has the _integer decomposition property_ ; (iii) the
_blocker_ of $H$ is _Mengerian_. One family of hypergraphs with Mengerian
duals is the family of _balanced_ hypergraphs (no submatrix of $M_{H}$ is the
incidence matrix of an odd cycle).
One motive to study cover-decomposition is to resolve the following conjecture
of Bang-Jensen and Yeo [7]: there is an integer $k$ such that every $k$-arc-
strongly-connected digraph contains two edge-disjoint spanning strongly-
connected subdigraphs. This would be a directed analogue of fact due to Nash-
Williams and Tutte that every $2k$-edge-connected graph contains $k$ disjoint
spanning trees, which can be restated as $\overline{\mathsf{p}}(r)\geq\lfloor
r/2\rfloor$ for the appropriately defined family.
Preliminary work on cover-decomposability of translates of convex polygons
appears in [26, 35, 32, 29, 4]. There is an unpublished proof [22] that
$\overline{\mathsf{p}}(\mathbb{R}^{2}|\textsc{Translates}(Unit-Disc),33)\geq
2$. On the other hand, unit balls in $\mathbb{R}^{3}$ or higher dimensions are
not cover-decomposable [23, 28].
Let $\mathbb{R}^{2}|{\textsc{Hpl}}$ denote the family of hypergraphs such that
the ground set is a finite set in $\mathbb{R}^{2}$, where this time edges are
intersections of $V$ with _halfspaces_. Recently Smorodinsky and Yuditsky [34]
proved
$\overline{\mathsf{p}}^{\prime}(\mathbb{R}^{2}|{\textsc{Hpl}},\delta)=\lceil\delta/2\rceil$
(improving upon [6]) and
$\overline{\mathsf{p}}(\mathbb{R}^{2}|{\textsc{Hpl}},r)\geq\lceil r/3\rceil.$
They note the latter is not tight, since Fulek [16] showed
$\overline{\mathsf{p}}(\mathbb{R}^{2}|{\textsc{Hpl}},3)=2$.
It is known that $\overline{\mathsf{p}}(\mathbb{R}^{2}|\textsc{Axis-Aligned-
Rectangles},r)\equiv 1$ [10] by probabilistic arguments.
We have considered geometric families where each instance has a discrete set
of points that must be covered (e.g. by translates of a fixed polygon). This
is called _total-cover-decomposability_ by Pálvölgyi [30, 31] and constrasts
with, “when the entire plane is covered $\delta(t)$ times, we can decompose
into $t$ covers of the entire plane.” He shows polygons which have the latter
property, but not the former. A complete classification of open polygons with
respect to total-cover-decomposability was obtained by Pálvölgyi in [31].
We mention two indecomposability constructions. First, [28] use the Hales-
Jewett theorem to show that
$\overline{\mathsf{p}}(\mathbb{R}^{2}|\textsc{Lines},r)\equiv 1$. Second, [31]
gives the following small indecomposable construction. The ground set of the
hypergraph is the set of all strings on at most $r-1$ B’s and at most $r-1$
R’s. We have a hyperedge for each sequence
$(\sigma_{1},\sigma_{2},\dotsc,\sigma_{r})$ of strings such that each
$\sigma_{i}$ is a prefix of $\sigma_{i+1}$, and such that all the suffices
$\sigma_{i+1}\backslash\sigma_{i}$ start with with same letter. E.g. a typical
edge for $r=4$ is $\\{R,RBR,RBRB,RBRBB\\}$. Such hypergraphs do not have
Property B; an example of their utility is that for every non-convex
quadrilateral $A$, $\mathbb{R}^{2}|\textsc{Translates}(A)$ contains these
hypergraphs and so is not cover-decomposable.
Here is another combinatorial question of Pálvölgyi [30]. Consider an $m\times
r$ matrix of positive integers which is weakly increasing from left to right
and top to bottom. Then think of each row as a set of size $r$, giving a
hyperedge on ground set $\mathbb{Z}$. Do these _shift chains_ have Property B
if $r\geq 4$? (A counterexample is known [30] for $r=3$.)
Two outstanding open problems are to determine whether
$\overline{\mathsf{p}}(\delta)=\Omega(\delta)$ for the family of unit discs,
and whether the family of all squares is cover-decomposable.
## Appendix B Small VC-Dimension
The VC-dimension of a set system $(V,\mathcal{E})$ is defined as follows. We
say $S\subseteq V$ is _shattered_ if every $T\subseteq S$ can be obtained as
an intersection of $S$ with some hyperedge, i.e. if $\\{S\cap E\mid
E\in\mathcal{E}\\}={\bf 2}^{S}$. Then, the VC-dimension equals the size of the
largest shattered set. The _dual VC-dimension_ of $(V,\mathcal{E})$ is the VC-
dimension of $(V,\mathcal{E})^{*}$.
To show that primal and dual VC-dimension 2 is not enough to ensure cover-
decomposability, it suffices to use the following construction of Pach, Tardos
and Tóth [28]. Let $T$ be a $k$-ary rooted tree, with $k$ levels of vertices;
so $T$ has $k^{k-1}$ leaves and $\sum_{i=0}^{k-1}k^{i}$ nodes in total. For
each non-leaf node $v$ we define its _sibling hyperedge_ to be the $k$-set
consisting of $v$’s children. For each leaf node $v$ we define its _ancestor
hyperedge_ to be the $k$-set consisting of the nodes on the path from $v$ to
the root node. We let $\mathrm{PTT}_{k}$ denote the hypergraph consisting of
all ancestor and sibling hyperedges. As shown in [28],
$\mathsf{p}(\mathrm{PTT}_{k})=1$. First, we prove Theorem 7.
###### Proof of Theorem 7.
We will show that $\mathrm{PTT}_{k}$ has VC-dimension and dual VC-dimension at
most 2. Thus $PTT_{k}$ verifies the first part of Theorem 7 (since $k=r$) and
$\mathrm{PTT}_{k}^{*}$ verifies the second part. A key observation is that in
$\mathrm{PTT}_{k}$,
any two distinct edges intersect in either 0 or 1 vertices. (1)
First, we bound the primal VC-dimension. Suppose for the sake of contradiction
that there is a shattered vertex set $\\{x,y,z\\}$ of size 3. Since the set is
shattered, there is a hyperedge $E$ containing all of $\\{x,y,z\\}$. But by
the definition of shattering, there must be another hyperedge $E^{\prime}\neq
E$ with $E^{\prime}\cap\\{x,y,z\\}=\\{x,y\\}$. This contradicts (1), so we are
done.
Second, we bound the dual VC-dimension. Suppose for the sake of contradiction
that $E,E^{\prime},E^{\prime\prime}$ are three hyperedges which are shattered
in the dual. This implies that $E\cap E^{\prime}\cap E^{\prime\prime}$ and
$(E\cap E^{\prime})\backslash E^{\prime\prime}$ are both nonempty. But this
would imply $|E\cap E^{\prime}|\geq 2$, contradicting (1). ∎
In the remainder of this section we show that set systems with unit VC-
dimension have large cover-decomposition and polychromatic numbers (Theorem
6). The following is a convenient way of looking at such hypergraphs.
###### Definition 17.
A hypergraph $(V,\mathcal{E})$ is called _cross-free_ if the following holds
for every pair $S,T\in\mathcal{E}$: at least one of $S\cap T,S\backslash
T,T\backslash S,$ or $V\backslash S\backslash T$ is empty.
Observe that a hypergraph has VC-dimension 1 if and only if the dual
hypergraph is cross-free. In the proofs of this section we will also use
_laminar_ hypergraphs, which are a subclass of the cross-free hypergraphs.
###### Definition 18.
A hypergraph $(V,\mathcal{E})$ is called _laminar_ if the following holds for
every pair $S,T\in\mathcal{E}$: at least one of $S\cap T,S\backslash T$, or
$T\backslash S$ is empty. (Equivalently, either $S\cap
T=\varnothing,S\subseteq T$, or $T\subseteq S$.)
We will need the following folklore fact (we provide a proof in Appendix D).
###### Lemma 19.
In a laminar hypergraph, $\mathsf{p}^{\prime}=\delta$.
By duality, the following theorem proves the first half of Theorem 6.
###### Proposition 20.
For the family of cross-free hypergraphs,
$\overline{\mathsf{p}}^{\prime}(\delta)=\lceil\delta/2\rceil$.
###### Proof.
First, for each $\delta$ we demonstrate a cross-free hypergraph with minimum
degree $\delta$ which cannot be decomposed into more than
$\lceil\delta/2\rceil$ covers. Take a hypergraph whose ground set $V$ has
$\delta+1$ elements, with one hyperedge $V\backslash\\{v\\}$ for each $v\in
V$. Since each set cover has at least 2 hyperedges, we have
$\overline{\mathsf{p}}^{\prime}\leq\lfloor(\delta+1)/2\rfloor$, as needed.
To prove the other direction
$\overline{\mathsf{p}}^{\prime}\geq\lfloor(\delta+1)/2\rfloor$ for all cross-
free hypergraphs, we use induction on $\delta$. Clearly the bound holds for
$\delta=0$ or $\delta=1$. For the inductive step, we have two cases. First, if
the cross-free hypergraph $(V,\mathcal{E})$ has two sets $S,T\in\mathcal{E}$
for which $S\cup T=V$, then this is a cover with maximum degree 2 at each
node. By induction
$\overline{\mathsf{p}}^{\prime}(V,\mathcal{E}\backslash\\{S,T\\})\geq\lceil(\delta-2)/2\rceil$
and combining this cover-decomposition with $\\{S,T\\},$ we are done. Second,
if the cross-free hypergraph has no such $S,T$, then in fact the cross-free
hypergraph is laminar, and we are done by Lemma 19. ∎
Again by duality, the following theorem proves the second half of Theorem 6.
###### Proposition 21.
For the family of cross-free hypergraphs, $\overline{\mathsf{p}}(r)=\lceil
r/2\rceil$.
###### Proof.
Consider again the hypergraph used in the first half of the proof of
Proposition 20. Note that it is self-dual, e.g. since its incidence matrix is
the all-ones matrix minus the identity matrix, which is symmetric. Thus, we
deduce $\overline{\mathsf{p}}(r)\leq\lfloor(r+1)/2\rfloor$ for all $r$.
Next, we prove the other direction. We must show for all $k$, that if every
hyperedge has size at least $2k-1$, then there is a polychromatic
$k$-colouring. A key observation is the following: if
$\mathcal{E}^{\mathrm{min}}$ denotes the inclusion-minimal elements of
$\mathcal{E}$, then a colouring is polychromatic for $(V,\mathcal{E})$ if and
only if it is polychromatic for $(V,\mathcal{E}^{\mathrm{\min}})$. So we may
reset $\mathcal{E}:=\mathcal{E}^{\mathrm{\min}}$, which does not affect $r$ or
cross-freeness. Moreover now $\mathcal{E}$ is a _clutter_ : there do not exist
two different hyperedges $A,B\in\mathcal{E}$ for which $A\subset B$.
###### Lemma 22.
In a cross-free clutter $(V,\mathcal{E})$, either all hyperedges are pairwise
disjoint, or for every two hyperedges $A,B$ we have $A\cup B=V$.
We provide a proof in Appendix D. Using the lemma, Proposition 21 now boils
down to two cases. The first case is that all hyperedges are pairwise
disjoint. In this case a polychromatic $k$-colouring is easy to obtain: for
each hyperedge $E$, just colour its $|E|\geq 2k-1\geq k$ elements in any way
that uses all $k$ colours.
Finally, we deal with the case that for every two hyperedges $A,B$ we have
$A\cup B=V$. Rewriting, this means that $\mathcal{E}$ is of the form
$\\{V\backslash S_{1},V\backslash S_{2},\dotsc,V\backslash S_{m}\\}$ with
$m\geq 2$ where the $S_{i}$ are pairwise disjoint, and $|V\backslash
S_{i}|\geq 2k-1$ for all $i$. Repeat the following for $j=1,\dotsc,k-1$:
colour any uncoloured vertex $v$ with colour $j$, find any uncoloured vertex
$v^{\prime}$ such that $v$ and $v^{\prime}$ do not lie in the same $S_{i}$,
and colour $v^{\prime}$ with colour $j$. To see this is possible, we use
induction to prove: $v$ and $v^{\prime}$ exist in each stage, and after stage
$j$ the number of uncoloured vertices in $V\backslash S_{i}$ is at least
$2k-1-2j$, for each $i$. After stage $k-1$ we colour all of the remaining
uncoloured vertices with colour $k$. It is easy to see each colour class
appears in each $V\backslash S_{i}$, so we are done. ∎
We remark that Theorem 6 may be viewed as a theorem for yet another family of
tree-related hypergraphs. Namely, Edmonds & Giles showed [12] a hypergraph $H$
is cross-free precisely when there exists a directed tree, such that each
element of $V(H)$ occurs exactly once as a label on a node of the tree (a
given node can have multiple labels), and $\mathcal{E}(H)$ consists of those
label sets on the half-tree “pointed to” by each directed edge of the tree.
## Appendix C Proof of Claim 14
Claim 14 is based off of a similar claim [15, §2.5] for Erdős-Renyi random
graphs; indeed, the results for neighbourhood hypergraphs in [15] can be used
to prove Theorem 2(b) for the case $R\sim\delta$, but we are rehashing the
details here in order to prove Theorem 2(b) for the broader range of $R$ and
$\delta$.
###### Proof.
(For simplicity we rename $R^{\prime}$ to $R$, etc. in this proof.) We recap
the goal: given $R,\delta\to\infty$ with $R\geq(1-o(1))\delta$ and
$\delta=\omega(\ln R)$, in a random hypergraph with $n=R^{2}\delta$ vertices
and $m=R\delta^{2}$ edges with each incidence occuring with independent
probability $p=\frac{1}{R\delta}$, we want to show a.a.s. that the minimum set
cover size is at least $\frac{1}{p}\ln(pn)(1-o(1))$. Equivalently, we will
show for every fixed $\epsilon>0$ that the minimum set cover size is a.a.s. at
least $\frac{1}{p}\ln(pn)(1-\epsilon)$. We will use the fact that $p=o(1)$.
Fix any collection of $s$ edges, where $s=\frac{1}{p}\ln(pn)(1-\epsilon)$. It
is a set cover with probability
$\displaystyle(1-(1-p)^{s})^{n}$
$\displaystyle\leq(1-\exp(-(1+\Theta(p))sp))^{n}$
$\displaystyle=(1-\exp(-(1+\Theta(p))(1-\epsilon)\ln(pn)))^{n}$
$\displaystyle=(1-(pn)^{-1+\epsilon-\Theta(p)})^{n}$
$\displaystyle\leq\exp(-n(pn)^{-1+\epsilon-\Theta(p)})$
$\displaystyle\leq\exp(-\frac{1}{p}(pn)^{\epsilon-\Theta(p)}).$
There are $\tbinom{m}{s}\leq m^{s}=\exp(s\ln m)$ such collections, so a.a.s.
none are set covers provided that
$s\ln m-\frac{1}{p}(pn)^{\epsilon-\Theta(p)}\to-\infty.$
In turn, expanding the definition of $s$, we need
$\frac{1}{p}(\ln(pn)(1-\epsilon)\ln m-(pn)^{\epsilon-\Theta(p)})\to-\infty.$
This is straightforward to check using again the fact that $p=o(1)$ and using
$\delta\leq(1+o(1))R$ to verify that $\ln m=\ln R^{2}\delta\leq(1+o(1))\ln
R^{3}$ is sub-polynomial in $np=R$. ∎
## Appendix D Additional Short Proofs
###### Proof of Proposition 13.
Chernoff bounds tell us that if we assign each edge uniformly at random to
$\mathcal{E}_{1}$ or $\mathcal{E}_{2}$, then for each vertex $v$, its degree
in $\mathcal{E}_{1}$ is at least $d/2-\lambda$ with probability at least
$1-\exp(-\lambda^{2}/2d)$. Thus using the LLL again, provided
$2\exp(-\lambda^{2}/2d)Rd\leq 1/\mathrm{e}$, with positive probability every
vertex has degree at least $d/2-\lambda$ in both $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$. Solving for $\lambda$, we see
$\lambda=\sqrt{2d\ln(2\mathrm{e}Rd)}$ satisfies the inequality, and proves the
proposition. ∎
###### Proof of Lemma 15.
The lemma is a straightforward generalization of [21, Lemma 4] for a similar
family of LPs except without covering constraints: multicommodity flow in
trees, i.e. packing paths subject to integral edge capacities. Essentially the
same proof can be used; alternatively we can use the following reduction.
To reduce this more general packing-covering lemma to the original packing
lemma, (i) contract all edges except those involved in a tight constraint;
(ii) change all remaining (tight) covering constraints to (tight) packing
constraints. Then as a technicality, since the lemma from [21] is written
assuming all variables are fractional, (iii) drop all variables whose values
are at 0 or 1, moreover for each dropped path variable with value 1, reduce
the capacity of each edge on that path by 1. It is not hard to see that an
extreme point solution to the packing-covering LP remains an extreme point
solution to the multicommodity flow obtained after (i)–(iii), since it is
feasible and it is the unique solution to a linearly independent family of
tight constraints. This gives the reduction. ∎
###### Proof of Lemma 19.
We use induction on $\delta$, with $\delta=0$ being trivial. For $\delta>0$
consider the collection $\mathcal{M}$ of inclusion-maximal hyperedges. These
hyperedges are disjoint and (since $\delta>0$) they form a cover. By induction
the hypergraph obtained by deleting $\mathcal{M}$ has $\delta-1$ disjoint set
covers, so we are done. ∎
###### Proof of Lemma 22.
From the definitions of “cross-free” and “clutter,” we first have that for
every $S,T\in\mathcal{E}$, either $S\cap T=\varnothing$ or $S\cup T=V$. If the
former holds for every pair $S,T$ then we are done. So assume that there are
two particular sets $A,B$ in the clutter for which $A\cap B=V$.
We claim that every other set $C$ of the clutter has $A\cup C=V$. Suppose
otherwise, then $A\cap C=\varnothing$; this implies $C\subsetneq B$, which
violates the clutter property. Likewise, $B\cup C=V$. Then repeating this
argument by induction, we see that for all $S,T\in\mathcal{E}$, $S\cup T=V$. ∎
## Appendix E Sensor Cover in Graphs
As an aside: we do not try to optimize the constant 8; and the problem of
finding a max-duration edge-cover schedule is $\mathsf{APX}$-hard: to see this
note Holyer [19] showed deciding whether a cubic graph has three disjoint
perfect matchings is $\mathsf{NP}$-hard, which easily implies it is
$\mathsf{NP}$-hard to determine if a cubic graph with $d={\bf 1}$ permits a
schedule with $T=3$.
###### Proof of Theorem 16.
To formalize the proof of Theorem 16, let $G=(V,E)$ be a graph where every
edge $e$ has a integral duration $d_{e}$. Given a schedule $S:E\to R$, it has
_coverage_ $T$ if for each vertex $v$ and each time $0\leq t\leq T$, some edge
$e\ni v$ has $S(e)\leq t\leq S(e)+d_{e}$. For conciseness, in the rest of the
proof we say just _degree_ in place of _duration-weighted degree_. As
mentioned earlier, let $\overline{\delta}$ denote the minimum degree,
$\overline{\delta}=\min\\{\sum_{e\ni v}d_{e}\mid v\in V\\}$.
For convenience, we scale down all durations by a factor of
$\overline{\delta}$, so the new graph has rational durations and minimum
degree equal to 1. Next, round down each edge duration to the nearest number
of the form $2^{-i}$. It is clearly enough to produce a schedule of coverage
$1/8$ in this graph. Observe that this duration-rounded graph has minimum
degree at least $1/2$.
We will update the graph in several steps. So it is convenient to let
$G_{0}=(V_{0},E_{0})$ denote the original graph, and $G=(V,E)$ be the graph
evolving over time. We will repeatedly _dedicate_ edges to vertices and remove
them from $G$, in such a way that each edge is dedicated at most once, to one
of its endpoints.
For any edge with $d_{e}\geq 1/8$ observe we satisfy the target coverage
requirement of $1/8$ for both of its endpoints by setting $S(e)=0$. For such
an edge $e=uv$, we thus may delete vertices $u$ and $v$ from $G$; any other
edge incident to one of these vertices, say $uw$, should be used to cover $w$,
so we dedicate $uw$ to $w$ and delete $uw$ from $G$. Such vertices $u,v$ are
called _deleted in preprocessing_.
Next, we consider separately the edge sets $E_{i}=\\{e\in E\mid
d_{e}=2^{-i}\\}$. Note that $E$ is partitioned into $\biguplus_{i\geq
4}E_{i}$. Whenever some $E_{i}$ has a cycle $C$, we dedicate the edges of $C$
to its vertices, one per vertex (using its a cyclic order) and then delete
$E(C)$ from $G$ (and from $E_{i}$). We repeat this cycle deletion until each
$E_{i}$ is a forest.
In the remaining graph, whenever there is a vertex $v$ of degree at most
$1/4$, delete $v$ and for each incident edge $vw$, dedicate $vw$ to $w$ and
delete $vw$. Such a vertex $v$ is _deleted because of degree_. We iterate this
procedure as long as possible, i.e. until all remaining vertices have degree
more than $1/4$.
To complete the algorithm description and the proof we will use two claims:
###### Claim 23.
For any vertex $v$ deleted due to degree, the total duration of edges
dedicated to $v$ is at least 1/8.
###### Claim 24.
Every vertex of $V_{0}$ is either deleted in preprocessing or deleted due to
degree.
If we can establish these claims we are done: the vertices deleted in
preprocessing are already satisfied, while for every other vertex $v$ its
dedicated edges have total duration at least 1/8; since these dedicated sets
are disjoint we can easily schedule the edges dedicated to each $v$ separately
in such a way that we have 1/8 of coverage at $v$.
###### Proof of Claim 23.
The key is that the amount of duration dedicated to $v$ is at least half of
the degree drop at $v$ (from the start of the algorithm until just before we
delete $v$). This is easy to verify from checking the situations under which
we delete/dedicate edges. The initial degree of $v$ was at least 1/2, but it
was at most 1/4 upon deletion, so the degree drop was at least $1/2-1/4=1/4$.
Consequently the total duration of edges dedicated to $v$ is at least half of
1/4, i.e. 1/8. ∎
###### Proof of Claim 24.
Suppose for the sake of contradiction that the algorithm terminates with a
non-empty set $U$ of vertices remaining. Each one has degree greater than 1/4.
Since each $E_{i}$ is a forest (and remains so under deletions), the number of
edges left in each $E_{i}$ at termination is at most $|U|-1$. By double-
counting, the sum of the degrees is
$\sum_{i\geq 4}|E_{i}|2\cdot 2^{-i}\leq\sum_{i\geq 4}(|U|-1)2\cdot
2^{-i}<|U|/4.$
This is a contradiction to the fact that there are $|U|$ vertices, each with
degree greater than 1/4. ∎
This completes the proof of Theorem 16. ∎
|
arxiv-papers
| 2010-09-30T14:13:19 |
2024-09-04T02:49:13.244507
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B\\'ela Bollob\\'as, David Pritchard, Thomas Rothvo{\\ss} and Alex Scott",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/1009.6144"
}
|
1009.6188
|
∎
11institutetext: Abhijeet Paul, Mathieu Luisier and Gerhard Klimeck
22institutetext: School of Electrical and Computer Engineering and
Network for Computational Nanotechnology
Purdue University, West Lafayette, USA 47907
Tel.: 1-765-40-43589 22email: abhijeet.rama@gmail.com
# Modified valence force field approach for phonon dispersion: from zinc-
blende bulk to nanowires
Methodology and computational details
Abhijeet Paul Mathieu Luisier Gerhard Klimeck
(Received: date / Accepted: date)
###### Abstract
The correct estimation of thermal properties of ultra-scaled CMOS and
thermoelectric semiconductor devices demands for accurate phonon modeling in
such structures. This work provides a detailed description of the modified
valence force field (MVFF) method to obtain the phonon dispersion in zinc-
blende semiconductors. The model is extended from bulk to nanowires after
incorporating proper boundary conditions. The computational demands by the
phonon calculation increase rapidly as the wire cross-section size increases.
It is shown that the nanowire phonon spectrum differ considerably from the
bulk dispersions. This manifests itself in the form of different physical and
thermal properties in these wires. We believe that this model and approach
will prove beneficial in the understanding of the lattice dynamics in the next
generation ultra-scaled semiconductor devices.
###### Keywords:
Dynamical matrix Nanowire Phonons Valence Force Field
###### pacs:
63.20.-e Phonons in crystal lattices 63.22.Gh Nanotubes and nanowires
††journal: Journal of Computational Electronics
## 1 Introduction
The lattice vibration modes known as ‘phonons’ determine many important
properties in semiconductors like, (i) the phonon limited low field carrier
mobility in mosFETs JAP_anantram ; buin_sinw_phonon , (ii) the lattice thermal
conductivity in semiconductors which plays an important role in thermoelectric
design Mingo_kappa ; mingo_ph ; thermal_cond_dim , and (iii) the structural
stability of ultra-thin semiconductor nanowires SINW_110_phonon . A physics-
based method to calculate the phonon dispersion in semiconductors is required
to understand and link all these issues. As the device size approach the
nanometer scale and as the number of atoms in the structure become countably
finite, a continuum material description is no longer accurate. This work
provides a complete and elaborate description of an atomistic phonon
calculation method based on ‘Valence Force Field’ (VFF) model Keating_VFF ;
VFF_mod_herman ; VFF_mod_zunger , a frozen phonon approach, with application
to bulk and nanowire structures.
A variety of methods have been reported in the literature for the calculation
of the phonon spectrum such as the Valence Force Field (VFF) method and its
variants VFF_mod_herman ; VFF_mod_zunger ; Keating_VFF ; McMurry_VFF , Bond
Charge Model (BCM) bcm_weber ; BCM_model , Density Functional Method
SINW_110_phonon ; jauho_method , etc. We focus on VFF methods in this work.
There are multiple reasons for using a VFF based model: (a) in covalent bonded
crystals, like Si, Ge, GaAs, simple VFF potentials are sufficient to match the
experimental data McMurry_VFF , (b) valence coordinates and hence the
potential energy (U) depend only on the relative positions of the atoms and
are independent of rigid translations and rotations of the solid, and (c) it
is easy to extend the model to confined ultra-scaled structures made of few
atoms since the interactions are at the atomic level.
The original Keating VFF model Keating_VFF describe the LA, LO and TO phonons
reasonably well in zinc-blende materials, however, it does not produce the
flatness in the TA branch in Si, Ge bcm_weber ; VFF_mod_herman . Also the
limitation of the model to correctly describe the elastic constants (C11, C12,
C44) in these materials also limits its use VFF_mod_herman . In order to
extend the available VFF models to nanostructures, we need to identify the
models which can correctly describe the phonons in the entire Brillouin zone
(BZ) in zinc-blende semiconductors. There are older works where as many as six
parameters six_param_VFF have been used in VFF to obtain the correct phonon
dispersions. However, the task of this work has been to obtain a VFF model
which (i) can capture the correct physics and (ii) is computationally not very
expensive. Hence such a model can be extended to nanostructures like
nanowires, ultra-thin-bodies, etc. To this end we have identified two VFF
models which satisy the requirements. The modified VFF (MVFF) model presented
here combines these two following models, (i) VFF model from Sui et. al
VFF_mod_herman which is suitable for non-polar materials like Si and Ge and
(ii) VFF model from Zunger et. al VFF_mod_zunger which is suitable for
treating polar materials like GaP, GaAs, etc. This extended model is called
the ‘MVFF model’ in this study.
The main focus of this work is to show the implementation of VFF models for
phonon calculation in zinc-blende (diamond) lattices. We present the details
on the atomic groups which make up the interactions, the application of
boundary conditions in the nanostructures, the eigen value problem, the
computational requirements and the evaluation of lattice properties. We
benchmarked the model for variety of zinc-blende materials like Si, Ge, GaP,
GaAs, etc. In this paper we present the results using Si (sometimes Ge too) as
a specific example. We also present a comparison of the Keating VFF model
Keating_VFF with the present MVFF model for Si to elucidate the differences
in physical results and their computational requirements.
Previous theoretical works have reported the calculation of phonon dispersions
in SiNWs using a continuum elastic model and Boltzmann transport equation
continuum_model , atomistic first principle methods like DFPT (Density
Functional Perturbation Theory) SINW_110_phonon ; jauho_method ; sinw_cv ;
strain_effect_1 and atomistic frozen phonon approaches like Keating-VFF
(KVFF) Mingo_kappa ; SINW_111 . Thermal conductivity in SiNWs has been studied
previously using the KVFF model mingo_ph ; strain_effect_2 .
This paper has been arranged in the following sections. The MVFF theory (a
‘frozen phonon’ method) for the phonon dispersion in zinc-blende
semiconductors is reviewed in Sec. 2. It provides details about the total
potential energy (U) of the crystals in the MVFF model (Sec. 2.1),
construction of the dynamical matrix (DM) (Sec. 2.2), application of boundary
conditions to the DM (Sec. 2.3), solution of the resulting eigen value problem
(Sec. 2.4), and calculation of sound velocity ($V_{snd}$) (Sec. 2.5), lattice
thermal conductance ($\sigma_{l}$) (Sec. 2.6) and the mode Gr$\ddot{u}$neisen
parameters (Sec. 2.7) using the phonon spectrum. The computational details for
the calculation of phonon dispersion are presented in Sec. 3. Different
aspects of the dynamical matrix like the size, fill-factor, sparsity pattern,
etc., are provided in Sec. 3.1. Timing analysis for the assembly of the DM are
shown in Sec.3.2. Section 4 presents, a benchmark of the MVFF results against
experimental data for different semiconductors (Sec. 4.1), a comparison of the
MVFF and Keating-VFF (KVFF) models (Sec.4.2), phonon spectrum in Si nanowires
(SiNW) with free and clamped boundary conditions (Sec. 4.3) and lattice
thermal conductance using SiNW phonon spectrum (Sec.4.4). Conclusions are
given in Sec.5.
## 2 Theory, Approach and Parameters
In a given system, the phonons are modeled by solving the equations of motion
of its atomic vibrations. Since VFF is a crystalline model, the dynamical
equation for each atom ‘i’ can be written as,
$m_{i}\frac{\partial^{2}}{\partial t^{2}}(\Delta R_{i})=F_{i}=-\frac{\partial
U}{\partial(\Delta R_{i})}$ (1)
where, $\Delta R_{i}$, $F_{i}$ and U are the vibration vector of atom ‘i’, the
total force on atom ‘i’ in the crystal, and the potential energy of the
crystal, respectively. Equation (1) indicates that the calculation of the
vibrational frequencies requires a good estimation of the potential energy of
the system. The next part discusses the calculation of U within the MVFF
model.
### 2.1 Crystal Potential Energy (U)
The MVFF method VFF_mod_herman ; VFF_mod_zunger ; Keating_VFF approximates
the potential energy U, for a zinc-blende (or diamond) crystal, based on the
nearby atomic interactions (short-range) VFF_mod_herman ; VFF_mod_zunger as,
Figure 1: The short range interactions used for the calculation of phonon
dispersion in zinc-blende semiconductors.(a) Bond stretching (b) Bond bending
(c) cross bond stretching (d) cross bond bending-stretching and (e) coplanar
bond bending interaction.
$\displaystyle U$ $\displaystyle\approx$ $\displaystyle\frac{1}{2}\sum_{i\in
N_{A}}\Bigg{[}\sum_{j\in nn(i)}U^{ij}_{bs}+\sum^{j\neq k}_{j,k\in
nn(i)}\big{(}U^{jik}_{bb}$ (2) $\displaystyle+U^{jik}_{bs-bs}+U^{jik}_{bs-
bb}\big{)}+\sum^{j\neq k\neq l}_{j,k,l\in COP_{i}}U^{jikl}_{bb-bb}\Bigg{]}$
where $N_{A}$, $nn(i)$, and $COP_{i}$ represent the total number of atoms in
one unitcell, the number of nearest neighbors for atom ‘i’, and the coplanar
atom groups for atom ‘i’, respectively. The first two terms in Eq. (2) are
from the original KVFF model Keating_VFF . The other interaction terms are
needed to accurately describe the phonon dispersion in the entire Brillioun
Zone (BZ). The terms $U^{ij}_{bs}$ and $U^{jik}_{bb}$ represent the elastic
energy from bond stretching and bond-bending between the atoms connected to
each other (Fig. 1.a,b). The terms $U^{jik}_{bs-bs}$, $U^{jik}_{bs-bb}$ and
$U^{jikl}_{bb-bb}$ represent the cross bond stretching VFF_mod_herman ;
VFF_mod_zunger , cross bond bending-stretching VFF_mod_zunger , and coplanar
bond bending VFF_mod_herman interactions, respectively (Fig. 1. c,d,e). The
functional dependence of each interaction term on the atomic positions is
given by,
$\displaystyle U^{ij}_{bs}$
$\displaystyle=\frac{3}{8}\alpha_{ij}\frac{(r^{2}_{ij}-d^{2}_{ij,0})^{2}}{\|d^{2}_{ij,0}\|}$
(3) $\displaystyle U^{jik}_{bb}$
$\displaystyle=\frac{3}{8}\beta_{jik}\frac{(\Delta\theta_{jik})^{2}}{\|d_{ij,0}\|\|d_{ik,0}\|}$
(4) $\displaystyle U^{jik}_{bs-bs}$
$\displaystyle=\frac{3}{8}\delta_{jik}\frac{(r^{2}_{ij}-d^{2}_{ij,0})(r^{2}_{ik}-d^{2}_{ik,0})}{\|d_{ij,0}\|\|d_{ik,0}\|}$
(5) $\displaystyle U^{jik}_{bs-bb}$
$\displaystyle=\frac{3}{8}\gamma_{jik}\frac{(r^{2}_{ij}-d^{2}_{ij,0})(\Delta\theta_{jik})}{\|d_{ij,0}\|\|d_{ik,0}\|}$
(6) $\displaystyle U^{jikl}_{bb-bb}$
$\displaystyle=\frac{3}{8}\sqrt{(\nu_{jik}\nu_{ikl})}\frac{(\Delta\theta_{jik})(\Delta\theta_{ikl})}{\sqrt{\|d_{ij,0}\|\|d^{2}_{ik,0}\|\|d_{kl,0}\|}}$
(7)
where $\Delta\theta_{jik}=r_{ij}\cdot r_{ik}-d_{ij,0}\cdot d_{ik,0}$, is the
angle deviation of the bond between atom ‘i’ and ’j’ and bond between atom ‘i’
and ‘k’. The term $r_{ij}$ ($d_{ij,0}$) is the non-ideal (ideal) bond vector
from atom ‘i’ to ‘j’. The coefficients $\alpha$, $\beta$, $\delta$, $\gamma$,
and $\nu$ determine the strength of the interactions used in the MVFF model
(like spring constants). They are used as fitting parameters to reproduce the
bulk phonon dispersion VFF_mod_herman ; VFF_mod_zunger . The unit of these
fitting parameters are in force per unit length (like $Nm^{-1}$). The value of
these strength parameters also changes according to the deviation of the bond
length and bond angle from their ideal values. This enables the inclusion of
the anharmonic properties of the lattice vibrations anharmonic in this model.
Hence, MVFF is sometimes referred to as ‘quasi-anharmonic’ model.
Interaction Terms: The primitive bulk unitcell used for phonon calculation is
made of two atoms (anion-cation pair for zinc-blende and 2 similar atoms for
diamond). The black dotted box with atom 1 and 2 represents the bulk primitive
unitcell in Fig. 2. The total number of terms in each interaction in Eq. (3-7)
for a bulk unitcell are provided in Table 1. Apart from the coplanar bond
bending interaction VFF_mod_herman all the other terms involve nearest
neighbor interactions . There are 21 coplanar (COP) groups present in a bulk
zinc-blende unitcell which are needed for the calculation of the phonon
dispersion. For clarity some of these COP groups shown using the number
combinations in the caption of Fig. 2. Each group consists of 4 atom arranged
as anion(A)-cation(C)-anion(A)-cation(C) (eg. 1(A)-2(C)-3(A)-4(C) in Fig. 2).
Details about the coplanar interaction groups are provided in Appendix A.
Figure 2: (Color online) Three co-planar atom groups (out of 21) shown in a bulk zinc-blende unitcell. The groups are (i) 1-2-3-4, (ii) 1-2-5-6 and (iii) 1-2-7-8. Atoms 1 and 2 form the bulk unitcell used in the calculations. Red (black) atoms are cations (anions). Table 1: Number of terms in different interactions of the MVFF model in a bulk zinc-blende unitcell (anion-cation pair) Interaction Type | Total terms (anion+cation)
---|---
Bond stretching (bs) | 8
Bond bending (bb) | 12
Cross bond stretching (bs-bs) | 12
Cross bond stretch-bend (bs-bb) | 12
Coplanar bond bending (bb-bb) | 21
### 2.2 Dynamical Matrix (DM)
The dynamical matrix captures the motion of the atoms under small restoring
force in a given system. In this Section we discuss the structure of this
matrix. The derivation of the DM from the equation of motion is given in
Appendix B. The DM calculation is based on the harmonic approximation (see
Appendix B). For the interaction between two atoms ‘i’ and ‘j’, the DM
component at atom ‘i’ is given by,
$\centering D(ij)=\begin{bmatrix}D^{ij}_{xx}&D^{ij}_{xy}&D^{ij}_{xz}\\\
D^{ij}_{yx}&D^{ij}_{yy}&D^{ij}_{yz}\\\
D^{ij}_{zx}&D^{ij}_{zy}&D^{ij}_{zz}\end{bmatrix}\@add@centering$ (8)
The 9 components of $D(ij)$ are defined as,
$\displaystyle D^{ij}_{mn}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}U_{elastic}}{\partial r^{i}_{m}\partial
r^{j}_{n}},$ $\displaystyle i,j\in N_{A}\quad\text{and}\quad m,n\in[x,y,z],$
where $N_{A}$ is the total number of atoms in the unitcell. For each atom the
size of $D(ij)$ is fixed to 3 $\times$ 3\. For $N_{A}$ atoms in the unitcell
the size of the dynamical matrix is $3N_{A}\times 3N_{A}$. However, the matrix
is mostly sparse. The sparsity pattern, fill factor, and other related
properties of the DM are discussed in Section 3.
Symmetry considerations in the DM: Under the harmonic approximation the
dynamical matrix exhibit symmetry properties that can be readily utilized to
reduce its assembly time. From software development point of view this is
crucial in optimizing matrix construction time, storage and compute times. Due
to the continuous nature of the potential energy U, we have,
$D^{ij}_{mn}=\frac{\partial^{2}U_{elastic}}{\partial r^{i}_{m}\partial
r^{j}_{n}}=\frac{\partial^{2}U_{elastic}}{\partial r^{j}_{n}\partial
r^{i}_{m}}=D^{ji}_{nm}$ (10)
A closer look at Eq. (10) shows the following symmetry relation,
$D(ij)=D(ji)^{\prime}\quad\forall i\neq j.$ (11)
This reduces the total number of calculations required to construct the
dynamical matrix and speeds up the calculations. Also if the matrix is stored
for repetitive use, then only one of the symmetry blocks needs to be stored.
This reduces the memory requirement in the software by a factor of 2. Further
reduction in the construction time of DM can be achieved depending on the type
of interaction, the symmetry of the crystal, and some implementation tricks
(not covered in this work, see Ref.DM_element_reduction for more discussion).
Also the knowledge about the underlying symmetry of the matrix can also help
in the selection of linear algebra approaches which can reduce the final
solution time (not covered in this work).
Table 2: Boundary conditions (BC) in DM based on the dimensionality of the structure Dimensionality | Periodic BC | Finite Edge BC
---|---|---
Bulk (3D) | 3 | 0
Thin Film (2D) | 2 | 1
Wire (1D) | 1 | 2
Quantum Dot (0D) | 0 | 3
### 2.3 Boundary conditions (BC)
To calculate the eigenmodes of the lattice vibration, it is important to apply
appropriate boundary conditions to the DM. In the case of bulk material, the
unitcell has periodic (Born-Von Karman) boundary conditions along all the
directions (x,y,z) VFF_mod_herman ; bcm_weber since the material is assumed
to have infinite extent in each direction. However, for nanostructures the
boundary conditions are different due to the finite extent of the material
along certain directions. The boundary conditions vary depending on the
dimensionality of the structure (1D, 2D or 3D, see Table 2) for which the
dynamical matrix is constructed. There are 2 types of boundary conditions; (i)
Periodic Boundary condition (PBC) which assumes infinite material extent in a
particular direction and (ii) Finite Edge boundary conditions (FEBC) such as
open or clamped, which assumes finite material extent in a particular
direction. Table 2 provides the boundary condition details depending on the
dimensionality of the structure used for phonon calculation.
The use of PBC has been discussed in many papers like VFF_mod_herman ;
VFF_mod_zunger ; bcm_weber . In this work we consider the boundary conditions
associated with geometrically confined nanostructures. The vibrations of the
surface atoms can vary from completely free (free BC) to damped oscillations
(damped BC). It is shown next that all these cases can be handled within one
single boundary condition.
Boundary conditions for nanostructures: The surface atoms (Fig. 3, hollow
atoms) of the nanostructures can vibrate in a very different manner compared
to the inner atoms (Fig. 3, filled atoms) since the surface atoms have
different number of neighbors and ambient environment compared to the inner
atoms. The degree of freedom of the surface atoms can be represented by a
direction dependent damping matrix $\Xi$, defined in Appendix C. In such a
case the dynamical matrix component between atom ‘i’ and ‘j’ ($D(ij)$) is
modified to,
$\tilde{D}(ij)=\Xi^{i}D(ij)\Xi^{j}$ (12)
Figure 3: Projected unitcell of a $\langle$100$\rangle$ oriented rectangular
SiNW shown with surface (hollow) and inner (gray filled) atoms.
### 2.4 Diagonalization of the dynamical matrix
After setting up the dynamical matrix with appropriate BCs the following
eigen-value problem must be solved,
$DQ(\lambda,q)=M\omega^{2}(\lambda,q)Q(\lambda,q),$ (13)
where, M is the atomic mass matrix. $\lambda$ and $q$ are the phonon
polarization and momentum vector respectively. The term $Q(\lambda,q)$ is a
column vector containing all the phonon eigen displacement modes
$u(\lambda,q)$ associated with the polarization $\lambda$ and momentum $q$.
For simplified numerical calculation slightly modifying Eq. (13) leads to,
$\overline{D}Q(\lambda,q)=\omega^{2}(\lambda,q)Q(\lambda,q)$ (14)
The detail for obtaining $\overline{D}$ is outlined in Appendix D. To obtain
Eq. (14) another step is needed. The time dependent vibration of each atom
($\Delta R(t)$) are represented as the linear combination of phonon eigen
modes of vibration $u(\lambda,q)$ (a complete basis set) as,
$\Delta R_{i}(t)=\sum_{q,P}u_{P}(\lambda,q)e^{i(q\cdot R_{i}-\omega t)}$ (15)
where, $P$ is the size of the basis set and $\omega$ the vibration frequency
of the modes. Using the result of Eq. (15) in the LHS of Eq. (1) yields,
$m_{i}\frac{\partial^{2}}{\partial t^{2}}\Delta
R_{i}(t)=-\omega^{2}\sum_{q,P}u_{P}(\lambda,q)e^{i(q\cdot R_{i}-\omega t)}$
(16)
After some mathematical manipulations and using Eq. (13) we obtain the final
eigen value problem given in Eq. (14).
### 2.5 Sound Velocity ($V_{snd}$)
A wealth of information can be extracted from the phonon spectrum of solids.
One important parameter is the group velocity ($V_{grp}$) of the acoustic
branches of the phonon dispersion which gives the velocity of sound
($V_{snd}$) in the solid. Depending on the acoustic phonon branch used for the
calculation of $V_{grp}$, the sound velocity can be either, (a) longitudinal
($V_{snd,l}$) or (b) transverse ($V_{snd,t}$). In solids, $V_{snd}$ is
obtained near the BZ center (for q $\rightarrow 0$) where $\omega\sim q$ .
Thus, $V_{snd}$ is given by,
$V_{snd}=\frac{\partial\omega(\lambda,q)}{\partial q}\Big{|}_{q\rightarrow
0},$ (17)
where $\lambda$ is the either the transverse or longitudinal polarization of
the phonon frequency.
### 2.6 Thermal Conductance ($\sigma_{l}$)
Another important physical property of the semiconductors that can be
extracted from the phonon spectrum is the lattice thermal conductivity. For a
small temperature gradient ($\Delta T$) at the two ends of a semiconductor,
the $\sigma_{l}$ is obtained using Landauer approach Land as outlined in
Refs. Mingo_kappa ; mingo_ph ; jauho_method . For 1D nanowires $\sigma_{l}$ at
a temperature ‘T’ is given by mingo_ph ; Wallace ,
$\sigma_{l,1D}=\frac{1}{2\pi}\cdot\int^{\omega_{fin}}_{0}\Pi(\omega)\frac{\partial}{\partial
T}\Big{[}\frac{1}{e^{\hbar\omega/k_{B}T}-1}\Big{]}\hbar\omega d\omega,$ (18)
where $\Pi(w)$ is the transmission of a phonon branch at frequency $\omega$,
$\hbar$ and $k_{B}$, the reduced Planck’s constant and Boltzmann constants,
respectively. Equation (18) is of general validity and involves a low
temperature approximation. Scattering causes the conductance to vary with the
nanowire’s length (L). In the case of ballistic thermal transport the
transmission ($\Pi(\omega)$) is always 1 for all the eigen frequencies.
### 2.7 Mode Gr$\ddot{u}$neisen Parameter ($\gamma_{i}$)
One of the advantages of using the MVFF model is its ability to keep track of
the phonon frequency shift under crystal stress. Under the action of
hydrostatic strain the crystal is compressed without changing its symmetry.
With pressure (P) the phonon frequency shifts, which is measured by a unitless
parameter called the mode ‘Gr$\ddot{u}$neisen Parameter’ given as
$\displaystyle\gamma_{i,q}$ $\displaystyle=$
$\displaystyle-\frac{\partial(ln(\omega_{i,q})}{\partial ln(V)},$ (19)
$\displaystyle=$
$\displaystyle\frac{B}{\omega_{i,q}}\cdot\frac{\partial\omega_{i,q}}{\partial
P},$ (20)
where, $\omega_{i,q}$ is the eigen frequency for the ith branch at $q$
momentum point. The terms B, P and V are the volume compressibility factor,
pressure on the system, and volume of the crystal, respectively. Theoretically
this parameter is extracted by calculating the eigen frequencies at ambient
condition (P = 0) and at small hydrostatic pressure ($\epsilon=\pm 0.02$) and
then taking the difference in the calculated frequencies. The modification of
the force constants under hydrostatic pressure is outlined in Ref.
VFF_mod_herman . The value of this parameter at high symmetry points
($\Gamma$, X, etc.) in the BZ can be measured experimentally by Raman
scattering spectroscopy gparam_1 .
In the remaining Sections we provide the computational details, show results
on phonon dispersion in bulk and nanowires and give some results on $V_{snd}$,
$\gamma_{i}$ and ballistic $\sigma_{l}$ in semiconductor nanowires.
Figure 4: Number of atoms per unitcell ($N_{A}$) with width (W) of
$\langle$100$\rangle$ oriented square SiNW. Figure 5: Sparsity pattern of the
dynamical matrix used in (a) Keating VFF model and (b) MVFF model. SiNW has W
= H = 2nm with 113 atoms in the unitcell.
## 3 Computational Details
This Section provides the computational details involved in obtaining the
phonon dispersion in semiconductor structures. Details about bulk and nanowire
(NW) structures are provided.
### 3.1 Dynamical matrix details
A primitive bulk zinc-blende unitcell has 2 atoms. This fixes the size of the
DM for the bulk structure to 6 $\times$ 6 ($3N_{A}\times 3N_{A}$) (see
Sec.2.2). However, for the case of nanowires, $N_{A}$ varies with shape, size
and orientation of the wire jauho_method . In this paper, all the results are
for square shaped SiNW with $\langle$100$\rangle$ orientation. Figure 4 shows
the variation in $N_{A}$ with the width (W) of Silicon NW (SiNW). The number
of atoms increase quadratically with W. For a 6nm $\times$ 6nm SiNW, $N_{A}$
is 1013 which means the size of DM is 3,039 $\times$ 3,039. Extrapolating the
$N_{A}$ data gives around 7,128 atoms for a 16nm $\times$ 16nm SiNW resulting
in a DM of size 21,384 $\times$ 21,384 (details in Appendix E). So, the
dynamical matrix size increases rapidly with increase in width.
Figure 6: Non zero (NZ) elements in the dynamical matrix and fill factor in
DM. Fill factor reduces as the wire unitcell size increases even though the
non-zero elements increase.
The increase of the DM size with wire cross-section imposes constraint on the
structure size which can be solved using the atomistic MVFF method. However,
the entire matrix is quite sparse which can be utilized significantly to
expand the physical size of the system that can be simulated by using
compressed matrix storage methods. The qualitative idea about the filling can
be observed from the sparsity pattern for a 2nm $\times$ 2nm SiNW dynamical
matrix as shown in Fig. 5. The quantitative analysis of the fill fraction of
the DM and the number of non-zero elements (NZ) in the DM are shown in Fig. 6.
The non-zero elements in the DM increase quadratically with W of SiNW. An
estimate for 16nm $\times$ 16nm SiNW gives about 800,117 non-zero elements
(details in Appendix E). However, to get an idea about the absolute filling of
the DM we define a term called the ‘fill-factor’ given as,
$\displaystyle fillfactor$ $\displaystyle=$ $\displaystyle
Total\,\,nonzero\,\,elements/Size\,\,of\,\,DM$ $\displaystyle=$
$\displaystyle\frac{NZ}{(3\times N_{A})^{2}}\propto\frac{1}{N_{A}}$
$\displaystyle since\;NZ\propto
N_{A}\;(see\;Appendix\;Eq.\,(\ref{NZ_with_NA}))$
Thus, fill factor varies inversely with number of atoms in the unitcell. The
relation of NZ with NA is provided in Appendix E.
The percentage fill factor of the DM reduces with increasing W of SiNW (Fig.
6). This value is $\sim$ 0.1% for a SiNW with W $\sim$ 25nm (Appendix E) . So
even though the non-zero elements increase with W, DM becomes sparser which
allows to store the DM in special compressed format like compressed row/column
scheme (CRS/CCS) CRS_CCS enabling better memory utilization.
Figure 7: (a) Time to assemble the DM ($time_{asm}$) with width (W) of $\langle$100$\rangle$ oriented SiNW. (b) Time to obtain the eigen solutions per k with W for a $\langle$100$\rangle$ oriented SiNW DM for 100% and 20% of the Eigen spectrum. The timing analysis is done on T6400 Intel processor with 2GHz speed. Entire Eigen spectrum along with eigen vectors are obtained using the ‘eig()’ function in MATLAB eig_matlab . The partial Eigen spectrum is calculated using the ‘eigs()’ function in MATLAB eigs_matlab . Table 3: Resource and timing estimate for larger $\langle$100$\rangle$ SiNW | | | | | $Time^{*}$
---|---|---|---|---|---
W | $N_{A}$ | NZ | % fill | $\overline{time_{asm}}^{*}$ | per k
(nm) | | | factor | (sec) | (hours)
16 | 7128 | 800117 | 0.423 | 238.48 | 33.2
20 | 11120 | 1252490 | 0.224 | 370.73 | 129.5
25 | 17346 | 1.96$\times 10^{6}$ | 0.101 | 576.91 | 505.2
∗Time estimates on an Intel T6400, 2GHz processor.
### 3.2 Timing analysis for the computation of DM
The numerical assembly of the DM takes a considerable time due to the many
interactions calculated in the MVFF model. The assembly time ($time_{asm}$)
increases as $N_{A}$ increases. To give an idea about the timing, the
dynamical matrix for SiNW with different W are constructed on a single CPU
(Intel T6400, 2GHz processor). The assembly time is calculated for each width
5 times to obtain a mean value for the $time_{asm}$. The error bar at each W
is the standard deviation from the mean $time_{asm}$ (Fig. 7a). In the present
case the assembly of the DM is done atom by atom which is useful for distorted
materials as well as alloys. The assembly time for the DM in single materials
can be reduced dramatically by the assumption of homogeneous bond lengths and
a matrix stamping technique nemo3d .
After the DM is assembled, it is solved to obtain the eigen modes of
oscillations. The time needed to diagonalize ($t_{diag}$) the DM, for each ‘k’
point, using the MATLAB ‘eig’ function eig_matlab is also calculated (on the
same processor). The $t_{diag}$ value varies as the sixth power of W as shown
in Fig. 7b. However, if only 20% percent of the Eigen values are calculated
the time requirement now goes by the fifth power (shown by the lower line in
Fig. 7b). The Eigen values in this case are calculated using the ‘eigs’
function in matlab eigs_method. The calculation of only 20% of the Eigen
spectrum reduces the per-k energy calculation time by $\sim$75% for a square
SiNW with W = H = 6nm. However, the possibility of using only the partial
spectrum for the evaluation of the important lattice parameters (like thermal
conductance, etc.) is not in the scope of present discussion. For the
calculation of physical quantities, in this paper, the complete Eigen spectrum
has been used.
Extrapolating the data for the computational and timing requirement obtained
for the smaller SiNWs, can provide some estimates about the size and time
requirement for larger SiNWs (Table 3). Analytical fits for the variation of
the size and time parameters with W are provided in Appendix E. The timing
requirements also help us in the resource requirement estimate for a future
Bandstructure Lab extension for phonons on nanoHUB.org bslab .
Table 4: Force constants (N/m) used for phonon dispersion calculation Material | Model | $\alpha$ | $\beta$ | $\delta$ | $\gamma$ | $\nu$
---|---|---|---|---|---|---
Si | MVFF VFF_mod_herman | 45.1 | 4.89 | 1.36 | 0 | 9.14
Si | KVFF Keating_VFF | 48.5 | 13.8 | 0 | 0 | 0
Ge | MVFF VFF_mod_herman | 37.8 | 4.24 | 0.49 | 0 | 7.62
Figure 8: Benchmark of simulated bulk phonon dispersion with experimental
phonon data for (a) Si and (b) Ge. Experimental data is obtained using neutron
scattering at 80K bulk_si_exp_phon .
## 4 Results
In this Section we show results for the phonon spectrum in bulk and confined
semiconductor structures using both the MVFF and KVFF models. Also some of the
physical properties extracted from the phonon dispersions are reported.
### 4.1 Experimental Benchmarking
The first step to check the correctness of the MVFF model is to compare the
simulated results with experimental data. Figure 8 shows the simulated and
experimental bulk_si_exp_phon bulk phonon dispersion for (a) Silicon and (b)
Germanium. The value of the strength parameters are provided in Table 4. A
very good agreement between the experimental and simulated data is obtained.
To further support the correctness of the MVFF model, $V_{snd}$ is calculated
in bulk Si and Ge along $\langle$100$\rangle$ direction (Table 5). The
extracted sound velocity compares very well with the experimental sound
velocity data ioffe_online (max error $\leq$ 10%).
Table 5: Sound Velocity in km/sec in Si, Ge bulk and square nanowires with W = H = 6nm. Material | Structure | $V_{snd}$ calc. | $V_{snd}$ Expt. ioffe_online
---|---|---|---
Si | Bulk $V_{l}$ [100] | 9.09 | 8.43 ($\sim$8%)
Bulk $V_{t}$ [100] | 5.71 | 5.84 ($\sim$2%)
NW $V_{l}$ | 6.51 | –
NW $V_{t}$ | 4.46 | –
Ge | Bulk $V_{l}$ [100] | 5.13 | 4.87 ($\sim$5%)
Bulk $V_{t}$ [100] | 3.36 | 3.57 ($\sim$6%)
NW $V_{l}$ | 3.70 | –
NW $V_{t}$ | 2.61 | –
The comparison of second order elastic constants for Si and Ge evaluated using
the MVFF model (using the formulation provided in Ref. VFF_mod_herman ) with
experimental data Madelung is provided in Table 6. The MVFF derived values
compare quite well with the experimental data Madelung .
Another comparison carried out to test the robustness of MVFF model is the
comparison of the mode Gr$\ddot{u}$neisen parameters at the high symmetry
$\Gamma$ and X point with the experimental data (see Table 7). The MVFF
results are in good comparison with the experimental data.
An advantage of using a higher order phonon model is that both phonon
dispersions as well the physical parameters can be matched to a good accuracy.
Hence, MVFF model captures the experimental phonon dispersion as well as the
elastic properties in bulk zinc-blende material very well.
Table 6: Elastic constants ($10^{10}Nm^{-2}$) obtained from the MVFF model compared with experimental data Madelung for Si and Ge. The corresponding errors in the theoretical values are also shown. Material | Model | C11 | C12 | C44
---|---|---|---|---
Si | MVFF | 16.80 | 6.47 | 7.63
Si | Expt. | 16.57 | 6.39 | 7.96
Error | | $\sim$1.4% | $\sim$1.2% | $\sim$4.14%
Ge | MVFF | 13.22 | 4.84 | 6.29
Ge | Expt. | 12.40 | 4.13 | 6.83
Error | | $\sim$6.6% | $\sim$17.2% | $\sim$8%
### 4.2 Comparison of VFF models
In this Section we compare the original Keating VFF model Keating_VFF with
the MVFF model to show the need for the more elaborate MVFF model. Both
computational requirement and the physical result comparisons are provided in
this Section. From computational point of view the DM for both the models are
quite different (Fig. 5). The difference in the sparsity pattern arises
because of the coplanar interaction present in the MVFF model which takes into
account interactions beyond the nearest neighbors. The KVFF model has fewer
non-zero elements compared to the MVFF model. The increase in number of NZ
elements increases more rapidly in the MVFF model vs. the KVFF model (Fig. 9).
The MVFF model required twice as many matrix elements compared to the KVFF
model for a 5nm $\times$ 5nm SiNW. Thus, MVFF model demands more storage
space.
Table 7: Comparison of the mode Gr$\ddot{u}$neisen Parameters for bulk Si using the two phonon models. $\gamma_{i}$ | MVFF | KVFF | Expt./Abinitio | Ref.
---|---|---|---|---
$\gamma^{\Gamma}_{LO,TO}$ | 1.05 | 0.81 | $0.98\pm 0.06$ | gparam_1
$\gamma^{\Gamma}_{TA}$ | -0.68 | -0.43 | -0.62 | VFF_mod_herman
$\gamma^{\Gamma}_{LA}$ | 0.95 | 0.7 | 0.85 | VFF_mod_herman
$\gamma^{X}_{LO,LA}$ | 1.08 | 0.816 | 1.03 | gparam_2
$\gamma^{X}_{TO}$ | 1.25 | 0.83 | $1.5\pm 0.2$ | gparam_1
$\gamma^{X}_{TA}$ | -1.58 | -0.33 | $-1.4\pm 0.3$ | gparam_1
A comparison of the bulk Si phonon dispersion from the two models is shown in
Fig. 10. The parameters for both the models are provided in Table 4.
Qualitatively MVFF shows a better match to the experimental data compared to
the KVFF model. There are few important points to be noted in the bulk phonon
dispersion. The KVFF model reproduces the acoustic branches very well near the
Brillouin zone (BZ) center but overestimates the values near the zone edge (at
X and L point in the BZ, Fig. 10). The MVFF model overcomes this shortcoming
and reproduces the acoustic branches very well in the entire BZ. Comparison of
sound velocity along the $\langle$100$\rangle$ direction for bulk Si obtained
from both the models show a very good match to the experimental data (Table
8).
Figure 9: Matrix size and number of non zero elements required by the two
models. MVFF has more elements needed for accurate phonon dispersion. Figure
10: Comparison of simulated phonon results with experimental data (at 80K
from bulk_si_exp_phon ) from the two phonon models (a) Keating VFF and (b)
Modified VFF. The KVFF model fails to reproduce many important features in the
experimental data as shown by the arrows in (a). The shortcomings are (i) over
estimation of the acoustic mode at X by $\sim$60%, (ii) acoustic branch over
estimated at L by $\sim$95% and failure to reproduce the correct value for the
optical branches altogether pointed in (iii) and (iv).
The KVFF model overestimates the optical phonon branch frequencies whereas the
MVFF model produces a very good match to the experimental data (Fig. 10) and
Table 8. The comparison of the optical frequency at the $\Gamma$ point reveals
that the KVFF model overshoots the experimental value by $\sim$ 7% whereas the
MVFF model is higher by only $\sim$ 0.6%.
The comparison of the mode Gr$\ddot{u}$neisen parameters for bulk Si using the
two models is shown in Table 7. KVFF gives wrong values of these parameters
compared to the experimental values. However, MVFF model is able to reproduce
the experimental value very well. This shows the importance of using a quasi-
harmonic model to correctly obtain the phonon frequency shifts VFF_mod_herman
; gparam_3 . A similar failure of the KVFF model for III-V zinc-blende
materials have been reported in Ref. olga_gparam .
The correct representation of bulk phonons is very important since this will
affect the phonon spectrum of the confined structures. At the same time the
physical properties like lattice thermal conductivity, phonon density of
states (DOS), etc. are also affected. Since the MVFF model matches the
experimental bulk phonon data more accurately, though at the expense of
additional calculations and storage, compared to the original KVFF model, we
believe that the MVFF model will give better results for phonon dispersion in
nanostructures.
### 4.3 Phonons in nanowires
After benchmarking the bulk phonon dispersion, the same parameters are used to
obtain the phonon spectrum in $\langle$100$\rangle$ square SiNW (Fig. 11). The
result for a 2nm $\times$ 2nm free-standing SiNW is shown in Fig. 11(a). Some
of the key features to notice in the phonon dispersion are, (i) presence of
two acoustic branches ($\omega(q)\sim$ q, 1,2 in 11(a)), (ii) two degenerate
modes (3,4 in 11(a)) with $\omega(q)\sim\,q^{2}$, which are called the
‘flexural modes’, typically observed in free-standing nanowires
SINW_110_phonon ; jauho_method ; SINW_111 and (iii) heavy mixing of the
higher energy sub-bands leaving no ‘proper’ optical mode. The features are
quite different from the bulk phonon spectrum. This will strongly affect other
physical properties of nanowires extracted using the phonon dispersion.
Table 8: Comparison of bulk parameters in Si for two models Model | $V^{bulk}_{l,100}$ | $V^{bulk}_{t,100}$ | $\omega_{opt}(\Gamma)$
---|---|---|---
(km/sec) | (km/sec) | (THz)
MVFF | 9.09 | 5.71 | 15.49
KVFF | 8.35 | 5.75 | 16.46
Expt. | 8.43 ioffe_online | 5.84 ioffe_online | 15.39 bulk_si_exp_phon
In Fig. 11b, we explore the effect of a substrate on which the nanowire may be
mounted. Only the bottom surface of the SiNW is clamped whereas the other
three sides have free boundary condition. Using the boundary condition method
discussed in sec.2.3, the phonon spectrum in a 2nm $\times$ 2nm
$\langle$100$\rangle$ SiNW are calculated with different damping values
($\Xi=1$ (free standing) and 0.1). The effect of damping is very prominent at
the BZ edge compared to the zone center. Zone edge frequencies decrease in
energy as the damping increases. A reduction of $\sim 2.11\times$ are observed
for the zone edge frequency for 1st branch at $\Xi=0.1$ (Fig. 11(b)) which
shows that the NW vibrational energy is decreasing more at higher momentum ‘q’
values. The first four branches are very strongly affected, however, the
higher phonon branches are less affected.
Figure 11: (a) Phonon dispersion in $\langle$100$\rangle$ oriented SiNW with W
= H = 2nm. For clarity only the lowest 40 sub-bands are shown. (b) Dependence
of phonon dispersion on the damping of vibration of the bottom surface atoms
for $\Xi$ = 1 and 0.1. Reduction of phonon energy at the Brillouin zone
boundary is stronger compared to the zone center.
### 4.4 Ballistic lattice thermal conductance ($\sigma^{bal}_{l}$) in SiNWs
The ballistic $\sigma_{l}$ for square SiNWs is calculated using their phonon
dispersions. The conductance is calculated using Eq. (18) assuming semi-
infinite extensions along the wire growth axis (X-axis) and CBC on the
periphery (Y and Z axis) of the wire (Fig.3) . Clamping the bottom surface
affects the $\sigma^{bal}_{l}$ stronger at higher temperature compared to the
lower temperature (Fig. 12a). Figure 12b shows the $\sigma^{bal}_{l}$ at 300K.
The reduction in $\sigma^{bal}_{l}$ from free-standing wire to a clamped wire
($\Xi$ = 0.1) is $\sim$13.1%. Hence, fixing the surface atoms have a strong
impact of the lattice thermal conductance in SiNW.
Figure 12: Ballistic lattice thermal conductivity ($\sigma^{bal}_{l}$ for a
2nm $\times$ 2nm $\langle$100$\rangle$ SiNW with different bottom surface
damping. $\sigma^{bal}_{l}$ drops as damping increases. Inset shows
$\sigma^{bal}_{l}$ at 100K, 200K and 300K. As the bottom surface changes from
free-standing to clamped ($\Xi$ = 0.1), $\sigma^{bal}_{l}$ reduces by
$\sim$12%, $\sim$12.6% and $\sim$13.1% at 100K, 200K and 300K, respectively .
## 5 Conclusions
The details for calculating the phonon dispersion in zinc-blende semiconductor
structures using the modified Valence Force Field (MVFF) method have been
outlined. The MVFF method has been applied to calculate the phonon spectra in
confined nanowire structures with varying boundary conditions. The methodology
and the computational requirements for the method has been provided.
Comparison of original Keating VFF with MVFF shows that, MVFF provides
accurate phonon dispersion but at the expense of higher computational demands.
Different VFF models can be used for obtaining the solution for physical
quantities depending on the type of application, size of the structure and the
computational resources available. We believe that the MVFF model will provide
better phonon dispersion in ultra-scaled nanostructures. The MVFF method will
be crucial in understanding and modeling the thermal properties of ultra-
scaled semiconductor devices.
## Acknowledgments
The authors would like to acknowledge the computational resources from
nanoHUB.org, an National Science Foundation (NSF) funded, NCN project.
Financial support from MSD Focus Center, one of six research centers funded
under the Focus Center Research Program (FCRP), a Semiconductor Research
Corporation (SRC) entity and by the Nanoelectronics Research Initiative (NRI)
through the Midwest Institute for Nanoelectronics Discovery (MIND) are also
acknowledged.
## APPENDICES
## Appendix A Details of bulk zinc-blende coplanar interaction
The coplanar interactions are important to obtain the flat nature of the
acoustic phonon branches in Si and Ge VFF_mod_herman . There are 21 such
interactions in a zinc-blende crystal. The normalized locations of all the
atoms involved in the coplanar interactions are shown in Table 9. The
corresponding groups used for bulk phonon dispersion calculations are given in
Table 10.
Table 9: Normalized atomic coordinates ($[\overline{x},\overline{y},\overline{z}]=[x,y,z]/a_{0}$) used for coplanar interaction calculation. No. | $\overline{x}$ | $\overline{y}$ | $\overline{z}$ | No. | $\overline{x}$ | $\overline{y}$ | $\overline{z}$
---|---|---|---|---|---|---|---
1* | 0 | 0 | 0 | 14 | 0 | 0.50 | -0.50
2* | 0.25 | 0.25 | 0.25 | 15 | -0.50 | -0.50 | 0
3 | 0.25 | -0.25 | -0.25 | 16 | -0.50 | 0 | 0.50
4 | -0.25 | 0.25 | -0.25 | 17 | 0 | -0.50 | 0.50
5 | -0.25 | -0.25 | 0.25 | 18 | 0.25 | 0.75 | 0.75
6 | 0 | 0.50 | 0.50 | 19 | -0.25 | 0.75 | 0.25
7 | 0.50 | 0 | 0.50 | 20 | -0.25 | 0.25 | 0.75
8 | 0.50 | 0.50 | 0 | 21 | 0.75 | 0.25 | 0.75
9 | 0 | -0.50 | -0.50 | 22 | 0.75 | -0.25 | 0.25
10 | 0.50 | -0.50 | 0 | 23 | 0.25 | -0.25 | 0.75
11 | 0.50 | 0 | -0.50 | 24 | 0.75 | 0.75 | 0.25
12 | -0.50 | 0 | -0.50 | 25 | 0.75 | 0.25 | -0.25
13 | -0.50 | 0.50 | 0 | 26 | 0.25 | 0.75 | -0.25
*Belong to the main bulk unitcell used for DM calculation.
Table 10: Atoms forming the coplanar interaction groups. 4 atoms in each group. No. | Members | No. | Members
---|---|---|---
1 | 2 | 1 | 3 | 9 | 12 | 1 | 2 | 6 | 18
2 | 2 | 1 | 4 | 12 | 13 | 1 | 2 | 7 | 21
3 | 2 | 1 | 5 | 15 | 14 | 1 | 2 | 8 | 24
4 | 3 | 1 | 2 | 6 | 15 | 6 | 2 | 7 | 22
5 | 3 | 1 | 4 | 13 | 16 | 6 | 2 | 8 | 25
6 | 3 | 1 | 5 | 16 | 17 | 7 | 2 | 6 | 19
7 | 4 | 1 | 2 | 7 | 18 | 7 | 2 | 8 | 26
8 | 4 | 1 | 3 | 10 | 19 | 8 | 2 | 6 | 20
9 | 4 | 1 | 5 | 17 | 20 | 8 | 2 | 7 | 23
10 | 5 | 1 | 2 | 8 | 21 | 5 | 1 | 4 | 14
11 | 5 | 1 | 3 | 11 | | | | |
*Atom numbers are same as shown in Table 9.
## Appendix B Derivation of dynamical matrix from the equation of motion
A crystal in equilibrium has zero total force. However, in the presence of
perturbations like lattice vibrations, etc. a small restoring force works on
the system. The total force ($F_{total}$) under small perturbation is given by
the Taylor series expansion as,
$\displaystyle F_{total}$ $\displaystyle=$ $\displaystyle-\sum_{i\in
N}\frac{\partial U}{\partial\Delta R_{i}}\,(=\,0\,at\,eqb.)$ (B.1)
$\displaystyle-\frac{1}{2}\sum_{i,j\in N}\frac{\partial^{2}U}{\partial\Delta
R_{i}\partial\Delta R_{j}}\cdot\Delta R_{j}+\ldots$
where, N represents all the atoms present in the system and U is the potential
energy of the system. In Eq. (B.1) first term in RHS is zero under
equilibrium. The next non-zero term is the second term in Eq. (B.1). Under
harmonic approximation, only the second term is considered and the higher
order (anharmonic) terms are neglected. Now combining Eq. (1) and Eq. (B.1)
one can obtain the following,
$\displaystyle F_{total}$ $\displaystyle=$ $\displaystyle\sum_{i\in
N}m_{i}\frac{\partial^{2}}{\partial t^{2}}\Delta R_{i}$ (B.2) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i,j\in N}\frac{\partial^{2}U}{\partial\Delta
R_{i}\partial\Delta R_{j}}\cdot\Delta R_{j}$ $\displaystyle=$ $\displaystyle
DR$ (B.3)
where, D is called the ‘Dynamical matrix’ and R is a column vector of
displacement for each atom given as,
$D=\begin{bmatrix}D(11)&D(12)&\cdots&D(1N)\\\ D(21)&D(22)&\cdots&D(2N)\\\
\vdots&\vdots&\ddots&\vdots\\\ D(N1)&D(N2)&\cdots&D(NN)\end{bmatrix}$ (B.4)
$R^{T}=\begin{bmatrix}\Delta R_{1}&\Delta R_{2}&\cdots&\Delta
R_{N}\end{bmatrix}$ (B.5)
Definition of $D(ij)$ is given in Eq. (2.2).
## Appendix C Treatment of surface atoms
The damped displacement of the surface atom ‘j’ can be represented the matrix
$\Xi^{j}$ given as,
$\Xi^{j}=\begin{bmatrix}\epsilon^{j}_{x}&0&0\\\ 0&\epsilon^{j}_{y}&0\\\
0&0&\epsilon^{j}_{z}\end{bmatrix}$ (C.1)
Taking into account the individual components the displacement vector for the
atom ‘j’ we obtain,
$\tilde{r}^{j}_{n}=\epsilon^{j}_{n}r^{j}_{n}\quad n\in[x,y,z]$ (C.2)
This modifies eqn.2.2 as,
$\tilde{D}^{ij}_{mn}=\epsilon^{j}_{n}D^{ij}_{mn}$ (C.3)
Combining Eq. (C.1) and (C.3) the dynamical matrix component between atom ‘i’
and ‘j’ can be represented as,
$\tilde{D}(ij)=\begin{bmatrix}\epsilon^{i}_{x}D^{ij}_{xx}\epsilon^{j}_{x}&\epsilon^{i}_{x}D^{ij}_{xy}\epsilon^{j}_{y}&\epsilon^{i}_{x}D^{ij}_{xz}\epsilon^{j}_{z}\\\
\epsilon^{i}_{y}D^{ij}_{yx}\epsilon^{j}_{x}&\epsilon^{i}_{y}D^{ij}_{yy}\epsilon^{j}_{y}&\epsilon^{i}_{y}D^{ij}_{yz}\epsilon^{j}_{z}\\\
\epsilon^{i}_{z}D^{ij}_{zx}\epsilon^{j}_{x}&\epsilon^{i}_{z}D^{ij}_{zy}\epsilon^{j}_{y}&\epsilon^{i}_{z}D^{ij}_{zz}\epsilon^{j}_{z}\end{bmatrix}$
(C.4)
which can be written in a compressed form as,
$\tilde{D}(ij)=\Xi^{i}D(ij)\Xi^{j}$ (C.5)
The value of $\epsilon_{x,y,z}\in[0,1]$, where completely free surface atoms
have value 1 and completely tied atoms have value 0.
## Appendix D Inclusion of mass in the Dynamical matrix
In Eq. (13) the mass of the atoms in on the RHS. It is convenient to include
the mass in DM itself. This modifies the the LHS of the equation. The modified
DM component between atom ‘i’ and ‘j’ thus, becomes,
$\tilde{D}(ij)=\begin{bmatrix}\frac{1}{\sqrt{m_{i}}}D^{ij}_{xx}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{xy}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{xz}\frac{1}{\sqrt{m_{j}}}\\\
\frac{1}{\sqrt{m_{i}}}D^{ij}_{yx}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{yy}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{yz}\frac{1}{\sqrt{m_{j}}}\\\
\frac{1}{\sqrt{m_{i}}}D^{ij}_{zx}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{zy}\frac{1}{\sqrt{m_{j}}}&\frac{1}{\sqrt{m_{i}}}D^{ij}_{zz}\frac{1}{\sqrt{m_{j}}}\end{bmatrix}$
(D.1)
here $m_{i}$ and $m_{j}$ are the masses of atom ‘i’ and ‘j’ respectively.
Eq.D.1 can be written in a compressed manner as,
$\bar{D}(ij)=M^{-1}_{i}D(ij)M^{-1}_{j},$ (D.2)
where $M_{i}$ is given as,
$M_{i}=\begin{bmatrix}\sqrt{m_{i}}&0&0\\\ 0&\sqrt{m_{i}}&0\\\
0&0&\sqrt{m_{i}}\end{bmatrix}$ (D.3)
## Appendix E Fitted analytical expressions for DM properties
* •
Atoms in a $\langle$100$\rangle$ SiNW unitcell: The $N_{A}$ data obtained for
the square wires till 6nm $\times$ 6nm can be fitted to a quadratic polynomial
given as,
$N_{A}(W)=27.57W^{2}+4.59W$ (E.1)
Using Eq. (E.1) for a 16nm $\times$ 16nm SiNW gives around 7128 atoms.
* •
Non-zero elements in a $\langle$100$\rangle$ SiNW DM: The data for non-zero
elements in the DM for SiNW with W till 6nm can be fitted to a quadratic
polynomial given by,
$NZ(W)=3156W^{2}-495.5W$ (E.2)
Using Eq. (E.2) for a 16nm $\times$ 16nm SiNW yields around 800117 non-zero
elements in the DM.
* •
Relation of NZ elements to $N_{A}$ in a $\langle$100$\rangle$ SiNW: The number
of non-zero (NZ) elements vary linearly with the number of atoms (Fig. 13).
Fitting the NZ elements with $N_{A}$ for each wire under study following
relations are obtained,
$\displaystyle NZ_{KVFF}$ $\displaystyle\approx$ $\displaystyle 109.3\times
N_{A}$ (E.3) $\displaystyle NZ_{MVFF}$ $\displaystyle\approx$ $\displaystyle
205.6\times N_{A}$ (E.4)
This shows that the number of NZ elements in MVFF method is roughly twice the
NZ elements in KVFF model.
Figure 13: Variation in the number of non-zero (NZ) elements with $N_{A}$ for
the two phonon models (1) Keating VFF and (2) Modified VFF. For both the cases
NZ varies linearly with $N_{A}$. MVFF has roughly twice the number of NZ
elements compared to KVFF model.
* •
Percentage fill factor for a $\langle$100$\rangle$ SiNW DM: The percentage
fill-factor can be derived using Eq. (E.1) and (E.2) which leads to the
following expression,
$\%fill-factor(W)\approx\frac{0.81}{W^{2}}\propto NA^{-1}$ (E.5)
Eqn.E.5 estimates a 16nm $\times$ 16nm SiNW DM is filled only 0.32%. This
shows that the DM matrix is very sparsely filled for larger wires.
* •
Mean assembly time for $\langle$100$\rangle$ SiNW DM: The data for mean
$time_{asm}$ of the DM for SiNW with W till 6nm can be fitted to a quadratic
polynomial given by,
$time_{asm}(W)=0.9079W^{2}-0.3789W\,secs$ (E.6)
Using Eq. (E.6) a 16nm $\times$ 16nm SiNW DM is estimated to be assembled on
single CPU in 226.35 secs.
* •
Eigen solution time per ‘k’ point for $\langle$100$\rangle$ SiNW DM: The eigen
values are obtained using the ‘eig’ solver in MATLAB eig_matlab . The time
needed for the solution of all the eigen values (with the eigen vectors) for
each momentum vector ‘k’ can be fitted to the following expression,
$time_{eigen}(W)=5.4\times 10^{-3}W^{6.1}\;secs$ (E.7)
Thus, the solution time goes with the sixth power of W. Also this expression
gives an estimate time for the solution of one k point using the MATLAB eig
solver (on a PC) as 1.19e5 seconds ( 1.4 days). Thus, for very large systems
parallel eigen solvers (like SCALAPACK slug ) as well as finding few eigen
solutions (using eigs or other sparse eigen solvers, like LAPACK, etc.) is a
feasible method.
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|
arxiv-papers
| 2010-09-30T16:33:26 |
2024-09-04T02:49:13.254219
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhijeet Paul and Mathieu Luisier and Gerhard Klimeck",
"submitter": "Abhijeet Paul",
"url": "https://arxiv.org/abs/1009.6188"
}
|
1009.6217
|
# A Patchy Cloud Model for the L to T Dwarf Transition
Mark S. Marley NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035,
U.S.A.; Mark.S.Marley@NASA.gov Didier Saumon Los Alamos National Laboratory,
Mail Stop F663, Los Alamos NM 87545; dsaumon@lanl.gov Colin Goldblatt
Astronomy Department & NASA Astrobiology Institute Virtual Planetary
Laboratory, University of Washington, Box 351580, Seattle WA 98195, USA;
cgoldbla@uw.edu
###### Abstract
One mechanism suggested for the L to T dwarf spectral type transition is the
appearance of relatively cloud-free regions across the disk of brown dwarfs as
they cool. The existence of partly cloudy regions has been supported by
evidence for variability in dwarfs in the late L to early T spectral range,
but no self-consistent atmosphere models of such partly cloudy objects have
yet been constructed. Here we present a new approach for consistently modeling
partly cloudy brown dwarfs and giant planets. We find that even a small
fraction of cloud holes dramatically alter the atmospheric thermal profile,
spectra, and photometric colors of a given object. With decreasing cloudiness
objects briskly become bluer in $J-K$ and brighten in $J$ band, as is observed
at the L/T transition. Model spectra of partly cloudy objects are similar to
our models with globally homogenous, but thinner, clouds. Hence spectra alone
may not be sufficient to distinguish partial cloudiness although variability
and polarization measurements are potential observational signatures. Finally
we note that partial cloud cover may be an alternative explanation for the
blue L dwarfs.
brown dwarfs — stars: atmospheres
## 1 INTRODUCTION
As brown dwarfs cool over time, their atmospheres undergo a sequence of
chemical and physical changes that result in an evolving emergent spectrum
and–consequently–varying spectral types. The most remarkable changes take
place at the L to T type transition. The latest L dwarfs have red near-
infrared colors, strong CO absorption in $K$ band and relatively shallow water
absorption bands modulating their spectra. Over a small effective temperature
range of only 100 to 200 K the spectrum rapidly changes to exhibit blue near-
infrared color, weakening CO absorption, strengthening $\rm CH_{4}$
absorption, and deeper water bands (see Kirkpatrick (2005) for a review).
The proximal cause of these changes is the loss of cloud opacity. As clouds
dissipate the visible atmosphere cools, bringing on the chemical change from
CH4 to CO. Without clouds providing a significant, nearly gray opacity, flux
can emerge through molecular opacity windows in the J and H bands with a
brightening leading to a blueward color shift (Dahn et al. 2010, Tinney et al.
2003, Vrba et al. 2004). Studies of $\rm L+T$ binary dwarfs (e.g., Liu et al.
2006; Looper et al. 2008; Stumpf et al. 2010 and references therein) show that
this brightening is an intrinsic signature of the transition and not the
manifestation of some other effect in color-magnitude diagrams (Stumpf et al.,
2010).
Two main underlying causes of this loss in cloud opacity have been suggested.
In one view the atmospheric dynamical state changes, resulting in larger
particle sizes that more rapidly ‘rain out’ of the atmosphere, leading to a
sudden clearing or collapse of the cloud (Knapp et al., 2004; Tsuji &
Nakajima, 2003; Tsuji et al., 2004). This view is supported by fits of spectra
to model spectra (Saumon & Marley, 2008) computed with the Ackerman & Marley
(2001) cloud model. In that formalism a tunable parameter, $f_{\rm sed}$
controls cloud particle sizes and optical depth. Larger $f_{\rm sed}$ yields
larger particles along with physically and optically thinner clouds. Cushing
et al. (2008) and Stephens et al. (2009) have demonstrated that progressively
later T dwarfs can be fit by increasing $f_{\rm sed}$ across the transition.
Late T dwarfs are generally best fit by models that neglect cloud opacity.
Burrows et al. (2006) also suggest changes in cloud particle sizes as a
possible mechanism.
The second view is inspired by thermal infrared images of the atmospheres of
Jupiter and Saturn at $\sim 5\,\mu$m (e.g, Westphal 1969; Westphal et al.
1974; Orton et al. 1996; Baines et al. 2005). In this spectral region gaseous
opacity is very low, allowing flux from deeply seated, warm atmospheric
regions to escape. Higher lying clouds, however, locally reduce the emergent
flux. As a result these planets take on a mottled appearance with bright high-
flux, low cloud opacity regions lying adjacent to cooler, darker, and cloudier
regions (optical depth varies from $\sim 0$ to $\sim 20$ between these regions
(Banfield et al., 1998)). Ackerman & Marley (2001) suggested that the arrival
of such cloud holes near the end of the L spectral sequence may be responsible
for the L to T transition. Burgasser et al. (2002) tested this hypothesis with
a simple ‘toy model’ by summing weighted contributions of the spectra of
cloudy and cloudless models. They showed that the observed $J$ band
brightening across the transition could arise from decreasing cloud coverage.
Further support for this hypothesis comes from observations of L and T dwarf
variability. While previous studies were somewhat equivocal (summarized in
Artigau et al. (2009)), two early T dwarfs have recently been shown to have
large near-infrared photometric variability (Artigau et al., 2009; Radigan et
al., 2010) consistent with surface variations in cloud coverage modulated by
rotation.
The approach to modeling holes of Burgasser et al. was highly simplistic. The
principal shortcoming being that it is not physically correct to combine the
contributions of separate cloudy and cloudless models. Deep in the atmosphere
of a brown dwarf the entropy in the convection zone must match that of the
interior. Thus the temperature at a given, deep, pressure level is expected to
be horizontally constant. However for a fixed $T_{\rm eff}$ a cloudy
atmosphere is everywhere hotter than a cloudless atmosphere. As an example
Figure 1 presents model atmosphere profiles for a uniform cloudy and a
cloudless atmosphere following the techniques of Marley et al. (2002) and
Saumon & Marley (2008). At depth, the cloudy profile is warmer than the
cloudless profile by over 400 K; the difference in some models is even
greater. Thus standard cloudy and cloudless models cannot simultaneously be
valid descriptions of the real atmosphere at two locations even though both
models are descriptions of an atmosphere with the same $T_{\rm eff}$. Clearly
a new technique for self-consistently treating partly cloudy atmospheres is
required. While we do not yet understand why cloud holes might appear, we here
present a new approach inspired from models of Earth’s atmosphere to model
their influence and apply our results to model the spectra and colors of L and
T dwarfs.
## 2 Modeling Partly Cloudy Skies
Instead of combining separately computed profiles for purely cloudy and cloud-
free dwarfs we wish to construct a single, global temperature-pressure profile
$T(P)$ that incorporates simultaneously the influences of both cloudy and
cloud-free regions on the energy balance of the atmosphere. The final profile
should conserve the total flux while allowing for nearby atmospheric regions
to have differing cloud – but not thermal – profiles. This conceptually allows
clouds to be displaced by winds, updrafts, or downdrafts and change location,
as long as the global mean cloud fraction is constant.
In three-dimensional terrestrial numerical weather prediction and climate
models, clouds are typically smaller than the adopted computational grid
scale. Various methods are used to treat this in the radiative transfer
calculations, one of which is to consider separate cloudy and cloud-free sub-
columns in the same model column, with a single $T(P)$ structure. This same
approach can be used in a single column model of the Earth, allowing the
global annual mean energy budget to be reproduced (Goldblatt & Zahnle 2010).
Physically, such considerations are most important on Earth in the tropics.
Taking the zonal mean $T(P)$ and moisture profile would give a local runaway
greenhouse due to the high water vapor content. However, dry areas in the
tropics caused by subsiding air act as “radiator fins”, allowing radiation
from the surface to escape to space (Peirrehumbert 1995). In brown dwarfs
atmospheres, cloud free areas would be analogous to the subsidence regions in
Earth’s tropics.
To implement this approach we set the parameter $0\leq h\leq 1$ to be the
fraction of the atmosphere described by the cloud free sub-column, from which
the local flux at some level in the atmosphere is $\cal F_{\rm hole}$. The
remaining fraction $(1-h)$ is the cloudy sub-column, with local flux $\cal
F_{\rm cloud}$. Both sub-columns share the same $T(P(z))$ profile but have
different emergent flux, in general ${\cal F_{\rm hole}}>{\cal F}_{\rm
cloud}$. We compute the total flux, ${\cal F_{\rm tot}}(z)$ through the
atmosphere, which is used in the radiative-convective equilibrium calculation,
as
${\cal F_{\rm tot}}(z)=h{\cal F}_{\rm hole}(z)+(1-h){\cal F}_{\rm cloud}(z).$
$None$
We stress that this is not a combination of separate models, but rather
conceptually represents two adjacent sub-columns in the atmosphere with the
same thermal profile and differing opacity. Our radiative-convective
equilibrium model is then employed to solve for a single $T(P)$ atmospheric
thermal profile (McKay et al. 1989; Marley & McKay 1999) that carries net flux
$\sigma T_{\rm eff}^{4}={\cal F_{\rm tot}}(z)$ through all statically stable
layers of the atmosphere. Layers that would be convectively unstable in pure
radiative-equilibrium are iteratively adjusted to follow an adiabat. The cloud
profile and atmospheric chemistry are updated along with the atmospheric
profile as it is converged to a solution, so the final thermal, chemical, and
cloud profiles are all mutually self-consistent.
A sampling of resulting model profiles is shown in Figure 1. The hottest
profile is for a global, homogeneous $f_{\rm sed}=2$ model while the coolest
model assumes no cloud opacity. Adding a small fraction of cloud free area to
the $f_{\rm sed}=2$ cloudy model efficiently cools the atmospheric profile. In
the example shown setting $h=0.25$ produces a model that lies over one-third
of the way between the fully cloudy ($f_{\rm sed}=2$) and the cloudless (nc)
models in the region above $T(P)=T_{\rm eff}$. Increasing the surface area
fraction of holes to $h=0.5$ produces an even cooler model. For comparison the
figure also presents a homogeneous cloudy model with a much higher
sedimentation efficiency, $f_{\rm sed}=4$. This model with globally low cloud
opacity follows a $T(P)$ profile very similar to the partly cloudy model with
thicker clouds ($f_{\rm sed}=2$) with 50% of cloud-free regions. We have found
that this is generally the case: a partly cloudy model presents a similar
thermal profile to a model with a thinner cloud, a point we return to in §4.
## 3 Photometry and Spectra of Partly Cloudy Dwarfs
The synthetic spectrum for a converged $T(P)$ atmospheric profile is obtained
from Equation (1). We compute absolute magnitudes and colors with the
ultracool dwarf evolution models of Saumon & Marley (2008). Figure 2 shows
synthetic near-infrared photometry from the partly cloudly models. For a fixed
$T_{\rm eff}$, as $h$ is increased the model colors briskly move to the blue
in $J-K$ and $J-H$. The $J$ band flux increases as $h$ increases from 0 to
0.75, but then dims slightly for cloud free models ($h=1$). This is because
atmospheres with even a small cloud cover are warmer in the atmospheric region
from which the $J$ band flux emerges. On the other hand, $M_{H}$ is relatively
constant across the transition from $h=0$ to $h=1$ at constant $T_{\rm eff}$
but shows a similar dimming as $h\rightarrow 1$.
The evolution of model dwarfs with fixed $f_{\rm sed}$ produces trajectories
that do not exhibit a rapid L to T transition as a global homogenous cloud
sinks too gradually below the photosphere (Burrows et al. 2006; Saumon &
Marley 2008, Fig. 4). With the Ackerman & Marley (2001) cloud model, the L to
T dwarf transition can only be modeled with an increase of the cloud
sedimentation parameter $f_{\rm sed}$. Figure 2 demonstrates that the
transition can also be described as a progressive increase in cloud-free areas
at a fixed $f_{\rm sed}$ and $T_{\rm eff}\sim 1200\,\rm K$. The L/T transition
dwarf colors and the J band brightening are well fit by this approach.
However, 1200 K is slightly cooler than the observed $T_{\rm eff}$ temperature
of the transition of $\sim 1300\,\rm K$ (Golimowski et al. 2004; Stephens et
al. 2009) and if $T_{\rm eff}$ falls appreciably across the transition the J
band bump would be weakened (Fig. 2). For a different choice of $f_{\rm sed}$
and model gravity a different transition $T_{\rm eff}$ would be expected.
As noted in Saumon & Marley, there is an offset of $\sim 0.3$ to the blue in
the $J-K$ color of models from the bulk of L and T dwarf photometry. This may
arise from shortcomings in the $K$ band pressure-induced opacity of molecular
hydrogen. Nonetheless, the trend of the late T dwarf $J-K$ color can be better
reproduced with models with $h\sim 0.5-0.75$. In the $M_{H}$ vs. $J-H$
diagram, the behavior of field L dwarfs and late T dwarfs are better
reproduced by the cloudy $f_{\rm sed}=2$ and the cloudless sequences,
respectively although the latter would also be better matched with partly
cloudy models with $h\sim 0.5-0.75$. This suggests the spectra of late T
dwarfs could be influenced by clouds, contrary to the usual assumption
(Burgasser et al., 2010).
## 4 Distinguishing Partly Cloudy Dwarfs
In Saumon & Marley (2008) we demonstrated that by increasing the cloud
sedimentation efficiency $f_{\rm sed}$ as a brown dwarf cools from $T_{\rm
eff}=1400$ to 1200 K, the predicted model colors reproduce those across the L
to T transition. In the previous section we likewise showed that increasing
fractional cloudiness – at fixed $f_{\rm sed}$ – has the same result. This
leads us to consider how to distinguish the two cases.
As shown in Figure 1 a partly cloudy $T(P)$ profile (based on a $f_{\rm
sed}=2$ cloud) can be nearly identical to a model with a thinner homogeneous
cloud but the spectrum from these two models are not necessarily the same
because the former uses Eq. (1) to compute the flux. In the partly cloudy case
some flux from deep, hot regions of the atmosphere (${\cal F}_{\rm hole}$ is
escaping through the clear regions that are otherwise totally shielded by the
cloud in the homogeneous case (see the middle panel of Figure 7 of Ackerman &
Marley (2001)). Thus we expect that even for identical $T(P)$ profiles the
emission spectra will differ.
Indeed that is the case as shown in Figure 4 which shows spectra computed from
profiles shown in Figure 1. Focusing on the $J$ band, which features the
lowest molecular opacity and the deepest atmospheric window in the near-
infrared (Ackerman & Marley, 2001), the greatest flux is found for the
cloudless model. The homogenous cloudy $f_{\rm sed}=2$ model is faintest, with
the $f_{\rm sed}=4$ case falling in between. Even though the partly cloudy
$h=0.5$ model has essentially the same thermal profile as the $f_{\rm sed}=4$
model, it is brighter in the $J$ band because some flux is escaping from
deeper in the atmosphere. Of course flux conservation requires that the partly
cloudy model must be fainter at other wavelengths, here in $K$ band. Thus the
model spectra are increasingly bluer in the near-infrared from $f_{\rm sed}=2$
to $f_{\rm sed}=4$ to the partly cloudy model. Figure 4 shows that the partly
cloudy model is very close to the $f_{\rm sed}=4$ homogeneous cloudy model at
wavelengths where the flux is low and emitted from the upper atmosphere, and
intermediate between $f_{\rm sed}=4$ and cloudless in the $JHK$ flux peaks
where part of the flux comes from the deeper atmosphere. Thus, for a given
$T_{\rm eff}$ and gravity, the presence of holes in the cloud cover have a
discernible effect on the near-infrared spectrum.
For a given observed spectrum, where $T_{\rm eff}$, gravity and composition
are not known a priori, how can we distinguish a partly from a homogeneous
cloudy atmosphere? We explored this problem by fitting the near-infrared
partly cloudy model spectra with our large library of cloudy models, using the
method of Cushing et al. (2008), and allowing $T_{\rm eff}$, gravity, and
$f_{\rm sed}$ to vary freely. We find that in general, the best fitting cloudy
model has the same gravity, the same $T_{\rm eff}$ (or slightly higher by
$\sim 100\,$K) and a higher $f_{\rm sed}$ depending on $h$. The fitted cloudy
spectra are close to the partly cloudy spectra, and some of the differences
can be attributed to the grid spacing of the cloudy models. It appears that
the $JHK$ colors of a partly cloudy model ($T_{\rm eff}$, $g$, $f_{\rm sed}$,
$h$) can be well matched with a cloudy model ($T_{\rm eff}^{\prime}$,
$g^{\prime}$, $f_{\rm sed}^{\prime}$). The differences in the near-infrared
spectrum are thus subtle and comparable to the differences found between
observed spectra and the best fitting models (e.g. Cushing et al. (2008);
Stephens et al. (2009)). Given that even with bolometric luminosity
measurements $T_{\rm eff}$ temperatures cannot yet be measured to much better
than $\sim 50\,\rm K$ precision, effective temperature cannot yet provide a
strong constraint on $h$. The most promising avenue to distinguish the two
models in the L/T transition spectroscopically appears to be strong spectral
features that are very sensitive to temperature and that are formed in the
deeper, hotter regions of the atmospheres, such as the 0.99$\,\mu$m band of
FeH and the alkali doublets of Na I ($1.14\,\mu$m) and K I ($1.175$ and
$1.25\,\mu$m). Those should become stronger as the cloud coverage decreases.
Given these considerations, variability may be the best method of
characterizing cloud patchiness in brown dwarfs. Partly cloudy dwarfs may be
variable if the lengthscales of the cloudy patches are large enough and if the
viewing inclination is not near pole on. Variations in the geometry of the
cloud cover (“weather”) would also lead to detectable variations in brightness
on time scales different from the rotation period. Indeed there is indication
that a sizable fraction of L dwarfs are variable (Gelino et al., 2002; Koen,
2003). Sample sizes are as yet too small to determine if blue L dwarfs or L/T
transition objects show greater variability than other L dwarfs, but
systematic studies would elucidate any trends.
Polarization is another possible way to distinguish partly from fully cloudy
objects. Cloud free brown dwarfs should not show appreciable polarization
(Sengupta & Marley, 2009). Sengupta & Marley (2010) demonstrated that very
rapidly rotating, low gravity dwarfs with homogenous clouds can show sizeable
polarization, but only for rotation periods less than about 4 hours. However
partly cloudy L dwarfs may show a polarization signal at $I$ band consistent
with that observed in some objects (Menard & Delfosse, 2004). Measurable
polarization in a slowly rotating or high gravity brown dwarf would point to
the presence of inhomogeneous cloud cover.
## 5 Discussion and Conclusions
There is growing evidence that the transition from the L to the T spectral
class happens over a small effective temperature range (e.g., Stephens et al.,
2009). It is difficult for any model of a globally uniform, homogenous cloud
to either sink out of sight or precipitate rapidly enough to account for the
observed rapid change in spectral properties. Instead the transition may arise
as holes appear in an otherwise uniform global cloud deck.
We have presented the first self-consistent method for computing one-
dimensional global atmospheric profiles appropriate to an atmosphere which has
both clear and cloudy patches. The model does not depend upon the physical
sizes of the patches, nor does it explain why patches might appear, but rather
parameterizes the global fraction of clear and cloudy regions. The resulting
$T(P)$ profiles converged to by our model are intermediate in temperature
between hot, cloudy models and colder, cloudless ones. We confirm the
conclusion of Burgasser et al. (2002), who used a more simplistic model to
argue that patchiness in brown dwarf clouds is a plausible mechanism for the L
to T type transition.
In addition to the L to T transition, patchy clouds have been suggested by
Folkes et al. (2007) as a possible mechanism to explain those L dwarfs that
have unusually blue near-infrared colors (Cruz et al., 2003; Knapp et al.,
2004; Burgasser et al., 2008). Burgasser et al. (2008) demonstrated that a
$T_{\rm eff}=1700\,\rm K$ model with thin ($f_{\rm sed}=4$) clouds provides a
good spectral match to the blue L dwarf 2MASS J1126–5003. We find that in this
effective temperature regime a model with thicker ($f_{\rm sed}=2$) and 50%
cloud coverage produces a comparable spectrum. Thus we agree with Folkes et
al. (2007) that partial cloudiness is potential alternative mechanism to thin
global clouds for the blue L dwarfs. Differences in cloud coverage might
result from different rotational rates, gravities, or even viewing geometries
(pole vs. equator).
The growing evidence for temporal photometric variability at the L to T
transition (Artigau et al., 2009; Radigan et al., 2010) supports the
plausibility of partial cloudiness being responsible for the change in colors
across the transition. Future 3-dimensional simulations of brown dwarf
atmospheres (e.g., Freytag et al., 2010) coupled with a new generation of
cloud models will likely help to elucidate the underlying mechanism
responsible for cloud fragmentation at $T_{\rm eff}\sim 1200$–1300 K. In the
meantime additional observations of brown dwarf photometric variability and
observations of L/T binaries will shed light on this poorly understood phase
of their evolution.
## 6 Acknowledgements
We thank M.C. Cushing for performing partly cloudy fits and the anonymous
referee for helpful comments. NASA provided support for this work via the
Planetary Atmospheres Program (MSM and CG), the Spitzer Space telescope
Theoretical Research Program (DS), and Astrobiology Institute’s Virtual
Planetary Laboratory Lead Team (CG).
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Figure 1: Model atmosphere temperature-pressure profiles with $T_{\rm
eff}=1400\,\rm K$, $\log g{\rm(cm/s^{2})}=5$ and solar metallicity. Solid
lines are for atmospheres with horizontally homogenous cloud cover (labeled
$f_{\rm sed}=2$ and 4) or no clouds (labeled nc). Two partly cloudy models
with hole fraction $h=0.25$ and 0.5 and based on a $f_{\rm sed}=2$ cloud model
are shown with dashed lines. The condensation curves of iron and forsterite
are shown with dotted lines.
Figure 2: Near-infrared model color-magnitude diagrams (MKO system). Red and
blue lines, respectively, show model colors for cloudy sequence with $f_{\rm
sed}=2$ and cloudless models (both $\log g=5$). Horizontal green lines connect
partly cloudy model colors with $T_{\rm eff}=1000$ to 2000 K in steps of 200K
(from bottom to top). Partly cloudy models (green dots) are for cloud-free
fractions $h$ of 0, 0.25, 0.5, 0.75 and 1, from right to left. Square symbols
show field dwarfs with M dwarfs in black, L dwarfs in red and T dwarfs in
blue. The photometry is primarily from Leggett et al. (2002) and Knapp et al.
(2004). Additional sources are given in Saumon & Marley (2008) (Fig. 7).
Figure 3: Absolute model fluxes ($d=10\,$pc) for various cloud treatments, all
with $T_{\rm eff}=1400\,$K, $\log g=5$ and solar metallicity. Homogeneous
cloudy spectra are shown for $f_{\rm sed}=2$ and 4 along with a cloudless
spectrum. The partly cloudy model with $h=0.5$ uses a $f_{\rm sed}=2$ cloud.
These spectra are computed from the thermal profiles shown in Figure 1.
|
arxiv-papers
| 2010-09-30T18:46:40 |
2024-09-04T02:49:13.263218
|
{
"license": "Public Domain",
"authors": "Mark S. Marley, Didier Saumon, Colin Goldblatt",
"submitter": "Mark S. Marley",
"url": "https://arxiv.org/abs/1009.6217"
}
|
1010.0106
|
b
# Rate analysis for a hybrid quantum repeater
Nadja K. Bernardes1 nadja.bernardes@mpl.mpg.de Ludmiła Praxmeyer2 Peter van
Loock1 peter.vanloock@mpl.mpg.de 1Optical Quantum Information Theory Group,
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1/Bau
26, 91058 Erlangen, Germany Institute of Theoretical Physics I, Universität
Erlangen-Nürnberg, Staudttr. 7/B2, 91058 Erlangen, Germany 2Institute of
Physics, Nicolaus Copernicus University, ul. Grudziadzka 5, 87-100 Torun,
Poland
###### Abstract
We present a detailed rate analysis for a hybrid quantum repeater assuming
perfect memories and using optimal probabilistic entanglement generation and
deterministic swapping routines. The hybrid quantum repeater protocol is based
on atomic qubit-entanglement distribution through optical coherent-state
communication. An exact, analytical formula for the rates of entanglement
generation in quantum repeaters is derived, including a study on the impacts
of entanglement purification and multiplexing strategies. More specifically,
we consider scenarios with as little purification as possible and we show that
for sufficiently low local losses, such purifications are still more powerful
than multiplexing. In a possible experimental scenario, our hybrid system can
create near-maximally entangled ($F=0.98$) pairs over a distance of 1280 km at
rates of the order of 100 Hz.
quantum repeaters
###### pacs:
0000
††preprint: PRE/003
## I Introduction
In most quantum information processes entanglement plays an essential role. It
enables us not only to teleport quantum information bennett , but also to
achieve perfectly secure quantum communication ekert . Unfortunately, the
quantum channels over which entanglement is distributed are in general noisy.
Owing to fundamental principles, common procedures used in classical
communication, such as amplification or cloning wootters ; dieks , cannot be
applied and therefore the fidelity of transmission will be limited by the
length of the channel. To avoid the exponential decay with the distance and be
able to perform long-distance quantum communication, quantum repeaters were
proposed briegel ; dur . Instead of distributing entanglement over long
distances, entanglement will be generated in smaller segments and a
combination of entanglement swapping zukowski and entanglement purification
bennett ; deutsch enables one to extend the entanglement over the entire
channel.
There are various promising proposals for implementing quantum repeaters. The
most prominent of these approaches use some heralding mechanism based on
single-photon detection to generate entangled pairs duan ; childress1 ;
childress2 . In these schemes, typically, high-fidelity entangled pairs are
created, while the success probabilities in the initial entanglement
distribution are very low. Other schemes, employing bright multiphoton
signals, are much more efficient, but have only modest initial fidelities,
since they are more sensitive to photon losses in the communication channel.
As a consequence, these coherent-state-based protocols require more
purification steps PvLa ; Ladd ; PvLb .
Schemes for practically implementing a quantum repeater are not
straightforward, not even for not too long distances such as a few hundred
kilometers. The steps of entanglement distillation and swapping require
advanced local quantum logic, such as two-qubit entangling gates; furthermore,
the typical duration to successfully generate an entangled pair imposes severe
constraints on the quantum memory decoherence times. Depending on how
imperfect these operations are and how short the quantum memory decoherence
time is, the final fidelity may still exponentially decay with the total
distance. Another important issue are the rates at which quantum transmission
succeeds. Assuming that this rate is mainly determined by the channel
transmissivity, the overall transmission rate will also decay exponentially,
unless sufficient quantum memories and/or quantum error detection mechanisms
are available.
In this paper, we will perform a study on the rates for a hybrid quantum
repeater PvLa ; Ladd ; PvLb . In this proposal, entanglement is created
between atomic qubits through an optical coherent state. In order to simplify
the analysis, we shall assume that perfect memories are available, that is,
memories with infinite decoherence times. Moreover, we will assume that
optimal entanglement generation probabilities and deterministic swapping
routines are available. More specifically, the generalized measurements on the
optical mode that lead to the initial, conditional entangled qubit pairs can
be as good and efficient as allowed quantum mechanically. We will derive exact
formulas for the time needed to generate an entangled pair and hence for the
final rates at a given target fidelity. As opposed to previous, mainly
numerical studies PvLa ; Ladd ; vanmeter ; munro , our rate calculations are
fully analytical. For this purpose, we shall first analyze the significance of
nested purifications and multiplexing with the aim of using as little as
possible of these experimentally demanding techniques. Our assumption of
perfect memories makes the use of probabilistic entanglement purification
preferable to alternate techniques based upon deterministic quantum error
correction (QEC) perseguers ; jiang ; fowler ; munro2 . These schemes, using
QEC codes, are experimentally more demanding due to their need of sufficient
spatial memory resources and more complicated quantum gates for encoding and
syndrome identification. In general, there will be a trade-off between the
requirements on the memory decoherence times and those on the quantum error
detection mechanism used to suppress the exponential fidelity decay. As
mentioned, in our case, we shall focus on a scenario where memories are ideal
and quantum error detection as simple as possible. The paper is structured as
follows. In Sec. II, we briefly describe the hybrid quantum repeater,
explaining how to generate, purify, and swap entanglement. In Sec. III, we
calculate the rates to generate an entangled pair over the entire distance for
a hybrid quantum repeater considering different strategies. We conclude in
Sec. IV and give more details on various formulas and derivations in the
appendix.
## II Hybrid Quantum Repeater
### II.1 Entanglement generation
In a hybrid quantum repeater scheme, the entanglement distribution mechanism
is based on dispersive light-matter interactions, obtainable from the Jaynes-
Cummings interaction Hamiltonian in the limit of large detuning schleich .
Such a Hamiltonian can be obtained by single electrons trapped in quantum dots
bracker or by neutral donor impurities in semiconductors strauf . It leads to
a conditional phase-rotation of the field mode,
$\hat{U}_{int}=e^{i\theta\hat{\sigma}_{z}\hat{a}^{\dagger}\hat{a}}$, where
$\theta$ is an effective interaction time, $\hat{\sigma}_{z}$ is the qubit
Pauli-Z operator and $\hat{a}$ ($\hat{a}^{\dagger}$) is the annihilation
(creation) operator of the electromagnetic field mode. Using a coherent state
$|\alpha\rangle$ as the probe beam and an electron-spin system in a cavity
(i.e., a two-level system or a “$\Lambda$-system” as an effective two-level
system), the total output state will be ideally described as
$\hat{U}_{int}\left[\frac{(|0\rangle+|1\rangle)}{\sqrt{2}}|\alpha\rangle\right]=\frac{|0\rangle|\alpha\rangle+|1\rangle|\alpha
e^{-i\theta}\rangle}{\sqrt{2}}.$ (1)
This interaction enables one to generate an entangled two-qubit state. First,
we let a bright coherent-state pulse, or “qubus”, which we denote as system
“B”, interact with an atomic qubit superposition state, system “A”, resulting
in the state (1). The coherent state is then sent through a lossy channel and
interacts in a second cavity with system “C”, resulting in an entangled state
between the two qubits (systems “A” and “C”) and the probe beam (system “B”).
This state, after local transformations, is given by PvLb
$\mu_{E}^{2}|\Phi^{+}\rangle\langle\Phi^{+}|+(1-\mu_{E}^{2})|\Phi^{-}\rangle\langle\Phi^{-}|$
(2)
where
$\displaystyle|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}|\sqrt{\eta}\alpha\rangle_{B}|\phi^{+}\rangle_{AC}+\frac{1}{2}e^{-i\eta\xi}|\sqrt{\eta}\alpha
e^{i\theta}\rangle_{B}|10\rangle_{AC}$
$\displaystyle+\frac{1}{2}e^{i\eta\xi}|\sqrt{\eta}\alpha
e^{-i\theta}\rangle_{B}|01\rangle_{AC},$
$\displaystyle|\Phi^{-}\rangle=\frac{1}{\sqrt{2}}|\sqrt{\eta}\alpha\rangle_{B}|\phi^{-}\rangle_{AC}-\frac{1}{2}e^{-i\eta\xi}|\sqrt{\eta}\alpha
e^{i\theta}\rangle_{B}|10\rangle_{AC}$
$\displaystyle+\frac{1}{2}e^{i\eta\xi}|\sqrt{\eta}\alpha
e^{-i\theta}\rangle_{B}|01\rangle_{AC},$
with the maximally entangled Bell states
$|\phi^{\pm}\rangle=(|00\rangle\pm|11\rangle)/\sqrt{2}$. Photon losses in the
channel are described by a beam splitter which transmits, on average, $\eta$
photons, $\xi\equiv\alpha^{2}\sin{\theta}$, and
$\mu_{E}=(1+e^{-(1-\eta)\alpha^{2}(1-\cos{\theta})})^{1/2}/\sqrt{2}$
($\alpha\in\mathbb{R}$). Considering a standard telecom fiber, where photon
loss is assumed to be 0.17 dB per km, the transmission parameter will be
$\eta(L,L_{att})=e^{-L/L_{att}}$, where $L$ is the total distance of the
channel and the attenuation length is assumed to be $L_{att}=25.5$ km.
Measuring the state of the qubus mode permits the preparation of a two-qubit
entangled state. One way to achieve this final step is through homodyne
detection. This is a very efficient and practical way, but the final
fidelities are rather modest (e.g. $F<0.8$ for 10 km). Another slightly less
practical way, but with a considerable improvement in the final fidelities, is
the unambiguous state discrimination (USD) approach. High initial fidelities
can then be achieved at the expense of lower entanglement generation rates. In
general, using the USD measurement, fidelities can be tuned in the whole range
$0.5<F<1$ for any given elementary distance $L_{0}$, with correspondingly
smaller success probabilities for larger fidelities. In any case, the USD
approach gives us ultimate bounds on the performance of the entanglement
generation procedure, when the quantum mechanically optimal USD is considered
PvLb .
In the USD approach, we must be able to distinguish between the state
$|\sqrt{\eta}\alpha\rangle$ and the set of states $\\{|\sqrt{\eta}\alpha
e^{i\theta}\rangle,|\sqrt{\eta}\alpha e^{-i\theta}\rangle\\}$. The fidelity
between these two density operators gives a lower bound to the failure
probability (the probability for obtaining an inconclusive measurement
outcome),
$P_{?}\geq
F=\sqrt{\langle\sqrt{\eta}\alpha|\hat{\rho}|\sqrt{\eta}\alpha\rangle},$ (3)
where
$\hat{\rho}=\frac{1}{2}(|\sqrt{\eta}\alpha
e^{i\theta}\rangle\langle\sqrt{\eta}\alpha e^{i\theta}|+|\sqrt{\eta}\alpha
e^{-i\theta}\rangle\langle\sqrt{\eta}\alpha e^{-i\theta}|).$
This leads to an optimal (minimal) failure probability
$P_{?}^{opt}=e^{-\eta\alpha^{2}(1-\cos{\theta})}.$ (4)
By looking at Eq. (2), it is possible to establish a connection between the
fidelity of the successfully created, entangled pair (a rank-2 mixture of the
$|\phi^{\pm}\rangle$ Bell states for an error-free identification of the state
$|\sqrt{\eta}\alpha\rangle$) and the optimal failure probability. Considering
that $\mu_{E}^{2}\equiv F$, this failure probability will be given by
$P_{?}^{opt}=(2F-1)^{\eta/(1-\eta)}.$ (5)
A practical implementation of a suboptimal USD measurement based upon linear
optics and photon detection can be found in PvLb . A protocol for implementing
the optimal USD is given in azuma .
Assuming that (5) is an optimal bound for the failure probability, the optimal
upper bound for the success probability to generate an entangled pair is
$P_{success}^{opt}=1-P_{?}^{opt}$. We will use this bound for the probability
of success in the following sections.
### II.2 Entanglement purification and swapping
Using the same interactions as presented in the preceding section, we are able
to perform entanglement swapping and entanglement purification. In either
case, local two-qubit gates are needed. Following Ref. PvLc , a measurement-
free, deterministic controlled-phase gate can be achieved with a sequence of
four conditional displacements of a coherent-state probe interacting with the
two spins. More specifically, by appropriate choice of $\beta_{1}$ and
$\beta_{2}$ ($\beta_{1}\beta_{2}=\pi/8$), up to a global phase and local
unitaries, the total unitary operator representing the controlled-phase gate
will be
$\hat{D}(i\beta_{2}\hat{\sigma}_{z2})\hat{D}(\beta_{1}\hat{\sigma}_{z1})\hat{D}(-i\beta_{2}\hat{\sigma}_{z2})\hat{D}(-\beta_{1}\hat{\sigma}_{z1}).$
(6)
Here the operator $\hat{D}(\beta)=e^{\beta\hat{a}^{\dagger}-\beta^{*}\hat{a}}$
describes a phase-space displacement of the probe by $\beta$. In fact, it can
be shown PvLc that the sequence of (6) can be achieved through uncontrolled
displacements and controlled rotations of the probe via the same Jaynes-
Cummings-type interaction as used for the entanglement generation. Another
scheme to obtain a two-qubit gate is proposed in Refs. PvLa ; Ladd . Although
in this scheme less conditional operations are needed, making it less
sensitive to losses, even without losses, a small amount of decoherence is
introduced due to the remaining entanglement between the probe and the spins
after the gate operation. This decoherence effect depends on $\theta$ and on
the initial probe state, and scales with $\alpha\theta^{2}$ (for an initial
coherent-state probe with amplitude $\alpha$), thus becoming negligible for
sufficiently small $\theta$ and $\alpha\theta\sim 1$.
Single-qubit rotations, measurements, and this controlled-phase gate are
sufficient resources to implement the standard purification protocol
introduced in Ref. deutsch . From Eq. (2), it is possible to see that, after
the USD measurement has taken place, the output state will be a rank-2 state
(a mixture of two Bell states). The probability of success of the entanglement
purification is
$P_{purification}=F^{2}+(1-F)^{2},$ (7)
and the final fidelity becomes
$F_{purification}=\frac{F^{2}}{F^{2}+(1-F)^{2}}.$ (8)
As was shown in Ref. dur , the purification protocol introduced by Deutsch et
al. deutsch is particularly efficient for these classes of states (i.e., it
is more efficient than for full-rank mixtures).
The same kind of operations as for purification are sufficient to implement
the entanglement swapping. For a state given as in Eq. (2), the swapping will
be deterministic ($P_{swapping}\equiv 1$) and its final fidelity is given by
$F_{swapping}={F^{2}+(1-F)^{2}}.$ (9)
In a more realistic approach, considering errors in the gates caused by local
losses, Eqs. (7-9) will no longer be valid. According to the analysis
presented in Ref. Louis , dissipation will be introduced in the probe mode
between and during each interaction with the spins. A coherent-state matrix
element $|\gamma\rangle\langle\beta|$ after dissipation changes to
$\langle\beta|\gamma\rangle^{1-T}|\gamma\sqrt{T}\rangle\langle\beta\sqrt{T}|$,
where $T$ is the transmission parameter walls . The interaction sequence (6)
will then change to terms that cause single-qubit dephasing, but also one term
which causes a dephasing on both qubits; this effect scales with
$(1-T)\alpha^{2}\sin{\theta}^{2}$ (for an initial coherent-state probe with
amplitude $\alpha$). For further details see Appendix A. The resulting gate is
extremely sensitive to losses. However, the aim of our work is a better
understanding of the building blocks in the hybrid quantum repeater, and hence
we shall minimize the effect of local losses by choosing in most parts of our
analysis a sufficiently high local transmission parameter. In those figures
where the parameter T is not explicitly given, it is assumed that the error is
at least as small as $1-T=0.001\%$. The main influence of photon losses then
occurs in the communication channel and therefore depends on the communication
distance.
## III Rate analysis
In this section, the rates of entanglement generation over the entire distance
for a hybrid quantum repeater will be calculated. The memories are considered
ideal, and the entanglement connection deterministic, such that the primary
source of errors will be photon losses in the channel.
Figure 1: (color online). Idealized quantum repeater. A total distance L is
divided in $2^{n}$ segments with length $L_{0}=L/2^{n}$. Initially,
entanglement is generated between neighboring repeater stations. The qubits at
the intermediate stations are then connected. Finally, entanglement over the
entire distance L is obtained.
Let us first consider a general quantum repeater as illustrated in Fig. (1). A
total distance L is divided in $2^{n}$ segments, each of length
$L_{0}=L/2^{n}$. First, entanglement is generated between the adjacent nodes,
which is accomplished with probability $P_{0}$. Then these segments are
connected, extending the entanglement from $L_{0}$ to $2L_{0}$. This step is
performed many times, until the terminal nodes, separated by $L=2^{n}L_{0}$,
are entangled. The initial distribution of entanglement over small segments
does not prevent the final fidelity from decaying exponentially. Even if the
connecting operations are perfect, we must take into account that the initial
entangled pairs are not perfectly entangled pairs, and this will cause a decay
of the final fidelity. As a consequence, purification and quantum memories are
essential for the full quantum repeater, but their implementations introduce a
series of experimental difficulties. There are various ways to optimize the
combinations of swapping and purification (e.g., considering residual
entanglement razavi1 ; razavi2 , multiplexing collins , blind connecting
measurements razavi2 ). However, these approaches are more significant for the
case of imperfect memories and probabilistic swappings, whereas our model uses
perfect memories and deterministic swappings.
In the following sections, we will focus on the effects of spatial
multiplexing compared to a parallel scheme for the hybrid quantum repeater. In
a parallel repeater, the $i$th memory pair in one segment interacts only with
the $i$th pair in neighboring segments, however, for the multiplexing scheme,
resources can be dynamically allocated. Moreover, it will be discussed what
the impact of purification on the rates is and how the two methods of
multiplexing and purification compare 111Note that there are different ways to
do and use multiplexing. The more conventional application is spatial
multiplexing, as typically used in discrete-variable single-photon-based
repeaters collins . In this case, the probability to successfully generate an
entangled pair, $P_{0}<1$, is fairly small and multiplexing means that
neighboring segments even from different, parallel repeater chains can be
connected immediately after the corresponding pairs have been created.
Another type of multiplexing is temporal munro2 . In this case, the initial
entanglement is generated almost instantly by transmitting in parallel
sufficiently many probe pulses (after their interactions with a corresponding
number of spins) to a receiving spin that interacts with these pulses almost
simultaneously to guarantee that at least one entangled pair is created with
an effective, near-unit success probability, $P_{0}^{\rm eff}\to 1$.
Experimentally, to make this kind of temporal multiplexing efficient,
sufficiently fast local interactions and gates are needed. As a result, when
also swapping and error detection are deterministic (by replacing purification
by error correction), the full repeater protocol becomes near-deterministic,
as opposed to our scheme which keeps both the initial entanglement generation
and the entanglement purification probabilistic.
In principle, our rate analysis could be directly combined with the ideas of
Ref. munro2 by simply substituting our $P_{0}$ by an effective probability
of, for instance, $P_{0}^{\rm eff}\equiv(1-(1-P_{0})^{n})$, where $n$ is the
number of cavities transmitting the probe pulses and $P_{0}$ in this formula
is still the same function of fidelity, losses, etc., as described in the main
text for our USD-based entanglement distribution. Then, $P_{0}^{\rm eff}\to 1$
can be achieved for sufficiently large $n$ even when the original $P_{0}$ is
as small as 10-20 %. However, as our goal is to keep the number of spatial
resources as small and the scheme as simple as possible, we will not consider
this type of temporal multiplexing here. Our setting contains perfect
memories, but modest spatial resources, such that $P_{0}^{\rm eff}\equiv
P_{0}$ remains of the order of 1-10 %..
### III.1 Entanglement generation and swapping
First, let us calculate the rate for generating an entangled pair for a
quantum repeater in parallel without any purification or dynamical allocation
of resources. For the ideal memory case, we calculate the simple case of
entanglement-length doubling ($n=1$) with a single memory per half node. Since
the $n$-level quantum repeater is the entanglement-length doubling of two
$(n-1)$-level systems, it is essential to understand the basic process with
$n=1$.
Even simpler, let us start with just one segment, $n=0$. The average time
necessary to generate an entangled pair at distance $L$ (in this case
$L=L_{0}$) is given by
$\left\langle T\right\rangle_{0}=\frac{T_{0}}{P_{0}},$ (10)
where $T_{0}=2L_{0}/c$ is the minimum time to successfully generate
entanglement over $L_{0}$, assuming that this is the time spent on classical
communication to verify the success of the entanglement generation over
$L_{0}$ (including the initial transmission of the probe beam) and $c$ is the
speed of light in an optical fiber ($2\times 10^{8}m/s$).
For two pairs, $n=1$, the average time necessary to generate an entangled pair
at distance $L$ is then given by collins
$\left\langle
T\right\rangle_{1}=\frac{T_{0}}{P_{0}}\frac{(3-2P_{0})}{(2-P_{0})}.$ (11)
Recall that our memories are assumed to be ideal such that one successfully
created pair can be kept until a second pair is created in the neighboring
segment. The result in Eq. (11) is equivalent to the problem of a
geometrically distributed random variable with success probability $P_{0}$.
For $2^{n}$ pairs next to each other in $2^{n}$ segments, the average time
needed to generate an entangled pair at distance $L$ is given by
$\left\langle T\right\rangle_{n}=T_{0}Z_{n}(P_{0}),$ (12)
where the average number of steps to successfully generate entanglement in all
$2^{n}$ pairs, $Z_{n}(P)$, is
$Z_{n}(P)=\sum^{2^{n}}_{j=1}\binom{2^{n}}{j}\frac{(-1)^{j+1}}{1-(1-P)^{j}},$
(13)
and $P$ is the probability of success. Detailed calculations are presented in
Appendix B.
The rate to successfully generate entanglement in all of $2^{n}$ pairs over L
and to eventually obtain one L-distant pair can now be written as
$R_{n}=\frac{1}{\left\langle T\right\rangle_{n}}=\frac{1}{T_{0}Z_{n}(P_{0})}.$
(14)
Figure 2: (color online). Rates for a hybrid quantum repeater over a total
distance $L=1280$ km with $L_{0}=20$ km without purification and without
multiplexing, but including perfect memories. Comparison between the exact
formula to generate entangled pairs (green line) as given by Eqs. (12-14) and
the approximated one (black dashed line) corresponding to
$(\frac{2}{3})^{n}\frac{P_{0}}{T_{0}}$.
Commonly in the literature sangouard , for small $P_{0}$, these rates are
approximated by $\frac{P_{0}}{T_{0}}\left(\frac{2}{3}\right)^{n}$. However, as
illustrated in Fig. (2) for a total distance of $L=1280$ km and $L_{0}=20$ km,
the approximate formula is underestimating the rates in some regimes by more
than $50\%$ for our case of the USD-based hybrid repeater.
Figure 3: (color online). Different strategies to connect memory elements
situated in different columns. The parallel architecture a) connects only
elements in the same chain. Contrary to it, multiplexing b) connects any
available entangled pairs.
### III.2 Multiplexing versus parallelization
How would the rates be affected, if multiplexing is introduced in the system?
In the multiplexed scheme, where more than one memory per half node exist, as
shown in Fig. (3), entangled pairs are connected not only with pairs in the
same chain (as in the parallel scheme). Instead, as soon as one entangled pair
is successfully generated in one of the columns, it can be connected to a
neighboring pair, no matter in which chain they are positioned. The rate to
generate one entangled pair over the total distance for $n=1$ and $r$ memories
per half node is given by
$R_{mult,1,r}=\frac{1}{T_{0}Z_{mult,1,r}(P_{0})}=\frac{1}{T_{0}}\frac{1-(1-P_{0})^{2r}}{1+2(1-P_{0})^{r}}.$
(15)
For further details see Appendix C. Plotting this rate for a total distance of
$L=40$ km and four memory pairs per segment, $r=4$, see Fig. (4), it is
possible to confirm that the multiplexed scheme performs better than the
parallel one. However, an experimental implementation of this type of spatial
multiplexing would require fairly demanding feedforward techniques on the
pulses that enact the gates for entanglement swapping. It is therefore more
practical and efficient to operate the repeater in parallel, provided perfect
memories are available, as it is the case in our scenario. In the following
sections, the influence of purification will be studied. By comparing
purification with multiplexing, we will justify why we shall not consider
multiplexing in the further analysis.
Figure 4: (color online). Rates for a total distance $L=40$ km, $L_{0}=20$ km.
Comparison between the parallel (red, dashed line) and the multiplexed scheme
(black, solid line) for 2 segments ($n=1$) and 4 memories per half node
($r=4$).
### III.3 Entanglement purification
Taking into account that purification is an expensive task, either in terms of
spatial or in terms of temporal resources, the most efficient and simplest
extension beyond those schemes described in the preceding sections would
employ just one round of purification at the first nesting level. Intuitively,
considering the purification procedure probabilistic and entanglement swapping
deterministic, performing the purification at the beginning will make better
use of the memories than by doing it at the end or somewhere in between; for a
more quantitative justification, see Appendix D. Starting again with $n=0$,
already initially there is a need to generate at least two pairs, such that
purification is possible. In this way, the average time needed to generate a
purified pair will be
$\left\langle
T\right\rangle_{purif,0}=\frac{T_{0}}{P_{0}P_{1}}\frac{(3-2P_{0})}{(2-P_{0})}.$
(16)
Here we employ the same purifying protocol as the one introduced in zeilinger
; in this case the probability of one round of purification to succeed,
$P_{1}$, is given by Eq. (7).
For $n>0$, the calculation of the rates is not straightforward. In this
regime, we have found an upper and a lower bound for the time needed to
generate $2^{n}$ purified pairs. The former times, which give us the lower
bounds for the rates, correspond to those cases when purification will start
only after all the pairs are successfully generated, even if two pairs in the
same column are already present after a shorter time. In this slowest case,
the average time needed to generate $2^{n}$ one-round purified pairs is given
by
$\left\langle T\right\rangle_{purif,upper,n}=T_{0}Z_{n+1}(P_{0})Z_{n}(P_{1}).$
(17)
For the most optimistic case, corresponding to the fastest possible way to
achieve purification, we imagine that if purification fails, it is not
necessary to start from the beginning, trying to generate new pairs again, but
we assume that the two pairs necessary for purification are still available,
imagining that they have not been destroyed. The average time needed to
generate $2^{n}$ one-round purified pairs is then given by
$\left\langle
T\right\rangle_{purif,lower,n}=T_{0}(Z_{n+1}(P_{0})+Z_{n}(P_{1})).$ (18)
Aiming to find a compromise between the upper and lower bounds for the rates,
the rates can be calculated in an approximate fashion, combining the ideas
from the recurrence and the exact formula. The average time needed to generate
one purified pair will be
$\left\langle T\right\rangle_{purif,approx,0}=\frac{T_{0}}{P_{L_{0}}},$ (19)
where $P_{L_{0}}$ is an effective probability, from (16),
$P_{L_{0}}={P_{0}P_{1}}\frac{(2-P_{0})}{(3-2P_{0})}$. From this, it follows
that
$\left\langle T\right\rangle_{purif,approx,n}=T_{0}Z_{n}(P_{L_{0}}).$ (20)
Figure 5: (color online). Rates for a hybrid quantum repeater over a total
distance $L=1280$ km and $L_{0}=20$ km. Comparison between the lower bound
(green, solid line), upper bound (red, dot dashed line) and the approximate
formula (blue, dashed line).
How are the upper bound, the lower bound, and the approximate times, and their
corresponding rates, related to each other? Fig. (5) shows for a total
distance $L=1280$ km, the rates in these three cases. The approximate formula
gives a result which always stays in the middle between the upper and the
lower bounds. The rates are then given by
$\displaystyle R_{purif,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\left\langle T\right\rangle_{purif,approx,n}}$
$\displaystyle=$ $\displaystyle\frac{1}{T_{0}Z_{n}(P_{L_{0}})}.$
Figure 6: (color online). Rates for total distance $L=320$ km (below), $L=640$
km (center), $L=1280$ km (above) with $L_{0}=20$ km. Comparison between the
rates for a hybrid quantum repeater with one round of purification (black,
dashed line), two rounds of purification (red, solid line), and three rounds
of purification (blue, dot dashed line) at the first nesting level.
There is one remaining question: do more rounds of purification at the first
nesting level always increase the rates for the hybrid quantum repeater?
Utilizing (III.3) we plotted in Fig. (6) the rates to generate an entangled
purified pair over different total distances applying one, two, or three
rounds of purification 222$P_{L_{0}}$ varies depending on the number of rounds
of purification. For one round,
$P_{L_{0},1purif}={P_{0}P_{1}}\frac{(2-P_{0})}{(3-2P_{0})}$. For two rounds,
$P_{L_{0},2purif}={P_{L_{0},1purif}P_{1}}\frac{(2-P_{L_{0},1purif})}{(3-2P_{L_{0},1purif})}$.
And for three rounds,
$P_{L_{0},3purif}={P_{L_{0},2purif}P_{1}}\frac{(2-P_{L_{0},2purif})}{(3-2P_{L_{0},2purif})}$..
We see that not always does the increase of rounds of purification result in
an increase of the rates. More specifically, for this scheme and these
distances, three rounds of purification only increase the rates for very high
final fidelities. It should be pointed out here that the number of initial
resources 333These initial resources can be either spatial or temporal, where
in the latter case, for every new round of purification, a fresh initial pair
is created (so-called entanglement pumping dur ). Such an approach requires a
minimum of spatial resources childress1 ; childress2 , however, a maximum of
time. This means one needs extremely good memories and, nonetheless, the total
rates remain fairly low. Even though in our setting we do assume perfect
memories, throughout we shall stick to the faster standard purification
methods at the first nesting level using a sufficient initial supply of
spatial resources, similar to Ref. PvLa . increases also with the number of
rounds of purification, and so, in a more appropriate analysis, keeping the
numbers of initial resources the same, the rates for multiple-rounds of
purification should even perform worse.
### III.4 Multiplexing versus purification
Figure 7: (color online). Rates to generate one entangled pair over a total
distance of $L=40$ km (below), $L=640$ km (center) and $L=1280$ km (above)
with $L_{0}=20$ km. Comparison between multiplexing for 2 memories per half
node (red solid line) and one round of purification in the first nesting level
(blue dashed line).
After analyzing the effects of multiplexing and purification independently, we
would like to compare both strategies. Although in Eq. (15) the rates for
$n=1$ and $r$ memories per half node have been calculated, for $n>1$, there is
not such an analytical formula. In this case, the rates were calculated
through an effective probability according to Eq. (III.3) with different
$P_{L_{0}}$ in each case. For the scheme with purification, we have
$P_{L_{0}}=P_{0}P_{1}\frac{(2-P_{0})}{(3-2P_{0})}$ and for multiplexing,
$P_{L_{0}}=\frac{1-(1-P_{0})^{2r}}{1+2(1-P_{0})^{r}}$. We kept the number of
initial resources equal in both schemes (two memories per half node). As it is
possible to observe in Fig. (7), even for a total distance as small as $L=40$
km, for sufficiently high fidelities, the scheme with just one round of
purification is providing higher rates. For longer distances, $L=640$ km and
$L=1280$ km, one round of purification is clearly more powerful than
multiplexing without purification.
One may argue that the effects of multiplexing become more significant in a
configuration with more memories per half node, i.e., bigger $r$. However, as
can be seen from Fig. (8), the rate to generate an entangled pair after one
round of purification (only two memories per half node) is higher than the
rate to generate an entangled pair for a scheme using multiplexing with 32
available memory pairs per segment, $r=32$, for fidelities larger than
$~{}0.95$. For multiplexing schemes with $r=16$, the purification is
performing better already for a final fidelity $F_{final}>0.84$. Taking into
account all these results and the fact that multiplexing is difficult to
implement, multiplexing will not be considered in our final analysis. It has
to be pointed out here that this argument is not valid anymore if the losses
considered in the controlled- phase gates are bigger than $0.01\%$ (see Sec.
II.B and App. A).
Figure 8: (color online). Rates to generate one entangled pair for a total
distance of $L=1280$ km with $L_{0}=20$ km. Comparison between one round of
purification in the first nesting level (blue dashed line) and multiplexing
(red solid line) for 32 memories per half node (above) and for 16 elements
memories per half node (below).
### III.5 Results
Figure 9: (color online). Rates for total distance $L=80$ km (below), $L=160$
km (center), $L=320$ km (above) with $L_{0}=20$ km. Comparison between direct
transmission with one round of purification (blue, thick line), direct
transmission without purification (red, dashed line), quantum relay (black,
dotted line), hybrid quantum repeater with one round of purification (purple,
thick line), and hybrid quantum repeater without purification (black, dot
dashed line).
In Fig. (9) the rates for total distances $L=80$ km, $160$ km, $320$ km are
shown, with $L_{0}$ always equal to $20$ km. We have compared the case where
one round of purification takes place at the beginning before the connecting
steps (solid lines) to the case with no purification (dashed/dotted lines). We
have further compared the rates for the hybrid quantum repeater to the rate
upper bounds for transmitting the state directly (i.e., without swapping) and
to the rates of a quantum relay scheme, where all entangled pairs have to be
successfully generated in each segment at the same time (i.e., without using
memories). Even for a relatively small total distance, $L=80$ km, the hybrid
quantum repeater is performing better than the quantum relay or direct
transmission. For larger distances, $L=160$ km and $320$ km, the difference
between the rates of those schemes is even bigger, and the rates for the
quantum relay and direct transmission are effectively zero.
Aiming to analyze the length dependence of the rates, in Fig. (10), we plotted
the rates for the hybrid quantum repeater scheme with perfect memories and two
rounds of purification in the first nesting level for a total distance of
$L=1280$ km, $2560$ km, $5120$ km and $10240$ km. We observe that the rates
only decrease inverse-linearly with the total distance. For ideal memories and
using purification, we show that it is possible to avoid the exponential decay
of the rates with the distance.
In Fig. (11) we can show the rates of the hybrid quantum repeater for $L=1280$
km with different losses in the local gates, T. We show that the protocol can
run with local gate errors, however, depending on T, the final fidelity and
the generation rates can decrease very fast. For $1-T=0.001\%$ the rates are
practically the same as for $T=1$. For $1-T=0.01\%$ and a final fidelity of
0.95, rates of near 16 pairs per second can be achieved. On the other hand,
for $1-T=0.1\%$ rates of the order of only 10 pairs per second with a final
fidelity below 0.84 can be achieved.
Figure 10: (color online). Rates for the hybrid quantum repeater with two
rounds of purification for total distance $L=1280$ km (blue, thick line),
$L=2560$ km (red, dashed line), $L=$ 5120 km (purple, dotted line), $L=$ 10240
km (black, thin line). Figure 11: (color online). Rates for the hybrid quantum
repeater with two rounds of purification for total distance $L=1280$ km for
different parameters of local losses in the controlled-phase gate, $1-T=0$
(blue, thick line), $1-T=0.001\%$ (red, dashed line), $1-T=0.01\%$ (pink,
dotted line), $1-T=0.1\%$ (black, thin line).
## IV Conclusion
In this work, we calculated analytically the rates for the so-called hybrid
quantum repeater. Comparing these rates with those obtainable through direct
entanglement distribution (without using swapping, purification, or memories)
and through quantum relay schemes (using swapping, but no purifications and no
memories), for sufficiently long distances, the hybrid quantum repeater scheme
leads to significantly better rates. In fact, only the hybrid quantum repeater
scales sub-exponentially with the channel length, while both direct
distribution and quantum relay keep the exponential decay with distance. We
found the rates for a system where the subroutines (generation, swapping, and
purification) can all be implemented through weak dispersive light-matter
interaction, as proposed in Ref. PvLa .
In the hybrid repeater scheme with homodyne measurement from Ref. PvLa , a
rate of 15 pairs per second with final $F=0.98$ was achieved using 16 qubits
per half node for a total distance of $L=1280$ km, segment lengths of
$L_{0}=10$ km, and local gate errors of $0.1\%$. In our case, by employing
generalized measurements (instead of homodyne projection measurements) for an
optimal unambiguous discrimination of coherent states (USD) as proposed in
Ref. PvLb , for just 4 qubits per half node, two rounds of purification in the
first nesting level, $L=1280$ km, $L_{0}=20$ km, and gate errors of $0.001\%$,
we achieved rates of the order of 100 pairs per second with $F=0.98$. Note
that in the scheme from Ref. PvLa the rates were calculated performing Monte
Carlo simulations where not only more rounds of purification in different
nesting levels are allowed, but also multiplexing. Compared to the banded
scheme from Ref. vanmeter for the same distances and final fidelities, we
achieve rates with similar order of magnitude, however, it should be pointed
out that Ref. vanmeter uses a scheme with initially 50 qubits per half node,
and correspondingly more rounds of purification in different nesting levels.
We achieve our rates (of nearly 100 Hz) in a fairly practical setting where
the spatial resources are reduced (e.g. only four cavities per half node),
quantum error detection is close-to-minimal (i.e., only two rounds of
purification at the very beginning), and without spatial multiplexing.
However, these results are only obtained under the basic assumptions of very
low local losses, optimal USD measurements for entanglement distribution,
deterministic entanglement swapping, and perfect quantum memories. In future
work, we aim to relax one or more of these requirements. In fact, our
analytically obtained rates can be certainly further improved, if we include
multiplexing as well as purification in different nesting levels, possibly
leading to an improvement of two orders of magnitude compared to the scheme
from Ref. PvLa , and similar to the results presented in Ref. munro .
Moreover, further improvements in the initial entanglement distribution can be
achieved, if instead of coherent states and USD measurements, squeezed states
and homodyne measurements (together with extra local phase-space
displacements) are utilized, as was recently shown in Ref. ludmila . Besides
its quantitative significance, the work presented here represents a first step
towards a better understanding of the importance of each of the building
blocks in the architecture of a hybrid quantum repeater.
### Acknowledgments
Support from the Emmy Noether Program of the Deutsche Forschungsgemeinschaft
is gratefully acknowledged. L.P. gratefully acknowledges financial support for
the Future and Emerging Technologies (FET) programme within the Seventh
Framework Programme for Research of the European Commission, under the FET-
Open grant agreement CORNER no. FP7-ICT-213681.
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## Appendix A
Purification and entanglement swapping can be performed utilizing controlled-
phase (CZ) gates, Hadamard operations, and measurements on the qubits.
Assuming that the CZ-gate is the only operation that leads to errors in those
schemes, we are able to calculate the final fidelity for the purification and
swapping, as well as the probability of purification, using the imperfect CZ-
gate obtained in Ref. Louis . These are as follows:
$\displaystyle
P_{purification}^{imp}=\frac{1}{2}+e^{\frac{\pi(T-1)}{\sqrt{T}}}(F-1)F+$
$\displaystyle\left(e^{\frac{\pi(T-1)\left(2-i\text{sech}\left(\frac{\log(T)}{2}\right)\right)}{2\sqrt{T}}}+e^{\frac{\pi(T-1)\left(2+i\text{sech}\left(\frac{\log(T)}{2}\right)\right)}{2\sqrt{T}}}\right)\times$
$\displaystyle\frac{(2(F-1)F+1)}{4},$ (22)
$\displaystyle
F_{purification}^{imp}=\frac{F+Fe^{-\frac{\pi\left(2-2T\right)}{\sqrt{T}}}}{4P_{purification}^{imp}}+$
$\displaystyle\frac{2Fe^{\frac{\pi(T-1)}{\sqrt{T}}}\left(\sin\left(\pi\left(\frac{3}{2}-\frac{2}{T+1}\right)\right)F+F-1\right)}{4P_{purification}^{imp}},$
(23)
$\displaystyle
F_{swapping}^{imp}=\frac{1}{4}+\frac{1}{4}e^{2\pi\sinh\left(\frac{\log(T)}{2}\right)}(1-2F)^{2}+$
$\displaystyle\frac{e^{\pi\sinh\left(\frac{\log(T)}{2}\right)}}{2}(2(F-1)F+1)\cos\left(\frac{\pi}{2}\tanh\left(\frac{\log(T)}{2}\right)\right).$
(24)
## Appendix B
Consider a coin, which, in one toss, gives “tail” as an outcome with
probability $p$ and “head” with probability $q=1-p$. Imagine we are just
interested in the tail’s outcomes. We will here calculate the number of times
we have to toss one or more coins to obtain one, two, three or more tails.
Note that the problem of calculating the average time necessary to have one
success event in this case is analogous to our problem of calculating the
average time to successfully generate one (or many) entangled pair(s) in
different segments of the repeater.
#### 1) one success
Imagine that a coin is flipped repeatedly until the first tail outcome. How
many times on average do we have to toss the coin until we get a tail? The
average time necessary to obtain one tail is:
$\sum_{k=1}^{\infty}kq^{k-1}p=\frac{1}{p}=:\mathcal{Z}_{1}(p).$ (25)
Note that the character $\mathcal{Z}$ is used here in Appendix B for the
average time to have tail as an outcome; when average times are calculated to
generate entanglement over repeater segments, the character $Z$ is used
instead in the main text and Appendix C.
#### 2) two successes
Now imagine we have two identical coins and we flip both coins at once until
we get a “tail and tail” outcome. If two tails are obtained in the same trial,
game is over. If “tail and head” result is obtained, we keep the tail coin
(simulating the situation of memory qubits in the repeater) and flip only the
other coin until the second tail is obtained. In this case, the average time
needed to obtain two tails is:
$\displaystyle\sum_{k=1}^{\infty}k(q^{2})^{k-1}p^{2}+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{2})^{k-1}2pq)(q^{l-1}p)=$
$\displaystyle\frac{1}{p}\frac{(3-2p)}{(2-p)}=:\mathcal{Z}_{2}(p).\,$ (26)
The first sum describes the cases when we had the head outcomes in both coins
$(k-1)$ times and then 2 tails were obtained at once. The second sum counts
the cases when at $k$-th trial a tail in one coin was obtained and stored and
then after another $l$ (single) trials we got the second success.
#### 3) three and more successes
For 3 successes (in this case 3 tails) in 3 identical coins, the average time
needed will be:
$\displaystyle\\!\\!\\!\\!\\!\\!\sum_{k=1}^{\infty}k(q^{3})^{k-1}p^{3}+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{3})^{k-1}3p^{2}q)(q^{l-1}p)$
$\displaystyle+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}(k+l)((q^{3})^{k-1}3pq^{2})((q^{2})^{l-1}p^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!+\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\sum_{m=1}^{\infty}(k+l+m)((q^{3})^{k-1}3pq^{2})((q^{2})^{l-1}2pq)(q^{m-1}p)$
$\displaystyle=\\!\\!\frac{p(19+3p(p-4))-11}{p(p-2)(p(p-3)+3)}=:\mathcal{Z}_{3}(p).\;\;$
(27)
In a similar way, we calculate average waiting times for $N$ successes (tails)
in $N$ coins and obtain the following recurrence formula:
$\mathcal{Z}_{N}(p)=\frac{1}{1-q^{N}}\left(1+\sum_{j=1}^{N-1}{N\choose
j}q^{j}p^{N-j}\mathcal{Z}_{j}(p)\right),$ (28)
where $\mathcal{{Z}}_{1}(p)=\frac{1}{p}$.
Recurrence formula (28) can be solved, and its solution is given by
$\displaystyle\mathcal{Z}_{N}(p)=\sum_{k=1}^{N}{N\choose
k}\frac{(-1)^{k+1}}{1-(1-p)^{k}}\;.$ (29)
By substituting $N$ by $2^{n}$ in Eq. (29), we can calculate the average
number of steps necessary to successfully generate entanglement in $2^{n}$
pairs, as in Eq. (13).
## APPENDIX C
In order to compare the multiplexing with the parallelization strategy, we
have to calculate the average times needed to successfully generate entangled
pairs over the corresponding segments. We will start our calculation here with
the simplest case (two columns and two rows, similar to Fig. 3 with two
memories per half node).
#### 1) two columns ($n=1$), two rows ($r=2$)
Imagine now that we have 4 coins to toss; coins that can be distinguished by
their positions, arranged into two columns, two rows each. We toss the coins
all at once in the first step, and in the next steps, we toss only those coins
that had a head outcome in the previous trial. The average number of steps
needed for at least one success (tail) in every column will obviously depend
on whether these successes are required to appear in the same row or whether
they are allowed to appear in two different rows. The first case corresponds
to parallelization, the second case to multiplexing. Probabilities and average
numbers of steps are calculated in a similar manner to Appendix B, taking into
account that successes have to be appropriately distributed among columns and
rows. In the following, we will not refer to coins anymore, but to repeater
segments and the probabilities to generate entanglement over these segments.
The probability that at least one entangled pair is created in every column in
the case of multiplexing, i.e., when it does not matter in which row the
entanglement was created, is given by:
$\displaystyle
P_{mult,1,2}=\sum_{k=1}^{\infty}(q^{4})^{k-1}(p^{4}+4p^{3}q+4p^{2}q^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}2p^{2}q^{2}\sum_{l=1}^{\infty}(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}(p^{3}+3p^{2}q+2pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}pq^{2}\sum_{m=1}^{\infty}(q^{2})^{m-1}(p^{2}+2pq)\;.$
$P_{mult,n,r}$ is the probability to create at least one entangled pair in
every column from the $2^{n}$ columns with $r$ rows, in the case of
multiplexing. The average number of steps needed for at least one success in
every column in this case can be calculated as follows:
$\displaystyle
Z_{mult,1,2}(q)=\sum_{k=1}^{\infty}(q^{4})^{k-1}k(p^{4}+4p^{3}q+4p^{2}q^{2})$
(31) $\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}2p^{2}q^{2}\sum_{l=1}^{\infty}(k+l)(q^{2})^{l-1}(p^{2}+2pq)+$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(k+l)(q^{3})^{l-1}(p^{3}+3p^{2}q+2pq^{2})+$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}pq^{2}\times$
$\displaystyle\sum_{m=1}^{\infty}(q^{2})^{m-1}(k+l+m)(p^{2}+2pq)$
$\displaystyle=$ $\displaystyle\frac{1+2q^{2}}{1-q^{4}}\;.$
The probability that at least two “parallel” entangled pairs are created is:
$\displaystyle
P_{parallel,1,2}=\sum_{k=1}^{\infty}(q^{4})^{k-1}(p^{4}+4p^{3}q+2p^{2}q^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4p^{2}q^{2}\sum_{l=1}^{\infty}(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}(p^{3}+3p^{2}q+pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}2pq^{2}\sum_{m=1}^{\infty}(q^{2})^{m-1}(p^{2}+2pq)\,.$
The corresponding average number of steps needed for parallel successes in two
columns ($n=1$) is given by:
$\displaystyle
Z_{parallel,1,2}(q)=\sum_{k=1}^{\infty}(q^{4})^{k-1}k(p^{4}+4p^{3}q+2p^{2}q^{2})$
(33) $\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4p^{2}q^{2}\sum_{l=1}^{\infty}(k+l)(q^{2})^{l-1}(p^{2}+2pq)$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k-1}4pq^{3}\sum_{l=1}^{\infty}(k+l)(q^{3})^{l-1}(p^{3}+3p^{2}q+pq^{2})$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}(q^{4})^{k}4pq^{3}\sum_{l=1}^{\infty}(q^{3})^{l-1}2pq^{2}\times$
$\displaystyle\sum_{m=1}^{\infty}(q^{2})^{m-1}(k+l+m)(p^{2}+2pq)$
$\displaystyle=$
$\displaystyle\frac{1+q+5q^{2}+4q^{4}}{1+q+q^{2}-q^{4}-q^{5}-q^{6}}\,.$
#### 2) two columns ($n=1$), $r>2$ rows
Calculations similar to those from Eq. (31) show that for 3, 4,…, r rows (see
Fig. 3 with $r$ memories per half node) the average number of steps needed for
at least one success in every column in the case of multiplexing is given by:
$\displaystyle Z_{mult,1,3}(q)=\frac{1+2q^{3}}{1-q^{6}},$ (34)
$\displaystyle Z_{mult,1,4}(q)=\frac{1+2q^{4}}{1-q^{8}},$ (35)
and
$\displaystyle Z_{mult,1,r}(q)=\frac{1+2q^{r}}{1-q^{2r}},$ (36)
respectively.
In the main text, Eq. (15) is, except for $T_{0}$, exactly the inverse of Eq.
(36) with $q=1-P_{0}$
## Appendix D
The choice of purifying the entangled pairs at the very beginning can be
intuitively justified through the probabilistic character of the purification
procedure and the deterministic entanglement swapping. Taking as much
advantage from the perfect memories as possible, it is reasonable to think
that the earlier the purification starts, the smaller the necessary times to
generate and purify an entangled pair will be. For a more quantitative
justification, let us compare the rates to generate a purified entangled pair
in two extreme cases. For purification occurring at the first nesting level,
the rate will be calculated as in Eq. (19). On the other hand, for
purification at the end (i.e., at the last nesting level), the rates will be
given by $R_{purif,n,end}=\frac{P_{1}}{T_{0}Z_{2n}(P_{0})}$. Recall that both
$P_{0}$ and $P_{1}$ are functions of the fidelity and that we are comparing
the rates for the same final fidelity. Hence these quantities will have
different values in each case. In Fig. (12) we illustrate that a scheme where
purification occurs in the first level performs better than one with
purification at the end.
Figure 12: (color online). Rates for the hybrid quantum repeater with one
round of purification at the first nesting level (blue dashed line) and at the
last nesting level (red line) for total distance $L=1280$ km and $L_{0}=20$
km.
|
arxiv-papers
| 2010-10-01T09:55:46 |
2024-09-04T02:49:13.277037
|
{
"license": "Public Domain",
"authors": "Nadja K. Bernardes, Ludmi{\\l}a Praxmeyer, and Peter van Loock",
"submitter": "Nadja Kolb Bernardes",
"url": "https://arxiv.org/abs/1010.0106"
}
|
1010.0115
|
# A modified proximity approach in the fusion of heavy-ions
Ishwar Dutt idsharma.pu@gmail.com Rajni Bansal Department of Physics, Panjab
University, Chandigarh -160 014, India.
###### Abstract
By using a suitable set of the surface energy coefficient, nuclear radius, and
universal function, the original proximity potential 1977 is modified. The
overestimate of the data by 4% reported in the literature is significantly
reduced. Our modified proximity potential reproduces the experimental data
nicely compared to its older versions.
###### pacs:
25.70.Jj, 24.10.-i.
## I Introduction
Recently, great theoretical and experimental efforts are taken to studying the
fusion of heavy nuclei leading to several new phenomena including the
understanding of the formation of neutron -rich and super heavy elements id1 ;
id2 . The precise knowledge of the interaction potential between two nuclei is
a difficult task and continuing efforts are needed in this direction. This
problem has been of very active research over the last three decades and
remains one of the most widely studied subject in low-energy heavy-ion physics
id1 ; id2 ; rkp1 ; blocki77 ; blocki81 ; wr94 ; ms2000 ; siwek04 ; wang06 ;
deni06 .
The total interaction potential is sum of the long range Coulomb repulsive
force and short range nuclear attractive force. The Coulomb part of the
interaction potential is well-known, whereas nuclear part is not clearly
understood. A large number of efforts have been made to giving simple and
accurate forms of the nuclear interaction potentials id1 ; id2 ; rkp1 ;
blocki77 ; blocki81 ; wr94 ; ms2000 ; siwek04 ; wang06 ; deni06 . Among such
efforts, proximity potential is well known for its simplicity and numerous
applications. Based upon the proximity force theorem blocki77 ; blocki81 , a
simple formula for ion-ion interaction potential as a function of the
separation between the surfaces of two approaching nuclei was presented
blocki77 ; blocki81 .
As pointed out by many authors ms2000 , original form of the proximity
potential 1977 overestimates the experimental data by 4% for fusion barrier
heights. In a recent study involving the comparison of 16 proximity
potentials, one of us and collaborators pointed out that proximity potential
1977 overestimates the experimental data by 6.7% for symmetric colliding
nuclei id1 . Similar results were obtained for asymmetric colliding nuclei id1
.
With the passage of time, several improvement/ modifications were made over
the original proximity potential 1977 to remove the gray part of the
potential. It includes either the better form of the surface energy
coefficient wr94 or the universal function and/or nuclear radius ms2000 . A
careful look reveals that these modifications/improvements are not able to
explain the experimental data id1 ; siwek04 . A deep survey also pointed out
that these technical parameters (i.e. surface energy coefficient, nuclear
radius, and universal function) were chosen quite arbitrarily in the
literature. Among them, the surface energy coefficient is available in a large
variety of forms from time to time id1 ; id2 . It affects the fusion barrier
heights and cross sections significantly id1 ; id2 . Also, nuclear radius is
available in large variety of forms id1 ; id2 . These forms varies either in
terms of its coefficients or either different mass or isospin dependence. The
third technical parameter i.e, the universal function, is also parametrized in
different forms id1 ; blocki77 ; ms2000 ; blocki81 . Unfortunately, no
systematic study is available in the literature, where one can explore the
role of these technical parameters in fusion barrier positions, heights, and
cross sections. Alternatively, a best set of the above-mentioned parameters is
still missing.
In the present study, our aim is to modify the original proximity potential
1977 by using a suitable set of the above-stated technical parameters
available in the literature. In addition, to compare the final outcome with
the huge amount of experimental data available since last three decades. The
choice of the potential and its form to be adopted is one of the most
challenging task when one wants to compare the experimental data with theory.
The present systematic study includes the reactions with combine mass between
A = 19 and A = 294 units. In total, 390 experimentally studied reactions with
symmetric as well as asymmetric colliding partners are taken into
consideration. Section II describes the Model in brief, Section III depicts
the Results and Summary is presented in Section IV.
## II The Model
The total ion-ion interaction potential $V_{T}(r)$ between two colliding
nuclei with charges $Z_{1}$ and $Z_{2}$, center separation $r$, and density
distribution assumed spherical, and frozen, is approximated as ms2000
$\displaystyle V_{T}(r)=V_{N}(r)+\frac{Z_{1}Z_{2}e^{2}}{r},$ (1)
where e is the charge unit. The above form of the Coulomb potential is
suitable when two approaching nuclei are well separated. The nuclear part of
the potential $V_{N}(r)$ is calculated in the framework of the proximity
potential 1977 blocki77 as
$V_{N}\left(r\right)=4\pi\overline{R}\gamma
b\Phi(\frac{r-C_{1}-C_{2}}{b}){~{}\rm MeV},$ (2)
where $\overline{R}=\frac{C_{1}C_{2}}{C_{1}+C_{2}}$ is the reduced radius.
Here $C_{i}$ denotes the matter radius and is calculated using relation ms2000
$C_{i}=c_{i}+\frac{N_{i}}{A_{i}}t_{i}~{}~{}~{}~{}(i=1,2),$ (3)
where $c_{i}$ denotes the half-density radii of the charge distribution and
$t_{i}$ is the neutron skin of the nucleus. To calculate $c_{i}$, we used the
relation given in Ref. ms2000 as
$c_{i}=R_{00i}\left(1-\frac{7}{2}\frac{b^{2}}{R_{00i}^{2}}-\frac{49}{8}\frac{b^{4}}{R_{00i}^{4}}+\cdots\cdots\right)~{}~{}~{}~{}~{}~{}~{}(i=1,2).$
(4)
Here, $R_{00}$ is the nuclear charge radius read as
$R_{00i}=1.2332A^{1/3}_{i}\left\\{1+\frac{2.348443}{A_{i}}-0.151541\left(\frac{N_{i}-Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm},$ (5)
where $N_{i}$ and $Z_{i}$ refer to neutron and proton contents of
target/projectile nuclei. This form of radius is taken from the recent work of
Royer and Rousseau gr09 and is obtained by analyzing as many as 2027 masses
with N, Z $\geq$ 8 and a mass uncertainty $\leq$ 150 keV. The neutron skin
$t_{i}$ used in Eq. (3) is calculated according to Ref. ms2000 .
The surface energy coefficient $\gamma$ was taken from the work of Myers and
Świa̧tecki ms66 and has the form
$\gamma=\gamma_{0}\left[1-k_{s}\left(\frac{N-Z}{A}\right)^{2}\right],$ (6)
where N and Z refer to the total neutrons and protons content. It is clear
from Eqs. (5) and (6) that both nuclear radius as well as surface energy
coefficient depend on the relative neutron excess. In the above formula,
$\gamma_{0}$ is the surface energy constant and $k_{s}$ is the surface-
asymmetry constant. Both constants were first parameterized by Myers and
Świa̧tecki ms66 by fitting the experimental binding energies. The first set
of these constants yielded values $\gamma_{0}$ and
$k_{s}=1.01734~{}\rm~{}MeV/fm^{2}$ and 1.79, respectively. In original
proximity version, $\gamma_{0}$ and $k_{s}$ were taken to be
$0.9517~{}\rm~{}MeV/fm^{2}$ and 1.7826 ms67 , respectively. Later on, these
values were revised in a large variety of forms depending upon the advancement
in the theory as well in experiments id1 ; id2 . In total, 14 such
coefficients are highlighted in Ref. id2 and the role of extreme 4 sets is
analyzed deeply. Out of them, two best sets of surface energy coefficients are
stressed. In the present study, we shall restrict to the latest set of
$\gamma$ values i.e. $\gamma_{0}$ =$1.25284{~{}\rm MeV/fm^{2}}$ and $k_{s}$ =
$2.345$ presented in Ref id2 . This particular set of values were obtained
directly from a least-squares adjustment to the ground-state masses of 1654
nuclei ranging from 16O to 263106 and fission-barrier heights mn95 .
The universal function $\Phi(\frac{r-C_{1}-C_{2}}{b})$ used in Eq. (1) has
been derived by several authors in different forms blocki77 ; ms2000 ;
blocki81 . In original proximity potential, $\Phi(\frac{r-C_{1}-C_{2}}{b})$
was parametrized in the cubic-exponential form blocki77
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l}-\frac{1}{2}\left(\xi-2.54\right)^{2}-0.0852\left(\xi-2.54\right)^{3},~{}\mbox{
for $\xi\leq 1.2511$ },\\\
-3.437\exp\left(-\frac{\xi}{0.75}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.2511$ },\end{array}\right.$ (7)
with $\xi$ = $(r-C_{1}-C_{2}$)/$b$. The surface width $b$ (i.e.
$b=\frac{\pi}{\sqrt{3}}a~{}{\rm with}~{}a=0.55~{}\rm fm)$ has been evaluated
close to unity. We labeled this universal function as $\Phi$-${1977}$.
Later on, Blocki et al., blocki81 modified the above form as
$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l}-1.7817+0.9270\xi+0.143\xi^{2}-0.09\xi^{3},~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\leq 0.0$ },\\\ -1.7817+0.9270\xi+0.01696\xi^{2}-0.05148\xi^{3},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $0.0\leq\xi\leq 1.9475$ },\\\
-4.41\exp\left(-\frac{\xi}{0.7176}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.9475$ }.\end{array}\right.$ (8)
In the present study, we use this form of universal function and marked it as
$\Phi$-${1981}$. By using the above stated parameters, we construct a new
proximity potential and labeled as Prox 2010. Along with the above modified
form, we shall also use the original proximity potential 1977 blocki77 and
its recently modified form proximity potential 2000 ms2000 . We labeled them
as Prox 1977 and Prox 2000, respectively.
## III Results and Discussions
By using the above new version of the proximity potential (Prox 2010) along
with its older versions (i.e. Prox 1977 and Prox 2000), fusion barriers are
calculated for 390 reactions by using the conditions:
$\frac{dV_{T}(r)}{dr}|_{r=R_{B}}=0,~{}~{}{\rm{and}}~{}~{}\frac{d^{2}V_{T}(r)}{dr^{2}}|_{r=R_{B}}\leq
0.$ (9)
The height of the barrier and position is marked, respectively, as $V_{B}$ and
$R_{B}$.
As one see from the preceding section, three factors govern the success of
proximity potential are (i) the surface energy coefficient, (ii) the universal
function, and (iii) nuclear radius. We analyzed the literature very carefully
and found that the latest information on these three factors can shape the new
proximity potential. Recently, the role of surface energy coefficient stated
above is studied in detail in Ref id2 . As for as radius is concern, we shall
restrict to its latest form given in Ref. gr09 . However, the role of the
third parameter i.e., the universal function in fusion barriers is analyzed in
Fig. 1. Here, we display $\Delta V_{B}~{}(\%)$ and $\Delta R_{B}~{}(\%)$
defined as
$\Delta V_{B}~{}(\%)=\frac{V_{B}^{theor}-V_{B}^{expt}}{V_{B}^{expt}}\times
100,$ (10)
and
$\Delta R_{B}~{}(\%)=\frac{R_{B}^{theor}-R_{B}^{expt}}{R_{B}^{expt}}\times
100,$ (11)
as a function of $Z_{1}Z_{2}$ using two sets of above mentioned universal
functions [Eqs. (7) and (8)]. It is clear from the figure that deviations are
significantly reduced by using $\Phi$-${1981}$ compared to its original form
$\Phi$-${1977}$. The universal function $\Phi$-${1981}$ reduces the average
deviation over 390 reactions by 1 % for fusion barriers. The experimental
values are taken directly from the literature id1 ; id2 ; ms2000 . Actually,
it is clear from the literature that no experiment can extract information
about the fusion barriers directly. All experiments measure the fusion
differential cross sections and then with the help of a theoretical model, one
can extract the fusion barriers.
In Fig. 2, we display the theoretical fusion barrier heights $V_{B}^{theor}$
(MeV) and positions $R_{B}^{theor}$ (fm) verses the corresponding experimental
values. We note from the figure that Prox 2010 potential reproduces the
experimental fusion barrier heights within 1.4%. This result is in close
agreement with other recently parametrized potentials presented in Ref. id1 .
However, the original form of the proximity potential 1977 presented in Ref.
id1 overestimates the data by 6.7% for symmetric colliding nuclei. However,
the fusion barrier positions show some scattering from the central line
(marked by shaded area). This scattering may be due to the variation in the
experimental setups and theoretical method one used to extract these values
Aljuwair84 ; Trotta01 .
We quantify our outcome in Figs. 3 and 4. In Fig. 3, the percentage deviations
between the theoretical and experimental values are presented. The original
proximity potential 1977 (Prox 1977) along with its recently modified form
(Prox 2000) are also displayed. We note from the upper panel of Fig. 3 that
Prox 2010 potential on average gives better results compared to its older
versions for fusion barrier heights. However, slight deviations are visible
for fusion barrier positions. This may be due to the fact that in the
proximity potential Prox 2010, we use the value of surface energy coefficient
that gives stronger attraction compared to one used in Prox 1977 and Prox 2000
potentials. Therefore, in Prox 2010 potential, the counterbalance between the
repulsive Coulomb and attractive nuclear part of the interaction potential
occurs at larger distances, and hence it pushes the barrier outwards. The
fusion barrier heights are reproduced within $\pm~{}5\%$ on average. On the
other hand, fusion barrier positions reproduced the experimental values within
$\pm~{}10\%$. Especially for the heavier colliding nuclei, we see that Prox
2010 potential reproduces the data much better on the average compared to
other versions. For lighter nuclei, however, small scattering is visible. This
could also be due to the uncertainty in the radius of the lighter colliding
nuclei. In Figs. 1-4, only 155 reactions are displayed to maintain the
clarity. The average deviation for the fusion barrier heights over 390
reactions is 0.77 % using our modified potential Prox 2010, whereas Prox 1977,
and Prox 2000 give 3.99 %, and 4.45 %, respectively. This shows that our
modified proximity explains the experimental data nicely.
In Fig. 4, we display the difference between the theoretical and
experimentally extracted fusion barriers. We further note that Prox 2010
potential gives better results. The difference especially for the heavy
systems is significantly reduced. This was the problem with original as well
as its recently modified form as pointed out by several authors ms2000 ;
siwek04 . It is clear from Figs. 3 and 4, that Prox 2010 potential is able to
reproduce the experimental data much better than its older versions. The small
difference is not significant because of the uncertainties in the analysis of
the experimental data.
Finally, we test our newly modified proximity potential Prox 2010 on fusion
probabilities. In Fig. 5, we display the fusion cross sections $\sigma_{fus}$
(in mb) as a function of the center-of-mass energy $E_{c.m.}$ (MeV) for the
reactions of ${}^{26}Mg+^{30}Si$ Morsad90 , ${}^{16}O+^{46}Ti$ Neto90 ,
${}^{48}Ca+^{48}Ca$ Stefanini09 , ${}^{12}C+^{92}Zr$ newton01 ,
${}^{40}Ca+^{58}Ni$ sikora79 , and ${}^{16}O+^{144}Sm$ leigh95 . The fusion
cross sections are calculated using well known Wong model wg72 . The older
versions of proximity potentials that is, Prox 1977 and Prox 2000 are also
displayed. It is clearly visible from the figure that Prox 2010 potential is
in good agreement, whereas, its older forms are far from the experimental
data. We further note that Prox 1977 and Prox 2000 potentials show similar
results. It means that no improvements is seen in Prox 2000 potential as was
claimed in Ref. ms2000 .
## IV Summary
In the present study, we present a best set of the surface energy coefficient,
the nuclear radius, and the universal function available in the literature. We
find that these parameters which were used quite arbitrarily in past years
affect the fusion barrier heights, positions, and cross sections
significantly. By using the above set of parameters, a new proximity potential
is constructed. Our newly constructed proximity potential Prox 2010 reproduces
the fusion barriers and cross sections better than its earlier versions.
## References
## References
* (1) I. Dutt and R. K. Puri Phys. Rev. C 81, 044615 (2010); 81, 064609 (2010).
* (2) I. Dutt and R. K. Puri Phys. Rev. C 81, 047601 (2010);81, 064608 (2010).
* (3) R. K. Puri _et al._ , Eur. Phys. J. A 23, 429 (2005); R. Arora _et al._ , ibid. 8, 103 (2000); R. K. Puri _et al._ , ibid. 3, 277 (1998); R. K. Puri _et al._ , Phys. Rev. C 45, 1837 (1992); ibid. 43, 315 (1991).
* (4) J. Blocki, J. Randrup, W. J. Świa̧tecki, and C. F. Tsang, Ann. Phys. (N.Y.) 105, 427 (1977).
* (5) J. Blocki and W. J. Świa̧tecki, Ann. Phys. (N.Y.) 132, 53 (1981).
* (6) W. Reisdorf, J. Phys. G: Nucl. Part. Phys. 20, 1297 (1994).
* (7) W. D. Myers and W. J. Świa̧tecki, Phys. Rev. C 62, 044610 (2000); and earlier references therein.
* (8) K. Siwek-Wilczyńska and J. Wilczyński, Phys. Rev. C 69, 024611 (2004).
* (9) A. Dobrowolski, K. Pomorski, and J. Bartel, Nucl. Phys. A 729, 713 (2003); V. I. Zagrebaev, Nucl. Phys. A 734, 164 (2004); N. Wang _et al._ , Phys. Rev. C 78, 054607 (2008).
* (10) V. Y. Denisov, Phys. Lett. B526, 315 (2002).
* (11) G. Royer and R. Rousseau, Eur. Phys. J. A 42, 541 (2009).
* (12) W. D. Myers and W. J. Świa̧tecki, Nucl. Phys. 81, 1 (1966).
* (13) W. D. Myers and W. J. Świa̧tecki, Ark. Fys. 36, 343 (1967).
* (14) P. Möller, J. R. Nix, W. D. Myers, and W. J. Światecki, At. Data Nucl. Data Tables 59, 185 (1995).
* (15) H. A. Aljuwair et al., Phys. Rev. C 30, 1223 (1984).
* (16) M. Trotta _et al._ , Phys. Rev. C 65, 011601(R) (2001).
* (17) A. Morsad,J. J. Kolata, R. J. Tighe, X. J. Kong, E. F. Aguilera, and J. J. Vega, Phys. Rev. C 41, 988 1990
* (18) R. L. Neto, J. C. Acquadro, P. R. S. Gomes, A. S. D. Toledo, C. F. Tenreiro, E. Crema, N. C. Filho, and M. M. Coimbra, Nucl. Phys. A512, 333 (1990).
* (19) A. M. Stefanini, _et al._ , Phys. Lett. B 679, 95 (2009).
* (20) J. O. Newton, C. R. Morton, M. Dasgupta, J. R. Leigh, J. C. Mein, D. J. Hinde, H. Timmers, and K. Hagino, Phys. Rev. C 64, 064608 (2001).
* (21) B. Sikora, J. Bisplinghoff, W. Scobel, M. Beckerman, and M. Blann, Phys. Rev. C 20, 2219 (1979).
* (22) J. R. Leigh _et al._ , Phys. Rev. C 52, 3151 (1995).
* (23) C. Y. Wong, Phys. Lett. B42, 186 (1972); C. Y. Wong, Phys. Rev. Lett. 31, 766 (1973).
Figure 1: The percentage deviation $\Delta V_{B}~{}(\%)$ and $\Delta
R_{B}~{}(\%$) as a function of $Z_{1}Z_{2}$ using two different sets of
universal functions [Eqs. (7) and (8)] implemented in the original proximity
potential Prox 1977. The experimental values are taken from Refs. id1 ; id2 ;
ms2000 . Figure 2: The fusion barrier heights $V_{B}$ (MeV) and positions
barriers $R_{B}$ (fm) as a function of the corresponding experimental values
using our modified proximity potentials Prox 2010. The experimental values are
taken from Refs. id1 ; id2 ; ms2000 Figure 3: The same as Fig.1, but for
different older proximity potentials along with our modified form i.e., Prox
1977, Prox 2000, and Prox 2010, respectively. Figure 4: The variation of
$\Delta V_{B}$ $(=V_{B}^{theor}-V_{B}^{expt})$ and $\Delta R_{B}$
$(=R_{B}^{theor}-R_{B}^{expt})$ as a function of $Z_{1}Z_{2}$ using Prox 1977,
Prox 2000, and Prox 2010 potentials. The experimental values are taken from
Refs. id1 ; id2 ; ms2000 . Figure 5: (Color online) The fusion cross sections
$\sigma_{fus}$ (mb) as a function of center-of-mass energy $E_{c.m.}$ using
older versions of proximity potential (Prox 1977 and Prox 2000) along with new
version (Prox 2010). The experimental data are taken from Morsad 1990 Morsad90
, Neto 1990 Neto90 , Stefanini Stefanini09 , Newton 2001 newton01 , Sikora
1979 sikora79 , and Leigh 1995 leigh95 .
|
arxiv-papers
| 2010-10-01T10:19:21 |
2024-09-04T02:49:13.286591
|
{
"license": "Public Domain",
"authors": "Ishwar Dutt and Rajni Bansal",
"submitter": "Ishwar Dutt",
"url": "https://arxiv.org/abs/1010.0115"
}
|
1010.0172
|
# Prime order automorphisms of Klein surfaces representable by rotations of
the euclidean space
Antonio F. Costa
UNED
Facultad de Ciencias
Senda del Rey, 9
28040 Madrid
Spain Cam Van Quach Hongler
Université de Genève
Section de mathematiques
2-4, rue du Lièvre
CH-1211 Genève 64
Switzerland
## 1 Introduction
A (closed) surface embedded in the Euclidean space inherits a conformal
structure so it is a Riemann surface. This is the easiest way to present
examples of Riemann surfaces. In 1882, F. Klein asked the following question
([K1]): is every Riemann surface conformally equivalent to an embedded surface
in the Euclidean space?. The positive answer was given by A. Garsia in 1960,
[G]. Several generalizations have been produced in [R1] and [Ko], [Ko2],
[Ko3].
In the study of moduli spaces of Riemann surfaces, the surfaces with
automorphisms play an important rôle: the branch loci for the orbifold
structure of the moduli space consists of such type of surfaces. A classical
and easy method to present examples of Riemann surfaces with automorphisms is
to consider embedded surfaces in the Euclidean space that are invariant by
isometries. R. Rüedy in [R2] describes the automorphisms of compact Riemann
surfaces that can be represented by rotations of the space acting on embedded
surfaces. Later, in [C], the anticonformal automorphisms of Riemann surfaces
representable by isometries on embedded Riemann surfaces were completely
characterized. Now we are interested in the presentation and visualization of
automorphisms of Riemann surfaces with boundary as embedded surfaces.
In order to consider bordered surfaces we shall use the theory of Klein
surfaces. A compact bordered (with non-empty boundary) orientable Klein
surface is a compact bordered and orientable surface endowed with a dianalytic
structure. Let $S$ be a bordered orientable Klein surface and $p$ a prime.
Assume that $f$ is an order $p$ automorphism of $S$. In this work we obtain
the conditions on the topological type of $(S,f)$ to be conformally equivalent
to $(S^{\prime},f^{\prime})$ where $S^{\prime}$ is a bordered orientable Klein
surface embedded in the Euclidean space and $f^{\prime}$ is the restriction to
$S^{\prime}$ of a prime order rotation. The boundary of $S^{\prime}$ is a link
in $\mathbb{R}^{3}$ and $S^{\prime}$ is a Seifert surface of $\partial
S^{\prime}$. Since the Klein structure follows from a result in [Ko], the main
difficulty for our task is the topological realization (Lemma 5).
Our results can help to visualize some automorphisms of Riemann surfaces that
are not easy to see representing such automorphisms as restrictions of
isometries of $\mathbb{S}^{4}$ or $\mathbb{R}^{4}$. In Section 3 we represent
two famous automorphisms using this method: the order seven automorphisms of
the Klein quartic and the Wiman surface.
## 2 Prime order automorphisms of Klein surfaces representable by rotations
A bordered orientable Klein surface is a compact surface with non-empty
boundary endowed with a dianalytic structure, i.e. an equivalence class of
dianalytic atlases. A dianalytic atlas is a set of complex charts with
analytic or antianalytic transition maps (see [AG]). If $S$ is an orientable
Klein surface with genus $g$ and $k$ boundary components, there is a complex
double $\widehat{S}$ of $S$ that is a Riemann surface with genus
$\widehat{g}=2g+k-1$. An automorphism $f$ of $S$ is an (auto)-homeomorphism of
$S$ such that locally on each chart of $S$ is analytic or antianalytic. If
$\widehat{g}>1$ then the automorphism $f$ has finite order. Thus we shall
assume $\widehat{g}>1$ and the orientability of the surfaces.
Let $S$ and $S^{\prime}$ be bordered orientable Klein surfaces that are
homeomorphic and let $f,f^{\prime}$ be automorphisms of $S$ and $S^{\prime}$
respectively. We say that $f$ and $f^{\prime}$ are topologically equivalent or
have the same topological type if there is a homeomorphism $h:S\rightarrow
S^{\prime}$, such that $f=h^{-1}\circ f^{\prime}\circ h$.
Two automorphims topologically equivalent have the same order and the same
number of fixed points. To avoid technical complications we shall consider
automorphisms of prime order $p>2$. In order to describe the topological types
of automorphisms of bordered orientable Klein surfaces we need two sets of
invariants.
###### Definition 1
Rotation indexes for the fixed points of an automorphism. Let $S$ be a
bordered orientable Klein surface and $f$ be an automorphism of $S$ of prime
order $p$. Assume $S$ is oriented. Let $\\{x_{1},...,x_{r}\\}$ be the set of
fixed points of $f$. Let $D_{1}$,…,$D_{r}$ be a set of disjoint discs in $S$
such that $x_{j}\in D_{j}$. There are charts
$c_{j}:D_{j}\rightarrow\mathbb{C}$ of $S$ (sending the orientation of $D_{j}$
on the canonical orientation of $\mathbb{C}$) such that $c_{j}\circ f\circ
c_{j}^{-1}:c_{j}(D_{j})\rightarrow c_{j}(D_{j})$ is $z\rightarrow e^{2\pi
iq_{j}/p}$, $-p/2<q_{j}<p/2$. The set of rotation indexes $\\{\epsilon
2q_{1}\pi/p,...,\epsilon 2q_{r}\pi/p\\}$, with $\epsilon=1$ or $-1$, is an
invariant for the topological type of $f$.
###### Definition 2
Rotation indexes for the boundary components. Let $S$ be a bordered orientable
Klein surface and $f$ be an automorphism of $S$ of order $p$. Let
$\\{C_{1},...,C_{k}\\}$ be the set of boundary connected components of $S$.
Assume that $S$ is oriented. There are homeomorphisms
$h_{j}:C_{j}\rightarrow\mathbb{S}^{1}=\\{z\in\mathbb{C}:\left\|z\right\|=1\\}$
preserving the orientation of $C_{j}$ induced by the orientation of $S$ such
that $h_{j}\circ f\circ h_{j}^{-1}:h_{j}(D_{j})\rightarrow h_{j}(D_{j})$ is
$z\rightarrow e^{2\pi iq_{j}/p}$, $-p/2<q_{j}<p/2$. The set of rotation
indexes $\\{\epsilon 2q_{1}\pi/p,...,\epsilon 2q_{k}\pi/p\\}$, with
$\epsilon=1$ or $-1$, is an invariant for the topological type of $f$.
The fact of $S$ be a Klein surface does not give an orientation to $S$ then
the sets $\\{2q_{1}\pi/p,...,2q_{k}\pi/p\\}$ and
$\\{-2q_{1}\pi/p,...,-2q_{k}\pi/p\\}$ must be considered equal.
###### Theorem 3 (Classification of automorphisms of Klein surfaces [Y])
Let $f$ and $f^{\prime}$ two orientation preserving automorphisms of prime
order $p$ of two homeomorphic bordered orientable Klein surfaces. The
automorphisms $f$ and $f^{\prime}$ are topologically equivalent if and only if
the set of rotation indexes for the fixed points and the boundary components
of $f$ are equal to the set of rotation indexes for the fixed points and the
boundary components of $f^{\prime}$.
Let $S$ be a bordered orientable surface embedded in the Euclidean space. The
Euclidean metric induces a structure of Klein surface on $S$. If $\rho$ is a
rotation in the space and $S$ is invariant by $\rho$, $\rho$ restricted to $S$
is an automorphism $\rho_{S}$ of the Klein surface $S$. If $h:S\rightarrow K$
is an isomorphism between Klein surfaces the automorphism $h\circ\rho_{S}\circ
h^{-1}$ is an automorphism representable as a rotation.
###### Theorem 4
Let $f$ be an automorphism of prime order $p$ of a bordered orientable Klein
surface (we consider that $f$ is orientation preserving if $p=2$). The
automorphism $f$ is representable as a rotation if and only if the set of
rotation indices of fixed points of $f$ is contained in $\\{\pm 2q\pi/p\\}$,
for some $q$ such that $0<q<p/2$.
Note that there are no restriction on the rotation indices of boundary
components.
If $f$ is representable as a rotation of angle $2q\pi/p$ then it is necessary
that the rotation indices of the fixed points of $f$ are $2q\pi/p$ or
$-2q\pi/p$. For the sufficiency in the Theorem first we need a topological
construction that we shall present in the following Lemma.
###### Lemma 5 (Topological construction)
Let $S$ be a bordered orientable compact surface and $h$ be an
autohomeomorphism of $S$ with order $p$ where $p$ is a prime (if $p=2$ we
assume also that $h$ is orientation preserving). Assume that the rotation
index of every fixed point of $h$ is equal to $\pm\frac{2\pi q}{p}$, where
$0<q<q/2$. Hence there is a surface $\Sigma$ embedded in $\mathbb{R}^{3}$ such
that $\Sigma$ is equivariant under a rotation $\rho$ of angle $\frac{2\pi
q}{p}$ and $(S,h)$ is topologically equivalent to $(\Sigma,\rho)$.
Proof. Consider $S/\left\langle h\right\rangle$ the orbit surface. Assume that
$S/\left\langle h\right\rangle$ has genus $g$ and $t+l$ boundary components
and the quotient orbifold $O(S/\left\langle h\right\rangle)$ has $r$ conic
points. Let $\omega:\pi_{1}O(S/\left\langle h\right\rangle)\rightarrow
C_{p}=\left\langle\gamma\right\rangle$ be the monodromy epimorphism of the
branched covering $S\rightarrow S/\left\langle h\right\rangle$. Corresponding
to $\omega$, there is a canonical presentation of $\pi_{1}O(S/\left\langle
h\right\rangle)$:
$\pi_{1}O(S/\left\langle h\right\rangle)=\left\langle
a_{i},b_{i},x_{i},e_{i},c_{i}:\Pi[a_{i},b_{i}]\Pi x_{i}\Pi
e_{i}=1,x_{i}^{p}=1,e_{i}c_{i}e_{i}^{-1}c_{i}=1,c_{i}^{2}=1\right\rangle$
such that: $\omega(a_{i})=\omega(b_{i})=\omega(c_{i})=1$,
$\omega(x_{j})=\gamma^{q}$, $j=1,...,n$, $\omega(x_{j})=\gamma^{p-q}$,
$j=n+1,...,r$, $\omega(e_{j})=\gamma^{q_{j}}$, $j=1,...,t$, $0<q_{i}<p,$
$\omega(e_{j})=1$, $j=t+1,...,t+l$. That means there are $n$ fixed points of
$h$ with rotation index $\frac{2\pi q}{p}$ and $r-n$ fixed points of $h$ with
rotation index $-\frac{2\pi q}{p}$ and the rotation index of the boundary
component $C_{i}$ of $S/\left\langle h\right\rangle$ corresponding to $e_{i}$
is $\frac{2\pi q_{i}}{p}$.
Consider the usual orientation of $\mathbb{R}^{3}$ with $Oz$ oriented in the
increasing way of $z$. Consider $r$ discs $D_{i},i=1,...r$
$D_{i}=\left\\{(x,y,i):\left\|(x,y)\right\|\leq 1\right\\}$
Assume that $D_{i}$ has the orientation producing in $\partial D_{i}$ an
orientation such that the linking number with $Oz$ is:
$\displaystyle Lk(\partial D_{i},Oz)$ $\displaystyle=1,i=1,...,n$
$\displaystyle Lk(\partial D_{i},Oz)$ $\displaystyle=-1,i=n+1,...,r$
Consider in
$R_{m}=\\{(x,y,z)\in\mathbb{R}^{3}:-m-1<z<-m\\}$
for $m\in\\{1,...,t-1\\}$, an embedded oriented annulus $C_{m}$ such that
$C_{m}\cap Oz=\varnothing$, $\partial C_{m}=C_{m}^{+}\cup C_{m}^{-}$ and
$Lk(C_{m}^{-},Oz)=q_{m}$
Figure 1: An annulus $C_{m}$ with $q_{m}=3$
Let us connect $D_{i}$ with $D_{i+1}$ by 1-handles $b_{i}$, $C_{i}$ with
$C_{i+1}$ by 1-handles $h_{i}$ (more precisely, $C_{i}^{+}$ to $C_{i+1}^{+}$)
and $D_{1}$ with $C_{1}$ by the 1-handle $h$ (more precisely, $\partial D_{1}$
with $C_{1}^{+}$) in such a way that:
$S_{0}={\textstyle\bigsqcup}D_{i},C_{i},b_{i},h_{i},h$
is an embedded surface homeomorphic to a sphere with $t$ holes. Furthermore
the 1-handles must connect the discs and the annuli such that $S_{0}$ has an
orientation which induces on $D_{i}$ and $C_{i}$ their desired orientations
and $S_{0}\cap Oz$ are exactly the centers of the discs $D_{i}$.
Figure 2: Example of $S_{0}$
Let $F$ be a surface with the topological type of genus $g$ with $l$ boundary
components and contained in
$\\{(x,y,z)\in\mathbb{R}^{3}:x>2\text{, }z<-t\\}$
and $Y$ an annulus with the boundaries on the boundaries of discs $\Delta_{1}$
in $S_{0}$ and $\Delta_{2}$ in $F$, and with $Y\cap Oz=\varnothing$. The
surface $(S_{0}-\Delta_{1})\sqcup Y\sqcup(F-\Delta_{1})$ will be denoted $M$.
Let
$f:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}/\left\langle\rho\right\rangle=\mathbb{R}^{3}$
be the quotient map given by the orbits of the rotation $\rho$ of angle
$\frac{2\pi q}{p}$ around the axis $Oz$. The surface $\Sigma$ that we are
looking for is $f^{-1}(M)$. Note that each boundary component $C_{j}^{-}$ of
$C_{j}$ remains as a boundary component of $M$, and since
$Lk(C_{j}^{-},Oz)=q_{j}$, the action of the rotation $\rho$ on
$f^{-1}(C_{j}^{-})$ is equivalent to the action of $z\rightarrow e^{2\pi
iq_{j}/p}$ on the unit cicle in complex plane.
Proof of the Theorem. Let $(\Sigma,\rho)$ be a topological realization of
$(S,h)$ given by the Lemma above. Using [Ko] we can deform
$\Sigma/\left\langle\rho\right\rangle$ to an embedded surface
$\Sigma^{\prime}$ that is conformally equivalent to $S/\left\langle
h\right\rangle$, and since the deformation can be chosen very close to
$\Sigma/\left\langle\rho\right\rangle$, we can consider that
$\Sigma^{\prime}\cap OZ=\varnothing$. If
$f:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}/\left\langle\rho\right\rangle=\mathbb{R}^{3}$,
$f^{-1}(\Sigma^{\prime})$ is the surface that we are looking for.
###### Remark 6
It is possible to extend the results in this Section to automorphisms of Klein
surfaces with boundary orientable or not. The result obtained in that
situation is as follows: Let $f$ be an automorphism of prime order $p$ of a
bordered Klein surface. The automorphism $f$ is representable as a rotation if
and only if the rotation indices of fixed points of $f$ are all equal to
$2q\pi/p$, for some $q$ such that $0<q<p/2$.
## 3 Examples: visualizing two famous automorphisms of order seven.
### 3.1 The order seven automorphism of the Klein quartic
In [K2], F. Klein introduces the a Riemann surface $S_{K}$ of genus $3$ with
automophisms group isomorphic to $PSL(2,7)$. As the title of [K2] suggests the
automorphisms of order seven of $S_{K}$ have a special importance and Klein
remarks that such an automorphism cannot be represented as the restriction of
a rotation on an embedded surface in the Euclidean space.
The surface $S_{K}$ is described as the quartic $x^{3}y+y^{3}z+z^{3}x=0$ with
the order seven automorphism $f$:
$x\rightarrow\zeta x$, $y\rightarrow\zeta^{2}y$, $z\rightarrow\zeta^{4}z$,
where $\zeta=e^{2\pi i/7}$.
The orbit space $S_{K}/\left\langle f\right\rangle$ is the Riemann sphere and
$S_{K}\rightarrow S_{K}/\left\langle f\right\rangle$ is a cyclic covering
branched on three points, i. e. the automorphism $f$ has three fixed points:
$p_{1}$, $p_{2}$ and $p_{3}$. Let $D_{1}$, $D_{2}$ and $D_{3}$ be three
disjoint discs in $S_{K}$ around $p_{1},p_{2}$ and $p_{3}$ respectively and we
shall consider that $D_{i}$ is equivariant by the action of $f$. Assume that
the action of $f$ on $D_{1}$ is equivalent to the rotation $z\rightarrow\zeta
z$, on $D_{2}$ is equivalent to the rotation $z\rightarrow\zeta^{2}z$ and on
$D_{3}$ is equivalent to the rotation $z\rightarrow\zeta^{4}z$.
Now consider the bordered orientable Klein surface $B_{K}=S_{K}-$
$\overset{\circ}{D}_{2}\cup\overset{\circ}{D}_{3}$. Then $(B_{K},f)$ is
representable by a rotation, i. e., $B_{K}$ is isomorphic to a surface
$B_{K}^{\prime}$ that is embedded in the Euclidean space in such a way that
$f$ produces on $B_{K}^{\prime}$ the restriction of a rotation of order seven
$\rho$. The Figure 3 shows the surface $B_{K}^{\prime}$ and has been
constructed using the method in Section 2. Note that the surface in the Figure
3 is topologically equivalent to $(B_{K},f)$ and the rotation indices in the
fixed points and boundaries coincide.
Figure 3: The order seven automorphism of the Klein quartic without two discs
embedded in the Euclidean space
Consider $\mathbb{S}^{4}$ as two dimension 4 balls $B_{1}$, $B_{2}$ such that
$\partial B_{1}=\partial B_{2}$. The surface $B_{K}^{\prime}$ is inside the
tridimensional sphere $\partial B_{1}=\partial B_{2}$ in such a way that the
rotation giving the order seven automorphism is the restriction of a rotation
of $\mathbb{S}^{4}$. Let $C_{1}$ and $C_{2}$ be the boundary components of
$B_{K}$. The cone on $C_{1}$ from the center $c_{1}$ of $B_{1}$ and the cone
on $C_{2}$ from the center $c_{2}$ of $B_{2}$, produce a genus $3$ surface
$S_{K}^{\prime}$ immersed in $\mathbb{S}^{4}$. There is a rotation $r$ of
$\mathbb{S}^{4}$ leaving invariant $S_{K}^{\prime}$ and such that
$(S_{K}^{\prime},r)$ is topologically equivalent to $(S_{K},f)$.
Furthermore, since $S_{K}^{\prime}-\\{c_{1},c_{2}\\}/\left\langle
r\right\rangle$ is a sphere without two points then the conformal structure
inherited from $\mathbb{S}^{4}/\left\langle r\right\rangle$ is conformally
equivalent to the Riemann sphere without two points. Such structure of
$S_{K}^{\prime}-\\{c_{1},c_{2}\\}/\left\langle r\right\rangle$ lifts in a
unique equivariant way to $S_{K}^{\prime}-\\{c_{1},c_{2}\\}$. Hence the
surface $S_{K}^{\prime}-\\{c_{1},c_{2}\\}$ with the conformal structure given
by $\mathbb{S}^{4}$ is conformally equivalent to the Klein quartic without two
points, so in this case is not necessary to deform the surface using [Ko].
### 3.2 The order seven automorphism of the Wiman surface of genus $3$
There is only one Riemann surface of genus $3$ and with an automorphism of
order $7$ different from the Klein quartic. This is the genus $3$ Wiman
surface $S_{A}$: $w^{2}=z^{7}-1$ [W]. In this case the order seven
automorphism $f_{A}$ has also three fixed points: $p_{1}$, $p_{2}$ and
$p_{3}$. Let $D_{1}$, $D_{2}$ and $D_{3}$ be three disjoint discs in $S_{A}$
around $p_{1},p_{2}$ and $p_{3}$ respectively and $D_{i}$ is equivariant by
the action of $f_{A}$. The action of $f_{A}$ on $D_{1}$ is equivalent to the
rotation $z\rightarrow\zeta z$, on $D_{2}$ is equivalent to the rotation
$z\rightarrow\zeta z$ and on $D_{3}$ is equivalent to the rotation
$z\rightarrow\zeta^{5}z$. The surface $B_{A}=S_{A}-\overset{\circ}{D}_{3}$ is
homeomorphic to a surface $B_{A}^{\prime}$ that is embedded in the Euclidean
space in such a way that $f_{A}$ induces on $B_{A}^{\prime}$ the restriction
of a rotation: see the Figure 4. In fact $B_{A}$ is the Seifert surface of
minimal genus of the link of the singularity $w^{2}-z^{7}=0$, that is the
torus knot $(2,7)$.
Figure 4: The order seven automorphism of the punctured Wiman surface embedded
in the Euclidean space
Let $B$ be the unit ball of $\mathbb{R}^{4}$ with center in $0$. Consider
$B_{A}^{\prime}$ in the tridimensional sphere $\partial B$ in such a way that
the rotation giving the order seven automorphism is the restriction of a
rotation of $\mathbb{R}^{4}$. Let $C$ be the boundary of $B_{A}$. The cone on
$C$ from $0$, produces a genus $3$ surface $S_{A}^{\prime}$ immersed in
$\mathbb{R}^{4}$. There is a rotation $r$ of $\mathbb{R}^{4}$ leaving
invariant $S_{A}^{\prime}$ and such that $(S_{A}^{\prime},r)$ is topologically
equivalent to $(S_{A},f)$.
Furthermore, and since $S_{A}^{\prime}\rightarrow S_{A}^{\prime}/\left\langle
r\right\rangle$ has three branched points, the surface
$S_{K}^{\prime}-\\{0\\}$ with the conformal structure induced by
$\mathbb{R}^{4}$ is conformally equivalent to the Wiman surface of genus $3$
without one point. Note that there is an automorphism of order two $h$ in
$S_{K}^{\prime}-\\{0\\}$ permuting the two fixed points of the automorphism of
order seven and with seven fixed points. The automorphism $h$ can be
represented also by a rotation indicating that the surface $S_{K}$ is
hyperelliptic and the product $h\circ r$ is the Wiman automorphism of order
$14$ of $S_{K}$.
Acknowlegments. We wish to thank Le Fonds National Suisse de la Recherche
Scientifique for its support.
## References
* [AG] Alling, N. L.; Greenleaf, Newcomb Foundations of the theory of Klein surfaces. Lecture Notes in Mathematics, Vol. 219. Springer-Verlag, Berlin-New York, 1971. ix+117 pp.
* [C] Costa, A. F.; Embeddable anticonformal automorphisms of Riemann surfaces. Comment. Math. Helv. 72 (1997), no. 2, 203–215.
* [G] Garsia, A. M.; Imbedding of Riemann Surfaces in Euclidean Space, Comment. Math. Helv. 35 (1961) 93-110.
* [K1] Klein, F.; Gesammelte mathematische Abhandlungen. (German) Erster Band: Liniengeometrie, Grundlegung der Geometrie, zum Erlanger Programm. Herausgegeben von R. Fricke und A. Ostrowski (von F. Klein mit ergänzenden Zusätsen verzehen). Reprint der Erstauflagen [Verlag von Julius Springer, Berlin, 1921]. Springer-Verlag, Berlin-New York, 1973. ii+xii+612 pp.
* [K2] Klein, F.; Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann. 14 (1878), no. 3, 428–471.
* [Ko] Ko, S.-K.; Embedding bordered Riemann surfaces in Riemannian Manifolds, J. Korean Math. Soc. 30 (1993) 465-484.
* [Ko2] Ko, S.-K.; Embedding compact Riemann surfaces in Riemannian manifolds. Houston J. Math. 27 (2001), no. 3, 541–577.
* [Ko3] Ko, S.-K.; Embedding bordered Riemann surfaces in 4-dimensional Riemannian manifolds. Houston J. Math. 33 (2007), no. 3, 649–661.
* [R1] Rüedy, R.; Embedding of Open Riemann Surfaces, Comment. Math. Helv. 46 (1971) 214-225
* [R2] Rüedy, R.; Symmetric Embeddings of Riemann Surfaces, Ann of Math. Studies, 409-418
* [Y] Yokoyama, Y.; Complete classification of periodic maps on compact surfaces. Tokyo J. Math. 15 (1992), pp. 247–279.
* [W] Wiman, A.: Über die hyperelliptischen curven und diejenigen vom geschlechte $p=3$ welche eindeutigen transformationen in sich zulassen. Bihang Till Kongl. Svenska Veienskaps-Akademiens Handlingar, (Stockholm 1895-6), bd.21, 1-23.
|
arxiv-papers
| 2010-10-01T14:29:55 |
2024-09-04T02:49:13.296898
|
{
"license": "Public Domain",
"authors": "Antonio F. Costa and Cam Van Quach Hongler",
"submitter": "Antonio F. Costa",
"url": "https://arxiv.org/abs/1010.0172"
}
|
1010.0299
|
# Poincaré functions with spiders’ webs
Helena Mihaljević-Brandt Mathematisches Seminar, Christian-Albrechts-
Universität zu Kiel, 24118 Kiel, Germany helenam@math.uni-kiel.de and Jörn
Peter Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24118
Kiel, Germany peter@math.uni-kiel.de
###### Abstract.
For a polynomial $p$ with a repelling fixed point $z_{0}$, we consider
_Poincaré functions_ of $p$ at $z_{0}$, i.e. entire functions $L$ which
satisfy $L(0)=z_{0}$ and $p(L(z))=L(p^{\prime}(z_{0})\cdot z)$ for all
$z\in{\mathbb{C}}$. We show that if the component of the Julia set of $p$ that
contains $z_{0}$ equals $\\{z_{0}\\}$, then the (fast) escaping set of $L$ is
a _spider’s web_ ; in particular it is connected. More precisely, we classify
all linearizers of polynomials with regards to the spider’s web structure of
the set of all points which escape faster than the iterates of the maximum
modulus function at a sufficiently large point $R$.
The second author has been supported by the Deutsche Forschungsgemeinschaft,
Be 1508/7-1.
## 1\. Introduction
Let $f$ be a transcendental entire function. With the fundamental work of
Eremenko [4], the _escaping set_
$\displaystyle\operatorname{I}(f):=\\{z\in{\mathbb{C}}:f^{n}(z)\to\infty\text{
as }n\to\infty\\}$
has become an intensively studied object in transcendental holomorphic
dynamics. Since then, much progress has been achieved in exploring the
topological and dynamical properties of the escaping set and some of its
subsets (for some results, see [9, 13, 14, 15, 16, 17]).
Rippon and Stallard discovered that the _fast escaping set_
$\operatorname{A}(f)$, which was originally introduced by Bergweiler and
Hinkkanen [2], shares many significant features with $\operatorname{I}(f)$. If
we set $M(f,r):=\max_{\left|z\right|=r}\left|f(z)\right|$ and choose any
constant $R$ such that
$\displaystyle\operatorname{M}(f,r)>r\ \text{whenever}\ r\geq R,$ (1.1)
the fast escaping set of $f$ can be described as
$\displaystyle\operatorname{A}(f)=\bigcup_{l\in{\mathbb{N}}}\operatorname{A}_{R}^{-l}(f),$
where $\operatorname{A}_{R}^{l}(f)$ are the so-called _level sets_ , defined
by
$\displaystyle\operatorname{A}_{R}^{l}(f):=\\{z\in{\mathbb{C}}:|f^{n+l}(z)|\geq\operatorname{M}^{n}(R),n\geq\max\\{0,-l\\}\\}.$
(Throughout the article $\operatorname{M}^{n}$ denotes the n-th iterate of the
maximum modulus function.)
Recently, Rippon and Stallard [16, 14] introduced the concept of an
_(infinite) spider’s web_. This is a connected set $E\subset{\mathbb{C}}$ with
the property that there exists a sequence of increasing simply-connected
domains $(G_{n})$ whose union is all of ${\mathbb{C}}$ such that $\partial
G_{n}\subset E$ for all $n$. Functions whose (fast) escaping set is a spider’s
web have some strong dynamical properties. For instance, every such function
has only bounded Fatou components and there exists no curve to $\infty$ on
which $f$ is bounded (compare [16]). In particular, the set of singular values
of $f$ must be unbounded. (For precise definitions see Section 2).
In [16], various sufficient criteria are presented such that
$\operatorname{I}(f)$ and $\operatorname{A}(f)$ is a spider’s web. Primarily,
this is the case whenever the set
$\displaystyle\operatorname{A}_{R}(f):=\operatorname{A}_{R}^{0}(f)$
is a spider’s web for any $R$ as in (1.1).
In this paper, we present a large and interesting class of functions whose
escaping set is a spider’s web, namely, Poincaré functions of certain
polynomials. To make this precise, let $p$ be a polynomial with a repelling
fixed point $z_{0}$ (i.e. $p(z_{0})=z_{0}$ and
$\left|p^{\prime}(z_{0})\right|>1$). Then there exists an entire function $L$
called a _Poincaré function_ or a _linearizer of $p$ at $z_{0}$_ which
satisfies
$\displaystyle L(0)=z_{0}\quad\text{and}\quad p(L(z))=L(p^{\prime}(z_{0})\cdot
z)\;\text{for all}\;z\in{\mathbb{C}}.$
In the above functional equation, we can iterate the function $p$; this
already indicates that the analysis of a linearizer strongly depends on the
dynamical properties of $p$. However, $L$ does not only depend on $p$ but also
on $z_{0}$ and $p^{\prime}(z_{0})$ which makes linearizers good candidates for
constructing functions with various interesting analytical properties (see
e.g. Section 3). Furthermore, they are naturally good candidates for
constructing gauge functions to estimate the Hausdorff measure of escaping and
Julia sets of exponential functions (see [11]).
It was conjectured by Rempe that the escaping set of a linearizer of a
quadratic polynomial for which the critical point escapes is a spider’s web.
In this article, we show that this is true; moreover, we classifiy all
linearizers of polynomials corresponding to whether the sets
$\operatorname{A}_{R}(L)$ are spiders’ webs or not.
###### Theorem 1.1.
Let $p$ be a polynomial of degree $d\geq 2$, let $z_{0}$ be a repelling fixed
point of $p$ and let $L$ be a linearizer of $p$ at $z_{0}$. If $R$ satisfies
(1.1) then $\operatorname{A}_{R}(L)$ is a spider’s web if and only if the
component of ${\mathcal{J}}(p)$ which contains $z_{0}$ equals $\\{z_{0}\\}$.
Since polynomials for which all critical points converge to $\infty$ have
totally disconnected Julia sets [5, p.85], we obtain, using [16, Theorem 1.4],
the following corollary which also implies Rempe’s conjecture.
###### Corollary 1.2.
Let $p$ be a polynomial of degree $d\geq 2$ for which all critical points
escape and let $L$ be a linearizer of $p$. Assume that $R$ satisfies (1.1).
Then each of the sets $\operatorname{A}_{R}(L)$, $\operatorname{A}(L)$ and
$\operatorname{I}(L)$ is a spider’s web. In particular, this is true whenever
$p(z)=z^{2}+c$ and $c$ lies outside the Mandelbrot set.
We believe that the dichotomy established in Theorem 1.1 for the sets
$\operatorname{A}_{R}(L)$ also extends to the sets $\operatorname{A}(L)$ and
$\operatorname{I}(L)$. However, we were not able to prove this. For the fast
escaping set, such a result would follow if every continuum in
$\operatorname{A}(f)$ (or every ’loop’) would be contained in some level set
$\operatorname{A}_{R}^{l}(f)$, which we also believe to be true (compare
Question $2$ and $3$ in [16]).
In the proof of Theorem 1.1, we establish spiders’ webs by proving that the
corresponding linearizers grow regularly and that there exist simple closed
curves arbitrary close to $0$ on which the minimum modulus grows fast enough.
Since the order of a linearizer of a quadratic polynomial is given by $\log
2/\log\left|p^{{}^{\prime}}(z_{0})\right|$, we obtain for any given
$\rho\in(0,\infty)$ a linearizer of order $\rho$ whose escaping set is a
spider’s web.
### Acknowledgements
We would like to thank Adam Epstein for drawing our attention to Poincaré
functions and for pointing out various interesting phenomena related to them.
Moreover, we want to thank Walter Bergweiler, Jean-Marie Bois, Lasse Rempe,
Phil Rippon and Gwyneth Stallard for many interesting discussions.
## 2\. Preliminaries
The complex plane, the Riemann sphere and the unit disk are denoted by
${\mathbb{C}}$, $\widehat{{\mathbb{C}}}:={\mathbb{C}}\cup\\{\infty\\}$ and
${\mathbb{D}}$, respectively. The circle at $0$ with radius $r$ will be
denoted by ${\mathbb{S}}_{r}$. We write ${\mathbb{D}}_{r}(z)$ for the
Euclidean disk of radius $r$ centred at $z$.
If not stated differently, we will assume throughout the article that
$f:{\mathbb{C}}\to{\mathbb{C}}$ is a non-constant, non-linear entire function;
so $f$ is either a polynomial of degree $\geq 2$ or a transcendental entire
map.
Let $C\subset{\mathbb{C}}$ be a compact set. The _maximum modulus_
$\operatorname{M}(f,C)$ and the _minimum modulus_ $\operatorname{m}(f,C)$ of
$f$ relative to $C$ are defined to be
$\displaystyle\operatorname{M}(f,C):=\max_{z\in
C}|f(z)|\quad\text{and}\quad\operatorname{m}(f,C):=\min_{z\in C}|f(z)|.$
In the case when $C={\mathbb{S}}_{r}$ we will simplify the notation by writing
$\operatorname{M}(f,r)$ and $\operatorname{m}(f,r)$ for
$\operatorname{M}(f,{\mathbb{S}}_{r})$ and
$\operatorname{m}(f,{\mathbb{S}}_{r})$, respectively. Finally, recall that the
_order_ of $f$ is defined as
$\displaystyle\rho(f):=\limsup_{r\to\infty}\frac{\log\log\operatorname{M}(f,r)}{\log
r}.$
### 2.1. Background on dynamics of entire maps
We denote by $\operatorname{Crit}(f):=\\{z\in{\mathbb{C}}:f^{\prime}(z)=0\\}$
the set of _critical points_ , by
${\mathcal{C}}(f):=f(\operatorname{Crit}(f))$ the set of _critical values_ ,
and by ${\mathcal{A}}(f)$ the set of all _(finite) asymptotic values_ of $f$.
The elements of
${\mathcal{S}}(f)=\overline{{\mathcal{C}}(f)\cup{\mathcal{A}}(f)}$ are called
_singular values_ of $f$, and ${\mathcal{S}}(f)$ can be characterized as the
smallest closed subset of ${\mathbb{C}}$ such that $f:{\mathbb{C}}\setminus
f^{-1}({\mathcal{S}}(f))\to{\mathbb{C}}\setminus{\mathcal{S}}(f)$ is a
covering map. If $f$ is a polynomial then ${\mathcal{A}}(f)=\emptyset$ and
${\mathcal{C}}(f)$ is finite, so in this case,
${\mathcal{S}}(f)={\mathcal{C}}(f)$. The _postsingular set_ of $f$ is defined
to be ${\mathcal{P}}(f):=\overline{\bigcup_{n\geq 0}f^{n}({\mathcal{S}}(f))}$.
Denote by $f^{n}$ the $n$-th iterate of $f$. A point $w\in{\mathbb{C}}$ is
said to be _exceptional under $f$_ if its backward orbit, i.e., the set of all
points $z$ which are mapped to $w$ by some $f^{n}$, is finite. The set of all
exceptional values of $f$ will be denoted by ${\mathcal{E}}(f)$. It is well
known that ${\mathcal{E}}(f)$ contains at most one point. We write
${\mathcal{O}}(f)$ for the set of all (finite) _omitted values_ of $f$.
If $z$ is a periodic point of $f$ of period $n$, we call
$\mu(z):=(f^{n})^{{}^{\prime}}(z)$ the _multiplier_ of $z$. A periodic point
$z$ is called _attracting_ if $0\leq|\mu(z)|<1$, _indifferent_ if $|\mu(z)|=1$
and _repelling_ if $|\mu(z)|>1$. An attracting periodic point $z$ is called
_superattracting_ if $\mu(z)=0$.
The _Fatou set_ ${\mathcal{F}}(f)$ of $f$ is the set of all points that have a
neighbourhood in which the iterates of $f$ form a normal family; the _Julia
set_ ${\mathcal{J}}(f)$ is defined to be
${\mathbb{C}}\setminus{\mathcal{J}}(f)$. For a point $z\in{\mathcal{J}}(f)$ we
denote by ${\mathcal{J}}_{z}(f)$ the component of ${\mathcal{J}}(f)$ that
contains $z$.
### 2.2. Poincaré functions
Let $z_{0}$ be a repelling fixed point of $f$ with multiplier $\lambda$. By
the Kœnigs Linearization Theorem [10, Theorem 8.2], there exists a holomorphic
function $l$ defined in a neighbourhood of $0$ such that $l(0)=z_{0}$ and
locally, $l^{-1}\circ f\circ l(z)=\lambda z$. It was observed already by
Poincaré that $l$ continues to a holomorphic function $L$ on the entire
complex plane, meaning there exists an entire map $L$ such that $L(0)=z_{0}$
and
$\displaystyle f(L(z))=L(\lambda z)$ (2.1)
for all $z\in{\mathbb{C}}$. Every such map is called _linearizer_ or _Poincaré
function_ of $f$ at $z_{0}$. A linearizer is unique up to a constant. More
precisely, if $L$ satisfies (2.1), then so does $L_{c}:z\mapsto L(cz)$ for
every $c\in{\mathbb{C}}^{*}$, and every solution of the equation (2.1) is of
this form. If $L_{1}$ and $L_{2}$ are two linearizers then we say that they
have the same _normalization_ if
$L_{1}^{{}^{\prime}}(0)=L_{2}^{{}^{\prime}}(0)$. We say that $L$ has the
_standard normalization_ if $L^{{}^{\prime}}(0)=1$.
###### Proposition 2.1.
Let $f_{1}$ and $f_{2}$ be entire functions, and assume that there exists a
conformal map $\varphi(z)=az+b$ such that
$\displaystyle f_{2}=\varphi^{-1}\circ f_{1}\circ\varphi$
everywhere in ${\mathbb{C}}$. If $L_{1}$ and $L_{2}$ are linearizers of
$f_{1}$ and $f_{2}$ at $z_{1}$ and $z_{2}=\varphi^{-1}(z_{1})$, respectively,
with the same normalization, then
$\displaystyle L_{2}=\varphi^{-1}\circ L_{1}\circ(\varphi-b),$
where $(\varphi-b)(z):=\varphi(z)-b=az$.
###### Proof.
Consider the function $\widetilde{L}(z):=\varphi^{-1}\circ
L_{1}\circ(\varphi-b)(z)$. Then $\widetilde{L}$ satisfies
$\displaystyle f_{2}\circ\widetilde{L}(z/\lambda)$ $\displaystyle=$
$\displaystyle f_{2}\circ\varphi^{-1}\circ
L_{1}\circ(\varphi-b)(z/\lambda)=\varphi^{-1}\circ f_{1}\circ
L_{1}(az/\lambda)$ $\displaystyle=$
$\displaystyle\varphi^{-1}L_{1}(az)=\widetilde{L}(z).$
Since $f_{1}$ and $f_{2}$ are conformally conjugate, the multipliers at
$z_{1}$ and $z_{2}$ coincide, hence $\widetilde{L}$ is a linearizer of $f_{2}$
at $\varphi^{-1}(z_{1})=z_{2}$. Furthermore,
$\widetilde{L}^{{}^{\prime}}(0)=L_{1}^{{}^{\prime}}(0)$, hence $L_{1}$ and
$\widetilde{L}$ have the same normalization, yielding $L_{2}=\widetilde{L}$. ∎
In many dynamical settings, conformal conjugacies produce no relevant
dynamical consequences, hence it is natural to ask the following: Assume that
$f_{1}$ and $f_{2}$ are as in Proposition 2.1 (so $f_{1}$ and $f_{2}$ are
conformally conjugate entire functions) and let $L_{1}$ be a linearizer of
$f_{1}$. Does there exist a linearizer $L_{2}$ of $f_{2}$ which is conformally
conjugate to $L_{1}$ (and hence has the same dynamics)? In general, the answer
is no. If namely such a linearizer $L_{2}$ would exist, then a corresponding
conjugacy, say $\psi$, would map ${\mathcal{S}}(L_{1})$ bijectively onto
${\mathcal{S}}(L_{2})$, which turns out to be equivalent to the condition
$\displaystyle\psi({\mathcal{P}}(f_{1}))={\mathcal{P}}(f_{2})$ (2.2)
(see Proposition 3.2). Since $\varphi$ conjugates $f_{1}$ and $f_{2}$, it
already satisfies (2.2), so in particular, the map $\psi^{-1}\circ\varphi$ is
a conformal automorphism of ${\mathbb{C}}$ that fixes the set
${\mathcal{P}}(f_{1})$.
Now if $Z$ is an arbitrary finite subset of ${\mathbb{C}}$ with at least two
elements, then
$G_{Z}:=\\{h(z)=az+b:a\in{\mathbb{C}}^{*},b\in{\mathbb{C}},h(Z)=Z\\}$ is a
finite group and one can easily check that the map
$G_{Z}\to{\mathbb{C}}^{*},az+b\mapsto a$ is an injective group-homomorphism.
Hence $G_{Z}$ is isomorphic to a finite subgroup of ${\mathbb{C}}^{*}$, which
must be a cyclic group generated by a root of unity. So every such $G_{Z}$ is
generated by a map of the form $z\mapsto\exp(2\pi ik/n)z+b$ with coprime $k$
and $n$ and $n\leq|Z|$. This allows to phrase necessary geometric conditions
on a finite set $Z$ such that $G_{Z}$ is not trivial. It is clear that such
conditions are rather strong; e.g., if $z\mapsto\exp(2\pi ik/n)z+b$ is a
generator of $G_{Z}$ and $p$ its (unique) fixed point in ${\mathbb{C}}$ then
all elements of $Z$ must lie on $r$ circles centred at $p$, where $r\cdot
n\leq|Z\setminus\\{p\\}|$. To give an explicit dynamical example, one can
consider the unique real parameter $c$, for which $f(z):=z^{2}+c$ has a
superattracting cycle of period three; one easily sees that
$G_{{\mathcal{P}}(f)}$ is trivial.
However, triviality of $G_{{\mathcal{P}}(f_{1})}$ implies $\psi\equiv\varphi$.
So if $\varphi(z)=az+b$, then by Proposition 2.1, every linearizer of $f_{2}$
is of the form
$\displaystyle L_{2}(z)=\varphi^{-1}\circ L_{1}\circ c(\varphi-b)$
for some $c\in{\mathbb{C}}^{*}$, and no such map can be conformally conjugate
to $L_{1}$ via $\varphi$ whenever $b\neq 0$ (and $c\neq 1$).
Before the end of this paragraph let us observe that one can iterate $f$
inside the functional equation and obtain
$\displaystyle f^{n}\circ L(z)=L\circ\lambda^{n}(z)$ (2.3)
as an iterated version of (2.1), where $\lambda^{n}$ denotes the function
$z\mapsto\lambda^{n}z$.
The growth of the function $f$ and a linearizer $L$ are related in the
following sense: If $f$ is transcendental entire then $L$ has infinite order.
If $f$ is a polynomial then $\rho(L)=\log d/\log|\lambda|$.
### 2.3. Polynomial dynamics near $\infty$ and repelling fixed points
If $p$ is a polynomial, the Julia set of $p$ is compact and
$\operatorname{I}(p)$ is an open connected subset of ${\mathcal{F}}(p)$;
moreover, it is simply-connected if and only if ${\mathcal{J}}(p)$ is
connected. This property is equivalent to the relation
${\mathcal{C}}(p)\cap\operatorname{I}(p)=\emptyset$ [10, Lemma 9.4, Theorem
9.5].
Near $\infty$, the iterates of a polynomial behave in the following simple
way.
###### Proposition 2.2.
Let $p(z)=\sum_{n=0}^{d}a_{n}z^{n}$ be a polynomial of degree $d\geq 2$. Then
for any $\varepsilon>0$ there exists $R_{\varepsilon}>0$ such that for every
$z$ with $|z|>R_{\varepsilon}$, we have
$\displaystyle(1-\varepsilon)|a_{d}|\cdot|z|^{d}\leq|p(z)|\leq(1+\varepsilon)|a_{d}|\cdot|z|^{d},$
and $R_{\varepsilon}\to\infty$ as $\varepsilon\to 0$.
If $\varepsilon$ is chosen small enough such that
$(1-\varepsilon)|a_{d}|R_{\varepsilon}^{d-1}>1$, then
$\displaystyle((1-\varepsilon)|a_{d}|)^{q_{n}(d)}\cdot|z|^{d^{n}}\leq|p^{n}(z)|\leq((1+\varepsilon)|a_{d}|)^{q_{n}(d)}\cdot|z|^{d^{n}}$
for all $n\in{\mathbb{N}}$ and all $z\in{\mathbb{C}}$ with
$|z|>R_{\varepsilon}$, where $q_{n}(z):=(z^{n}-1)/(z-1)=z^{n-1}+\dots+z+1$.
###### Proof.
The first statement is elementary and well-known.
Note that we have chosen $\varepsilon$ sufficiently small such that
$|z|>R_{\varepsilon}$ implies $|p(z)|>R_{\varepsilon}$. We will prove the
statement inductivly. So for $n=1$ we have $q_{1}(z)=1$ and the claim follows
from the first part. For the iterate $p^{n+1}(z)=p(p^{n}(z))$ we then obtain
$\displaystyle|p(p^{n}(z))|$ $\displaystyle\leq$
$\displaystyle(1+\varepsilon)|a_{d}||p^{n}(z)|^{d}\leq(1+\varepsilon)|a_{d}|\left[((1+\varepsilon)|a_{d}|)^{q_{n}(d)}|z|^{d^{n}}\right]^{d}$
$\displaystyle=$ $\displaystyle((1+\varepsilon)|a_{d}|)^{d\cdot
q_{n}(d)+1}|z|^{d^{n+1}}=((1+\varepsilon)|a_{d}|)^{q_{n+1}(d)}|z|^{d^{n+1}}$
as well as
$\displaystyle|p(p^{n}(z))|$ $\displaystyle\geq$
$\displaystyle(1-\varepsilon)|a_{d}||p^{n}(z)|^{d}\geq(1-\varepsilon)|a_{d}|\left[((1-\varepsilon)|a_{d}|)^{q_{n}(d)}|z|^{d^{n}}\right]^{d}$
$\displaystyle=$ $\displaystyle((1-\varepsilon)|a_{d}|)^{d\cdot
q_{n}(d)+1}|z|^{d^{n+1}}=((1-\varepsilon)|a_{d}|)^{q_{n+1}(d)}|z|^{d^{n+1}}.$
∎
Near a repelling fixed point of $p$, we can make the following statement on
the escaping set $\operatorname{I}(p)$.
###### Proposition 2.3.
Let $z_{0}$ a repelling fixed point of $p$. For every $\delta>0$ there exists
a simple closed curve
$\gamma_{\delta}\subset{\mathbb{D}}_{\delta}(z_{0})\cap\operatorname{I}(p)$
around $z_{0}$ if and only if ${\mathcal{J}}_{z_{0}}(p)=\\{z_{0}\\}$.
###### Proof.
Let us first assume that for every $\delta>0$ there exists a simple closed
curve $\gamma_{\delta}\subset\operatorname{I}(p)$ around $z_{0}$ such that
$\operatorname{dist}(z_{0},\gamma_{\delta})<\delta$. Then
${\mathcal{J}}_{z_{0}}(p)$ is contained in the interior of every
$\gamma_{\delta}$, hence it must consist of a single point.
If ${\mathcal{J}}_{z_{0}}(p)=\\{z_{0}\\}$, then for every $\delta>0$ there
exist open, non-empty disjoint sets $U_{\delta}$ and $V_{\delta}$ such that
${\mathcal{J}}(p)\subset U_{\delta}\cup V_{\delta}$,
${\mathcal{J}}_{z_{0}}(p)\subset U_{\delta}$ and
$\operatorname{dist}(z_{0},U_{\delta})<\delta/2$. Furthermore, we can assume
$U_{\delta}$ to be connected; otherwise, we replace $U_{\delta}$ by the
connected component of $U_{\delta}$ that contains $z_{0}$, which is also an
open set. By the Plane Separation Theorem [19, Chapter VI, Theorem 3.1], there
exists a simple closed curve $S_{\delta}$ which separates $z_{0}$ from
${\mathcal{J}}(p)\cap V_{\delta}$ such that
$S_{\delta}\cap{\mathcal{J}}(p)=\emptyset$ and every point in
${\mathcal{J}}(p)\cap U_{\delta}$ is at distance less than $\delta/2$ from
$S_{\delta}$. Hence, $\operatorname{dist}(S_{\delta},z_{0})<\delta$ and
$S_{\delta}\subset{\mathcal{F}}(p)$. Moreover, the component of
${\mathcal{F}}(p)$ which contains $S_{\delta}$ must be $\operatorname{I}(p)$
since every bounded component of the Fatou set is simply-connected. ∎
## 3\. The set of singular values of a linearizer
If not stated differently, we will assume throughout this section that $f$ is
an entire function, $z_{0}$ a repelling fixed point of $f$ and $L$ a
linearizer of $f$ at $z_{0}$. We begin with a simple connection between
exceptional values of $f$ and omitted values of $L$.
###### Proposition 3.1.
The sets ${\mathcal{O}}(L)$ and
${\mathcal{E}}(f)\setminus\left\\{z_{0}\right\\}$ are equal.
###### Proof.
Since $L(0)=z_{0}$, the point $z_{0}$ is never an omitted value of $L$. If
$a\in{\mathbb{C}}\setminus{\mathcal{E}}(f)$, then the backward orbit of $a$
has infinitely many elements. Since $L$ omits at most one finite value, the
backward orbit of $a$ under $f$ intersects $L({\mathbb{C}})$, i.e., there
exists $n\in{\mathbb{N}}$ and $w\in{\mathbb{C}}$ with $L(w)\in f^{-n}(a)$.
This means $a=f^{n}(L(w))=L(\lambda^{n}w)$, so $a\notin{\mathcal{O}}(L)$. This
proves
${\mathcal{O}}(L)\subset{\mathcal{E}}(f)\setminus\left\\{z_{0}\right\\}$.
Now let $a\in{\mathbb{C}}\setminus{\mathcal{O}}(L)$. If $a=z_{0}$, then we are
done. So suppose that $a\neq z_{0}$. Then there exists $z\neq 0$ with
$L(z)=a$. By the iterated functional equation, $L(z/\lambda^{j})\in
f^{-j}(a)$. Since $z\neq 0$ and $L$ is injective in a neighborhood of $0$, the
backward orbit of $a$ under $f$ has infinitely many elements. ∎
Next, we will show that the postsingular set of $f$ and the set of singular
values $L$ coincide. This seems to be well-known (and to us, the main parts of
the proof have been presented by A. Epstein), but we could not find a
reference, which is why we include a proof.
###### Proposition 3.2.
The following relations are true:
* $(i)$
${\mathcal{C}}(L)=\bigcup_{n\geq
0}f^{n}({\mathcal{C}}(f))\setminus{\mathcal{E}}(f)$.
* $(ii)$
${\mathcal{S}}(L)={\mathcal{P}}(f)$.
###### Proof.
Let $w=L(z)\in{\mathcal{C}}(L)$, in particular $w\notin{\mathcal{O}}(L)$.
Since $L^{\prime}(0)\neq 0$, we have $w\neq z_{0}$. It follows from
Proposition 3.1 that $w\notin{\mathcal{E}}(f)$. Differentiating the iterated
functional equation yields
$\displaystyle 0=(f^{n})^{\prime}(L(z/\lambda^{n}))\cdot
L^{\prime}(z/\lambda^{n})\cdot\frac{1}{\lambda^{n}}.$
Denote by $\operatorname{Crit}(f)$ the set of critical points of $f$. Since
$L^{\prime}(z/\lambda^{n})\neq 0$ if $n$ is large enough, it follows that
$L(z/\lambda^{n})\in\operatorname{Crit}(f^{n})$. Since
$\operatorname{Crit}(f^{n})=\bigcup_{k=0}^{n-1}f^{k}(\operatorname{Crit}(f))$
by the chain rule, there exists some $k\leq n-1$ with
$L(z/\lambda^{n})=f^{k}(y)$, where $y\in\operatorname{Crit}(f)$. It follows
that
$\displaystyle w=L(z)=f^{n}(L(z/\lambda^{n}))=f^{n}(f^{k}(y))=f^{n+k}(y),$
i.e., $w\in\bigcup_{n\geq 0}f^{n}({\mathcal{C}}(f))$.
For the other inclusion, let $w\in
f^{n}({\mathcal{C}}(f))\setminus{\mathcal{E}}(f)$. We want to show that there
exists some $z\in L^{-1}(w)$ with $L^{\prime}(z)=0$. Again, we differentiate
the iterated functional equation and obtain
$\displaystyle L^{\prime}(z)=(f^{n+1})^{\prime}(L(z/\lambda^{n+1}))\cdot
L^{\prime}(z/\lambda^{n+1})\cdot\frac{1}{\lambda^{n+1}}$
for all $z\in{\mathbb{C}}$. There exists some $y\in\operatorname{Crit}(f)$
such that $w=f^{n+1}(y)$. Clearly, $y\notin{\mathcal{E}}(f)$ since
$w\notin{\mathcal{E}}(f)$. By Proposition 3.1, we have
$y\notin{\mathcal{O}}(L)$, so there exists $z\in{\mathbb{C}}$ with
$y=L(z/\lambda^{n+1})$. It follows by the chain rule that $L^{\prime}(z)=0$,
and we have $w=f^{n+1}(y)=f^{n+1}(L(z/\lambda^{n+1}))=L(z)$, which finishes
the proof of (i).
We now prove (ii). For the composition $f\circ L$ one obtains
$\displaystyle{\mathcal{S}}(f\circ
L)=S(f|_{f({\mathbb{C}})})\cup\overline{f({\mathcal{S}}(L))}={\mathcal{S}}(f)\cup
f({\mathcal{S}}(L)),$
since every Picard value of $f$ is also a singular value of $f$. Let us
abbreviate $S:={\mathcal{S}}(f)\cup f({\mathcal{S}}(L))$. Since the
composition is a covering map, it follows from (2.1) that must be a covering
map as well. Hence
$\displaystyle{\mathcal{S}}(f)\cup
f({\mathcal{S}}(L))=S\supset{\mathcal{S}}(L\circ\lambda)={\mathcal{S}}(L).$
The argument is commutative with respect to (2.1), so we obtain the opposite
inclusion, yielding the equality ${\mathcal{S}}(L)={\mathcal{S}}(f)\cup
f({\mathcal{S}}(L))$. But for a point $w\in{\mathcal{S}}(f)$, this implies
that $w\in{\mathcal{S}}(L)$, and so $f(w)\in
f({\mathcal{S}}(L))\subset{\mathcal{S}}(L)$. By proceeding inductively, it
follows for every $n\in{\mathbb{N}}$ that $f^{n}(w)\in{\mathcal{S}}(L)$, hence
${\mathcal{P}}(f)\subset{\mathcal{S}}(L)$.
Let $w\in{\mathbb{C}}\setminus{\mathcal{P}}(f)$. Then there exists a disk
$D\ni w$ such that all inverse branches of all iterates of $f$ exist in $D$.
Let $v\in D$ and $z\in L^{-1}(v)$, and define $z_{n}:=z/\lambda^{n}$ and
$v_{n}:=L(z_{n})$. Let $g_{n}$ be the branch of $(f^{n})^{-1}$ such that
$g_{n}(v)=v_{n}$ and let $D_{n}:=g_{n}(D)$. By the Shrinking Lemma in [8], it
follows that the diameter of the domains $D_{n}$ converges to $0$ (Actually,
the statement in [8] is not phrased such that it completely covers our setting
but the proof gives what we require). We choose a domain $U$ in which $L$ is
injective. Then for $n$ large enough, $D_{n}$ lies in $L(U)$. Let $T$ be the
branch of $L^{-1}$ that maps $D_{n}$ into $U$. Then we have
$\displaystyle L\circ(\lambda^{n}\circ T\circ g_{n})(z)=f^{n}\circ
L\circ\underbrace{(T\circ g_{n})(z)}_{\in U}=(f^{n}\circ g_{n})(z)=z.$
Since $z$ is an arbitrarily chosen preimage of an arbitrary point in $D$, all
inverse branches of $L$ can be defined in $D$. Hence
$w\in{\mathbb{C}}\setminus{\mathcal{S}}(L)$. ∎
If $f$ is a polynomial then ${\mathcal{A}}(L)$ is contained in the union of
attracting and parabolic periodic cycles and the accumulation points of
recurrent critical points in ${\mathcal{J}}(f)$ [3, Theorem 1]. Depending on
the location of the repelling fixed point $z_{0}$ relative to
${\mathcal{F}}(f)$, we can exclude certain attracting cycles of $f$ as
asymptotic values for $L$.
###### Proposition 3.3.
Let $f$ be a polynomial and let $w\in{\mathcal{A}}(L)$. If $w$ is an
attracting periodic point of $f$, then $z_{0}$ lies in the boundary of the
immediate attracting basin of $w$.
###### Proof.
Let $w$ be an attracting periodic point of $f$ of period $k$ and assume that
$w$ is an asymptotic value of $L$. Then there exists a path $\gamma$ to
$\infty$ for which $\lim_{t\to\infty}L(\gamma(t))=w$. Since
$w\in{\mathcal{F}}(f)$ and ${\mathcal{F}}(f)$ is open, we can assume that
$L(\gamma)\subset{\mathcal{F}}(f)$. It follows from (2.3) that every path
$\gamma_{n}(t):=\lambda^{-n}\cdot\gamma(t)$ is again an asymptotic path for
$L$. Moreover, the limit of $L$ along $\gamma_{nk}$ is contained in
$f^{-nk}(w)$. On the other hand, every such limit point must lie in the set of
attracting periodic points [3, Theorem 1], hence it follows that
$\lim_{t\to\infty}L(\gamma_{nk}(t))=w$. Furthermore, for every $\varepsilon>0$
there exists $N_{\varepsilon}\in{\mathbb{N}}$ such that for all $n\geq
N_{\varepsilon}$, the curve $L(\gamma_{nk})$ intersects
${\mathbb{D}}_{\varepsilon}(z_{0})$. Hence $z_{0}\in\partial A^{*}(w)$. ∎
Recall that a point $z\in{\mathcal{J}}(f)$ is called a _buried point_ if it
does not belong to the boundary of any Fatou component (other that
$\operatorname{I}(f)$).
###### Corollary 3.4.
If $f$ is a polynomial and $z_{0}$ is a buried point (of $f$) then $L$ has no
asymptotic values.
Linearizers can be very useful to construct entire or meromorphic functions
whose set of singular values satisfies certain conditions. For instance, in
[9], there was given an example of an entire function of finite order with no
asymptotic values and only finitely many critical values such that the
ramification degree on its Julia set was unbounded; the constructed function
was a linearizer of a certain hyperbolic quadratic polynomial. Here we want to
show another interesting example that can be constructed using linearizers, in
this case of a transcendental entire function $f$.
Let $f(z):=\mu\exp(z)$ where $\mu\in{\mathbb{C}}$ is chosen such that
$\bigcup_{n\geq 0}f^{n}(0)$ is dense in ${\mathbb{C}}$. The existence of such
parameters is well-known. By [7, Theorem 2], the function $f$ has infinitely
many fixed points. Since ${\mathcal{S}}(f)=\\{0\\}$, at most one of them is
non-repelling [1, Theorem 7], so we can pick a repelling fixed point $z_{0}$
of $f$. Let $L$ be a linearizer of $f$ at $z_{0}$. It follows from the
functional equation that $0$ is an omitted value of $L$. By Proposition 3.2,
every point $w_{n}:=f^{n}(0)$ is an asymptotic value of $L$. It is also not
hard to check that $L$ has a direct singularity lying over each of the points
$w_{n}$. (For a clarification of terminology, see e.g. [3]; our last claim
also follows from [3, Theorem 1.4], which is formulated for linearizers of
rational maps only, but extends to linearizers of transcendental entire maps
with the same proof.) Hence $L$ is a map for which the set of projections of
direct singularities (or direct asymptotic values) is dense in ${\mathbb{C}}$.
This is optimal, since by a theorem of Heins [6], the set of projections of
direct singularities is always countable.
## 4\. Maximum and minimum modulus estimates
In the remaining part of the article we prove Theorem 1.1. From now on, we
consider an arbitrary but fixed polynomial $p$ of degree $d\geq 2$, hence $p$
can be written as
$\displaystyle p(z)=\sum_{i=0}^{d}a_{i}z^{i}=a_{0}+a_{1}z\dots+a_{d}z^{d},\
\;a_{d}\neq 0.$
For every $\varepsilon>0$ we pick a constant $R_{\varepsilon}\geq 1$ for which
the conclusion of Proposition 2.2 is satisfied, and such that
$\varepsilon_{1}<\varepsilon_{2}$ implies
$R_{\varepsilon_{1}}>R_{\varepsilon_{2}}$. We assume that $p$ has a repelling
fixed point $z_{0}$ with multiplier $\lambda$, and we denote by $L$ a
linearizer of $p$ at $z_{0}$. We also pick a constant $R_{L}\geq 1$ such that
$\operatorname{M}(L,s)>s$ for all $s\geq R_{L}$.
###### Lemma 4.1 (Regularity of growth).
Let $\varepsilon>0$, $r>\max\\{R_{\varepsilon},R_{L}\\}$ and define
$k_{\varepsilon}:=\log((1-\varepsilon)|a_{d}|)$ and
$K_{\varepsilon}:=\log((1+\varepsilon)|a_{d}|)$. Then
$\displaystyle\prod_{i=0}^{n-1}\left(\\!d+\frac{\log
k_{\varepsilon}}{\log\operatorname{M}(L,|\lambda|^{i}r)}\right)\\!\leq\\!\frac{\log\operatorname{M}(L,|\lambda|^{n}r)}{\log\operatorname{M}(L,r)}\\!\leq\\!\prod_{i=0}^{n-1}\\!\left(\\!d+\frac{\log
K_{\varepsilon}}{\log\operatorname{M}(L,|\lambda|^{i}r)}\right)$
holds for all $n\in{\mathbb{N}}$.
###### Proof.
Let $r$ be as assumed, and let $\tilde{z}\in{\mathbb{S}}_{r}$ be a point for
which $L(\tilde{z})\geq L(z)$ for all $z\in{\mathbb{S}}_{r}$. Let
$\tilde{w}:=L(\tilde{z})$. Then $|\tilde{w}|=\operatorname{M}(L,r)$ and it
follows from the functional equation (2.1) and Proposition 2.2 that
$\displaystyle\log\operatorname{M}(L,|\lambda|r)$ $\displaystyle=$
$\displaystyle\log\operatorname{M}(p\circ
L,r)=\log\operatorname{M}(p,L({\mathbb{S}}_{r}))\geq\log p(\tilde{w})$
$\displaystyle\geq$
$\displaystyle\log\left((1-\varepsilon)|a_{d}|\cdot|\tilde{w}|^{d}\right)=k_{\varepsilon}+d\cdot\log\operatorname{M}(L,r),$
and
$\displaystyle\log\operatorname{M}(L,|\lambda|r)$ $\displaystyle=$
$\displaystyle\log\operatorname{M}(p,L({\mathbb{S}}_{r}))\leq\log\operatorname{M}(p,\operatorname{M}(L,r))$
$\displaystyle\leq$
$\displaystyle\log\left((1+\varepsilon)|a_{d}|\cdot\operatorname{M}(L,r)^{d}\right)=K_{\varepsilon}+d\cdot\log\operatorname{M}(L,r).$
Hence,
$\displaystyle\left(\frac{k_{\varepsilon}}{\log\operatorname{M}(L,r)}+d\right)\leq\frac{\log\operatorname{M}(L,|\lambda|r)}{\log\operatorname{M}(L,r)}\leq\left(\frac{K_{\varepsilon}}{\log\operatorname{M}(L,r)}+d\right).$
The statement now follows immediately from the fact that
$\displaystyle\frac{\log\operatorname{M}(L,|\lambda|^{n}r)}{\log\operatorname{M}(L,r)}=\frac{\log\operatorname{M}(L,|\lambda|^{n}r)}{\log\operatorname{M}(L,|\lambda|^{n-1}r)}\cdot\;\dots\;\cdot\frac{\log\operatorname{M}(L,|\lambda|r)}{\log\operatorname{M}(L,r)}.$
∎
###### Lemma 4.2.
For every $k\in{\mathbb{N}}$ there exists $R_{k}>0$ such that for all
$R>R_{k}$, $m\leq d^{k}$ and $n>k$,
$\displaystyle\operatorname{M}(L,r_{n})>r_{n+1}^{m},$
where the sequence $(r_{n})$ is defined by
$\displaystyle r_{n}:=|\lambda|^{n}\cdot\operatorname{M}^{n}(L,R).$
Moreover, we can choose
$R_{1}=2\cdot\max\bigg{\\{}\log|a_{d}|,\log\frac{2}{|a_{d}|},\log|\lambda|\bigg{\\}}$.
###### Proof.
Let $\varepsilon\in(0,1/2)$ be arbitrary but fixed, and let
$R>\max\\{R_{L},R_{\varepsilon}\\}$. It follows from Lemma 4.1 with
$r=\operatorname{M}^{n}(L,R)$ that
$\displaystyle\log\operatorname{M}(L,r_{n})$
$\displaystyle=\log\operatorname{M}(L,|\lambda|^{n}\operatorname{M}^{n}(L,R))$
$\displaystyle\geq\prod_{i=0}^{n-1}\\!\left(d+\frac{\log
k_{\varepsilon}}{\log\operatorname{M}(L,|\lambda|^{i}\operatorname{M}^{n}(L,R))}\right)\\!\cdot\log\operatorname{M}(L,\operatorname{M}^{n}(L,R))$
$\displaystyle\geq\left(d-\frac{|\log k_{\varepsilon}|}{\log
R}\right)^{n}\cdot\log\operatorname{M}^{n+1}(L,R).$
By definition,
$\displaystyle\log r_{n+1}^{m}$
$\displaystyle=m\log(|\lambda|^{n+1}\operatorname{M}^{n+1}(L,R))$
$\displaystyle=m(n+1)\log|\lambda|+m\log\operatorname{M}^{n+1}(L,R).$
Define $c_{R}:=\frac{|\log k_{\varepsilon}|}{\log R}$. We want to show that
there exists $R_{k}$ such that when $R>R_{k}$, $m\leq d^{k}$ and $n\geq k+1$,
then
$\displaystyle\log\operatorname{M}^{n+1}(L,R)\cdot((d-c_{R})^{n}-m)>m(n+1)\log|\lambda|.$
Obviously, it is sufficient if the wanted constant $R_{k}$ satisfies
$\displaystyle\log
R_{k}\cdot((d-c_{R_{k}})^{n}-d^{k})>d^{k}(n+1)\log|\lambda|$
for all $n\geq k+1$, and this is certainly true when we choose $R_{k}$
sufficiently large. We will omit the details since they follow from elementary
calculus; however, one can prove inductively that every $R_{k}$ with $\log
R_{k}>\max\\{2|\log k_{\varepsilon}|,\frac{2k}{d}|\log
k_{\varepsilon}|,\frac{\sqrt{\operatorname{e}}\log|\lambda|}{(2-\sqrt{\operatorname{e}})(k+2)}\\}$
is sufficiently large. Hence for $k=1$ we can choose
$R_{1}=2\max\\{|\log|a_{d}||,|\log\frac{1}{2}|a_{d}||,\log|\lambda|)$ since
$|\log k_{\varepsilon}|=|\log((1-\varepsilon)|a_{d}|)|$ and
$\varepsilon\in(0,1/2)$, and since
$\frac{\sqrt{\operatorname{e}}}{3\cdot(2-\sqrt{\operatorname{e}})}<2$. ∎
###### Lemma 4.3 (growth of minimum modulus).
Suppose that ${\mathcal{J}}_{z_{0}}(p)=\\{z_{0}\\}$ and let
$m\in{\mathbb{N}}_{>1}$. Then there exists $R_{m}>0$ with the following
property: For every $r>R_{m}$ there is a simple closed curve $\Gamma^{r}$
separating ${\mathbb{S}}_{r}$ and ${\mathbb{S}}_{r^{m}}$ such that
$\displaystyle\operatorname{m}(L,\Gamma^{r})>\operatorname{M}(L,r).$
###### Proof.
Let $D$ be a disk around $0$ such that $L|_{D}$ is conformal. Let $\delta>0$
be sufficiently small such that ${\mathbb{D}}_{\delta}(z_{0})\subset L(D)$. By
Proposition 2.3 there exists a simple closed curve
$\gamma_{\delta}\subset{\mathbb{D}}_{\delta}(z_{0})\cap\operatorname{I}(p)$
which surrounds $z_{0}$. Since such a curve exists in the intersection of
every arbitrarily small neighbourhood of $z_{0}$ and $\operatorname{I}(p)$, we
can assume w.l.o.g. that $D={\mathbb{D}}$. Let
$\Gamma_{\delta}=L^{-1}(\gamma_{\delta})\cap{\mathbb{D}}$. Then
$\Gamma_{\delta}$ is a simple closed curve surrounding $0$. Define
$\displaystyle
s:=\min_{z\in\Gamma_{\delta}}|z|=\operatorname{dist}(0,\Gamma_{\delta})\quad\text{and}\quad
t:=\max_{z\in\Gamma_{\delta}}|z|.$
Obviously, both $s$ and $t$ are finite and positive constants.
Let $r>\left(\frac{|\lambda|\cdot t}{s}\right)^{\frac{1}{m-1}}$ be an
arbitrary but fixed number. We define $l(r)$ to be the unique integer for
which
$\displaystyle|\lambda|^{l(r)-1}\leq r<|\lambda|^{l(r)}.$
Similarly, for the external radius $t$ of the curve $\Gamma_{\delta}$ we
denote by $l(t)$ the unique natural number for which
$\displaystyle t\cdot|\lambda|^{l(t)}\leq r^{m}<t\cdot|\lambda|^{l(t)+1}.$
(Note that the lower bound for $r$ implies that $s\cdot|\lambda|^{l(t)}>r$.)
By taking logarithms we obtain the equivalent equations
$\displaystyle l(r)-1\leq\frac{\log r}{\log|\lambda|}<l(r)$
and
$\displaystyle l(t)\leq\frac{m\cdot\log r-\log t}{\log|\lambda|}<l(t)+1.$
A combination of these two inequalities yields
$\displaystyle m\cdot l(r)-\left(\frac{\log
t}{\log|\lambda|}+m+1\right)<l(t)<m\cdot l(r)-\frac{\log t}{\log|\lambda|}.$
(4.1)
Let us fix an $\varepsilon\in(0,1/2)$. Let $j\in{\mathbb{N}}$ be minimal with
the property that $p^{j}(\gamma_{\delta})\subset\\{z:|z|>R_{\varepsilon}\\}$.
Note that there is a unique integer $j$ with this property since
$\gamma_{\delta}$ is a compact subset of $\operatorname{I}(p)$. We define
$\displaystyle\Gamma^{r}:=\\{z\in{\mathbb{C}}:\lambda^{-l(t)}\cdot
z\in\Gamma_{\delta}\\}.$
Observe that $\Gamma^{r}$ separates ${\mathbb{S}}_{r}$ and
${\mathbb{S}}_{r^{m}}$. In order to simplify the calculations, let us consider
the logarithms of the minimum and maximum modulus. Using Proposition 2.2,
these can be estimated in the following way:
$\displaystyle\log\operatorname{m}(L,\Gamma^{r})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\log\operatorname{m}(p^{l(t)}\circ
L,\Gamma_{\delta})=\log\operatorname{m}(p^{l(t)},\gamma_{\delta})\geq\log\operatorname{m}(p^{l(t)-j},R_{\varepsilon})$
$\displaystyle\geq$
$\displaystyle\\!\\!\log\\{((1-\varepsilon)|a_{d}|)^{q_{l(t)-j}(d)}\cdot
R_{\varepsilon}^{d^{l(t)-j}}\\}$ $\displaystyle=$
$\displaystyle\\!\\!q_{l(t)-j}(d)\cdot\log((1-\varepsilon)|a_{d}|)+d^{l(t)-j}\cdot\log
R_{\varepsilon},$ $\displaystyle\log\operatorname{M}(L,r)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\log\operatorname{M}(p^{l(r)},L({\mathbb{S}}_{r\cdot|\lambda|^{-l(r)}}))\leq\log\operatorname{M}(p^{l(r)},R_{\varepsilon})$
$\displaystyle\leq$
$\displaystyle\\!\\!\log\\{((1+\varepsilon)|a_{d}|)^{q_{l(r)}(d)}\cdot
R_{\varepsilon}^{d^{l(r)}}\\}$ $\displaystyle\leq$
$\displaystyle\\!\\!q_{l(r)}(d)\cdot\log((1+\varepsilon)|a_{d}|)+d^{l(r)}\cdot\log
R_{\varepsilon}.$
Equation (4.1) yields the relation $m\cdot l(r)-C<l(t)<m\cdot l(r)+c$ with the
constants $C:=\log t/\log|\lambda|+m+1$ and $c:=\log t/\log|\lambda|$.
Furthermore, by Proposition 2.2 we can estimate the polynomials
$q_{n+1}(d)=d^{n}+\ldots+d+1=(d^{n+1}-1)/(d-1)$ by $d^{n}\leq q_{n+1}(d)\leq
d^{n+1}$. Together, we obtain
$\displaystyle\log\operatorname{m}(L,\Gamma^{r})$ $\displaystyle>$
$\displaystyle d^{m\cdot
l(r)-C-j-1}\cdot\log((1-\varepsilon)|a_{d}|)+d^{m\cdot l(r)-C-j}\cdot\log
R_{\varepsilon}$ $\displaystyle=$ $\displaystyle d^{m\cdot
l(r)}\cdot\frac{\log((1-\varepsilon)|a_{d}|R_{\varepsilon}^{d})}{d^{C+j+1}},$
$\displaystyle\log\operatorname{M}(L,r)$ $\displaystyle\leq$ $\displaystyle
d^{l(r)}\cdot\log((1+\varepsilon)|a_{d}|)+d^{l(r)}\cdot\log R_{\varepsilon}$
$\displaystyle=$ $\displaystyle
d^{l(r)}\cdot\log((1+\varepsilon)|a_{d}|R_{\varepsilon})$
as new lower and upper bounds for the minimum and maximum modulus,
respectively. Since $m\geq 2$, it is sufficient to find a constant $R_{m}$
such that for all $r>R_{m}$,
$\displaystyle
d^{2l(r)}\cdot\frac{\log((1-\varepsilon)|a_{d}|R_{\varepsilon}^{d})}{d^{C+j+1}}$
$\displaystyle>d^{l(r)}\cdot\log((1+\varepsilon)|a_{d}|R_{\varepsilon})$
$\displaystyle\Longleftrightarrow$ $\displaystyle d^{l(r)}$
$\displaystyle>\frac{\log((1+\varepsilon)|a_{d}|R_{\varepsilon})}{\log((1-\varepsilon)|a_{d}|R_{\varepsilon}^{d})}\cdot
d^{C+j+1}=:l_{\varepsilon}.$
Hence $R_{m}:=\max\bigg{\\{}\left(\frac{|\lambda|\cdot
t}{s}\right)^{\frac{1}{m-1}},|\lambda|^{\frac{\log l_{\varepsilon}}{\log
d}}\bigg{\\}}$ is sufficiently large. ∎
###### Proof of Theorem 1.1.
Let us start with the case when ${\mathcal{J}}_{z_{0}}(p)\neq\\{z_{0}\\}$.
Assume that $A_{R}(L)$ is a spider’s web for some sufficiently large $R$. By
definition, there exists a sequence of bounded simply-connected domains
$G_{n}$ such that $G_{n}\subset G_{n+1}$, $\partial G_{n}\subset A_{R}(L)$ for
$n\in{\mathbb{N}}$, and $\bigcup G_{n}={\mathbb{C}}$. We can assume w.l.o.g.
that every $G_{n}$ contains $0$ (since this is true anyway for all
sufficiently large $n$).
By Proposition 2.3, for every $n\in{\mathbb{N}}$, the curve $L(\partial
G_{n})$ intersects the filled Julia set of $p$. Let $K>0$ be the radius of the
smallest disk around $0$ which contains the (filled) Julia set of $p$. Then
there exists a sequence of points $w_{n}\in\partial G_{n}$ such that
$|L(w_{n})|\leq K$. But this contradicts the assumption that all points
$z\in\partial G_{n}$ satisfy $|L(z)|\geq\operatorname{M}(L,R)$.
Let us now consider the situation when ${\mathcal{J}}_{z_{0}}(p)=\\{z_{0}\\}$.
By [16, Theorem 8.1] it is sufficient to find a sequence of bounded simply-
connected domains $G_{n}$ such that for all (sufficiently large) $n$,
$\displaystyle
G_{n}\supset\\{z\in{\mathbb{C}}:|z|<\operatorname{M}^{n}(L,R)\\}$
and
$\displaystyle G_{n+1}\text{ is contained in a bounded component of
}{\mathbb{C}}\setminus L(\partial G_{n}).$
Let $R_{1}$ be the constant from Lemma 4.2, and set
$R:=\max\\{R_{L},R_{1}\\}$. For $n\in{\mathbb{N}}$ let
$r_{n}:=|\lambda|^{n}\operatorname{M}^{n}(L,R)$ (see also Lemma 4.2). By Lemma
4.3, there exists a simple closed curve $\Gamma^{r_{n}}$ separating
${\mathbb{S}}_{r_{n}}$ and ${\mathbb{S}}_{r_{n}^{d}}$ such that
$\operatorname{m}(L,\Gamma^{r_{n}})>\operatorname{M}(L,r_{n})$. We define
$G_{n}$ to be the interior of $\Gamma^{r_{n}}$. Then every $G_{n}$ is a
bounded simply-connected domain with
$\displaystyle
G_{n}\supset\\{z\in{\mathbb{C}}:|z|<r_{n}\\}\supset\\{z\in{\mathbb{C}}:|z|<\operatorname{M}^{n}(L,R)\\}.$
Furthermore, it follows from Lemma 4.2 with $m=d$ that
$\displaystyle\operatorname{m}(L,\partial
G_{n})=\operatorname{m}(L,\Gamma^{r_{n}})>\operatorname{M}(L,r_{n})>r_{n+1}^{d}>\max_{z\in\partial
G_{n+1}}|z|,$
hence $G_{n+1}$ is contained in a bounded component of ${\mathbb{C}}\setminus
L(\partial G_{n})$ and the claim follows. ∎
Note that Corollary 1.2 is an immediate consequence of Theorem 1.1.
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* [14] P. J. Rippon and G. M. Stallard, ‘On questions of Fatou and Eremenko’, _Proc. Amer. Math. Soc._ 133, no. 4 (2005), 1119–1126.
* [15] P. J. Rippon and G. M. Stallard, ‘Escaping points of entire functions of small growth’, Math. Z. 261, no. 3 (2009), 557–570, arXiv:math.DS/0801.3605.
* [16] P. J. Rippon and G. M. Stallard, ‘Fast escaping points of entire functions’, _Preprint_(2010)
* [17] G. Rottenfusser, J. Rückert, L. Rempe D. Schleicher, ‘Dynamic rays of bounded type entire functions’, to appear in Ann. Math., arXiv:math.DS/0704.3213.
* [18] G. Valiron, ‘Fonctions analytiques’, Presses Universitaires de France (1954).
* [19] G. T. Whyburn, ‘Analytic Topology’, American Mathematical Society Colloquium Publications, Vol. XXVIII (1942).
|
arxiv-papers
| 2010-10-02T07:30:06 |
2024-09-04T02:49:13.309756
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Helena Mihaljevi\\'c-Brandt, J\\\"orn Peter",
"submitter": "Helena Mihaljevic-Brandt",
"url": "https://arxiv.org/abs/1010.0299"
}
|
1010.0337
|
# Hamiltonian Vector Fields on Multiphase Spaces
of Classical Field Theory ††thanks: Work partially supported by CNPq (Conselho
Nacional de Desenvolvimento Científico e Tecnológico), Brazil
Michael Forger 1 and Mário Otávio Salles 2 Work done in partial fulfillment of
the requirements for the degree of Doctor in Science
(${}^{1}\,$ Departamento de Matemática Aplicada,
Instituto de Matemática e Estatística,
Universidade de São Paulo,
Caixa Postal 66281,
BR–05315-970 São Paulo, S.P., Brazil
${}^{2}\,$ Faculdade de Tecnologia de Carapicuiba
Av. Francisco Pignatari, 650,
BR–06390-310 Carapicuiba, S.P., Brazil )
###### Abstract
We present a classification of hamiltonian vector fields on multisymplectic
and polysymplectic fiber bundles closely analogous to the one known for the
corresponding dual jet bundles that appear in the multisymplectic and
polysymplectic approach to first order classical field theories.
Universidade de São Paulo
RT-MAP-0802
August 2024
## 1 Introduction
The quest for a fully covariant hamiltonian formulation of classical field
theory has a long history. In particular, the search for a first order
formalism analogous to that for classical mechanics in terms of concepts from
symplectic geometry has stimulated the development of new geometric tools
usually referred to as “multisymplectic” or “polysymplectic” structures whose
real significance is emerging only gradually. In fact, for many years there
has not even been a convincing general definition, although a standard class
of examples in terms of duals of jet bundles has long been known and widely
used.111This situation has been strikingly similar to that in classical
mechanics before it was realized that symplectic manifolds, rather than just
cotangent bundles, provide an adequate framework if one wants to accomodate
phenomena such as half-integral spin within classical mechanics. This defect
has recently been overcome [1], and it has been realized that both
multisymplectic and polysymplectic structures (which are not the same thing)
play an important role in the formalism; in particular, a multisymplectic
structure always induces a special kind of polysymplectic structure by means
of a construction called the “symbol”. But the fact that both types come
together to form a pair has been anticipated almost 20 years ago [2], when it
became apparent that the covariant hamiltonian formulation of first order
classical field theories requires the simultaneous use of two types of
“multiphase space” that we shall refer to as “ordinary multiphase space” and
“extended multiphase space”, respectively.
To be more precise, let us briefly recall the cornerstones of the construction
of the two types of multiphase space for first order lagrangian field
theories; for more details, the reader is referred to [2, 3, 4]. The starting
point is the choice of a fiber bundle $E$ over the space-time manifold $M$
called the configuration bundle because its sections represent the basic
fields of the theory at hand. Next, one takes the first order jet bundle $JE$
of $E$ to accomodate first order derivatives of these fields: this is an
affine bundle over $E$ and is also the domain of definition of the lagrangian.
Besides, one also considers the linearized first order jet bundle $\vec{J}E$
of $E$: this is a vector bundle over $E$ defined as the difference vector
bundle of $JE$. Finally, as in mechanics, one uses appropriate versions of the
Legendre transformation induced by the given lagrangian to pass to the
(twisted) affine dual
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ of $JE$ and to
the (twisted) linear dual
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ of
$\vec{J}E$: the former is the extended multiphase space and the latter is the
ordinary multiphase space of the theory. Note that the former is an affine
line bundle over the latter and that the hamiltonian obtained from the given
lagrangian through Legendre transformation is not a function but rather a
section of this affine line bundle, so both of these multiphase spaces are
essential ingredients for defining the concept of a hamiltonian system in
field theory! Moreover, it is well known that
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ carries a
naturally defined multisymplectic form $\,\omega$. However, what does not seem
to have been so widely noticed is the fact that
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ carries a
naturally defined polysymplectic form $\,\hat{\omega}$ – even though this form
already appears explicitly in Ref. [5]. As has been shown more recently [1],
it can be derived from the multisymplectic form $\,\omega$ on
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ by taking its
symbol, which turns out to be degenerate precisely along the fibers of the
aforementioned affine line bundle, and then passing to the corresponding
quotient of $J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ by
the kernel of $\,\omega$, which is precisely
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$. Note
that this polysymplectic form $\,\hat{\omega}$ on
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ is
canonical, whereas the form $\,\omega_{\mathscr{H}}$ on
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ obtained
as the pull-back of $\,\omega$ by means of a hamiltonian section
$\;\mathscr{H}:\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E\rightarrow
J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E\;$ is not, since
it depends on the choice of hamiltonian.222It should be noted that the form
$\,\omega_{\mathscr{H}}$ is closed and non-degenerate but not multisymplectic
in the sense of the definition given in Ref. [1].
In terms of adapted local coordinates $(x^{\mu},q^{i},p\>\\!_{i}^{\mu},p)$ for
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ and
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu})$ for
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$, induced
by local coordinates $x^{\mu}$ for $M$, local coordinates $q^{i}$ for the
typical fiber $Q$ of $E$ and a local trivialization of $E$ [4], we have
$\begin{array}[]{c}\mbox{extended multiphase space
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$}\\\\[2.84526pt]
\mbox{adapted local coordinates
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu},p)$}\\\\[2.84526pt] \mbox{multisymplectic
form}\quad\omega~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x\end{array}$
(1)
and
$\begin{array}[]{c}\mbox{ordinary multiphase space
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$}\\\\[2.84526pt]
\mbox{adapted local coordinates
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu})$}\\\\[2.84526pt] \mbox{polysymplectic
form}\quad\hat{\omega}~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,\,{\scriptstyle\otimes}\,\,d^{\,n}x_{\mu}\end{array}$
(2)
where $p\,$ is (except for a sign) a scalar energy variable and $d^{\,n}x$ is
the (local) volume form induced by the $x^{\mu}$ while $d^{\,n}x_{\mu}$ is the
(local) $(n\\!-\\!1)$-form obtained by contracting $d^{\,n}x$ with
$\,\partial_{\mu}\equiv\partial/\partial x^{\mu}$:
$d^{\,n}x_{\mu}~{}=~{}i_{\partial_{\mu}}\,d^{\,n}x~{}.$
The same picture prevails in the general case if we replace adapted local
coordinates by Darboux coordinates; see [1]. _Extended multiphase space is
multisymplectic, ordinary multiphase space is polysymplectic._
A crucial role in the development of the hamiltonian formalism is played by
the notion of a hamiltonian vector field. According to the picture outlined
above, this comes in two variants: a multisymplectic one and a polysymplectic
one. We shall deal with the two versions separately, beginning with the
pertinent definitions.
## 2 The multisymplectic case
According to Ref. [1], a multisymplectic fiber bundle of rank $N$ can be
defined as a fiber bundle $P$ over an $n$-dimensional base manifold $M$
equipped with a closed, non-degenerate $(n+1)$-form $\,\omega$ on its total
space $P$ which (a) is $(n-1)$-horizontal, i.e., such that its contraction
with any three vertical vector fields vanishes, and (b) admits a
multilagrangian distribution, i.e., an isotropic vector subbundle $L$ of the
vertical bundle $VP$ of $P$ of codimension $N$ and dimension $Nn+1$. (It then
turns out that $P$ has dimension $(N+1)(n+1)$.) Assuming this distribution to
be involutive, which is automatic as soon as $\,n\geqslant 3\,$ but has to be
imposed as a separate condition when $\,n=2\,$, Darboux’s theorem assures that
there exist local coordinates, called canonical local coordinates or Darboux
coordinates, in which $\,\omega$ assumes the form
$\omega~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x~{}.$
(3)
Locally, $\,\omega$ is exact, i.e.,
$\omega\;=\;-\,d\theta~{},$ (4)
where $d$ denotes the exterior derivative, with
$\theta~{}=~{}p\>\\!_{i}^{\mu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,+\;p\;d^{\,n}x~{}.$
(5)
The standard example is that of the extended multiphase space
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ mentioned above,
for which $\,\omega$ is also globally exact, i.e., the so-called
multicanonical form $\theta$ in equations (4) and (5) is globally defined, and
$L$ is the vector subbundle of $VP$ generated by the vector fields
$\partial/\partial p\>\\!_{i}^{\mu}$ and $\partial/\partial p$, that is, the
vertical bundle for the projection of
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ onto $E$ (with
respect to which $J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$
is a vector bundle).
Given this situation, we say that a vector field $X$ on $P$ is _locally
hamiltonian_ if $i_{X}\omega$ is closed, or equivalently, if
$L_{X}\omega~{}=~{}0~{}.$ (6)
It is called _globally hamiltonian_ if $i_{X}\omega$ is exact, that is, if
there exists an $(n-1)$-form $f$ on $P$ such that
$i_{X}\omega~{}=~{}df~{}.$ (7)
In this case, $f$ is said to be a _hamiltonian form associated with_ $X$.
Finally, when $\,\omega$ is exact and given by equation (4), $X$ is called
_exact hamiltonian_ if
$L_{X}\theta~{}=~{}0~{}.$ (8)
The main theorem states that these vector fields can be classified in terms of
their components with respect to canonical local coordinates, which are given
by the expansion
$X~{}=~{}X^{\mu}\,\frac{\partial}{\partial
x_{\phantom{i}}^{\mu}}\,+\,X^{i}\,\frac{\partial}{\partial
q^{i}}\,+\,X_{i}^{\mu}\,\frac{\partial}{\partial
p\>\\!_{i}^{\mu}}\,+\,X_{0}\,\frac{\partial}{\partial p}~{},$ (9)
whereas, locally, the hamiltonian form corresponding to such a vector field,
which is determined up to an arbitrary closed form, can be assumed to have an
expansion of the form
$f~{}=~{}f^{\mu}\;d^{\,n}x_{\mu}\,+\,{\textstyle\frac{1}{2}}\,f_{i}^{\mu\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}~{},$
(10)
where
$d^{\,n}x_{\mu\nu}~{}=~{}i_{\partial_{\nu}}i_{\partial_{\mu}}\,d^{\,n}x~{}.$
An easy calculation gives
$\begin{array}[]{rcl}i_{X}\omega\\!\\!&=&\\!\\!X^{\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}\,-\,X_{i}^{\mu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,+\,X^{i}\;dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\\\\[5.69054pt]
&&\\!\\!\mbox{}+\,X^{\mu}\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\,X_{0}\;d^{\,n}x~{},\end{array}$
(11)
and in the exact case
$i_{X}\theta~{}=~{}(p\>\\!_{i}^{\mu}X^{i}\,+\,p\,X^{\mu})\;d^{\,n}x_{\mu}\,-\,p\>\\!_{i}^{\mu}X^{\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}~{}.$
(12)
These formulas constitute the starting point for the proof of the following
###### Theorem 1
A vector field $X$ on $P$ is locally hamiltonian if and only if its components
$X^{\mu}$, $X^{i}$, $X_{i}^{\mu}$ and $X_{0}$ with respect to canonical local
coordinates, as defined by equation (9), satisfy the following conditions:
1. 1.
the coefficients $X^{\mu}$ and $X^{i}$ are independent of the multimomentum
variables $p\>\\!_{k}^{\kappa}$ and of the energy variable $p$, with the
coefficients $X^{\mu}$ depending only on the local coordinates $x^{\kappa}$ of
the base manifold $M$ as soon as $\,N>1$,
2. 2.
the remaining coefficients $X_{i}^{\mu}$ and $X_{0}$ can be expressed in terms
of the previous ones and of new coefficients $f_{0}^{\mu}$ which are also
independent of the multimomentum variables $p\>\\!_{k}^{\kappa}$ and of the
energy variable $p$, according to
$X_{i}^{\mu}~{}=\;-\,p\;\frac{\partial X^{\mu}}{\partial
q^{i}}\,-\,p\>\\!_{j}^{\mu}\;\frac{\partial X^{j}}{\partial
q^{i}}\,+\,p\>\\!_{i}^{\nu}\;\frac{\partial X^{\mu}}{\partial
x^{\nu}}\,-\,p\>\\!_{i}^{\mu}\;\frac{\partial X^{\nu}}{\partial
x^{\nu}}\,+\,\frac{\partial f_{0}^{\mu}}{\partial q^{i}}~{},$ (13)
(the first term being absent as soon as $\,N>1$) and
$X_{0}~{}=\;-\,p\;\frac{\partial X^{\mu}}{\partial
x^{\mu}}\,-\,p\>\\!_{i}^{\mu}\;\frac{\partial X^{i}}{\partial
x^{\mu}}\,+\,\frac{\partial f_{0}^{\mu}}{\partial x^{\mu}}~{}.$ (14)
The components of the corresponding hamiltonian form $f$ are given by
$f^{\mu}~{}=~{}p\>\\!_{i}^{\mu}X^{i}\,+\,p\,X^{\mu}\,+\,f_{0}^{\mu}~{},$ (15)
and
$f_{i}^{\mu\nu}~{}=~{}p\>\\!_{i}^{\nu}X^{\mu}\,-\,p\>\\!_{i}^{\mu}X^{\nu}~{}.$
(16)
In addition, when $\,\omega$ is exact and given by equation (4), $X$ is exact
hamiltonian if and only if the coefficients $f_{0}^{\mu}$ vanish.
This theorem has been explicitly stated in Ref. [6] and an explicit proof of a
more general theorem (where vector fields are replaced by multivector fields)
can be found in Ref. [7].
## 3 The polysymplectic case
According to Ref. [1], a polysymplectic fiber bundle of rank $N$ can be
defined as a fiber bundle $P$ over an $n$-dimensional base manifold $M$
equipped with a vertically closed, non-degenerate vertical $2$-form
$\,\hat{\omega}$ on its total space $P$ which (a) takes values in (the pull-
back $\pi^{*}\hat{T}$ to $P$ of) some given $\hat{n}$-dimensional coefficient
vector bundle $\hat{T}$ over $M$ and (b) admits a polylagrangian distribution,
i.e., an isotropic vector subbundle $L$ of the vertical bundle $VP$ of $P$ of
codimension $N$ and dimension $N\hat{n}$. (It then turns out that $P$ has
dimension $N\hat{n}+N+\hat{n}$.) Assuming this distribution to be involutive,
which is automatic as soon as $\,\hat{n}\geqslant 3\,$ but has to be imposed
as a separate condition when $\,\hat{n}=2\,$, Darboux’s theorem assures that
given any basis $\,\\{\,\hat{e}_{a}\,/\,1\leqslant a\leqslant\hat{n}\,\\}\,$
of local sections of $\hat{T}$, there exist local coordinates, called
canonical local coordinates or Darboux coordinates, in which $\,\hat{\omega}$
assumes the form
$\hat{\omega}~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{a}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{}.$
(17)
Locally, $\,\hat{\omega}$ is vertically exact, i.e.,
$\hat{\omega}\;=\;-\,d_{V}\hat{\theta}~{},$ (18)
where $d_{V}$ denotes the vertical exterior derivative, with
$\hat{\theta}~{}=~{}p\>\\!_{i}^{a}\;dq^{i}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{}.$
(19)
The standard example is that of the ordinary multiphase space
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ mentioned
above, with
$\,\hat{T}=\raisebox{0.86108pt}{${\textstyle\bigwedge}$}^{n-1}T^{*}M\,$ and
$\,e_{a}=d^{\,n}x_{\mu}$, for which $\,\hat{\omega}$ is also globally
vertically exact, i.e., the so-called polycanonical form $\hat{\theta}$ in
equations (18) and (19) is globally defined, and $L$ is the vector subbundle
of $VP$ generated by the vector fields $\partial/\partial p\>\\!_{i}^{\mu}$,
that is, the vertical bundle for the projection of
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ onto $E$
(with respect to which
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ is a
vector bundle).
Given this situation, we say that a vertical vector field $X$ on $P$ is
_locally hamiltonian_ if $i_{X}\hat{\omega}$ is vertically closed, or
equivalently, if
$L_{X}\hat{\omega}~{}=~{}0~{}.$ (20)
It is called _globally hamiltonian_ if $i_{X}\hat{\omega}$ is vertically
exact, that is, if there exists a section $f$ of the vector bundle
$\pi^{*}\hat{T}$ over $P$ such that
$i_{X}\hat{\omega}~{}=~{}d_{V}f~{}.$ (21)
In this case, $f$ is said to be a _hamiltonian section associated with_ $X$.
Finally, when $\,\hat{\omega}$ is vertically exact and given by equation (18),
$X$ is called _exact hamiltonian_ if
$L_{X}\hat{\theta}~{}=~{}0~{}.$ (22)
The main theorem states that these vector fields can be classified in terms of
their components with respect to canonical local coordinates, which are given
by the expansion
$X~{}=~{}X^{i}\,\frac{\partial}{\partial
q^{i}}\,+\,X_{i}^{a}\,\frac{\partial}{\partial p\>\\!_{i}^{a}}$ (23)
whereas, locally, the hamiltonian section corresponding to such a vector
field, which is determined up to (the pull-back to $P$ of) an arbitrary
section of $\hat{T}$, can be assumed to have an expansion of the form
$f~{}=~{}f^{a}\;\hat{e}_{a}~{}.$ (24)
An easy calculation gives
$i_{X}\hat{\omega}~{}=\;-\,X_{i}^{a}\;dq^{i}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}\,+\,X^{i}\;dp\>\\!_{i}^{a}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{},$
(25)
and in the exact case
$i_{X}\hat{\theta}~{}=~{}X^{i}p\>\\!_{i}^{a}\;\hat{e}_{a}~{}.$ (26)
These formulas constitute the starting point for the proof of the following
###### Theorem 2
A vector field $X$ on $P$ is locally hamiltonian if and only if its components
$X^{i}$ and $X_{i}^{\mu}$ with respect to canonical local coordinates, as
defined by equation (23), satisfy the following conditions:
1. 1.
the coefficients $X^{i}$ are independent of the multimomentum variables
$p\>\\!_{k}^{\kappa}$,
2. 2.
the remaining coefficients $X_{i}^{a}$ can be expressed in term of the
previous ones and of new coefficients $f_{0}^{a}$ which are also independent
of the multimomentum variables $p\>\\!_{k}^{\kappa}$, according to
$X_{i}^{a}~{}=\;-\,p\>\\!_{j}^{a}\;\frac{\partial X^{j}}{\partial
q^{i}}\,+\,\frac{\partial f_{0}^{a}}{\partial q^{i}}~{}.$ (27)
The components of the corresponding hamiltonian section $f$ are given by
$f^{a}~{}=~{}p\>\\!_{i}^{a}X^{i}\,+\,f_{0}^{a}~{},$ (28)
In addition, when $\,\hat{\omega}$ is vertically exact and given by equation
(18), $X$ is exact hamiltonian if and only if the coefficients $f_{0}^{a}$
vanish.
The proof of this theorem is entirely analogous to that of the previous one,
except that it is much simpler. The essence of the argument can already be
found in Ref. [8, 9], but the proper global context of the result is not
adequately treated there.
## 4 Outlook
The analogous problem of determining hamiltonian vector fields with respect to
the form $\,\omega_{\mathscr{H}}$ on ordinary multiphase space mentioned in
the introduction has been addressed and solved in Ref. [10], but the results
are somewhat complicated and not very enlightening. We now believe this to be
related to the fact that, according to the structurally natural definition
given in Ref. [1], $\omega_{\mathscr{H}}$ is _not_ multisymplectic.
One problem that, for the time being, remains open is to give a global,
coordinate independent formulation of the results of Theorems 1 and 2. This
question is presently under investigation.
## References
* [1] M. Forger & L. Gomes: _Multisymplectic and Polysymplectic Structures on Fiber Bundles_ , arXiv:0708.1586.
* [2] J.F. Cariñena, M. Crampin & L.A. Ibort: _On the Multisymplectic Formalism for First Order Field Theories_ , Diff. Geom. Appl. 1 (1991) 345-374.
* [3] M.J. Gotay, J. Isenberg, J.E. Marsden & R. Montgomery: _Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory_, arXiv:physics/9801019v2.
* [4] M. Forger & S.V. Romero: _Covariant Poisson Brackets in Geometric Field Theory_ , Commun. Math. Phys. 256 (2005) 375-410, arXiv:math-ph/0408008.
* [5] C. Günther: _The Polysymplectic Hamiltonian Formalism in Field Theory and Calculus of Variations I: the Local Case_ , J. Diff. Geom. 25 (1987) 23-53.
* [6] M. Forger & H. Römer: _A Poisson Bracket on Multisymplectic Phase Space_ , Rep. Math. Phys. 48 (2001) 211-218, arXiv:math-ph/0009037.
* [7] M. Forger, C. Paufler & H. Römer: _Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory_ , J. Math. Phys. 46 (2005) 112903, 29 pp., arXiv:math-ph/0407057.
* [8] I. Kanatchikov: _On Field Theoretic Generalizations of a Poisson Algebra_ , Rep. Math. Phys. 40 (1997) 225-234, arXiv:hep-th/9710069.
* [9] I. Kanatchikov: _Canonical Structure of Classical Field Theory in the Polymomentum Phase Space_ , Rep. Math. Phys. 41 (1998) 49-90, arXiv:hep-th/9709229.
* [10] M.O. Salles: _Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos_ , PhD thesis, IME-USP, June 2004.
|
arxiv-papers
| 2010-10-02T14:07:40 |
2024-09-04T02:49:13.320933
|
{
"license": "Public Domain",
"authors": "Michael Forger and M\\'ario Ot\\'avio Salles",
"submitter": "Mario Salles MOS",
"url": "https://arxiv.org/abs/1010.0337"
}
|
1010.0367
|
# Atomistic modeling of the
phonon dispersion and lattice properties of
free-standing $\langle$100$\rangle$ Si nanowires
Abhijeet Paul, Mathieu Luisier and Gerhard Klimeck School of Electrical and
Computer Engineering and Network for Computational Nanotechnology,
Purdue University, West Lafayette, IN, USA- 47906, e-mail: paul1@purdue.edu
###### Abstract
Phonon dispersions in $\langle$100$\rangle$ silicon nanowires (SiNW) are
modeled using a Modified Valence Force Field (MVFF) method based on atomistic
force constants. The model replicates the bulk Si phonon dispersion very well.
In SiNWs, apart from four acoustic like branches, a lot of flat branches
appear indicating strong phonon confinement in these nanowires and strongly
affecting their lattice properties. The sound velocity ($V_{snd}$) and the
lattice thermal conductance ($\kappa_{l}$) decrease as the wire cross-section
size is reduced whereas the specific heat ($C_{v}$) increases due to increased
phonon confinement and surface-to-volume ratio (SVR).
## I Introduction
Silicon nanowires (SiNWs) are playing a vital role in areas ranging from CMOS
[1] to thermo-electricity [2]. The finite extent and increased surface-to-
volume ratio (SVR) in these nanowires result in very different phonon
dispersions compared to the bulk materials [3, 4]. This work investigates the
effect of geometrical confinement on the phonon dispersion, the sound velocity
($V_{snd}$), the specific heat ($C_{v}$), and the lattice thermal conductance
($\kappa_{l}$) in $\langle$100$\rangle$ SiNWs using a Modified Valence Force
Field (MVFF) model based on atomistic force constants.
Previous theoretical works have reported the calculation of phonon dispersions
in SiNWs using a continuum elastic model and Boltzmann transport equation [4],
atomistic first principle methods like DFPT (Density Functional Perturbation
Theory) [5, 6, 7, 8] and atomistic frozen phonon approaches like Keating-VFF
(KVFF) [9, 10]. Thermal conductivity in SiNWs has been studied previously
using the KVFF model [11, 12]. The continuum approaches fail to capture the
atomistic effects like anisotropy in $C_{v}$ for nanowires, etc., whereas the
first principle models cannot be extended to very large structures due to
their heavy computational requirement. The VFF model overcomes these
shortcoming by capturing the atomistic effects and solving phonon dispersions
and transport in realistic structures [9, 11]. However, a simple KVFF model
does not reproduce the bulk dispersions very well and hence cannot be used
confidently for nanostructures [13]. The MVFF model overcomes the verifiable
shortcomings in bulk. We use the MVFF model for phonon calculations in SiNWs
and believe them to be more robust and predictive than the KVFF model [13].
Figure 1: The short range interactions used for Phonon dispersion
calculations. The interactions are (a) bond-stretching($\alpha$), (b) bond-
bending($\beta$) (c) bond stretch coupling($\gamma$) (d) bond bending-
stretching coupling ($\delta$) (e) coplanar bond bending coupling($\lambda$).
Figure 2: Projected unitcell structure of a $\langle$100$\rangle$ oriented
Silicon nanowire. The white atoms (for visual guidance) show the freely
vibrating surface atoms. Grey colored atoms are inside the wire
This paper is organized in the following sections. Section II briefly presents
the phonon model and its application to SiNWs. Also the computation of
different wire lattice properties is outlined there. The results on phonon
dispersion and other lattice properties are presented in Sec. III. Conclusions
are summarized in Sec. IV
## II Theory and Approach
### II-A Modified VFF method
The Valence Force Field (VFF) model is a force constant based atomic potential
(U) calculation method. The MVFF model we present here is based on the
combination of two different extended VFF models, (a) VFF model from Sui et.
al. [14] which is suited for non-polar materials like Si, Ge and (b) VFF model
from Zunger et. al. [15] which is suited for polar materials like GaP, GaAs,
etc. These methods have successfully modelled the phonon dispersion in various
semiconductor materials [16, 14, 15]. The force constants represent the
various kinds of interaction between the atoms. For phonon modeling in diamond
(and zinc-blende) lattices we consider the following interactions (Fig. 1),
(a) bond-stretching ($\alpha$), (b) bond-bending ($\beta$) (c) bond stretch
coupling ($\gamma$) (d) bond bending-stretching coupling ($\delta$) and (e)
co-planar bond bending coupling ($\lambda$). The first two terms (a and b in
Fig. 1) are from the original Keating model [16], which are not sufficient to
reproduce the bulk phonon dispersion accurately in zinc-blende semiconductors
[14, 13]. Hence, higher order interactions (term c, d and e in Fig. 1) are
included in the original KVFF model. The force constants in the MVFF model
depend on the bond length and angle variations from the ideal value [14]. Due
to this reason MVFF is also known as the ‘quasi-anharmonic model’.
The motion of atoms in the semiconductor structure is captured by a dynamical
matrix (DM). The dynamical matrix is assembled using the second derivative
(Hessian) of the crystal potential energy (U). The bulk DM has periodic
boundary conditions along all the directions (Born-von Karman condition, Fig.
1) due to the assumed infinite material extent along all the directions. For
free-standing SiNW, periodic boundary conditions are applied only along the
length of the wire (X-axis) assuming an infinitely long wire, whereas the
surface atoms (Y-Z axis) are free to vibrate as illustrated in Fig. 2). The
complete calculation details are provided in Ref. [13].
### II-B Lattice property calculations
A wealth of information can be extracted from the phonon spectrum of solids.
The description of the few important ones are provided below.
Sound Velocity: One important parameter is the group velocity ($V_{grp}$) of
the acoustic branches of the phonon dispersion. Near the Brillouin Zone (BZ)
center $V_{grp}$ characterizes the velocity of sound ($V_{snd}$) in the solid.
Depending on the acoustic phonon branch used for the calculation of $V_{grp}$,
the sound velocity can be either (a) longitudinal ($V_{snd,l}$) or (b)
transverse ($V_{snd,t}$). Thus, $V_{snd}$ is given by,
$V_{sndl/t}=V_{grp}=\frac{\partial\omega(\lambda,q)}{\partial
q}\Big{|}_{q\leftarrow 0},$ (1)
where, $\lambda$ and q are the phonon polarization (for longitudinal or
transverse direction) and wave vector respectively.
Figure 3: Comparison of the MVFF method (lines) and the experimental data
(dots) [17] for bulk Silicon phonon dispersion at 80K.
Lattice thermal properties: In a semiconductor where its two ends are
maintained at a small temperature difference ($\Delta T$), its specific heat
($C_{v}$) and thermal conductance ($\kappa_{l}$) can be evaluated from the
phonon dispersion using the Landauer’s formula [18]. The temperature dependent
specific heat is given by[19, 7],
$C_{v}(T)=k_{B}\sum_{n,q}\Big{[}\frac{\Big{(}\frac{\hbar\omega(n,q)}{k_{B}T}\Big{)}^{2}\cdot\exp\Big{(}\frac{-\hbar\omega(n,q)}{k_{B}T}\Big{)}}{(1-\exp\big{(}\frac{-\hbar\omega(n,q)}{k_{B}T)}\big{)}^{2}}\Big{]},$
(2)
where $k_{B}$, $\hbar$, T, and n are Boltzmann’s constant, reduced Planck’s
constant, mean temperature, and number of sub-bands, respectively. The
temperature dependent lattice thermal conductance ($\kappa_{l}$) for a 1D
conductor is given by [11, 6, 19],
$\kappa_{l}(T)=\hbar\int^{\omega_{max}}_{0}M(\omega)\cdot\omega\cdot\frac{\partial}{\partial
T}\Big{[}(\exp\big{(}\frac{\hbar\omega}{k_{B}T})-1\big{)}^{-1}\Big{]}d\omega,$
(3)
where, $M(\omega)$ is the number of modes at a given frequency $\omega$. In
the next section we present the results on phonon dispersion in SiNWs and the
lattice thermal properties.
TABLE I: Force constants used for Silicon in MVFF model Sp.const(N/m) | $\alpha$ | $\beta$ | $\gamma$ | $\delta$ | $\lambda$
---|---|---|---|---|---
Silicon | 45.1 | 4.89 | 0 | 1.36 | 9.14
## III Results and Discussion
### III-A Experimental benchmark
The MVFF model has been bench marked by calculating the phonon dispersions in
different bulk zinc-blende semiconductors. One of the representative
calculation is shown in Fig. 3 for bulk Si. The model accurately reproduces
the important features of the experimental phonon dispersion of Si (at 80K)
[17] in the entire BZ. The force constant values for Si are listed in Table.I.
The bond stretch coupling parameter ($\gamma$) is zero for Si. This term
contributes significantly for splitting the longitudinal optical (LO) and
longitudinal acoustic (LA) phonon branches near the BZ edge only in III-V
materials like GaAs, GaP [15].
### III-B Nanowire phonon dispersions
Figure 4: Effect of cross-section size on the phonon dispersion in
$\langle$100$\rangle$ SiNW. The wire dimensions are (W = H) (a) 2nm (b) 4nm,
and (c) 6nm. As the cross-section size increases the number of phonon modes
within a given energy range increase as shown here. The geometrical
quantization increases the separation of bands in smaller wires. Figure 5:
Eigen modes of the atomic displacement shown in a SiNW with W, H = 1.63nm. The
atomic displacement are shown at $q_{x}\sim$ 0 for (a) mode = 1, (b) mode = 2.
These modes are called flexural modes and cause bending of the NW. (c) mode =
3, called the longitudinal mode causes atomic motion along the wire axis. (d)
mode = 4, called the torsional mode, observed only in wires, represents the
extra rotational degree of freedom.
The phonon dispersion is calculated in free-standing $\langle$100$\rangle$
SiNW where the surface atoms are allowed to vibrate freely (Fig. 2). The
dispersion in SiNWs with cross-section size (W=H) (a) 2nm, (b) 4nm, and (c)
6nm are compared in Fig. 4. The number of sub-bands increases as the wire
cross-section size increases due to an increased number of atoms (Fig. 4).
This results in a higher degree of vibrational freedom. A comparison of the
number of sub-bands within 10 meV of energy for the different SiNWs considered
here shows that the 2nm $\times$ 2nm structure has $\sim$ 40 sub-bands, the
4nm $\times$ 4nm has $\sim$ 90 sub-bands and the 6nm $\times$ 6nm has nearly
110 sub-bands (Fig. 4). The increased separation of phonon sub-bands in
smaller wires is a consequence of the geometrical confinement. The higher sub-
bands have mixed acoustic and optical branch like properties. Many of the sub-
bands in this regime have group velocities close to zero which again indicates
a strong phonon confinement in thin nanowires. In SiNWs these flat phonon
branches appear due to the zone-folding of acoustic branches from the bulk
phonon-dispersion (Fig. 3).
Eigen modes of oscillation: The first four vibrational Eigen modes are shown
in Fig. 5 (arrows show the displacement direction). The first two modes (Fig.
5 a, b) are the flexural modes responsible for NW bending perpendicular to the
NW axis. The Eigen frequencies for the lowest two phonon bands near the zone
center are $\propto q^{2}$. The next mode (Fig. 5 c) is the longitudinal mode.
The mode no. 4 is torsional mode which represents the extra degree of
rotational movement in free standing wires (Fig. 5 d). Mode no. 3 and 4 behave
like the bulk acoustic branches ($\propto q$ near the zone center) [10]. Mode
1 and 2 appear due to the free-standing nature of the nanowires which are not
available in bulk. Hence, new modes as well as new phonon dispersion branches
appear in SiNW which are very different from their bulk counterpart. This also
has a strong impact on the lattice properties of the wires which is discussed
in the following part.
### III-C Lattice properties in SiNW
#### Sound velocity variation ($V_{snd}$)
The sound velocity ($V_{snd}$) is another way to compare the vibrational modes
in SiNWs. $V_{snd}$ is obtained using the sub-bands 3 and 4 (transverse and
longitudinal modes) of the phonon dispersion. $V_{snd}$ shows a reduction with
decreasing wire cross-section size (acoustic mode softening) [10] (Fig. 6).
This happens since the acoustic branches get flatter in small wires due to
phonon confinement. This strongly affects the thermal properties of the SiNWs
since the acoustic branches are mainly responsible for heat conduction. A
smaller sound velocity also results in smaller thermal conductance (discussed
later).
Figure 6: Sound Velocity ($V_{snd}$) in $\langle$100$\rangle$ SiNW with free
boundary. As a reference the bulk $V_{snd}$ is shown along the
$\langle$100$\rangle$ direction [20]. Increasing the wire diameter increases
the $V_{snd}$ since the vibrational degree of freedom increases. Figure 7:
Variation of the specific heat ($C_{v}$) in Silicon nanowires with width. As a
reference the specific heat for bulk Si is shown by a dashed line. SiNWs have
larger $C_{v}$ than bulk Si [7]. The specific heat increases as the cross-
section size of the wire decreases.
#### SiNW specific heat ($C_{V}$) variation
The specific heat of silicon nanowires increases with their decreasing cross-
section size (Fig. 7). This can be attributed to two reasons, (i) phonon
confinement due to small cross-section and (ii) an increased surface-to-volume
ratio (SVR) in smaller wires. Geometrical confinement separates the phonon
bands in energy with decreasing wire cross-section size (Fig. 4) which makes
only the few lower energy bands active at a given temperature (see Eq.2). This
explains the increasing energy needed to raise the temperature of the smaller
wires. Also the increasing SVR results in a higher partial phonon DOS
associated with the wire surface, which further enhances the specific heat
with decreasing wire cross-section [7].
Figure 8: Variation in the ballistic lattice thermal conductance
($\kappa_{l}$) (a) with temperature for different cross-section size SiNW and
(b) at room temperature (300K) for different NW width size.
#### Lattice thermal conductance ($\kappa_{l}$) variation
The modified phonon dispersion in SiNWs significantly affects the ballistic
$\kappa_{l}$ [11]. The ballistic $\kappa_{l}$ increases with temperature as
shown in Fig. 8a. The spread of the Bose-Einstein distribution
($F_{BD}=[\exp(\hbar\omega/k_{B}T)-1]^{-1}$ in energy increases with
increasing temperature. This results in a larger number of modes contributing
to the heat transfer [6] (see Eq. (3)). This explains the increased ballistic
$\kappa_{l}$ at elevated temperatures. Larger wires show more $\kappa_{l}$
since, (i) they have more phonon sub-bands resulting in more number of modes
(see Eq. (3)) and (ii) they have a higher acoustic velocity which is
responsible for larger heat conduction (Fig. 6). The ballistic $\kappa_{l}$ of
2nm $\times$ 2nm Si nanowire reduces by a factor of 8 as compared to the 6nm
$\times$ 6nm SiNW (Fig. 8b).
An important point to note here is that $\kappa_{l}$ is expected to decrease
further in smaller wires due to phonon scattering by other phonons, interfaces
and boundaries [11] which is neglected in the present study. The main message
here is that even ballistic phonons show a reduction in thermal conductance
with narrow SiNWs, which comes from (i) a modification of the phonon
dispersion (ii) phonon confinement effects in SiNWs.
## IV Conclusions
A modified atomistic force constant based model has been developed for the
calculation of phonon dispersion in zinc-blende lattices. This model shows a
very good agreement with experimental bulk phonon data. An important impact of
device miniaturization is the presence of surfaces (and hence higher surface-
to-volume ratio) which affects the phonon dispersion in SiNW. The clear
demarcation of phonon bands as acoustic and optical branches become vague in
SiNW. This manifests itself in the form of modified lattice properties in
SiNWs compared to bulk Si. We find that decreasing wire cross-section reduces
the sound velocity and the lattice thermal conductance whereas the specific
heat increases. The geometry dependent calculation of phonon dispersion in
small nanowires is therefore important to properly understand their thermal
properties.
## Acknowledgments
The authors would like to acknowledge the computational resources from
nanoHUB.org, an National Science Foundation (NSF) funded, NCN project.
Financial support from MSD Focus Center, one of six research centers funded
under the Focus Center Research Program (FCRP), a Semiconductor Research
Corporation (SRC) entity, by the Nanoelectronics Research Initiative (NRI)
through the Midwest Institute for Nanoelectronics Discovery (MIND) and NSF
PetaApps, grant number OCI-0749140, are also acknowledged.
## References
* [1] “ITRS Report 2009,” ITRS, Tech. Rep., 2009. [Online]. Available: www.itrs.net
* [2] A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” _Nature_ , vol. 451, no. 7175, pp. 163–167, 2008.
* [3] A. Buin, A. Verma, and M. Anantram, “Carrier-Phonon interaction in small cross-sectional silicon nanowires,” _Journal of Applied Physics_ , vol. 104, p. 053716, 2008.
* [4] J. Zou and A. Balandin, “Phonon heat conduction in a semiconductor nanowire,” _Journal of Applied Physics_ , vol. 89, no. 5, pp. 2932–2938, 2001.
* [5] H. Peelaers, B. Partoens, and F. M. Peeters, “Phonon Band Structure of Si Nanowires: A Stability Analysis,” _Nano Letters_ , vol. 9, no. 1, pp. 107–111, 2009.
* [6] T. Markussen, A.-P. Jauho, and M. Brandbyge, “Heat Conductance Is Strongly Anisotropic for Pristine Silicon Nanowires,” _Nano Letters_ , vol. 8, no. 11, pp. 3771–3775, 2008.
* [7] Y. Zhang, J. X. Cao, Y. Xiao, and X. H. Yan, “Phonon spectrum and specific heat of silicon nanowires,” _Journal of Applied Physics_ , vol. 102, no. 10, p. 104303, 2007.
* [8] X. Li, K. Maute, M. L. Dunn, and R. Yang, “Strain effects on the thermal conductivity of nanostructures,” _Phys. Rev. B_ , vol. 81, no. 24, p. 245318, Jun 2010.
* [9] N. Mingo and L. Yang, “Phonon transport in nanowires coated with an amorphous material: An atomistic Green’s function approach,” _Phys. Rev. B_ , vol. 68, no. 24, p. 245406, Dec 2003.
* [10] T. Thonhauser and G. D. Mahan, “Phonon modes in Si [111] nanowires,” _Phys. Rev. B_ , vol. 69, no. 7, p. 075213, Feb 2004.
* [11] N. Mingo, L. Yang, D. Li, and A. Majumdar, “Predicting the Thermal Conductivity of Si and Ge Nanowires,” _Nano Letters_ , vol. 3, no. 12, pp. 1713–1716, 2003.
* [12] H. Zhao, Z. Tang, G. Li, and N. R. Aluru, “Quasiharmonic models for the calculation of thermodynamic properties of crystalline silicon under strain,” _Journal of Applied Physics_ , vol. 99, no. 6, p. 064314, 2006.
* [13] A. Paul, M. Luisier, and G. Klimeck, “Modified valence force field approach for phonon dispersion: from zinc-blende bulk to nanowires Methodology and computational details,” _http://arxiv.org/abs/1009.6188_ , 2010.
* [14] Z. Sui and I. P. Herman, “Effect of strain on phonons in Si, Ge, and Si/Ge heterostructures,” _Phys. Rev. B_ , vol. 48, no. 24, pp. 17 938–17 953, 1993.
* [15] H. Fu, V. Ozolins, and Z. Alex, “Phonons in GaP quantum dots,” _Phys. Rev. B_ , vol. 59, no. 4, pp. 2881–2887, 1999.
* [16] P. N. Keating, “Effect of Invariance Requirements on the Elastic Strain Energy of Crystals with Application to the Diamond Structure,” _Phys. Rev._ , vol. 145, no. 2, pp. 637–645, 1966.
* [17] G. Nilsson and G. Nelin, “Study of the Homology between Silicon and Germanium by Thermal Neutron Spectrometry,” _Phys. Rev. B_ , vol. 6, no. 10, pp. 3777–3786, 1972.
* [18] R. Landauer, “Spatial variation of currents and fields due to localized scatterers in metallic conduction,” _IBM J. Res. Dev._ , vol. 1, no. 3, pp. 223–231, 1957.
* [19] D. C. Wallace, “Thermodynamics of crystals,” _Courier Dover Publications_ , 1998.
* [20] M. G. Holland, “Analysis of Lattice Thermal Conductivity,” _Phys. Rev._ , vol. 132, no. 6, pp. 2461–2471, Dec 1963.
|
arxiv-papers
| 2010-10-02T22:07:40 |
2024-09-04T02:49:13.328909
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhijeet Paul and Mathieu Luisier and Gerhard Klimeck",
"submitter": "Abhijeet Paul",
"url": "https://arxiv.org/abs/1010.0367"
}
|
1010.0375
|
# Fractional Hadamard transform with continuous variables in the context of
quantum optics††thanks: Project supported by the National Natural Science
Foundation of China (Grant Nos 10775097 and 10874174) and the Research
Foundation of the Education Department of Jiangxi Province of China.
Li-yun Hu, Xue-xiang Xu, and Shan-jun Ma
College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China Corresponding author. E-mail: hlyun2008@126.com;
hlyun@jxnu.edu.cn.
###### Abstract
We introduce the quantum fractional Hadamard transform with continuous
variables. It is found that the corresponding quantum fractional Hadamard
operator can be decomposed into a single-mode fractional operator and two
single-mode squeezing operators. This is extended to the entangled case by
using the bipartite entangled state representation. The new transformation
presents more flexibility to represent signals in the fractional Hadamard
domain with extra freedom provided by an angle and two-squeezing parameters.
Keywords: fractional Hadamard transform, fractional Hadamard operator, the
additivity of operator
PACC: 0367, 4250
## 1 Introduction
Fractional Fourier transform (FrFT) is a generalization of the ordinary
Fourier transform, which has been used in signal processing and image
manipulations [1, 2]. The concept of the FrFT was originally described by
Condon [3] and was later introduced for signal processing by Namias [4] as a
Fourier transform of fractional order. The 1-dimension FrFT of $\alpha$-order
is defined in Refs.[5, 6] as
$g\left(x^{\prime}\right)=\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\sin\alpha}}\int
e^{-\frac{\mathtt{i}\left(x^{\prime
2}+x^{2}\right)}{2\tan\alpha}+\frac{\mathtt{i}xx^{\prime}}{\sin\alpha}}f\left(x\right)\mathtt{d}x.$
(1)
The usual Fourier transform is a special case with order $\alpha=\pi/2$. On
the other hand, many orthogonal transform have been successfully used in
signal processing, such as discrete cosine transform [7], discrete Hartley
transform [8] and Hadamard transform.
Hadamard transform is not only an important tool in classical signal
processing, but also is of great importance for quantum computation
applications [9]. This transform, used to go from the position basis
$\left|x\right\rangle$ to the momentum basis, is defined as [10, 11]
$\mathcal{F}\left|x\right\rangle=\frac{1}{\sqrt{\pi}\sigma}\int_{-\infty}^{\infty}e^{2\mathtt{i}xy/\sigma^{2}}\left|y\right\rangle\mathtt{d}y,$
(2)
where $\sigma$ is the scale length (also makes the expression in the
exponential dimensionless), $\left|x\right\rangle$ and $\left|y\right\rangle$
are the eigenvector of coordinate operator $X.$ In Ref.[12], the explicit form
of $\mathcal{F}$ has been derived by using the technique of integration within
an ordered product (IWOP) of operators [13, 14, 15], and it is found that it
can be decomposed into a single-mode squeezing operator and a position-
momentum mutual transform operator, i.e., $\mathcal{F=}S_{1}^{-1}(-1)^{i\pi
a^{{\dagger}}a/2}$. In addition, the two-mode Hadamard transform with
continuous variables is also introduced by using the bipartite entangled state
representation, whose Hadamard operator involves a two-mode squeezing operator
and a mutual transform operator.
In this paper, we shall introduce the continuous fractional Hadamard transform
(CFrHT), which is a generalization of the usual Hadamard transform in Eq.(2).
The development of the CFrHT is based upon the same spirit of continuous
fractional Fourier transform (CFrFT). Then the CFrHT operator (CFrHTO) is
derived by using the IWOP technique, and its properties are analyzed. It is
found that the CFrHTO can be decomposed into a single-mode fractional operator
$e^{\mathtt{i}\alpha a^{\dagger}a}$ and two single-mode squeezing operators.
On the other hand, since the publication of the paper of Einstein, Podolsky
and Rosen (EPR) in 1935 [16], arguing the incompleteness of quantum mechanics,
the conception of entanglement has become more and more fascinating and
important as it plays a central role in quantum imformation and quantum
computation, we also shall introduce the two-mode CFrHO in bipartite entangled
state representation, which turns out to involve the two fractional operators
and two two-mode squeezing operators.
Our work is arranged as follows. In section 2, for the single-mode case, the
normally ordered fractional Hadamard operator is derived by using the IWOP
technique. The properties of fractional Hadamard operator is discussed in
section 3, such as the unitarity, the decomposition of the CFrHO and its
transform relation. Then the single-mode case is extended to two-mode case in
section 4 and some similar discussions to singled-mode case are presented.
Section 5 is devoted to exploring the measurements for the output states from
the CFrHT. Conclusions are involved in the last section.
## 2 Normally Ordered Fractional Hadamard Operator
In this section, we first introduce the continuous fractional Hadamard
transform (CFrHT), i.e.,
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|x\right\rangle=\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\mu\nu\sin\alpha}}\int_{-\infty}^{\infty}\exp\left\\{-\frac{\mathtt{i}\left(x^{2}/\mu^{2}+y^{2}/\nu^{2}\right)}{2\tan\alpha}+\frac{\mathtt{i}xy}{\mu\nu\sin\alpha}\right\\}\left|y\right\rangle\mathtt{d}y,$
(3)
where $\mu,\nu$ are the scale length (also make the expression in the
exponential dimensionless), $\alpha$ is an angle, and
$\mathcal{H}^{\alpha}\left(\mu,\nu\right)$ is called the CFrHT operator
(CFrHO). In particular, when $\alpha=\pi/2$ and $\mu=\nu=\sigma/\sqrt{2},$
Eq.(3) just reduces to Eq.(2).
In order to find the explicit expression of the CFrHO, multiplying Eq.(3) by
the bra $\int\mathtt{d}x\left\langle x\right|$ from the rights in two-side,
where $\left|y\right\rangle$ and $\left|x\right\rangle$ are coordinate
eigenvectors, $X\left|x\right\rangle=x\left|x\right\rangle$, and
$\left|x\right\rangle=\pi^{-1/4}\exp\left\\{-\frac{x^{2}}{2}+\sqrt{2}xa^{\dagger}-\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle,$ (4)
we can recast the CFrHO $\mathcal{H}_{\alpha}\left(\mu,\nu\right)$ into the
following integral form,
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)=\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\mu\nu\sin\alpha}}\int_{-\infty}^{\infty}\exp\left\\{-\frac{\mathtt{i}\left(x^{2}/\mu^{2}+y^{2}/\nu^{2}\right)}{2\tan\alpha}+\frac{\mathtt{i}xy}{\mu\nu\sin\alpha}\right\\}\left|y\right\rangle\left\langle
x\right|\mathtt{d}x\mathtt{d}y.$ (5)
Then using the vacuum projector’s normal ordering form
$\left|0\right\rangle\left\langle 0\right|=\colon
e^{-a^{{\dagger}}a}\colon$(where the symbol $\colon$ $\colon$ denotes the
normally ordering) and the IWOP technique to directly perform the integration,
we finally obtain
$\displaystyle\mathcal{H}_{\alpha}\left(\mu,\nu\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\mu\nu\sin\alpha}}\int_{-\infty}^{\infty}\colon\exp\left\\{-\frac{A}{2\mu^{2}}x^{2}+\sqrt{2}xa+\frac{\mathtt{i}xy}{\mu\nu\sin\alpha}\right.$
(6)
$\displaystyle-\left.\frac{B}{2\nu^{2}}y^{2}+\sqrt{2}ya^{\dagger}-\frac{\left(a^{{\dagger}}+a\right)^{2}}{2}\right\\}\colon\mathtt{d}x\mathtt{d}y$
$\displaystyle=$ $\displaystyle\sqrt{\frac{2\mu\nu
e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{u\sin\alpha}}\exp\left\\{\left(\frac{\nu^{2}A}{u}-\frac{1}{2}\right)a^{{\dagger}2}\right\\}$
$\displaystyle\times\exp\left\\{a^{\dagger}a\ln\frac{\mathtt{i}2\mu\nu}{u\sin\alpha}\right\\}\exp\left\\{\left(\frac{\mu^{2}B}{u}-\frac{1}{2}\right)a^{2}\right\\},$
where we have set $\allowbreak
A=\mathtt{i}\cot\alpha+\mu^{2},B=\mathtt{i}\cot\alpha+\nu^{2},u=\csc^{2}\alpha+AB,$and
used the operator identity in the last step of Eq.(6),
$\exp\left\\{fa^{\dagger}a\right\\}=\colon\exp\left\\{\left(e^{f}-1\right)a^{\dagger}a\right\\}\colon,$
(7)
Eq.(6) is the normally ordered form of the CFrHO. In particular, when
$\alpha=\pi/2$ and $\mu=\nu=\sigma/\sqrt{2},$ leading to
$A=B=\sigma^{2}/2,u=1+\sigma^{4}/4,$ then Eq.(6) becomes
$\displaystyle\mathcal{H}_{\pi/2}\left(\sigma/\sqrt{2},\sigma/\sqrt{2}\right)$
$\displaystyle=$
$\displaystyle\frac{2\sigma}{\sqrt{\sigma^{4}+4}}\exp\left\\{\frac{\sigma^{4}-4}{\sigma^{4}+4}\frac{a^{\dagger
2}}{2}\right\\}$ (8)
$\displaystyle\times\exp\left\\{a^{\dagger}a\ln\frac{4\mathtt{i}\sigma^{2}}{\sigma^{4}+4}\right\\}\exp\left\\{\frac{\sigma^{4}-4}{\sigma^{4}+4}\frac{a^{2}}{2}\right\\},$
which is just the result Eq.(7) in Ref.[12].
## 3 Properties of Fractional Hadamard Operator
From Eq.(5) one can see that the CFrHO is a unitary one, i.e.,
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)=\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)\mathcal{H}_{\alpha}\left(\mu,\nu\right)=1.$
In fact, uisng Eq.(5) and the orthogonality of coordinate state, $\left\langle
x^{\prime}\right.\left|x\right\rangle=\delta\left(x-x^{\prime}\right)$, we
have
$\displaystyle\mathcal{H}_{\alpha}\left(\mu,\nu\right)\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\mu\nu\sin\alpha}\int_{-\infty}^{\infty}\exp\left\\{\allowbreak\frac{\mathtt{i}\left(y^{\prime
2}-y^{2}\right)}{2\nu^{2}\tan\alpha}+\mathtt{i}x\frac{y-y^{\prime}}{\mu\nu\sin\alpha}\right\\}\left|y\right\rangle\left\langle
y^{\prime}\right|\mathtt{d}x\mathtt{d}y\mathtt{d}y^{\prime}$ (9)
$\displaystyle=$
$\displaystyle\frac{1}{\mu\nu\sin\alpha}\int_{-\infty}^{\infty}\delta\left(\frac{y-y^{\prime}}{\mu\nu\sin\alpha}\right)\exp\left\\{\allowbreak\frac{\mathtt{i}\left(y^{\prime
2}-y^{2}\right)}{2\nu^{2}\tan\alpha}\right\\}\left|y\right\rangle\left\langle
y^{\prime}\right|\mathtt{d}y\mathtt{d}y^{\prime}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left|y\right\rangle\left\langle
y\right|\mathtt{d}y=\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)\mathcal{H}_{\alpha}\left(\mu,\nu\right)=1.$
In order to see clearly its transform relation under the CFrHO, next we
examine its decomposition. Performing the change of variables,
$x/\mu\rightarrow x,$ $y/\nu\rightarrow y$, we can be recast Eq.(5) into the
following form,
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)=\sqrt{\frac{\mu\nu
e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\sin\alpha}}\int_{-\infty}^{\infty}\exp\left\\{-\frac{\mathtt{i}\left(x^{2}+y^{2}\right)}{2\tan\alpha}+\frac{\mathtt{i}xy}{\sin\alpha}\right\\}\left|\nu
y\right\rangle\left\langle\mu x\right|\mathtt{d}x\mathtt{d}y.$ (10)
By noticing that the single-mode squeezing operator $S_{1}$ [17] has its
natural expression in coordinate representation [13], i.e.,
$S_{1}\left(\mu\right)=\frac{1}{\sqrt{\mu}}\int_{-\infty}^{\infty}\mathtt{d}x\left|\frac{x}{\mu}\right\rangle\left\langle
x\right|,$ (11)
which leads to $\left|\nu
y\right\rangle=\frac{1}{\sqrt{\nu}}S_{1}^{-1}\left(\nu\right)\left|y\right\rangle,$
$\left\langle\mu x\right|=\frac{1}{\sqrt{\mu}}\left\langle
x\right|S_{1}\left(\mu\right),$ so Eq.(10) can be decomposed into
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)=S_{1}^{-1}\left(\nu\right)\mathcal{F}_{\alpha}S_{1}\left(\mu\right)=S_{1}^{-1}\left(\nu\right)e^{\mathtt{i}\alpha
a^{\dagger}a}S_{1}\left(\mu\right),$ (12)
where $\mathcal{F}_{\alpha}$ is given by
$\mathcal{F}_{\alpha}\equiv\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\sin\alpha}}\int_{-\infty}^{\infty}e^{-\frac{\mathtt{i}(x^{2}+y^{2})}{2\tan\alpha}+\frac{\mathtt{i}xy}{\sin\alpha}}\left|y\right\rangle\left\langle
x\right|\mathtt{d}x\mathtt{d}y=e^{\mathtt{i}\alpha a^{\dagger}a},$ (13)
this integral result can be obtained by using a similar way to deriving
Eq.(6). Thus we see that the CFrHO can be decomposed as a fractional operator
and two-single-mode squeezing operators.
Using the decomposition of the CFrHO in Eq.(12), and noticing that
$S_{1}\left(\mu\right)XS_{1}^{-1}\left(\mu\right)=\mu X,$
$S_{1}\left(\mu\right)PS_{1}^{-1}\left(\mu\right)=P/\mu,$ and
$e^{\mathtt{i}\alpha a^{\dagger}a}ae^{-\mathtt{i}\alpha
a^{\dagger}a}=ae^{-\mathtt{i}\alpha},$ which leads to
$e^{\mathtt{i}\alpha a^{\dagger}a}Xe^{-\mathtt{i}\alpha
a^{\dagger}a}=\allowbreak X\cos\alpha+P\sin\alpha,\text{ }e^{\mathtt{i}\alpha
a^{\dagger}a}Pe^{-\mathtt{i}\alpha a^{\dagger}a}=\allowbreak
P\cos\alpha-X\sin\alpha,$ (14)
thus we have
$\displaystyle\mathcal{H}_{\alpha}\left(\mu,\nu\right)X\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)$
$\displaystyle=$ $\displaystyle\mu
S_{1}^{-1}\left(\nu\right)\left(X\cos\alpha+P\sin\alpha\right)S_{1}\left(\nu\right)$
(15) $\displaystyle=$ $\displaystyle\frac{\mu}{\nu}X\cos\alpha+\mu\nu
P\sin\alpha,$
$\displaystyle\mathcal{H}_{\alpha}\left(\mu,\nu\right)P\mathcal{H}_{\alpha}^{\dagger}\left(\mu,\nu\right)$
$\displaystyle=$
$\displaystyle\frac{\allowbreak\nu}{\mu}P\cos\alpha-\frac{X}{\mu\nu}\sin\alpha,$
(16)
from which we see that the CFrHO plays the role of combining the coordinate
operator $X$ and momentum operator $P$ in a certain way (15)-(16), i.e.,
including the squeezing and the rotation. In paticular, when
$\alpha=\frac{\pi}{2},$ Eqs.(15)-(16) become
$\mathcal{H}_{\frac{\pi}{2}}\left(\mu,\nu\right)X\mathcal{H}_{\frac{\pi}{2}}^{\dagger}\left(\mu,\nu\right)=\mu\nu
P,\text{
}\mathcal{H}_{\frac{\pi}{2}}\left(\mu,\nu\right)P\mathcal{H}_{\frac{\pi}{2}}^{\dagger}\left(\mu,\nu\right)=-\frac{X}{\mu\nu},$
(17)
i.e., the mutual exchanging of coordinate-momentum operators.
On the other hand, there is a most important feature of the FrFT is that the
FrFT obeys the additivity rule, i.e., two successive FrFT of order $\alpha$
and $\beta$ makes up the FrFT of order $\alpha+\beta$. Then a question
naturally arises: Is the two successive CFrHOs still a CFrHO? To answer this
question, we examine the direct product
$\mathcal{H}_{\alpha}\left(\mu,\nu\right)\otimes\mathcal{H}_{\beta}\left(\mu^{\prime},\nu^{\prime}\right)$.
Using Eq.(12) it is easily seen that when $\mu=$ $\nu^{\prime}$ there is an
additivity of operator as follows
$\displaystyle\mathcal{H}_{\alpha}\left(\mu,\nu\right)\otimes\mathcal{H}_{\beta}\left(\mu^{\prime},\mu\right)$
$\displaystyle=$ $\displaystyle S_{1}^{-1}\left(\nu\right)e^{\mathtt{i}\alpha
a^{\dagger}a}S_{1}\left(\mu\right)S_{1}^{-1}\left(\mu\right)e^{\mathtt{i}\beta
a^{\dagger}a}S_{1}\left(\mu^{\prime}\right)$ (18) $\displaystyle=$
$\displaystyle\mathcal{H}_{\alpha+\beta}\left(\mu^{\prime},\nu\right),$
which can be seen as the additivity property of the CFrHOs. Here it should be
pointed out that the condition of additivitive operator for the CFrHOs is that
the parameter $\mu$ of the prior cascade operator should be equal to the
parameter $\nu^{\prime}$ of the next one, i.e., $\mu=\nu^{\prime}.$ This can
be clearly seen from the viewpoint of classical optics transform.
## 4 Two-mode CFrHT
Next, we shall extend the single-mode CFrHT to two-mode case by using the
entangled state representation [18],
$\left|\eta\right\rangle=\exp\left\\{-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$
(19)
where $\left|\eta=\eta_{1}+\mathtt{i}\eta_{2}\right\rangle$ is the common
eigenvector of two-particle’s relative coordinate $X_{1}-X_{2}$ and total
momentum $P_{1}+P_{2},$
$\left(X_{1}-X_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(20)
and $\left|\eta\right\rangle$ possesses the completeness and the
orthogonality,
$\int_{-\infty}^{\infty}\frac{\mathtt{d}^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1,\text{
}\left\langle\eta\right|\left.\eta^{\prime}\right\rangle=\pi\delta\left(\eta-\eta^{\prime}\right)\delta\left(\eta^{\ast}-\eta^{\prime\ast}\right).$
(21)
In a similar way to introducing Eq.(3), we examine the following transform,
$\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)\left|\eta\right\rangle=\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\mu\nu\sin\alpha}\int\frac{\mathtt{d}^{2}\eta^{\prime}}{\pi}e^{-\frac{\mathtt{i}\left(\allowbreak\left|\eta^{\prime}\right|^{2}/\nu^{2}+\left|\eta\right|^{2}/\mu^{2}\allowbreak\right)}{2\tan\alpha}+\frac{\mathtt{i}\left(\eta^{\prime}{}^{\ast}\allowbreak\eta+\eta^{\ast}\allowbreak\eta^{\prime}\right)}{2\mu\nu\sin\alpha}}\left|\eta^{\prime}\right\rangle.$
(22)
Using Eq.(21), one can see that $\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)$
is a unitary operator, i.e.,
$\mathcal{H}_{\alpha}^{C}\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}=\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}\mathcal{H}_{\alpha}^{C}=1$.
Here we should emphasize that, the exponential item in the right hand side of
Eq.(22) can be decomposed into a direct product of two exponential items in
the right hand side of Eq.(3), but $\left|\eta\right\rangle$ is an entangled
state (not the direct product of two single-mode coordinate states, which can
be seen clearly from its Schmidt decomposition [19]), thus this is a
nontrivial extension from single-mode case to two-mode case.
Performing a similar procedure to single-mode case, i.e., noticing that the
two-mode squeezing operator has its natural expression in the entangled state
representation,
$S_{2}\left(\mu\right)=\exp\left[\left(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}\right)\ln\mu\right]=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta}{\mu}\right\rangle\left\langle\eta\right|,$
(23)
which leads to
$\frac{1}{\nu}\left|\frac{\eta^{\prime}}{\nu}\right\rangle=S_{2}\left(\nu\right)\left|\eta^{\prime}\right\rangle,\frac{1}{\mu}\left|\frac{\eta}{\mu}\right\rangle=S_{2}\left(\mu\right)\left|\eta\right\rangle,$
then using the completeness of $\left|\eta\right\rangle$ and
$\left|00\right\rangle\left\langle 00\right|=\colon
e^{-a^{{\dagger}}a-b^{{\dagger}}b}\colon$ and the orthogonality in Eq.(21) we
can further decompose the operator
$\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)$ into the following form,
$\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)=S_{2}^{\dagger}\left(\nu\right)\mathcal{F}_{\alpha}^{C}S_{2}\left(\mu\right)=S_{2}^{\dagger}\left(\nu\right)\exp\left\\{\mathtt{i}\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right\\}S_{2}\left(\mu\right),$
(24)
where the operatror $\mathcal{F}_{\alpha}^{C}$ is given by
$\displaystyle\mathcal{F}_{\alpha}^{C}$ $\displaystyle=$
$\displaystyle\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\sin\alpha}\int\frac{\mathtt{d}^{2}\eta^{\prime}\mathtt{d}^{2}\eta}{\mu^{2}\nu^{2}\pi}e^{-\frac{\mathtt{i}\left(\allowbreak\left|\eta^{\prime}\right|^{2}/\nu^{2}+\left|\eta\right|^{2}/\mu^{2}\allowbreak\right)}{2\tan\alpha}+\frac{\mathtt{i}\left(\eta^{\prime}{}^{\ast}\allowbreak\eta+\eta^{\ast}\allowbreak\eta^{\prime}\right)}{2\mu\nu\sin\alpha}}\left|\frac{\eta^{\prime}}{\nu}\right\rangle\left\langle\frac{\eta}{\mu}\right|$
(25) $\displaystyle=$
$\displaystyle\exp\left\\{\mathtt{i}\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right\\}.$
Thus we see that the two-mode CFrHO can be decomposed into the form in
Eq.(24), i.e., two fractional operators and two two-mode squeezing operators.
This is a convient expression for further deriving the transforms and the
condition of additivitive operator. In fact, using Eqs.(24), (14) and
Eqs.(20), (21) leading to
$\displaystyle
S_{2}\left(\mu\right)\left(X_{1}-X_{2}\right)S_{2}^{\dagger}\left(\mu\right)$
$\displaystyle=$ $\displaystyle\mu\left(X_{1}-X_{2}\right),\text{
}S_{2}\left(\mu\right)\left(P_{1}+P_{2}\right)S_{2}^{\dagger}\left(\mu\right)=\mu\left(P_{1}+P_{2}\right),$
(26) $\displaystyle
S_{2}\left(\mu\right)\left(X_{1}+X_{2}\right)S_{2}^{\dagger}\left(\mu\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\mu}\left(X_{1}+X_{2}\right),\text{
}S_{2}\left(\mu\right)\left(P_{1}-P_{2}\right)S_{2}^{\dagger}\left(\mu\right)=\frac{1}{\mu}\left(P_{1}-P_{2}\right),$
(27)
we have
$\displaystyle\mathcal{H}_{\alpha}^{C}\left(X_{1}-X_{2}\right)\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}$
$\displaystyle=$ $\displaystyle\mu
S_{2}^{\dagger}\left(\nu\right)\mathcal{F}_{\alpha}^{C}\left(X_{1}-X_{2}\right)\left[\mathcal{F}_{\alpha}^{C}\right]^{{\dagger}}S_{2}\left(\nu\right)$
(28) $\displaystyle=$ $\displaystyle\mu
S_{2}^{\dagger}\left(\nu\right)\left(\left(X_{1}-X_{2}\right)\cos\alpha+\left(P_{1}-P_{2}\right)\sin\alpha\right)S_{2}\left(\nu\right)$
$\displaystyle=$
$\displaystyle\frac{\mu}{\nu}\left(X_{1}-X_{2}\right)\cos\alpha+\mu\nu\left(P_{1}-P_{2}\right)\sin\alpha,$
$\displaystyle\mathcal{H}_{\alpha}^{C}\left(X_{1}+X_{2}\right)\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}$
$\displaystyle=$
$\displaystyle\frac{\nu}{\mu}\left(X_{1}+X_{2}\right)\cos\alpha+\frac{1}{\mu\nu}\left(P_{1}+P_{2}\right)\sin\alpha,$
(29)
and
$\displaystyle\mathcal{H}_{\alpha}^{C}\left(P_{1}-P_{2}\right)\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}$
$\displaystyle=$
$\displaystyle\frac{\nu}{\mu}\left(P_{1}-P_{2}\right)\cos\alpha-\frac{1}{\mu\nu}\left(X_{1}-X_{2}\right)\sin\alpha,$
(30)
$\displaystyle\mathcal{H}_{\alpha}^{C}\left(P_{1}+P_{2}\right)\left[\mathcal{H}_{\alpha}^{C}\right]^{{\dagger}}$
$\displaystyle=$
$\displaystyle\frac{\mu}{\nu}\left(P_{1}+P_{2}\right)\cos\alpha-\mu\nu\left(X_{1}+X_{2}\right)\sin\alpha.$
(31)
From Eqs.(28)-(31) it is easy to see that when $\alpha=\pi/2,$ the role of
$\mathcal{H}_{\pi/2}^{C}$ is just exchanging $\left(X_{1}-X_{2}\right)$ and
$\left(P_{1}-P_{2}\right)$, $\left(X_{1}+X_{2}\right)$ and
$\left(P_{1}+P_{2}\right);$ while for $\alpha=\pi,$ $\mathcal{H}_{\pi}^{C}$
can be seen as an identity operator.
In addition, from the decomposition (24) one can see that the direct product
$\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)\otimes\mathcal{H}_{\beta}^{C}\left(\mu^{\prime},\nu^{\prime}\right)$
satisfies the additivity rule when $\mu=$ $\nu^{\prime}$, i.e.,
$\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)\otimes\mathcal{H}_{\beta}^{C}\left(\mu^{\prime},\nu^{\prime}\right)=\mathcal{H}_{\alpha+\beta}^{C}\left(\mu^{\prime},\nu\right).$
(32)
## 5 Measurements for the output states from the CFrHT
The measurement for quantum state plays an important role in quantum
computation and quantum imfromation. When a quantum state
$\left|f\right\rangle$ is transformed by the CFrHO, then what is the
measurement result with continuous orthogonal basis? For single-mode case, the
output state from the CFrHT is
$\left|g\right\rangle_{out}=\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|f\right\rangle$.
The measurement basis is choosen as a coordiante eigenvector, then the
measurement result is given by
$\displaystyle\left\langle x\right.\left|g\right\rangle_{out}$
$\displaystyle=$ $\displaystyle\left\langle
x\right|\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|f\right\rangle$ (33)
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\mathtt{d}x^{\prime}\left\langle
x\right|\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|x^{\prime}\right\rangle\left\langle
x^{\prime}\right.\left|f\right\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\mu\nu\sin\alpha}}\int_{-\infty}^{\infty}f\left(x^{\prime}\right)e^{-\frac{\mathtt{i}\left(x^{\prime
2}/\mu^{2}+x^{2}/\nu^{2}\right)}{2\tan\alpha}+\frac{\mathtt{i}x^{\prime}x}{\mu\nu\sin\alpha}}\mathtt{d}x^{\prime},$
which just corresponds to a generalized fractional Fourier transform of wave
function $f\left(x^{\prime}\right)=\left\langle
x^{\prime}\right.\left|f\right\rangle.$
For two-mode case, the measurement result by two-mode entangled state Bell
basis is
$\displaystyle\left\langle\eta^{\prime}\right.\left|g\right\rangle_{out}$
$\displaystyle=$
$\displaystyle\left\langle\eta^{\prime}\right|\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)\left|f\right\rangle$
(34) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{\mathtt{d}^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|\mathcal{H}_{\alpha}^{C}\left(\mu,\nu\right)\left|\eta\right\rangle\left\langle\eta\right.\left|f\right\rangle$
$\displaystyle=$
$\displaystyle\frac{e^{\mathtt{i}\left(\frac{\pi}{2}-\alpha\right)}}{2\mu\nu\sin\alpha}\int_{-\infty}^{\infty}\frac{\mathtt{d}^{2}\eta}{\pi}e^{-\frac{\mathtt{i}\left(\allowbreak\left|\eta^{\prime}\right|^{2}/\nu^{2}+\left|\eta\right|^{2}/\mu^{2}\allowbreak\right)}{2\tan\alpha}+\frac{\mathtt{i}\left(\eta^{\prime}{}^{\ast}\allowbreak\eta+\eta^{\ast}\allowbreak\eta^{\prime}\right)}{2\mu\nu\sin\alpha}}f\left(\eta\right),$
which is just a generalized complex fractional Fourier transform, and the wave
function $f\left(\eta\right)$ is the projection of quantum state
$\left|f\right\rangle$ on $\left\langle\eta\right|$. From Eqs.(33) and (34) we
can clearly see that the generalized FrFT of the wavefunction for any quantum
state $\left|f\right\rangle$ in coordinate/entangled state corresponds to the
wavefunction of Hadamard-transformed
($\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|f\right\rangle$) in
coordinate/entangled state. In other words, the generalized FrFT of
wavefunction is just the matrix element of CFrHO in $\left\langle x\right|$
($\left\langle\eta^{\prime}\right|$) and $\left|f\right\rangle$.
## 6 Conclusion
Based on quantum Hadamard transform, we have introduced the quantum fractional
Hadamard transform with continuous variables. It is found that the
corresponding quantum fractional Hadamard operator can be decomposed into a
single-mode fractional operator $e^{\mathtt{i}\alpha a^{\dagger}a}$ and two
single-mode squeezing operators. The two-mode fractional Hadamard transform is
also introduced by using the bipartite entangled state representation. It is
shown that the corresponding transform operatror involves two single-mode
fractional operators and two two-mode squeezing operators. For any quantum
state vector $\left|f\right\rangle$, the measurement results for the
transformed quantum state (for instance
$\left|g\right\rangle_{out}=\mathcal{H}_{\alpha}\left(\mu,\nu\right)\left|f\right\rangle)$
by continuous coordinate state $\left|x\right\rangle$ (or bipartite entangled
state $\left|\eta\right\rangle)$ just corresponds to a generalized (complex)
fractional Fourier transform. In addition, the new transformation gives us
more flexibility to represent signals in the fractional Hadamard domain with
extra freedom provided by an angle, and two-squeezing parameters. For more
discussions about the optical transforms in the context of quantum optics and
the discrete fractional Hadmard transform, we refer to Refs.[20, 21, 22].
## References
* [1] L. B. Almeida 1994 IEEE _Trans. Signal Process_. 42 3084
* [2] Tao R, Deng B, Zhang W Q, et. al. 2008 _IEEE Trans. Signal Process._ 56 158
* [3] Condon E U 1937 Proc. Nat. Acad. Sci. 23 158
* [4] Namias V 1980 _J. Inst. Math. Appl._ 25 241
* [5] McBride A C and Kerr F H 1987 _IMA J. Appl. Math._ 39 159
* [6] Kerr F H 1988 _J. Math. Anal. Appl._ 136 404
* [7] Pei S C, Yeh M H 2001 _IEEE Trans. Signal Process._ 49 1198
* [8] Pei S C, Tseng C C, Yeh M H and Shyu J J 1998 _IEEE Trans. Circuit Ssytem II_ 45 665
* [9] Nielsen M A and Chuang I L 2000 The Quantum Computation and Quantum Information, Cambridge University Press, Cambridge
* [10] Braunstein S L 1998 _Phys. Rev. Lett._ 80 4084
* [11] Parker S, Bose S and Plenio M B 2000 _Phys. Rev. A._ 61 032305
* [12] Fan H Y and Guo Q 2008 _Commun. Theor. Phys._ 49 859
* [13] Fan H Y, H. R. Zaidi and J. R. Klauder 1987 Phys. Rev. D 35 1831
* [14] Fan H Y 2003 J. Opt. B: Quantum. Semiclass. Opt. 5 R147
* [15] Hu L Y and Fan H Y 2009 _Commun. Theor. Phys._ (Beijing, China) 52 1071; Hu L Y and Fan H Y 2009 Chin. Phys. B 18 4657
* [16] Einstein A, Podolsky B and Rosen N 1935 _Phys. Rev._ 47 777
* [17] Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics, Cambridge Press, (London)
* [18] Fan H Y and Klauder J R 1994 Phys. Rev. A 49 704; Fan H Y and Fan Y 1996 Phys. Rev. A 54 958
* [19] Preskill J 1998 Quantum Information and Computation, California Institute of Technology; Hu L Y and Lu H L. 2007 Chin. Phys. 16 2200
* [20] Hu L Y and Fan H Y 2008 _J. Mod. Opt._ 55 1835; Fan H Y and Hu L Y 2009 _Chin. Phys. B_ 18 0611
* [21] Tao R, Lang J, Wang Y 2009 Opt. Commun. 282 1531
* [22] Hu L Y and Fan H Y 2009 Int. J. Theor. Phys. DOI 10.1007/s10773-009-0008-z
|
arxiv-papers
| 2010-10-03T02:49:50 |
2024-09-04T02:49:13.336773
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu, Xue-xiang Xu, and Shan-jun Ma",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1010.0375"
}
|
1010.0377
|
# Quantum Optical Version of Classical Optical Transformations and Beyond
Hong-yi Fan1 and Li-yun Hu2
1Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China;
Department of Material Science and Engineering, University of Science and
Technology of China, Hefei, Anhui 230026, China
2College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China Corresponding author. E-mail address: hlyun2008@126.com
(L.Y. Hu)
###### Abstract
By virtue of the newly developed technique of integration within an ordered
product (IWOP) of operators, we explore quantum optical version of classical
optical transformations such as optical Fresnel transform, Hankel transform,
fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-
Hadmard combinatorial transform etc. In this way one may gain benefit for
developing classical optics theory from the research in quantum optics, or
vice-versa. We can not only find some new quantum mechanical unitary operators
which correspond to the known optical transformations, deriving a new theorem
for calculating quantum tomogram of density operators, but also can reveal
some new classical optical transformations. For examples, we find the
generalized Fresnel operator (GFO) to correspond to the generalized Fresnel
transform (GFT) in classical optics. We derive GFO’s normal product form and
its canonical coherent state representation and find that GFO is the loyal
representation of symplectic group multiplication rule. We show that GFT is
just the transformation matrix element of GFO in the coordinate representation
such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian
beam propagation is directly demonstrated in the context of quantum optics.
Especially, the introduction of quantum mechanical entangled state
representations opens up a new area to finding new classical optical
transformations. The complex wavelet transform and the condition of mother
wavelet are studied in the context of quantum optics too. Throughout our
discussions, the coherent state, the entangled state representation of the
two-mode squeezing operators and the technique of integration within an
ordered product (IWOP) of operators are fully used. All these confirms Dirac’s
assertion: “ $...$for a quantum dynamic system that has a classical analogue,
unitary transformation in the quantum theory is the analogue of contact
transformation in the classical theory”.
Keywords: Dirac’s symbolic method; IWOP technique; entangled state of
continuum variables; entangled Fresnel transform; Collins formula; Generalized
Fresnel operator; complex wavelet transform; complex Wigner transform; complex
fractional Fourier transform; symplectic wavelet transform; entangled
symplectic wavelet transform; Symplectic-dilation mixed wavelet transform;
fractional Radon transform; new eigenmodes of fractional Fourier transform
###### Contents
1. 1 Introduction
2. 2 Some typical classical optical transformations
3. 3 The IWOP technique and two mutually conjugate entangled states
1. 3.1 The IWOP technique
2. 3.2 The IWOP technique for deriving normally ordered Gaussian form of the completeness relations of fundamental quantum mechanical representations
3. 3.3 Single-mode Wigner operator
4. 3.4 Entangled state $\left|\eta\right\rangle$ and its Fourier transform in complex form
5. 3.5 Two-mode Wigner operator in the $\left|\eta\right\rangle$ representation
4. 4 Two deduced entangled state representations and Hankel transform
1. 4.1 Deduced entangled states
2. 4.2 Hankel transform between two deduced entangled state representations
3. 4.3 Quantum optical version of classical circular harmonic correlation
5. 5 Single-mode Fresnel operator as the image of the classical Optical Fresnel Transform
1. 5.1 Single-mode GFO gained via coherent state method
2. 5.2 Group Multiplication Rule for Single-mode GFO
6. 6 Quantum Optical ABCD Law for optical propagation —single-mode case
7. 7 Optical operator method studied via GFO’s decomposition
1. 7.1 Four fundamental optical operators derived by decomposing GFO
2. 7.2 Alternate decompositions of GFO
3. 7.3 Some optical operator identities
8. 8 Quantum tomography and probability distribution for the Fresnel quadrature phase
1. 8.1 Relation between Fresnel transform and Radon transform of WF
2. 8.2 Another new theorem to calculating the tomogram
9. 9 Two-mode GFO and Its Application
1. 9.1 Two-mode GFO gained via coherent state representation
2. 9.2 Quantum Optical ABCD Law for two-mode GFO
3. 9.3 Optical operators derived by decomposing GFO
1. 9.3.1 GFO as quadratic combinations of canonical operators
2. 9.3.2 Alternate decompositions of GFO and new optical operator identities
4. 9.4 Quantum tomography and probability distribution for the Fresnel quadrature phase—two-mode entangled case
1. 9.4.1 $\left|\eta\right\rangle_{s,r\text{ }s,r}\left\langle\eta\right|$ as Radon transform of the entangled Wigner operator
2. 9.4.2 Inverse Radon transformation
10. 10 Fractional Fourier Transformation (FrFT) for 1-D case
1. 10.1 Quantum version of FrFT
2. 10.2 On the Scaled FrFT Operator
3. 10.3 An integration transformation from Chirplet to FrFT kernel
11. 11 Complex Fractional Fourier Transformation
1. 11.1 Quantum version of CFrFT
2. 11.2 Additivity property and eigenmodes of CFrFT
3. 11.3 From Chirplet to CFrFT kernel
1. 11.3.1 New complex integration transformation
2. 11.3.2 Complex integration transformation and complex Weyl transformation
4. 11.4 Squeezing for the generalized scaled FrFT
1. 11.4.1 Quantum correspondence of the scaled FrFT
2. 11.4.2 The Scaled CFrFT
3. 11.4.3 Entangled scaled FrFT
12. 12 Adaption of Collins diffraction formula and CFrFT
1. 12.1 Adaption of the Collins formula to CFrFT
2. 12.2 Adaption of the additivity property of CFrFT to the Collins formula for two successive Fresnel diffractions
13. 13 The Fractional Radon transform
14. 14 Wavelet transformation and the IWOP technique
1. 14.1 Quantum optical version of classical WTs
2. 14.2 The condition of mother wavelet in the context of quantum optics
3. 14.3 Quantum mechanical version of Parseval theorem for WT
4. 14.4 Inversion formula of WT
5. 14.5 New orthogonal property of mother wavelet in parameter space
6. 14.6 WT and Wigner-Husimi Distribution Function
15. 15 Complex Wavelet transformation in entangled state representations
1. 15.1 CWT and the condition of Mother Wavelet
2. 15.2 Parseval Theorem in CWT
3. 15.3 Orthogonal property of mother wavelet in parameter space
4. 15.4 CWT and Entangled Husimi distribution
1. 15.4.1 CWT and its quantum mechanical version
2. 15.4.2 Relation between CWT and EHDF
16. 16 Symplectic Wavelet transformation (SWT)
1. 16.1 Single-mode SWT
1. 16.1.1 Properties of symplectic-transformed—translated WT
2. 16.1.2 Relation between $W_{\psi}f\left(r,s;\kappa\right)$ and optical Fresnel transform
2. 16.2 Entangled SWT
3. 16.3 Symplectic-dilation mixed WT
17. 17 Fresnel-Hadamard combinatorial transformation
1. 17.1 The Hadamard-Fresnel combinatorial operator
2. 17.2 The properties of Hadamard-Fresnel operator
## 1 Introduction
The history of quantum mechanics records that from the very beginning the
founders of the quantum theory realized that there might exist formal
connection between classical optics and quantum mechanics. For example,
Schrödinger considered that classical dynamics of a point particle should be
the “geometrical optics” approximation of a linear wave equation, in the same
way as ray optics is a limiting approximation of wave optics; Schrödinger also
searched for some quantum mechanical state which behaves like a classical
‘particle’, and this state was later recognized as the coherent state [1, 2,
3], which plays an essential role in quantum optics theory and laser physics;
As Dirac wrote in his famous book $<$Principles of Quantum Mechanics$>$[4]:
“$\cdot\cdot\cdot$ for a quantum dynamic system that has a classical analogue,
unitary transformation in the quantum theory is the analogue of contact
transformation in the classical theory”. According to Dirac, there should
exist a formal correspondence between quantum optics unitary-transform
operators and classical optics transformations. Indeed, in the last century
physicists also found some rigorous mathematical analogies between classical
optics and quantum mechanics, i.e. the similarity between the optical
Helmholtz equation and the time-independent Schrödinger equation; Since 1960s,
the advent of a laser and the appearance of coherent state theory of radiation
field [1, 2, 5], quantum optics has experienced rapid development and achieved
great success in revealing and explaining the quantum mechanical features of
optical field and non-classical behavior (for instance, Hanbury-Brown-Twiss
effect, photon antibunching, squeezing, sub-Poissonian photon statistics) of
photons in various photon-atom interactions [6]. The relationship between
classical and quantum coherence has been discussed in the book of Mandel and
Wolf [6]; The Hermite-Gauss or the Laguerre-Gauss modes of a laser beam are
described using the bosonic operator algebra by Nienhuis and Allen [7]. In
addition, displaced light beams refracted by lenses according to the law of
geometrical optics, were found to be the paraxial optics analog of a coherent
state. Besides, phase space correspondence between classical optics and
quantum mechanics, say for example, the Wigner function theory, is inspected
in the literature [8].
On the other hand, classical optics, which tackles vast majority of physical-
optics experiments and is based on Maxwell’s equations, has never ceased its
own evolving steps, physicists have endeavored to develop various optical
transforms in light propagation through lens systems and various continuous
media. The two research fields, quantum optics and classical optics, have
their own physical objects and conceptions. From the point of view of
mathematics, classical optics is framed in the group transform and associated
representations on appropriate function space, while quantum optics deals with
operators and state vectors, and their overlap seems little at first glance.
It seems to us that if one wants to further relate them to each other, one
needs some new theoretical method to ”bridge” them. For example, what is the
quantum mechanical unitary operator corresponding to the Fresnel transform in
Fourier optics? Is there any so-called Fresnel operator as the image of
classical generalized Fresnel transform? Since generalized Fresnel transforms
are very popularly used in optical instrument design and optical propagation
through lenses and various media, it is worth of studying these transforms in
the context of quantum optics theory, especially based on coherent state,
squeezed state [9, 10] and the newly invented entangled state theory [11, 12,
13, 14].
Fortunately, the recently developed technique of integration within an ordered
product (IWOP) of operators [15, 16, 17] is of great aid to studying quantum
optical version of classical optical transformations. Using the IWOP technique
one may gain benefit for classical optics from quantum optics’ research, or
vice-versa.
Our present Review is arranged as follows: in section 2 we briefly recall the
classical diffraction theory [18, 19], this is preparing for later sections in
which we shall show that most frequently employed classical optical transforms
have their counterparts in quantum optics theory. In section 3 we introduce
the IWOP technique and demonstrate that the completeness relation of
fundamental quantum mechanical representations can be recast into normally
ordered Gaussian operator form. Using the IWOP technique we can directly
perform the asymmetric ket-bra integration
$\mu^{-1/2}\int_{-\infty}^{\infty}dq\left|q/\mu\right\rangle\left\langle
q\right|$ in the coordinate representation, which leads to the normally
ordered single-mode squeezing operator, this seems to be a direct way to
understanding the squeezing mechanism as a mapping from the classical scaling
$q\rightarrow q/\mu$. In section 4 with the help of IWOP technique and based
on the concept of quantum entanglement of Einstein-Podolsky-Rosen [20] we
construct two mutually conjugate entangled states of continuum variables,
$\left|\eta\right\rangle$ versus $\left|\xi\right\rangle,$ and their deduced
entangled states (or named correlated-amplitude—number-difference entangled
states), they are all qualified to make up quantum mechanical representations.
It is remarkable that using the IWOP technique to performing the asymmetric
ket-bra integration $\mu^{-1}\int
d^{2}\eta\left|\eta/\mu\right\rangle\left\langle\eta\right|$ leads to the two-
mode normally ordered two-mode squeezing operator, this implies that the two-
mode squeezed state is simultaneously an entangled state. We point out that
the entangled state $\left|\eta\right\rangle$ also embodies entanglement in
the aspect of correlative amplitude and the phase. We are also encouraged that
the overlap between two mutually conjugate deduced entangled states is just
the Bessel function— the optical Hankel transform kernel [21], which again
shows that the new representations in the context of physics theory match
beautiful mathematical formalism exactly. We then employ the deduced entangled
states to derive quantum optical version of classical circular harmonic
correlation. Section 5 is devoted to finding a quantum operator which
corresponds to the optical Fresnel transform, with use of the coherent state
representation and by projecting the classical sympletic transform
$z\rightarrow sz-rz^{\ast}$ ($\left|s\right|^{2}-\left|r\right|^{2}=1)$ in
phase space onto the quantum mechanical Hilbert space, we are able to
recognize which operator is the single-mode Fresnel operator (FO). It turns
out that the 1-dimensional optical Fresnel transform is just the matrix
element of the Fresnel operator $F$ in the coordinate eigenstates. Besides,
the coherent state projection operator representation of FO constitutes a
loyal realization of symplectic group, which coincides with the fact that two
successive optical Fresnel transforms make up a new Fresnel transform. Then in
Section 6 based on the coherent state projection representation of FO, we
prove $ABCD$ rule for optical propagation in the context of quantum optics. In
section 7 the quadratic operator form of FO is also presented and the four
fundamental optical operators are derived by decomposing the FO. In section 8
we discuss how to apply the Fresnel operator to quantum tomography theory, by
introducing the Fresnel quadrature phase $FXF^{\dagger}=X_{F},$ we point out
that Wigner operator’s Radon transformation is just the pure state projection
operator $\left|x\right\rangle_{s,rs,r}$ $\left\langle x\right|$, where
$\left|x\right\rangle_{s,r}=F\left|x\right\rangle$ and $\left|x\right\rangle$
is the position eigenstate, so the probability distribution for the Fresnel
quadrature phase is the Radon transform of the Wigner function. Moreover, the
tomogram of quantum state $\left|\psi\right\rangle$ is just the squared
modulus of the wave function ${}_{s,r}\left\langle
x\right|\left.\psi\right\rangle.$ This new relation between quantum tomography
and optical Fresnel transform may provide experimentalists to figure out new
approach for testing tomography. In addition, we propose another new theorem
for calculating tomogram, i.e., the tomogram of a density operator $\rho$ is
equal to the marginal integration of the classical Weyl correspondence
function of $F^{\dagger}\rho F$. In section 9 by virtue of the coherent state
and IWOP method we propose two-mode generalized Fresnel operator (GFO), in
this case we employ the entangled state representation to relate the 2-mode
GFO to classical transforms, since the 2-mode GFO is not simply the direct
product of two 1-mode GFOs. The corresponding quantum optics $ABCD$ rule for
two-mode case is also proved. The 2-mode GFO can also be expressed in
quadratic operators form in entangled way. The relation between optical FT and
quantum tomography in two-mode case is also revealed. In section 10 we propose
a kind of integration transformation,
$\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)\equiv
f\left(x,y\right),$ which is invertible and obeys Parseval theorem.
Remarkably, it can convert chirplet function to the kernel of fractional
Fourier transform (FrFT). This transformation can also serve for solving some
operator ordering problems. In section 11 we employ the entangled state
representation to introduce the complex FrFT (CFrFT), which is not the direct
product of two independent 1-dimensional FrFT transform. The eigenmodes on
CFrFT is derived. New eigen-modes for light propagation in graded-index medium
and the fractional Hankel transform are presented. The Wigner transform theory
is extended to the complex form and its relation to CFrFT is shown; The
integration transformation in section 10 is also extended to the entangled
case. In section 12 we shall treat the adaption problem of Collins diffraction
formula to the CFrFT with the use of two-mode (3 parameters) squeezing
operator and in the entangled state representation of continuous variables, in
so doing the quantum mechanical version of associated theory of classical
diffraction and classical CFrFT is obtained, which connects classical optics
and quantum optics in this aspect. In section 13 we introduce a convenient way
for constructing the fractional Radon transform. the complex fractional Randon
transform is also proposed; In sections 14 and 15 we discuss quantum optical
version of classical wavelet transforms (WTs), including how to recast the
condition of mother wavelet into the context of quantum optics; how to
introduce complex wavelet transform with use of the entangled state
representations. Some properties, such as Parseval theorem, Inversion formula,
and orthogonal property, the relation between WT and Wigner-Husimi
distribution function are also discussed. In section 16, we generalize the
usual wavelet transform to symplectic wavelet transformation (SWT) by using
the coherent state representation and making transformation $z\rightarrow
s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)$
($\left|s\right|^{2}-\left|r\right|^{2}=1)$ in phase space. The relation
between SWT and optical Fresnel transformation is revealed. Then the SWT is
extended to the entangled case by mapping the classical mixed transformation
$\left(z,z^{\prime}\right)\rightarrow\left(sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right)$
in 2-mode coherent state $\left|z,z^{\prime}\right\rangle$ representation. At
the end of this section, we introduce a new symplectic-dilation mixed WT by
employing a new entangled-coherent state representation
$\left|\alpha,x\right\rangle$. The corresponding classical optical transform
is also presented. In the last section, we introduce the Fresnel-Hadamard
combinatorial operator by virtue of the IWOP technique and
$\left|\alpha,x\right\rangle$. This unitary operator plays the role of both
Fresnel transformation for mode $\frac{a_{1}-a_{2}}{\sqrt{2}}$ and Hadamard
transformation for mode $\frac{a_{1}+a_{2}}{\sqrt{2}},$ respectively, and the
two transformations are combinatorial. All these sections are used to prove
the existence of a one-to-one correspondence between quantum optical operators
that transform state vectors in Hilbert space and the classical optical
transforms that change the distribution of optical field.
## 2 Some typical classical optical transformations
Here we briefly review some typical optical transforms based on light
diffraction theory. These transformations, as one can see in later sections,
are just the correspondence of some representation transformations between
certain quantum mechanical states of which some are newly constructed.
It was Huygens who gave a first illustrative explanation to wave theory by
proposing every point in the propagating space as a sub-excitation source of a
new sub-wave. An intuitive theory mathematically supporting Huygens’ principle
is the scalar diffraction approximation, so named because optical fields
(electromagnetic fields) actually are vector fields, whereby the theory is
valid approximately. This theory is based on the superposition of the combined
radiation field of multiple re-emission sources initiated by Huygens. Light
diffraction phenomena has played an important role in the development of the
wave theory of light, and now underlies the Fourier optics and information
optics. The formulation of a diffraction problem essentially considers an
incident free-space wave whose propagation is interrupted by an obstacle or
mask which changes the phase and/or amplitude of the wave locally by a well
determined factor [22]. A more rigorous, but still in the scheme of scalar
wave, derivation has been given by Kirchhoff who reformulated the diffraction
problem as a boundary-value problem, which essentially justifies the use of
Huygens principle. The Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) diffraction
formula is practically reduced to the Fresnel integral formula in paraxial and
far-field approximation [18, 19] that reads:
$U_{2}\left(x_{2},y_{2}\right)=\frac{\exp\left(ikz\right)}{i\lambda
z}\int\int_{-\infty}^{\infty}U_{1}\left(x_{1},y_{1}\right)\exp\left\\{i\frac{k}{2z}\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}\right]\right\\}dx_{1}dy_{1},$
(1)
where $U_{1}\left(x_{1},y_{1}\right)$ is the optical distribution of a
2-dimensional light source and $U_{2}\left(x_{2},y_{2}\right)$ is its image on
the observation plane, $\lambda$ is the optical wavelength,
$k=\frac{2\pi}{\lambda}$ is the wave number in the vacuum and $z$ is the
propagation distance. When
$z^{2}\gg\frac{k}{2}\left(x_{1}^{2}+y_{1}^{2}\right)_{\max},$ (2)
is satisfied, Eq. (1) reduces to
$\displaystyle U_{2}\left(x_{2},y_{2}\right)$
$\displaystyle=\frac{\exp\left(ikz\right)\exp\left[i\frac{k}{2z}\left(x_{1}^{2}+y_{1}^{2}\right)\right]}{i\lambda
z}$
$\displaystyle\times\int\int_{-\infty}^{\infty}U_{1}\left(x_{1},y_{1}\right)\exp\left[-i\frac{2\pi}{\lambda
z}\left(x_{1}x_{2}+y_{1}y_{2}\right)\right]dx_{1}dy_{1},$ (3)
which is named the Fraunhofer diffraction formula.
The Fresnel integral is closely related to the fractional Fourier transform
(FrFT), actually, it has been proved that the Fresnel transform can be
interpreted as a scaled FrFT with a residual phase curvature [23]. The FrFT is
a very useful tool in Fourier optics and information optics. This concept was
firstly introduced in 1980 by Namias [24] but not brought enough attention
until FrFT was defined physically, based on propagation in quadratic graded-
index media (GRIN media). Mendlovic and Ozaktas [25, 26] defined the
$\alpha$th FrFT as follows: Let the original function be input from one side
of quadratic GRIN medium, at $z=0$. Then, the light distribution observed at
the plane $z=z_{0}$ corresponds to the $\alpha$ equal to the ($z_{0}/L$)th
fractional Fourier transform of the input fraction, where
$L\equiv(\pi/2)(n_{1}/n_{2})^{1/2}$ is a characteristic distance. The FrFT can
also be implemented by lenses. Another approach for introducing FrFT was made
by Lohmann who pointed out the algorithmic isomorphism among image rotation,
rotation of the Wigner distribution function [29], and fractional Fourier
transforming [27]. Lohmann proposed the FrFT as the transform performed on a
function that leads to a rotation with an angle of the associated Wigner
distribution function, in this sense, the FrFT bridges the gap between
classical optics and optical Wigner distribution theory. Recently, the FrFT
has been paid more and more attention within different contexts of both
mathematics and physics [24, 25, 26, 27, 28]. The FrFT is defined as
$\displaystyle\mathcal{F}_{\alpha}\left[U_{1}\right]\left(x_{2},y_{2}\right)=\frac{e^{i(1-\alpha)\frac{\pi}{2}}}{2\sin\left(\frac{\pi}{2}\alpha\right)}\exp\left[-\frac{i\left(x_{2}^{2}+y_{2}^{2}\right)}{2\tan\left(\frac{\pi}{2}\alpha\right)}\right]$
$\displaystyle\times\int\int_{-\infty}^{\infty}\frac{dx_{1}dy_{1}}{\pi}\exp\left[-\frac{i\left(x_{1}^{2}+y_{1}^{2}\right)}{2\tan\left(\frac{\pi}{2}\alpha\right)}\right]\exp\left[\frac{i\left(x_{2}x_{1}+y_{2}y_{1}\right)}{\sin\left(\frac{\pi}{2}\alpha\right)}\right]U_{1}\left(x_{1},y_{1}\right).$
(4)
We can see that $F_{0}$ is the identity operator and $F_{\pi/2}$ is just the
Fourier transform. The most important property of FRFT is that $F_{\alpha}$
obeys the semigroup property, i.e. two successive FrFTs of order $\alpha$ and
$\beta$ makes up the FrFT of order $\alpha+\beta.$ A more general form
describing the light propagation in an optical system characterized by the
$\left[A,B;C,D\right]$ ray transfer matrix is the Collins diffraction integral
formula [30],
$\displaystyle U_{2}\left(x_{2},y_{2}\right)$
$\displaystyle=\frac{k\exp\left(ikz\right)}{2\pi
Bi}\int\int_{-\infty}^{\infty}dx_{1}dy_{1}U_{1}\left(x_{1},y_{1}\right)$
$\displaystyle\times\exp\left\\{\frac{ik}{2B}\left[A\left(x_{1}^{2}+y_{1}^{2}\right)-2\left(x_{1}x_{2}+y_{1}y_{2}\right)+D\left(x_{2}^{2}+y_{2}^{2}\right)\right]\right\\}dx_{1}dy_{1},$
(5)
where $AD-BC=1$ if the system is lossless. One can easily find the similarity
between Collins formula and the FrFT by some scaling transform and relating
the $\left[A,B,C,D\right]$ matrix to $\alpha$ in the FrFT [31]. Note that
$M=\left(\begin{array}[c]{cc}A&B\\\ C&D\end{array}\right)$ is a ray transfer
matrix describing optical systems belonging to the unimodular symplectic
group. When treating the light propagation in optical elements in near-axis
approximation, matrices $M$ representing linear transformations are a
convenient mathematical tool for calculating the fundamental properties of
optical systems, which is the origin of the name of matrix optics. In
cylindrical coordinates the Collins formula is expressed as [30, 32]
$U_{2}\left(r_{2},\varphi\right)=\frac{i}{\lambda
B}{\int}_{0}^{\infty}{\int}_{0}^{2\pi}\exp\left\\{-\frac{i\pi}{\lambda
B}\left[Ar_{1}^{2}+Dr_{2}^{2}-2r_{1}r_{2}\cos\left(\theta-\varphi\right)\right]\right\\}U_{1}\left(r_{1},\theta\right)r_{1}dr_{1}d\theta$
(6)
where $x_{1}=r_{1}\cos\theta,$ $y_{1}=r_{1}\sin\theta,$
$x_{2}=r_{2}\cos\varphi$ and $y_{2}=r_{2}\sin\varphi.$ When
$U_{1}\left(r_{1},\theta\right)$ has rotational symmetry
$U_{1}\left(r_{1},\theta\right)=u_{1}\left(r_{1}\right)\exp\left(im\theta\right),\text{
}U_{2}\left(r_{2},\varphi\right)=u_{2}\left(r_{2}\right)\exp\left(im\varphi\right),$
(7)
then (6) becomes
$u_{2}\left(r_{2}\right)=\frac{2\pi}{\lambda
B}\exp\left[i\left(1+m\right)\frac{\pi}{2}\right]{\int}_{0}^{\infty}\exp\left[-\frac{i\pi}{\lambda
B}\left[Ar_{1}^{2}+Dr_{2}^{2}\right]\right]J_{m}\left(\frac{2\pi
r_{1}r_{2}}{\lambda B}\right)u_{1}\left(r_{1}\right)r_{1}dr_{1},$ (8)
where we have used the $m$-order Bessel function
$J_{m}\left(x\right)=\frac{1}{2\pi}{\int}_{0}^{2\pi}\exp\left[ix\cos\theta+im\left(\theta-\frac{\pi}{2}\right)\right]d\theta.$
(9)
When $A=0$, (8) reduces to the standard Hankel transform (up to a phase
factor)
$u_{2}\left(r_{2}\right)\rightarrow\frac{2\pi}{\lambda
B}{\int}_{0}^{\infty}J_{m}\left(\frac{2\pi r_{1}r_{2}}{\lambda
B}\right)u_{1}\left(r_{1}\right)r_{1}dr_{1},$ (10)
The compact form of one-dimensional Collins formula is
$g\left(x_{2}\right)=\int_{-\infty}^{\infty}\mathcal{K}^{M}\left(x_{2},x_{1}\right)f\left(x_{1}\right)dx_{1},$
(11)
where the transform kernel is
$\mathcal{K}^{M}\left(x_{2},x_{1}\right)=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right],$
(12)
$M$ is the parameter matrix $\left[A,B,C,D\right]$. Eq. (12) is called
generalized Fresnel transform [33, 34, 35, 36]. In the following sections we
will show how we find the quantum optical counterpart for those
transformations of classical optics. For this purpose in the next chapter we
introduce the IWOP technique to demonstrate how Dirac’s symbolic method can be
developed and be applied to quantum optics theory. Also, we briefly review
some properties of the entangled state [11, 12, 13, 14] and reveal the
connection between the mutual transform generated by these entangled states
and the Hankel transform in classical optics.
## 3 The IWOP technique and two mutually conjugate entangled states
### 3.1 The IWOP technique
The history of mathematics tells us that whenever there appears a new
important mathematical symbol, there coexists certain operational rules for
it, the quantum mechanical operators in ket-bra projective form (the core of
Dirac’s symbolic method) also need their own operational rules. The
terminology “symbolic method” was first shown in the preface of Dirac’s book
$<$The Principle of Quantum Mechanics$>$: “The symbolic method, which deals
directly in an abstract way with the quantities of fundamental
importance$\cdot\cdot\cdot$, however, seems to go more deeply into the nature
of things. It enables one to express the physical law in a neat and concise
way, and will probably be increasingly used in the future as it becomes better
understood and its own special mathematics gets developed.” [4] Then two
questions naturally arise: How to better understand the symbolic method? How
to develop Dirac’s symbolic method, especially its mathematics? We noticed
that Newton-Leibniz integration rule only applies to commuting functions of
continuum variables, while operators made of Dirac’s symbols (ket versus bra,
e.g., $\left|q/\mu\right\rangle\left\langle q\right|$ of continuous parameter
$q$) in quantum mechanics are usually not commutative. Therefore integrations
over the operators of type $\left|\ \right\rangle\left\langle\ \right|$ (where
ket- and bra- state vectors need not to be Hermitian-conjugate to each other)
can not be directly performed by the Newton-Leibniz rule. Thus we invented an
innovative technique of integration within an ordered product (IWOP) of
operators that made the integration of non-commutative operators possible. The
core of IWOP technique is to arrange non-commutable quantum operators within
an ordered product (say, normal ordering) in a way that they become
commutable, in this sense the gap between q-numbers and c-numbers is
”narrowed”. However, the nature of operators which which are within : : is not
changed, they are still q-numbers, not c-numbers. After the integration over
c-numbers within ordered product is performed, we can get rid of the normal
ordering symbol after putting the integration result in normal ordering.
[37].The IWOP technique thus bridges this mathematical gap between classical
mechanics and quantum mechanics, and further reveals the beauty and elegance
of Dirac’s symbolic method and transformation theory. This technique develops
symbolic method significantly, i. e. makes Dirac’s representation theory and
the transformation theory more plentiful, and consequently to be better
understood. The beauty and elegance of Dirac’s symbolic method are further
revealed. Various applications of the IWOP technique, including constructing
the entangled state, developing the nonlinear coherent state theory, Wigner
function theory, etc. are found; many new unitary operators and operator-
identities as well as new quantum mechanical representations can be derived
too, which are partly summarized in the Review Articles [12].
We begin with listing some properties of normal product of operators which
means all the bosonic creation operators $a^{\dagger}$ are standing on the
left of annihilation operators $a$ in a monomial of $a^{\dagger}$ and $a$.
1\. The order of Bose operators $a$ and $a^{\dagger}$ within a normally
ordered product can be permuted. That is to say, even though
$\left[a,a^{\dagger}\right]=1$, we can have $\colon aa^{\dagger}\colon=\colon
a^{\dagger}a\colon=a^{\dagger}a,$ where $:$ $:$ denotes normal ordering.
2\. $c$-numbers can be taken out of the symbol $:$ $\colon$ as one wishes.
3\. The symbol $:$ : which is within another symbol $:$ $\colon$ can be
deleted.
4\. The vacuum projection operator $|0\rangle\langle 0|$ has the normal
product form
$|0\rangle\langle 0|=\colon e^{-a^{\dagger}a}\colon.$ (13)
5\. A normally ordered product can be integrated or differentiated with
respect to a $c$-number provided the integration is convergent.
### 3.2 The IWOP technique for deriving normally ordered Gaussian form of the
completeness relations of fundamental quantum mechanical representations
As an application of IWOP, (in the following, unless particularly mentioned,
we take $\hbar=\omega=m=1$ for convenience.) Using the Fock representation of
the coordinate eigenvector $Q|q\rangle=q|q\rangle,$
($Q=(a+a^{\dagger})/\sqrt{2}$)
$|q\rangle=\pi^{-1/4}e^{-\frac{q^{2}}{2}+\sqrt{2}qa^{\dagger}-\frac{a^{\dagger
2}}{2}}|0\rangle,$ (14)
we perform the integration below
$\displaystyle S_{1}$
$\displaystyle\equiv\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\mu}}|\frac{q}{\mu}\rangle\langle
q|$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}e^{-\frac{q^{2}}{2\mu^{2}}+\sqrt{2}\frac{q}{\mu}a^{\dagger}-\frac{a^{\dagger
2}}{2}}|0\rangle\langle 0|e^{-\frac{q^{2}}{2}+\sqrt{2}qa-\frac{a^{2}}{2}}.$
(15)
Substituting (13) into (15) we see
$S_{1}=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}e^{-\frac{q^{2}}{2\mu^{2}}+\sqrt{2}\frac{q}{\mu}a^{\dagger}-\frac{a^{\dagger
2}}{2}}\colon e^{-a^{\dagger}a}\colon
e^{-\frac{q^{2}}{2}+\sqrt{2}qa-\frac{a^{2}}{2}}.$ (16)
Note that on the left of $\colon e^{-a^{+}a}\colon$ are all creation
operators, while on its right are all annihilation operators, so the whole
integral is in normal ordering, thus using property 1 we have
$S_{1}=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}\colon
e^{-\frac{q^{2}}{2}(1+\frac{1}{\mu^{2}})+\sqrt{2}q(\frac{a^{\dagger}}{\mu}+a)-\frac{1}{2}(a+a^{\dagger})^{2}}\colon.$
(17)
As $a$ commutes with $a^{\dagger\text{ }}$ within $:$ $:$, so
$a^{\dagger\text{ }}$and $a$ can be considered as if they were parameters
while the integration is performing. Therefore, by setting $\mu=e^{\lambda}$,
sech$\lambda=\frac{2\mu}{1+\mu^{2}},$
tanh$\lambda=\frac{\mu^{2}+1}{\mu^{2}-1},$ we are able to perform the
integration and obtain
$\displaystyle S_{1}$
$\displaystyle=\sqrt{\frac{2\mu}{1+\mu^{2}}}\colon\exp\left\\{\frac{\left(\frac{a^{\dagger}}{\mu}+a\right)^{2}}{1+\frac{1}{\mu^{2}}}-\frac{1}{2}\left(a+a^{\dagger}\right)^{2}\right\\}\colon$
$\displaystyle=\left(\operatorname*{sech}\lambda\right)^{1/2}e^{-\frac{a^{\dagger
2}}{2}\tanh\lambda}\colon
e^{\left(\operatorname*{sech}\lambda-1\right)a^{\dagger}a}\colon
e^{\frac{a^{2}}{2}\tanh\lambda},$ (18)
which is just the single-mode squeezing operator in normal ordering appearing
in many references. It is worth mentioning that we have not used the SU(1,1)
Lie algebra method in the derivation. The integration automatically arranges
the squeezing operator in normal ordering. Using
$\displaystyle e^{\lambda a^{\dagger}a}$
$\displaystyle=\sum_{n=0}^{\infty}e^{\lambda n}|n\rangle\langle
n|=\sum_{n=0}^{\infty}e^{\lambda n}\frac{a^{\dagger n}}{n!}\colon
e^{-a^{\dagger}a}\colon a^{n}$
$\displaystyle=\colon\exp[\left(e^{\lambda}-1\right)a^{\dagger}a]\colon,$ (19)
Eq. (18) becomes
$\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\mu}}|\frac{q}{\mu}\rangle\langle
q|=e^{-\frac{a^{+2}}{2}\tanh\lambda}e^{(a^{+}a+\frac{1}{2})\ln\operatorname*{sech}\lambda}e^{\frac{a^{2}}{2}\tanh\lambda}.$
(20)
This shows the classical dilation $q\rightarrow\frac{q}{\mu}$ maps into the
normally ordered squeezing operator manifestly. It also exhibits that the
fundamental representation theory can be formulated in not so abstract way, as
we can now directly perform the integral over ket-bra projection operators.
Moreover, the IWOP technique can be employed to perform many complicated
integrations for ket-bra projection operators.
There is a deep ditch between quantum mechanical operators ($q$-numbers)
theory and classical numbers ($c$-numbers) theory. The IWOP technique arranges
non-commutable operators within an ordered product symbol in a way that they
become commutable, in this sense the ‘ditch’ between $q$-numbers and
$c$-numbers is ”shoaled”. However, the nature of operators are not changed,
they are still $q-$numbers, not $c$-numbers. After the integration over
$c$-numbers within ordered product is performed, we can finally get rid of the
normal ordering symbol by using (19).
When $\mu=1,$ Eq. (20) becomes
$\displaystyle\int_{-\infty}^{\infty}dq|q\rangle\langle q|$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi}}\colon
e^{-q^{2}+2q(\frac{a+a^{\dagger}}{\sqrt{2}})-\frac{1}{2}(a+a^{\dagger})^{2}}\colon$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi}}\colon
e^{-\left(q-Q\right)^{2}}\colon=1,\;\;$ (21)
a real simple Gaussian integration! This immediately leads us to put the
completeness relation of the momentum representation into the normally ordered
Gaussian form
$\int_{-\infty}^{\infty}dp|p\rangle\langle
p|=\int_{-\infty}^{\infty}\frac{dp}{\sqrt{\pi}}\colon
e^{-\left(p-P\right)^{2}}\colon=1,$ (22)
where $P=\left(a-a^{\dagger}\right)/(i\sqrt{2}),$ and $\left|p\right\rangle$
is the momentum eigenvector $P\left|p\right\rangle=p\left|p\right\rangle$,
$\left|p\right\rangle=\pi^{-\frac{1}{4}}\exp\left[-\frac{1}{2}p^{2}+i\sqrt{2}pa^{\dagger}+\frac{1}{2}a^{\dagger
2}\right]\left|0\right\rangle.$ (23)
In addition, we should notice that $\left|q\right\rangle$ and
$\left|p\right\rangle$ are related by the Fourier transform (FT), i.e.
$\left\langle
p\right|\left.q\right\rangle=\frac{1}{\sqrt{2\pi}}\exp\left(-iqp\right),$ the
integral kernel of the Fraunhofer diffraction formula in 1-dimensional is such
a FT, so FT in classical optics has its correspondence in quantum mechanical
representations’ transform. This enlightens us that in order to find more
general analogy between unitary operators in quantum optics and
transformations in classical optics we should construct new representations
for quantum optics theory, and these are the bi-partite entangled state and
many-particle entangled state. These ideal states can be implemented by
optical devices and optical network [38]. In the following we focus on the bi-
partite entangled state.
### 3.3 Single-mode Wigner operator
When we combine (21) and (22) we can obtain
$\pi^{-1}\colon
e^{-\left(q-Q\right)^{2}-\left(p-P\right)^{2}}\colon\equiv\Delta\left(q,p\right),$
(24)
which is just the normally ordered Wigner operator since its marginal
integration gives $|q\rangle\langle q|$ and $|p\rangle\langle p|$
respectively, i.e.,
$\displaystyle\int_{-\infty}^{\infty}dq\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{\sqrt{\pi}}\colon
e^{-\left(p-P\right)^{2}}\colon=|p\rangle\langle p|,$ (25)
$\displaystyle\int_{-\infty}^{\infty}dp\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{\sqrt{\pi}}\colon
e^{-\left(q-Q\right)^{2}}\colon=|q\rangle\langle q|.$ (26)
Thus the Wigner function of quantum state $\rho$ can be calculated as
$W(q,p)=$Tr$[\rho\Delta\left(q,p\right)]$. On the other hand, the Wigner
operator (24) can be recast into the coherent state representation,
$\Delta\left(q,p\right)\rightarrow\Delta\left(\alpha,\alpha^{\ast}\right)=\int\frac{d^{2}z}{\pi}\left|\alpha+z\right\rangle\left\langle\alpha-z\right|e^{\alpha
z^{\ast}-\alpha^{\ast}z},$ (27)
where $\left|z\right\rangle$ is a coherent state. In fact, using the IWOP
technique we can obtain
$\displaystyle\Delta\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\int\frac{d^{2}z}{\pi}\colon\exp\\{-\left|z\right|^{2}+\left(\alpha+z\right)a^{{\dagger}}+\left(\alpha^{\ast}-z^{\ast}\right)a$
$\displaystyle+\alpha
z^{\ast}-\alpha^{\ast}z-\left|\alpha\right|^{2}\\}\colon$
$\displaystyle=\frac{1}{\pi}\colon\exp\left\\{-2\left(a-\alpha\right)\left(a^{{\dagger}}-\alpha^{\ast}\right)\right\\}\colon,$
(28)
which is the same as (24).
### 3.4 Entangled state $\left|\eta\right\rangle$ and its Fourier transform
in complex form
The concept of quantum entanglement was first employed by Einstein, Rosen and
Poldosky (EPR) to challenge that quantum mechanics is incomplete when they
observed that two particles’ relative position $Q_{1}-Q_{2}$ and the total
momentum $P_{1}+P_{2}$ are commutable. Hinted by EPR, the bipartite entangled
state $\left|\eta\right\rangle$ is introduced as [39, 40]
$\left|\eta\right\rangle=\exp\left[-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle.$
(29)
$\left|\eta=\eta_{1}+\mathtt{i}\eta_{2}\right\rangle$ is the common eigenstate
of relative coordinate $Q_{1}-Q_{2}$ and the total momentum $P_{1}+P_{2}$,
$\left(Q_{1}-Q_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\text{
}\,\text{\
}\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(30)
where $Q_{i}=(a_{j}+a_{j}^{\dagger})/\sqrt{2},\
P_{j}=(a_{j}-a_{j}^{\dagger})/(\mathtt{i}\sqrt{2}),$ $j=1,2.$ Using the IWOP
technique, we can immediately prove that $\left|\eta\right\rangle$ possesses
the completeness relation
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=\int\frac{d^{2}{\eta}}{\pi}\colon
e^{-\left[\eta^{\ast}-(a_{1}^{\dagger}-a_{2})\right]\left[\eta-(a_{1}-a_{2}^{\dagger})\right]}\colon=1,\,d^{2}\eta=d\eta_{1}d\eta_{2},$
(31)
and orthonormal relation
$\left\langle\eta\right|\left.\eta^{\prime}\right\rangle=\pi\delta(\eta_{1}-\eta_{1}^{\prime})\delta(\eta_{2}-\eta_{2}^{\prime}).$
(32)
The Schmidt decomposition of $\left|\eta\right\rangle$ is
$\left|\eta\right\rangle=e^{-i\eta_{2}\eta_{1}}\int_{-\infty}^{\infty}dx\left|q\right\rangle_{1}\otimes\left|q-\sqrt{2}\eta_{1}\right\rangle_{2}e^{i\sqrt{2}\eta_{2}x},$
(33)
The $\left|\eta\right\rangle$ state can also be Schmidt-decomposed in momentum
eigenvector space as
$\left|\eta\right\rangle=e^{i\eta_{1}\eta_{2}}\int_{-\infty}^{\infty}dp\left|p\right\rangle_{1}\otimes\left|\sqrt{2}\eta_{2}-p\right\rangle_{2}e^{-i\sqrt{2}\eta_{1}p}.$
(34)
The $\left|\eta\right\rangle$ is physically appealing in quantum optics
theory, because the two-mode squeezing operator has its natural representation
on $\left\langle\eta\right|$ basis [40]
${\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right|=e^{a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda}e^{(a_{1}^{{}^{\dagger}}a_{1}+a_{2}^{{}^{\dagger}}a_{2}+1)\ln\operatorname*{sech}\lambda}e^{-a_{1}a_{2}\tanh\lambda},\;\mu=e^{\lambda},$
(35)
The proof of (35) is proceeded by virtue of the IWOP technique
$\displaystyle{\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right|$
$\displaystyle={\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\colon\exp\left\\{-\frac{|\eta|^{2}}{2}\left(1+\frac{1}{\mu^{2}}\right)+\eta\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\right.$
$\displaystyle+\left.\eta^{\ast}\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)+a_{1}^{\dagger}a_{2}^{{}^{\dagger}}+a_{1}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right\\}$
$\displaystyle=\frac{2\mu}{1+\mu^{2}}\colon\exp\left\\{\frac{\mu^{2}}{1+\mu^{2}}\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)-\left(a_{1}-a_{2}^{{}^{\dagger}}\right)\left(a_{1}^{{}^{\dagger}}-a_{2}\right)\right\\}\colon$
$\displaystyle=e^{a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda}e^{(a_{1}^{{}^{\dagger}}a_{1}+a_{2}^{{}^{\dagger}}a_{2}+1)\ln\operatorname*{sech}\lambda}e^{-a_{1}a_{2}\tanh\lambda}\equiv
S_{2},$ (36)
so the necessity of introducing $\left|\eta\right\rangle$ into quantum optics
is clear. $S_{2}$ squeezes $\left|\eta\right\rangle$ in the manifest way
$S_{2}\left|\eta\right\rangle=\frac{1}{\mu}\left|\eta/\mu\right\rangle,\text{
\ }\mu=e^{\lambda}\ ,$ (37)
and the two-mode squeezed state itself is an entangled state which entangles
the idle mode and signal mode as an outcome of a parametric-down conversion
process [41].
We can also introduce the conjugate state of $\left|\eta\right\rangle$ [42],
$\left|\xi\right\rangle=\exp\left[-\frac{1}{2}\left|\xi\right|^{2}+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle,\text{
}\xi=\xi_{1}+i\xi_{2},$ (38)
which obeys the eigen-equations
$\left(Q_{1}+Q_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{1}\left|\xi\right\rangle,\,\left(P_{1}-P_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{2}\left|\xi\right\rangle.$
(39)
Because
$\left[\left(Q_{1}-Q_{2}\right),\left(P_{1}-P_{2}\right)\right]=2\mathtt{i},$
so we name the conjugacy between $\left|\xi\right\rangle$ and
$\left|\eta\right\rangle.$ The completeness and orthonormal relations of
$\left|\xi\right\rangle$ are
$\displaystyle\int\frac{d^{2}\xi}{\pi}\left|\xi\right\rangle\left\langle\xi\right|$
$\displaystyle=\int\frac{d^{2}{\xi}}{\pi}\colon
e^{-\left[\xi^{\ast}-(a_{1}^{\dagger}+a_{2})\right]\left[\xi-(a_{1}+a_{2}^{\dagger})\right]}\colon=1,$
(40) $\displaystyle\left\langle\xi\right|\left.\xi^{\prime}\right\rangle$
$\displaystyle=\pi\delta(\xi_{1}-\xi_{1}^{\prime})\delta(\xi_{2}-\xi_{2}^{\prime}),\text{
}d^{2}\xi=d\xi_{1}d\xi_{2},$ (41)
respectively. $\left|\eta\right\rangle$ and $\left|\xi\right\rangle$ can be
related to each other by
$\left\langle\eta|\xi\right\rangle=\frac{1}{2}\exp\left(\frac{\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right),$
(42)
since $\xi^{\ast}\eta-\xi\eta^{\ast}$ is a pure imaginary number, Eq. (42) is
the Fourier transform kernel in complex form (or named entangled Fourier
transform, this concept should also be extended to multipartite entangled
states.) It will be shown in later sections that departing from entangled
states $\left|\eta\right\rangle$ and $\left|\xi\right\rangle$ and the
generalized Fresnel operator a new entangled Fresnel transforms in classical
optics can be found.
### 3.5 Two-mode Wigner operator in the $\left|\eta\right\rangle$
representation
Combining (31) and (40) we can construct the following operator
$\displaystyle\frac{1}{\pi^{2}}\colon
e^{-\left[\sigma^{\ast}-(a_{1}^{\dagger}-a_{2})\right]\left[\sigma-(a_{1}-a_{2}^{\dagger})\right]-\left[\gamma^{\ast}-(a_{1}^{\dagger}+a_{2})\right]\left[\gamma-(a_{1}+a_{2}^{\dagger})\right]}\colon$
$\displaystyle=\Delta\left(\alpha,\alpha^{\ast}\right)\otimes\Delta\left(\beta,\beta^{\ast}\right)\equiv\Delta\left(\sigma,\gamma\right),$
(43)
where
$\sigma=\alpha-\beta^{\ast},\;\gamma=\alpha+\beta^{\ast}.$ (44)
Eq. (43) is just equal to the direct product of two single-mode Wigner
operators. It is convenient to express the Wigner operator in the
$\left|\eta\right\rangle$ representation as [43]
$\Delta\left(\sigma,\gamma\right)=\int\frac{d^{2}\eta}{\pi^{3}}\left|\sigma-\eta\right\rangle\left\langle\sigma+\eta\right|e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}.$
(45)
For two-mode correlated system, it prefers to using
$\Delta\left(\sigma,\gamma\right)$ to calculate quantum states’ Wigner
function. For example, noticing
$\left\langle\eta\right|\left.00\right\rangle=\exp\\{-\left|\eta\right|^{2}/2\\},$
the two-mode squeezed states’ Wigner function is
$\displaystyle\left\langle
00\right|S_{2}^{{\dagger}}\left(\mu\right)\Delta\left(\sigma,\gamma\right)S_{2}\left(\mu\right)\left|00\right\rangle$
$\displaystyle=\left\langle
00\right|\mu^{2}\int\frac{d^{2}\eta}{\pi^{3}}\left|\mu\left(\sigma-\eta\right)\right\rangle\left\langle\mu\left(\sigma+\eta\right)\right|e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}\left|00\right\rangle$
$\displaystyle=\pi^{-2}\exp\left[-\mu^{2}\left|\sigma\right|^{2}-\left|\gamma\right|^{2}/\mu^{2}\right].$
(46)
## 4 Two deduced entangled state representations and Hankel transform
### 4.1 Deduced entangled states
Starting from the entangled state $\left|\eta=re^{i\theta}\right\rangle$ and
introducing an integer $m$, we can deduce new states [44],
$\left|m,r\right\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\left|\eta=re^{i\theta}\right\rangle
e^{-im\theta},$ (47)
which is worth of paying attention because when we operate the number-
difference operator,
$D\equiv a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}$ (48)
on $\left|\eta\right\rangle,$ using Eq.(29) we see
$D\left|\eta\right\rangle=\left(\eta
a_{1}^{{}^{\dagger}}+\eta^{\ast}a_{2}^{{}^{\dagger}}\right)\left|\eta\right\rangle=-i\frac{\partial}{\partial\theta}\left|\eta\right\rangle,\;\eta=|\eta|e^{i\theta},$
(49)
so the number-difference operator corresponds to a differential operation
$i\frac{\partial}{\partial\theta}$ in the
$\left\langle\eta\right|\;$representation, this is a remarkable property of
$\left\langle\eta\right|$. It then follows
$D\left|m,r\right\rangle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-im\theta}\left(-i\frac{\partial}{\partial\theta}\left|\eta=re^{i\theta}\right\rangle\right)=m\left|m,r\right\rangle.$
(50)
On the other hand, by defining
$K\equiv(a_{1}-a_{2}^{\dagger})(a_{1}^{\dagger}-a_{2}),$ (51)
we see $\left[D,K\right]=0,$ and $\left|m,r\right\rangle$ is its eigenstate,
$\ K\left|m,r\right\rangle=r^{2}\left|m,r\right\rangle,$ (52)
where $K$ is named correlated-amplitude operator since
$K\left|\eta\right\rangle=|\eta|^{2}\left|\eta\right\rangle.$ Thus we name
$\left|m,r\right\rangle$ correlated-amplitude—number-difference entangled
states. It is not difficult to prove completeness and orthonormal property of
$\left|m,r\right\rangle$,
$\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle\left\langle
m,r\right|=1,\;$ (53) $\left\langle
m,r\right|\left.m^{\prime},r^{\prime}\right\rangle=\delta_{m,m^{\prime}}\frac{1}{2r}\delta\left(r-r^{\prime}\right).$
(54)
On the other hand, from $\left|\xi\right\rangle$ we can derive another state
$\left|s,r^{\prime}\right\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\left|\xi=r^{\prime}e^{i\varphi}\right\rangle
e^{-is\varphi},$ (55)
which satisfies
$D\left|\xi\right\rangle=\left(a_{1}^{\dagger}\xi-
a_{2}^{\dagger}\xi^{\ast}\right)\left|\xi\right\rangle=-i\frac{\partial}{\partial\varphi}\left|\xi=r^{\prime}e^{i\varphi}\right\rangle.$
(56)
So $D$ in $\left\langle\xi=r^{\prime}e^{i\varphi}\right|$ representation is
equal to $i\frac{\partial}{\partial\varphi}$. Consequently,
$D\left|s,r^{\prime}\right\rangle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-is\theta}\left(-i\frac{\partial}{\partial\theta}\left|\xi=r^{\prime}e^{i\theta}\right\rangle\right)=s\left|s,r^{\prime}\right\rangle.$
(57)
Note $\left[D,(a_{1}^{\dagger}+a_{2})(a_{1}+a_{2}^{\dagger})\right]=0$ and
$(a_{1}^{\dagger}+a_{2})(a_{1}+a_{2}^{\dagger})\left|s,r^{\prime}\right\rangle=r^{\prime
2}\left|s,r^{\prime}\right\rangle.$ (58)
$\left|s,r^{\prime}\right\rangle$ is qualified to be a new representation
since
$\sum_{s=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left|s,r^{\prime}\right\rangle\left\langle
s,r^{\prime}\right|=1,\text{ }\left\langle
s,r^{\prime}\right|\left.s^{\prime},r^{\prime\prime}\right\rangle=\delta_{s,s^{\prime}}\frac{1}{2r^{\prime}}\delta\left(r^{\prime}-r^{\prime\prime}\right).$
(59)
### 4.2 Hankel transform between two deduced entangled state representations
Since $\left|\xi\right\rangle$ and $\left|\eta\right\rangle$ are mutual
conjugate, $\left|s,r^{\prime}\right\rangle$ is the conjugate state of
$\left|m,r\right\rangle$. From the definition of $\left|m,r\right\rangle$ and
$\left|s,r^{\prime}\right\rangle$ and (42) we calculate the overlap [21]
$\displaystyle\left\langle s,r^{\prime}\right|\left.q,r\right\rangle$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}d\varphi
e^{is\varphi}\left\langle\xi=r^{\prime}e^{i\varphi}\right|\int_{0}^{2\pi}d\theta\left|\eta=re^{i\theta}\right\rangle
e^{-im\theta}$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{is\varphi-
im\theta}\exp\left[irr^{\prime}\sin\left(\theta-\varphi\right)\right]d\theta
d\varphi$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{is\varphi-
im\theta}\sum_{l=-\infty}^{\infty}J_{l}\left(rr^{\prime}\right)e^{il\left(\theta-\varphi\right)}$
$\displaystyle=\frac{1}{2}\sum_{l=-\infty}^{\infty}\delta_{l,m}\delta_{l,s}J_{l}\left(rr^{\prime}\right)=\frac{1}{2}\delta_{s,m}J_{s}\left(rr^{\prime}\right),$
(60)
where we have identified the generating function of the s-order Bessel
function $J_{l},$
$e^{ix\sin t}=\sum_{l=-\infty}^{\infty}J_{l}\left(x\right)e^{ilt},\;$ (61)
and
$J_{l}\left(x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{l}}{k!\left(l+k\right)!}\left(\frac{x}{2}\right)^{l+2k}.$
(62)
Eq. (60) is remarkable, because $J_{s}\left(rr^{\prime}\right)$ is just the
integral kernel of Hankel transform. In fact, if we define
$\left\langle m,r\right|\left.g\right\rangle\equiv g\left(m,r\right),\text{
}\left\langle
s,r^{\prime}\right|\left.g\right\rangle\equiv\mathcal{G}\left(s,r^{\prime}\right),$
(63)
and use (53) as well as (60), we obtain
$\displaystyle\mathcal{G}\left(s,r^{\prime}\right)$
$\displaystyle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left\langle
s,r^{\prime}\right|\left.m,r\right\rangle\left\langle
m,r\right|\left.g\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}d\left(r^{2}\right)J_{s}\left(rr^{\prime}\right)g\left(s,r\right)\equiv\mathcal{H}\left[g\left(s,r\right)\right],$
(64)
which is just the Hankel transform of $g\left(m,r\right)$ (or it can be
regarded as a simplified form of the Collins formula in cylindrical
coordinate, see (10)). The inverse transform of (64) is
$\displaystyle g\left(m,r\right)$ $\displaystyle=\left\langle
m,\mathfrak{r}\right|\sum_{s=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left|s,r^{\prime}\right\rangle\left\langle
s,r^{\prime}\right|\left.g\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}d\left(r^{\prime
2}\right)J_{q}\left(rr^{\prime}\right)\mathcal{G}\left(m,r^{\prime}\right)\equiv\mathcal{H}^{-1}\left[\mathcal{G}\left(m,r^{\prime}\right)\right].$
(65)
Now we know that the quantum optical image of classical Hankel transform just
corresponds to the representation transformation between two mutually
conjugate entangled states $\left\langle s,r^{\prime}\right|$ and
$\left|m,r\right\rangle,$ this is like the case that the Fourier transform
kernel is just the matrix element between the coordinate state and the
momentum state, a wonderful result unnoticed before. Therefore the bipartite
entangled state representations’ transforms, which can lead us to the Hankel
transform, was proposed first in classical optics, can find their way back in
quantum optics.
### 4.3 Quantum optical version of classical circular harmonic correlation
From Eq.(47) we can see that its reciprocal relation is the circular harmonic
expansion,
$\left|\eta=re^{i\theta}\right\rangle=\sum_{m=-\infty}^{\infty}\left|m,r\right\rangle
e^{im\theta},$ (66)
or correlated-amplitude—number-difference entangled state
$\left|m,r\right\rangle$ can be considered as circular harmonic decomposition
of $\left|\eta=re^{i\theta}\right\rangle.$ Let $g\left(r,\theta\right)$, a
general 2-dimensional function expressed in polar coordinates, be periodic in
the variable $\theta,$ it can be looked as the wavefunction of the state
vector $\left|g\right\rangle$ in the $\left\langle\eta=re^{i\theta}\right|$
representation
$g\left(r,\theta\right)=\left\langle\eta=re^{i\theta}\right.\left|g\right\rangle,\text{
}\text{\ }$ (67)
using (66) we have
$g\left(r,\theta\right)=\sum_{m=-\infty}^{\infty}g_{m}\left(r\right)e^{-im\theta},\text{
\ }g_{m}\left(r\right)=\left\langle m,r\right.\left|g\right\rangle,$ (68)
$g_{m}\left(r\right)$ is the wavefunction of $\left|g\right\rangle$ in
$\left\langle m,r\right|$ representation. By using (53) and noticing that It
then follows from (49)
$e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}\left|\eta=re^{i\theta}\right\rangle=e^{-\alpha\frac{\partial}{\partial\theta}}\left|\eta=re^{i\theta}\right\rangle=\left|\eta=re^{i\left(\theta-\alpha\right)}\right\rangle,$
(69)
so
$e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}$
behaves a rotation operator in $\left|\eta\right\rangle$ representation, we
see that the expectation value of
$e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}$
in $\left|g\right\rangle$ is
$\displaystyle\pi\left\langle
g\right|e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{{\dagger}}}a_{2})}\left|g\right\rangle$
$\displaystyle=\pi\left\langle
g\right|\int\frac{d^{2}\eta}{\pi}e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}\left|\eta\right\rangle\left\langle\eta\right.\left|g\right\rangle$
$\displaystyle=\int_{0}^{\infty}rdrd\theta\int_{0}^{2\pi}\left\langle
g\right.\left|\eta^{\prime}=re^{i\left(\theta-\alpha\right)}\right\rangle\left\langle\eta=re^{i\theta}\right.\left|g\right\rangle$
$\displaystyle=\int_{0}^{\infty}rdr\int_{0}^{2\pi}g^{\ast}\left(r,\theta-\alpha\right)g\left(r,\theta\right)d\theta\equiv
R_{\alpha},$ (70)
which is just the cross-correlation between $g\left(r,\theta\right)$ and an
angularly rotated version of the same function,
$g^{\ast}\left(r,\theta-\alpha\right)$. On the other hand, using (53) we have
$\left|g\right\rangle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle\left\langle
m,r\right|\left.g\right\rangle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle
g_{m}\left(r\right).$ (71)
Substituting (71) into (70) and using the eigenvector equation (50) as well as
(54) we obtain
$\displaystyle R_{\alpha}$
$\displaystyle=\pi\sum_{m^{\prime}=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left\langle
m^{\prime},r^{\prime}\right|g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle
g_{m}\left(r\right)$
$\displaystyle=\pi\sum_{m^{\prime}=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-im\alpha}\int_{0}^{\infty}d\left(r^{2}\right)g_{m}\left(r\right)\delta_{m,m^{\prime}}\frac{1}{2r}\delta\left(r-r^{\prime}\right)$
$\displaystyle=2\pi\sum_{m=-\infty}^{\infty}e^{-im\alpha}\int_{0}^{\infty}r|g_{m}\left(r\right)|^{2}dr,$
(72)
from which we see that each of the circular harmonic components of the
crosscorrelation undergoes a different phase shift $-m\alpha,$ so $R_{\alpha}$
is not rotation invariant. However, when we consider only one harmonic
component
$R_{\alpha,M}=2\pi
e^{-iM\alpha}\int_{0}^{\infty}r|g_{M}\left(r\right)|^{2}dr,$ (73)
is extracted digitally, then from the phase associated with this component it
is possible to determine the angular shift that one version of the object has
undergone. When an optical filter that is matched to $R_{\alpha,M}$ of a
particular object is constructed, then if that some object is entered as an
input to the system with any angular rotation, a correlation peak of strength
proportional to $\int_{0}^{\infty}r|g_{M}\left(r\right)|^{2}dr$ will be
produced, independent of rotation. Hence an optical correlator can be
constructed that will recognize that object independent of rotation [19].
So far we have studied the circular harmonic correlation in the context of
quantum optics, we have endowed the crosscorrelation $R_{\alpha}$ with a
definite quantum mechanical meaning, i.e. the overlap between $\left\langle
g\right|$ and the rotated state
$e^{i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\left|g\right\rangle,$
in the entangled state representation.
Note that Fourier-based correlators is also very sensitive to magnification,
however, the magnitude of Mellin transform is independent of scale-size
changes in the input [19]. Now we examine when $\left|g\right\rangle$ is both
rotated and squeezed (by a two-mode squeezing operator
$S_{2}\left(\lambda\right)=\exp[\lambda(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})])$,
then from (37) and (67) we have
$S_{2}\left(\lambda\right)\left|g\right\rangle=\int\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right.\left|g\right\rangle=\int\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle
g\left(r,\theta\right),$ (74)
it follows the overlap between $\left\langle g\right|$ and the state
$e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}S\left(\lambda\right)\left|g\right\rangle,$
$\displaystyle W_{\alpha,\lambda}$ $\displaystyle\equiv\pi\left\langle
g\right|e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}S\left(\lambda\right)\left|g\right\rangle$
$\displaystyle=\left\langle
g\right|\int\frac{d^{2}\eta}{\mu}e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\left|\eta/\mu\right\rangle
g\left(r,\theta\right)$
$\displaystyle=\int_{0}^{\infty}\frac{rdr}{\mu}\int_{0}^{2\pi}\left\langle
g\right.\left|\eta^{\prime}=e^{i\left(\theta-\alpha\right)}r/\mu\right\rangle
g\left(r,\theta\right)d\theta$
$\displaystyle=\int_{0}^{\infty}\frac{rdr}{\mu}\int_{0}^{2\pi}g^{\ast}\left(r/\mu,\theta-\alpha\right)g\left(r,\theta\right)d\theta,$
(75)
which corresponds to the crosscorrelation arising from combination of
squeezing and rotation (joint transform correlator). On the other hand, from
(47) and (37) we see
$S_{2}\left(\lambda\right)\left|m,r\right\rangle=\frac{1}{2\pi\mu}\int_{0}^{2\pi}d\theta\left|\eta=\frac{r}{\mu}e^{i\theta}\right\rangle
e^{-im\theta}=\frac{1}{\mu}\left|m,\frac{r}{\mu}\right\rangle,$ (76)
and therefore
$\displaystyle W_{\alpha,\lambda}$
$\displaystyle=\frac{\pi}{\mu}\sum_{m^{\prime}=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left\langle
m^{\prime},r^{\prime}\right|g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,\frac{r}{\mu}\right\rangle
g_{m}\left(r\right)$
$\displaystyle=\frac{\pi}{\mu}\sum_{m^{\prime}=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-im\alpha}\int_{0}^{\infty}d\left(r^{2}\right)g_{m}\left(r\right)\delta_{m,m^{\prime}}\frac{1}{2r^{\prime}}\delta\left(\frac{r}{\mu}-r^{\prime}\right)$
$\displaystyle=\frac{2\pi}{\mu}\sum_{m=-\infty}^{\infty}e^{-im\alpha}\int_{0}^{\infty}rg_{m}\left(r\right)g_{m^{\prime}}^{\ast}\left(re^{-\lambda}\right)dr,\text{
\ }$ (77)
from which one can see that to achieve simultaneous scale and rotation
invariance, a two-dimensional object $g\left(r,\theta\right)$ should be
entered into the optical system in a distorted polar coordinate system, the
distortation arising from the fact that the radial coordinate is stretched by
a logarithmic transformation $\left(\lambda=-\ln\mu\right)$, which coincides
with Ref. [45]. The quantum optical version is thus established which is a new
tie connecting Fourier optics and quantum optics [46].
At the end of this section, using the two-variable Hermite polynomials’
definition [47]
$H_{m,n}\left(\xi,\xi^{\ast}\right)=\sum_{l=0}^{\min\left(m,n\right)}\frac{m!n!\left(-1\right)^{l}\xi^{m-l}\xi^{\ast
n-l}}{l!\left(m-l\right)!\left(n-l\right)!},$ (78)
which is quite different from the product of two single-variable Hermite
polynomials, and its generating function formula is
$\sum_{m,n=0}^{\infty}\frac{t^{m}t^{\prime
n}}{m!n!}H_{m,n}\left(\xi,\xi^{\ast}\right)=\exp\left[-tt^{\prime}+t\xi+t^{\prime}\xi^{\ast}\right],$
(79)
and noting
$H_{m,n}(\xi,\xi^{\ast})=e^{i(m-n)\varphi}H_{m,n}(r^{\prime},r^{\prime}),$ we
can directly perform the integral in (55) and derive the explicit form of
$\left|s,r^{\prime}\right\rangle,$
$\displaystyle\left|s,r^{\prime}\right\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\exp\\{-r^{\prime 2}/2+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-is\varphi-
a_{1}^{\dagger}a_{2}^{\dagger}\\}\left|00\right\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-r^{\prime
2}/2}\int_{0}^{2\pi}d\varphi\sum\limits_{m,n=0}^{\infty}\frac{a_{1}^{\dagger
m}a_{2}^{\dagger
n}}{m!n!}H_{m,n}(\xi,\xi^{\ast})e^{-is\varphi}\left|00\right\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-r^{\prime
2}/2}\int_{0}^{2\pi}d\varphi\sum\limits_{m,n=0}^{\infty}\frac{1}{\sqrt{m!n!}}H_{m,n}(r^{\prime},r^{\prime})e^{i\varphi\left(m-n-s\right)}\left|m,n\right\rangle$
$\displaystyle=e^{-r^{\prime
2}/2}\sum\limits_{n=0}^{\infty}\frac{1}{\sqrt{\left(n+s\right)!n!}}H_{n+s,n}(r^{\prime},r^{\prime})\left|n+s,n\right\rangle,$
(80)
which is really an entangled state in two-mode Fock space. Eqs. (79) and (80)
will be often used in the following discussions.
In the following we concentrate on finding the generalized Fresnel operators
in both one- and two- mode cases with use of the IWOP technique.
## 5 Single-mode Fresnel operator as the image of the classical Optical
Fresnel Transform
In this section we shall mainly introduce so-called generalized Fresnel
operators (GFO) (in one-mode and two-mode cases both) [48] and some
appropriate quantum optical representations (e.g. coherent state
representation and entangled state representation) to manifestly link the
formalisms in quantum optics to those in classical optics. In so doing, we
find that the various transforms in classical optics are just the result of
generalized Fresnel operators inducing transforms on appropriate quantum state
vectors, i.e. classical optical Fresnel transforms have their counterpart in
quantum optics. Besides, we can study the important $ABCD$ rule obeyed by
Gaussian beam propagation (also the ray propagation in matrix optics) [49] in
the domain of quantum optics.
### 5.1 Single-mode GFO gained via coherent state method
For the coherent state $\left|z\right\rangle$ in quantum optics [1, 2]
$\left|z\right\rangle=\exp\left[za^{\dagger}-z^{\ast}a\right]|0\rangle\equiv\left|\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle,$ (81)
which is the eigenstate of annihilation operator $a,$
$a\left|z\right\rangle=z\left|z\right\rangle$, using the IWOP and (13), we can
put the over-completeness relation of $\left|z\right\rangle$ into normal
ordering
$\int\frac{d^{2}{z}}{\pi}\left|z\right\rangle\left\langle
z\right|=\int\frac{d^{2}{z}}{\pi}\colon
e^{-\left(z^{\ast}-a^{\dagger}\right)\left(z-a\right)}\colon=1.$ (82)
the canonical form of coherent state $\left|z\right\rangle$ is expressed as
$\left|z\right\rangle=\left|p,q\right\rangle=\exp\left[i\left(pQ-
qP\right)\right]|0\rangle\equiv\left|\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle,$ (83)
where $z=\left(q+\mathtt{i}p\right)/\sqrt{2}$. It follows that $\left\langle
p,q\right|Q\left|p,q\right\rangle=q,$ $\left\langle
p,q\right|P\left|p,q\right\rangle=p,$ this indicates that the states
$\left|p,q\right\rangle$ generate a canonical phase-space representation for a
state $\left|\Psi\right\rangle,$ $\Psi\left(p,q\right)=\left\langle
p,q\right|\left.\Psi\right\rangle.$ Thus the coherent state is a good
candidate for providing with classical phase-space description of quantum
systems. Remembering that the Fresnel transform’s parameters
$\left(A,B,C,D\right)$ are elements of a ray transfer matrix $M$ describing
optical systems, $M$ belongs to the unimodular symplectic group, and the
coherent state $\left|p,x\right\rangle$ is a good candidate for providing with
classical phase-space description of quantum systems, we naturally think of
that the symplectic transformation $\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\ p\end{array}\right)$ in
classical phase space may mapping onto a generalized Fresnel operator in
Hilbert space through the coherent state basis. Thus we construct the
following ket-bra projection operator
$\iint\limits_{-\infty}^{\infty}dxdp\left|\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right|$ (84)
as the GFO. In fact, using notation of $\left|z\right\rangle$ (coherent
state), and introducing complex numbers $s,r,$
$s=\frac{1}{2}\left[A+D-i\left(B-C\right)\right],\;r=-\frac{1}{2}\left[A-D+i\left(B+C\right)\right],\;|s|^{2}-|r|^{2}=1,$
(85)
from (83) we know
$\displaystyle\left|\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle$
$\displaystyle=\left|\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle\equiv\left|sz-rz^{\ast}\right\rangle$
(94) $\displaystyle=\exp\left[-\frac{1}{2}|sz-rz^{\ast}|^{2}+(sz-
rz^{\ast})a^{\dagger}\right]|0\rangle,$ (95)
$\left(\begin{array}[c]{cc}s&-r\\\ -r^{\ast}&s^{\ast}\end{array}\right)$ is
still a symplectic group element, so (84) becomes [50]
$F_{1}\left(s,r\right)=\sqrt{s}\int\frac{d^{2}z}{\pi}\left|sz-
rz^{\ast}\right\rangle\left\langle z\right|,\;\;$ (96)
where the factor $\sqrt{s}$ is attached for anticipating the unitarity of the
operator $F_{1}.$ Eq. (84) tells us that $c$-number transform
$\left(q,p\right)\rightarrow\left(Aq+Bp,Cq+Dp\right)$ in coherent state basis
maps into $F_{1}\left(s,r\right)$. Now we prove $F_{1}\left(s,r\right)$ is
really the FO we want. Using the IWOP technique and Eq. (95) and (13) we can
perform the integral
$\displaystyle F_{1}\left(s,r\right)$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\colon\exp\left[-\left|s\right|^{2}\left|z\right|^{2}+sza^{\dagger}+z^{\ast}\left(a-ra^{\dagger}\right)+\frac{r^{\ast}s}{2}z^{2}+\frac{rs^{\ast}}{2}z^{\ast
2}-a^{\dagger}a\right]\colon$
$\displaystyle=\frac{1}{\sqrt{s^{\ast}}}\exp\left(-\frac{r}{2s^{\ast}}a^{\dagger
2}\right)\colon\exp\left\\{\left(\frac{1}{s^{\ast}}-1\right)a^{\dagger}a\right\\}\colon\exp\left(\frac{r^{\ast}}{2s^{\ast}}a^{2}\right)$
$\displaystyle=\exp\left(-\frac{r}{2s^{\ast}}a^{\dagger
2}\right)\exp\left\\{\left(a^{\dagger}a+\frac{1}{2}\right)\ln\frac{1}{s^{\ast}}\right\\}\exp\left(\frac{r^{\ast}}{2s^{\ast}}a^{2}\right),$
(97)
where we have used the mathematical formula [51]
$\displaystyle\int\frac{d^{2}z}{\pi}\exp\\{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}+fz^{2}+gz^{\ast 2}\\}$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left\\{\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right\\},$
(98)
with the convergent condition $\operatorname{Re}(\zeta\pm f\pm
g)<0,\operatorname{Re}(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g})<0.$ It then
follows
$\left\langle
z\right|F_{1}\left(s,r\right)\left|z^{\prime}\right\rangle=\frac{1}{\sqrt{s^{\ast}}}\exp\left[-\frac{\left|z\right|^{2}+\left|z^{\prime}\right|^{2}}{2}-\frac{rz^{\ast
2}}{2s^{\ast}}+\frac{r^{\ast}z^{\prime
2}}{2s^{\ast}}+\frac{z^{\ast}z^{\prime}}{s^{\ast}}\right].$ (99)
Then using
$\left\langle
x_{i}\right|\left.z\right\rangle=\pi^{-1/4}\exp\left(-\frac{x_{i}^{2}}{2}+\sqrt{2}x_{i}z-\frac{z^{2}}{2}-\frac{\left|z\right|^{2}}{2}\right).$
(100)
and the completeness relation of coherent state as well as (85) we obtain the
matrix element of $F_{1}\left(s,r\right)$ ($\equiv F_{1}\left(A,B,C\right)$)
in coordinate representation $\left\langle x_{i}\right|$,
$\displaystyle\left\langle
x_{2}\right|F_{1}\left(s,r\right)\left|x_{1}\right\rangle$
$\displaystyle=\int\frac{d^{2}z}{\pi}\left\langle
x_{2}\right|\left.z\right\rangle\left\langle
z\right|F_{1}\left(s,r\right)\int\frac{d^{2}z^{\prime}}{\pi}\left|z^{\prime}\right\rangle\left\langle
z^{\prime}\right|\left.x_{1}\right\rangle$ $\displaystyle=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right]\equiv\mathcal{K}\left(x_{2},x_{1}\right),$
(101)
which is just the kernel of generalized Fresnel transform
$\mathcal{K}\left(x_{2},x_{1}\right)$ in (12). The above discussions
demonstrate how to transit classical Fresnel transform to GFO through the
coherent state and the IWOP technique.
Now if we define $g\left(x_{2}\right)=\left\langle
x_{2}\right|\left.g\right\rangle$, $f\left(x_{1}\right)=\left\langle
x_{1}\right|\left.f\right\rangle$ and using Eq. (17), we can rewrite Eq. (11)
as
$\displaystyle\left\langle x_{2}\right|\left.g\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{1}\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|x_{1}\right\rangle\left\langle
x_{1}\right|\left.f\right\rangle$ $\displaystyle=\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|f\right\rangle,$ (102)
which is just the quantum mechanical version of GFO. Therefore, the
1-dimensional GFT in classical optics corresponds to the 1-mode GFO
$F_{1}\left(A,B,C\right)$ operating on state vector $\left|f\right\rangle$ in
Hilbert space, i.e.
$\left|g\right\rangle=F_{1}\left(A,B,C\right)\left|f\right\rangle$. One merit
of GFO is: using coordinate-momentum representation transform we can
immediately obtain GFT in “frequency” domain, i.e.
$\displaystyle\left\langle
p_{2}\right|F\left|p_{1}\right\rangle=\int_{-\infty}^{\infty}dx_{1}dx_{2}\left\langle
p_{2}\right|\left.x_{2}\right\rangle\left\langle
x_{2}\right|F\left|x_{1}\right\rangle\left\langle
x_{1}\right.\left|p_{1}\right\rangle$ $\displaystyle=\frac{1}{\sqrt{2\pi
iB}}\int_{-\infty}^{\infty}\frac{dx_{1}dx_{2}}{2\pi}\exp\left[\frac{iA}{2B}\left(x_{1}^{2}+\frac{x_{1}}{A}\left(Bp_{1}-2x_{2}\right)\right)+\frac{iD}{2B}x_{2}^{2}-ip_{2}x_{2}\right]$
$\displaystyle=\frac{1}{\sqrt{2\pi
i\left(-C\right)}}\exp\left[\frac{i}{2\left(-C\right)}\left(Dp_{1}^{2}-2p_{2}p_{1}+Ap_{2}^{2}\right)\right].$
(103)
Obviously, $F_{1}\left(A,B,C\right)$ induces the following transform
$F_{1}^{-1}\left(A,B,C\right)\left(\begin{array}[c]{c}Q\\\
P\end{array}\right)F_{1}\left(A,B,C\right)=\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}Q\\\ P\end{array}\right).$ (104)
### 5.2 Group Multiplication Rule for Single-mode GFO
Because two successive optical Fresnel transforms is still a Fresnel
transform, we wonder if the product of two GFO is still a GFO. On the other
hand, we have known that the GFO is the image of the symplectic transform
$z\rightarrow sz-rz^{\ast},$ we expect that the product of two symplectic
transforms maps into the GFO which is just the product of two GFOs. If this is
so, then correspondence between GFT and GFO is perfect. Using (96),
$\left\langle
z\right.\left|z^{\prime}\right\rangle=\exp\left[-\frac{1}{2}\left(|z|^{2}+|z^{\prime}|^{2}\right)+z^{\ast}z^{\prime}\right]$
and the IWOP technique we can directly perform the following integrals
$\displaystyle F_{1}\left(s,r\right)F_{1}\left(s^{\prime},r^{\prime}\right)$
$\displaystyle=\sqrt{ss^{\prime}}\int\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left|sz-
rz^{\ast}\right\rangle\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle\left\langle
z^{\prime}\right|$
$\displaystyle=\frac{1}{\sqrt{s^{\prime\prime\ast}}}\exp\left[-\frac{r^{\prime\prime}}{2s^{\prime\prime\ast}}a^{\dagger
2}\right]\colon\exp\left\\{\left(\frac{1}{s^{\prime\prime\ast}}-1\right)a^{\dagger}a\right\\}\colon\exp\left[\frac{r^{\prime\prime\ast}}{2s^{\prime\prime\ast}}a^{2}\right]$
$\displaystyle=\sqrt{s^{\prime\prime}}\int\frac{d^{2}z}{\pi}\left|s^{\prime\prime\ast}z-r^{\prime\prime}z^{\ast}\right\rangle\left\langle
z\right|=F_{1}\left(s^{\prime\prime},r^{\prime\prime}\right),$ (105)
where we have set
$s^{\prime\prime}=ss^{\prime}+rr^{\prime\ast},\;r^{\prime\prime}=r^{\prime}s+rs^{\prime\ast},$
(106)
or
$M^{\prime\prime}\equiv\left(\begin{array}[c]{cc}s^{\prime\prime}&-r^{\prime\prime}\\\
-r^{\ast\prime\prime}&s^{\ast\prime\prime}\end{array}\right)=\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{cc}s^{\prime}&-r^{\prime}\\\
-r^{\prime\ast}&s^{\prime\ast}\end{array}\right)=MM^{\prime},\
\left|s^{\prime\prime}\right|^{2}-\left|r^{\prime\prime}\right|^{2}=1,$ (107)
from which we see that it is just the mapping of the above ($A,B,C,D)$
matrices multiplication. Hence
$F_{1}\left(s,r\right)F_{1}\left(s^{\prime},r^{\prime}\right)$ is the loyal
representation of the product of two symplectic group elements shown in (107).
The above discussion actually reveals an important property of coherent
states, though two coherent state vectors are not orthogonal, but the equation
$\sqrt{ss^{\prime}}\int\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left|sz-
rz^{\ast}\right\rangle\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle\left\langle
z^{\prime}\right|=\sqrt{s^{\prime\prime}}\int\frac{d^{2}z}{\pi}\left|s^{\prime\prime\ast}z-r^{\prime\prime}z^{\ast}\right\rangle\left\langle
z\right|$ (108)
seems as if their overlap $\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle$ was
a $\delta$-function. The coherent state representation for GFOs’ product may
be visualized very easily, but it achieves striking importance, because it
does not change its form when treating symplectic transform according to
$z\rightarrow sz-rz^{\ast}$.
As a result of this group multiplication rule of GFO, we immediately obtain
$\displaystyle\mathcal{K}^{M^{\prime\prime}}\left(x_{2},x_{1}\right)=\left\langle
x_{2}\right|F_{1}\left(s^{\prime\prime},r^{\prime\prime}\right)\left|x_{1}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{3}\left\langle
x_{2}\right|F_{1}\left(s,r\right)\left|x_{3}\right\rangle\left\langle
x_{3}\right|F_{1}\left(s^{\prime},r^{\prime}\right)\left|x_{1}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{3}\mathcal{K}^{M}\left(x_{2},x_{3}\right)\mathcal{K}^{M^{\prime}}\left(x_{3},x_{1}\right),$
(109)
provided that the parameter matrices
$\left(s^{\prime\prime},r^{\prime\prime}\right)$ satisfy (106). Thus by virtue
of the group multiplication property of GFO we immediately find the successive
transform property of GFTs.
## 6 Quantum Optical ABCD Law for optical propagation —single-mode case
In classical optics, ray-transfer matrices,
$N=\left(\begin{array}[c]{cc}A&B\\\ C&D\end{array}\right),$ $AD-BC=1$, have
been used to describe the geometrical formation of images by a centered lens
system. For an optical ray (a centered spherical wavefront) passing through
optical instruments there is a famous law, named ABCD law, governing the
relation between input ray $\left(r_{1},\alpha_{1}\right)$ and output ray
$\left(r_{2},\alpha_{2}\right),$ i.e.
$\left(\begin{array}[c]{c}r_{2}\\\
\alpha_{2}\end{array}\right)=N\left(\begin{array}[c]{c}r_{1}\\\
\alpha_{1}\end{array}\right),$ (110)
where $r_{1}$ is the ray height from the optical axis, and $\alpha_{1}$ is
named the optical direction-cosine, $r_{1}/\alpha_{1}\equiv R_{1}$ specifies
the ray’s wavefront shape. Eq. (110) implies
$R_{2}\equiv\frac{r_{2}}{\alpha_{2}}=\frac{AR_{1}+B}{CR_{1}+D}.$ (111)
This law is the core of matrix optics, since it tells us how the curvature of
a centered spherical wavefront changes from one reference plane to the next.
Besides, the multiplication rule of matrix optics implies that if the ray-
transfer matrices of the $n$ optical components are
$N_{1},N_{2},N_{3},\cdots,N_{n}$, respectively, then the whole system is
determined by a matrix $N=N_{1}N_{2}N_{3}\cdots N_{n}.$
One of the remarkable things of modern optics is the case with which
geometrical ray-transfer methods, constituting the matrix optics, can be
adapted to describe the generation and propagation of Laser beams. In 1965
Kogelnik [52] pointed out that propagation of Gaussian beam also obeys ABCD
law via optical diffraction integration, i.e. the input light field
$f\left(x_{1}\right)$ and output light field $g\left(x_{2}\right)$ are related
to each other by so-called Fresnel integration [19]
$g\left(x_{2}\right)=\int_{-\infty}^{\infty}\mathcal{K}\left(A,B,C;x_{2},x_{1}\right)f\left(x_{1}\right)dx_{1},$
where
$\mathcal{K}\left(A,B,C;x_{2},x_{1}\right)=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right].$
The ABCD law for Gaussian beam passing through an optical system is [53]
$q_{2}=\frac{Aq_{1}+B}{Cq_{1}+D},$ (112)
where $q_{1}$ $(q_{2})$ represents the complex curvature of the input (output)
Gaussian beam, Eq. (112) has the similar form as Eq. (111). An interesting and
important question naturally arises [54]: Does ABCD law also exhibit in
quantum optics? Since classical Fresnel transform should have its quantum
optical counterpart?
To see the ABCD law more explicitly, using Eq.(85) we can re-express Eq.(97)
as
$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\sqrt{\frac{2}{A+D+i\left(B-C\right)}}\colon\exp\left\\{\frac{A-D+i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{\dagger
2}\right.$
$\displaystyle\left.+\left[\frac{2}{A+D+i\left(B-C\right)}-1\right]a^{\dagger}a-\frac{A-D-i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{2}\right\\}\colon,$
(113)
and the multiplication rule for $F_{1}$ is
$F\left(A^{\prime},B^{\prime},C^{\prime},D^{\prime}\right)F\left(A,B,C,D\right)=F\left(A^{\prime\prime},B^{\prime\prime},C^{\prime\prime},D^{\prime\prime}\right),$
where
$\left(\begin{array}[c]{cc}A^{\prime\prime}&B^{\prime\prime}\\\
C^{\prime\prime}&D^{\prime\prime}\end{array}\right)=\left(\begin{array}[c]{cc}A^{\prime}&B^{\prime}\\\
C^{\prime}&D^{\prime}\end{array}\right)\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right).$ (114)
Next we directly use the GFO to derive ABCD law in quantum optics. From
Eq.(114) we see that the GFO generates
$F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{\frac{2}{A+iB-i\left(C+iD\right)}}\exp\left\\{\frac{A-D+i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle,$ (115)
if we identify
$\frac{A-D+i\left(B+C\right)}{A+D+i\left(B-C\right)}=\frac{q_{1}-i}{q_{1}+i},$
(116)
then
$F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{-\frac{2/\left(C+iD\right)}{q_{1}+i}}\exp\left[\frac{q_{1}-i}{2\left(q_{1}+i\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (117)
The solution of Eq.(116) is
$q_{1}\equiv-\frac{A+iB}{C+iD}.$ (118)
Let $F_{1}\left(A,B,C\right)\left|0\right\rangle$ expressed by (117) be an
input state for an optical system which is characteristic by parameters
$A^{\prime},B^{\prime},C^{\prime},D^{\prime},$ then the quantum optical ABCD
law states that the output state is
$F_{1}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{\frac{-2/\left(C^{\prime\prime}+iD^{\prime\prime}\right)}{q_{2}+i}}\exp\left[\frac{q_{2}-i}{2\left(q_{2}+i\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (119)
which has the similar form as Eq.(117), where
$\left(C^{\prime\prime},D^{\prime\prime}\right)$ is determined by Eq.(114),
and
$\bar{q}_{2}=\frac{A^{\prime}\bar{q}_{1}+B^{\prime}}{C^{\prime}\bar{q}_{1}+D^{\prime}},\text{
\ }\bar{q}_{j}\equiv-q_{j},\text{ \ }\left(j=1,2\right)$ (120)
which resembles Eq.(112).
Proof:
According to the multiplication rule of two GFOs and Eqs.(113)-(114) we have
$\displaystyle
F_{1}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{1}\left(A,B,C\right)\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{2}{A^{\prime\prime}+D^{\prime\prime}+i\left(B^{\prime\prime}-C^{\prime\prime}\right)}}\exp\left\\{\frac{A^{\prime\prime}-D^{\prime\prime}+i\left(B^{\prime\prime}+C^{\prime\prime}\right)}{2\left[A^{\prime\prime}+D^{\prime\prime}+i\left(B^{\prime\prime}-C^{\prime\prime}\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{2}{A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)-iC^{\prime}\left(A+iB\right)-iD^{\prime}\left(C+iD\right)}}$
$\displaystyle\times\exp\left\\{\frac{A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)+iC^{\prime}\left(A+iB\right)+iD^{\prime}\left(C+iD\right)}{2\left[A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)-iC^{\prime}\left(A+iB\right)-iD^{\prime}\left(C+iD\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{-2/\left(C+iD\right)}{A^{\prime}q_{1}-B^{\prime}-i\left(C^{\prime}q_{1}-D^{\prime}\right)}}\exp\left\\{\frac{A^{\prime}q_{1}-B^{\prime}+i\left(C^{\prime}q_{1}-D^{\prime}\right)}{2\left[A^{\prime}q_{1}-B^{\prime}-i\left(C^{\prime}q_{1}-D^{\prime}\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle.$ (121)
Using Eq.(118) we see
$\frac{2/\left(C+iD\right)}{C^{\prime}q_{1}-D^{\prime}}=-2/\left(C^{\prime\prime}+iD^{\prime\prime}\right),$
together using Eq.(120) we can reach Eq.(119), thus the law is proved. Using
Eq. (117) we can re-express Eq.(120) as
$q_{2}=-\frac{A^{\prime}(A+iB)+B^{\prime}(C+iD)}{C^{\prime}(A+iB)+D^{\prime}(C+iD)}=-\frac{A^{\prime\prime}+iB^{\prime\prime}}{C^{\prime\prime}+iD^{\prime\prime}},$
(122)
which is in consistent to Eq.(118). Eqs. (117)-(122) are therefore self-
consistent.
As an application of quantum optical ABCD law, we apply it to tackle the time-
evolution of a time-dependent harmonic oscillator whose Hamiltonian is
$H=\frac{1}{2}e^{-2\gamma t}P^{2}+\frac{1}{2}\omega_{0}^{2}e^{2\gamma
t}Q^{2},\text{ \ \ }\hbar=1,$ (123)
where we have set the initial mass $m_{0}=1,$ $\gamma$ denotes damping. Using
$u\left(t\right)=e^{\frac{i\gamma}{2}Q^{2}}e^{-\frac{i\gamma
t}{2}\left(QP+PQ\right)}\ $to perform the transformation
$\displaystyle u\left(t\right)Qu^{-1}\left(t\right)$ $\displaystyle=e^{-\gamma
t}Q,$ $\displaystyle u\left(t\right)Pu^{-1}\left(t\right)$
$\displaystyle=e^{\gamma t}P-\gamma e^{\gamma t}Q,$ (124)
then $i\frac{\partial\left|\psi\left(t\right)\right\rangle}{\partial
t}=H\left|\psi\left(t\right)\right\rangle\ $leads to
$i\frac{\partial\left|\phi\right\rangle}{\partial
t}=\mathcal{H}\left|\phi\right\rangle,$
$\left|\phi\right\rangle=u\left(t\right)\left|\psi\left(t\right)\right\rangle,$
$H\rightarrow\mathcal{H}=u\left(t\right)Hu^{-1}\left(t\right)-iu\left(t\right)\frac{\partial
u^{-1}\left(t\right)}{\partial
t}=\frac{1}{2}P^{2}+\frac{1}{2}\omega^{2}Q^{2},.$ (125)
where $\omega^{2}=\omega_{0}^{2}-\gamma^{2}.$ $\mathcal{H}$ does not contain
$t$ explicitly. The dynamic evolution of a mass-varying harmonic oscillator
from the Fock state $\left|0\right\rangle$ at initial time to a squeezed state
at time $t$ is
$\left|\psi\left(t\right)\right\rangle_{0}=u^{-1}\left(t\right)\left|0\right\rangle=e^{\frac{i\gamma
t}{2}\left(QP+PQ\right)}e^{-\frac{i\gamma}{2}Q^{2}}\left|0\right\rangle,$
(126)
if we let $A=D=1,B=0,C=-\gamma;$ and $A^{\prime}=e^{-\gamma
t},D^{\prime}=e^{\gamma t},B^{\prime}=C^{\prime}=0,$ then
$q_{1}=\frac{1}{\gamma-i},$ $q_{2}=\frac{e^{-2\gamma t}}{\gamma-i}$, according
to Eq.(119) we directly obtain
$u^{-1}\left(t\right)\left|0\right\rangle=\sqrt{\frac{2e^{-\gamma
t}}{e^{-2\gamma t}+i\gamma+1}}\exp\left[\frac{e^{-2\gamma
t}-1-i\gamma}{2\left(e^{-2\gamma t}+1+i\gamma\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (127)
so the time evolution of the damping oscillator embodies the quantum optical
ABCD law.
## 7 Optical operator method studied via GFO’s decomposition
Fresnel diffraction is the core of Fourier optics [19, 30, 35, 36], Fresnel
transform is frequently used in optical imaging, optical propagation and
optical instrument design. The GFT represents a class of optical transforms
which are of great importance for their applications to describe various
optical systems. It is easily seen that when we let the transform kernel
$\mathcal{K}\left(x_{2},x_{1}\right)=\exp\left(ix_{2}x_{1}\right)$, the GFT
changes into the well-known Fourier transform, which is adapted to express
mathematically the Fraunhofer diffraction. And if
$\mathcal{K}\left(x_{2},x_{1}\right)=\exp[i\left(x_{2}-x_{1}\right)^{2}]$, the
GFT then describes a Fresnel diffraction. In studying various optical
transformations one also proposed so-called optical operator method [55] which
used quantum mechanical operators’ ordered product to express the mechanism of
optical systems, such that the ray transfer through optical instruments and
the diffraction can be discussed by virtue of the commutative relations of
operators and the matrix algebra. Two important questions thus naturally
arises: how to directly map the classical optical transformations to the
optical operator method? How to combine the usual optical transformation
operators, such as the square phase operators, scaling operator, Fourier
transform operator and the propagation operator in free space, into a concise
and unified form? In this section we shall solve these two problems and
develop the optical operator method onto a new stage.
### 7.1 Four fundamental optical operators derived by decomposing GFO
The GFO $F_{1}\left(A,B,C\right)$ can also be expressed in the form of
quadratic combination of canonical operators $Q$ and $P$ [56], i.e.,
$F_{1}\left(A,B,C\right)=\exp\left(\frac{iC}{2A}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
A\right)\exp\left(-\frac{iB}{2A}P^{2}\right),$ (128)
where we have set $\hbar=1,$ $A\neq 0.$ To confirm this, we first calculate
matrix element
$\displaystyle\left\langle
x\right|F_{1}\left(A,B,C\right)\left|p\right\rangle$
$\displaystyle=\exp\left(\frac{iC}{2A}x^{2}\right)\left\langle
x\right|\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
A\right)\left|p\right\rangle\exp\left(-\frac{iB}{2A}p^{2}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi
A}}\exp\left(\frac{iC}{2A}x^{2}-\frac{iB}{2A}p^{2}+\frac{ipx}{A}\right),$
(129)
where we have used the squeezing property
$\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
A\right]\left|p\right\rangle=\frac{1}{\sqrt{A}}\left|p/A\right\rangle.$ (130)
It then follows from (129) and $AD-BC=1$, we have
$\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|x_{1}\right\rangle=\int_{-\infty}^{\infty}dp\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|p\right\rangle\left\langle
p\right|\left.x_{1}\right\rangle=\mathcal{K}^{M}\left(x_{2},x_{1}\right).$
(131)
Thus $F_{1}\left(A,B,C\right)$ in (128) is really the expected GFO. Next we
directly use (84) and the canonical operator $\left(Q,P\right)$ representation
(128) to develop the optical operator method.
By noticing the matrix decompositions [31]
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
C/A&1\end{array}\right)\left(\begin{array}[c]{cc}A&0\\\
0&A^{-1}\end{array}\right)\left(\begin{array}[c]{cc}1&B/A\\\
0&1\end{array}\right),$ (132)
and comparing (84) and (128) as well as using (114) we know
$F_{1}\left(A,B,C\right)=F_{1}\left(1,0,C/A\right)F_{1}\left(A,0,0\right)F_{1}\left(1,B/A,0\right),$
(133)
where
$\displaystyle F_{1}\left(1,0,C/A\right)$
$\displaystyle=\frac{\sqrt{2+iC/A}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}1&0\\\
C/A&1\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (140)
$\displaystyle=\exp\left(\frac{iC}{2A}Q^{2}\right),$ (141)
which is named quadrature phase operator; and
$\displaystyle F_{1}\left(1,B/A,0\right)$
$\displaystyle=\frac{\sqrt{2-iB/A}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}1&B/A\\\
0&1\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (148)
$\displaystyle=\exp\left(-\frac{iB}{2A}P^{2}\right),$ (149)
which is named Fresnel propagator in free space; as well as
$\displaystyle F_{1}\left(A,0,0\right)$
$\displaystyle=\frac{\sqrt{A+A^{-1}}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}A&0\\\
0&A^{-1}\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (156)
$\displaystyle=\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln A\right],$ (157)
which is named scaling operator (squeezed operator [9, 10]). When
$A=D=0,B=1,C=-1,$ from (113) we see
$\displaystyle F_{1}\left(0,1,-1\right)$
$\displaystyle=\sqrt{-i}\int\frac{dxdp}{2\pi}\left|\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (164)
$\displaystyle=\exp\left[-\left(a^{\dagger}a+\frac{1}{2}\right)\ln i\right]$
$\displaystyle=\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right],$
(165)
which is named the Fourier operator, since it quantum mechanically transforms
[57]
$\displaystyle\exp\left[i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]Q\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]$
$\displaystyle=P,$
$\displaystyle\exp\left[i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]P\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]$
$\displaystyle=-Q.$ (166)
### 7.2 Alternate decompositions of GFO
Note that when $A=0$, the decomposition (128) is not available, instead, from
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)^{-1}=\left(\begin{array}[c]{cc}D&-B\\\
-C&A\end{array}\right),$ (167)
and (113), (114) and (128) we have
$F_{1}^{-1}\left(A,B,C\right)=\exp\left(-\frac{iC}{2D}Q^{2}\right)\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
D\right]\exp\left(\frac{iB}{2D}P^{2}\right),$ (168)
it then follows
$F_{1}\left(A,B,C\right)=\exp\left(-\frac{iB}{2D}P^{2}\right)\exp\left[\frac{i}{2}\left(QP+PQ\right)\ln
D\right]\exp\left(\frac{iC}{2D}Q^{2}\right),\text{ }D\neq 0.$ (169)
Besides, when we notice
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
D/B&1\end{array}\right)\left(\begin{array}[c]{cc}B&0\\\
0&1/B\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
A/B&1\end{array}\right),$ (170)
and
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&A/C\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}-1/C&0\\\
0&-C\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{cc}1&D/C\\\
0&1\end{array}\right),$ (171)
we have another decomposition for $B\neq 0,$
$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\exp\left(\frac{iD}{2B}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
B\right)$
$\displaystyle\times\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]\exp\left(\frac{iA}{2B}Q^{2}\right),\text{
}$ (172)
and for $C\neq 0$
$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\exp\left(-\frac{iA}{2C}P^{2}\right)\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln\left(\frac{-1}{C}\right)\right]$
$\displaystyle\times\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]\exp\left(-\frac{iD}{2C}P^{2}\right).$
(173)
### 7.3 Some optical operator identities
For a special optical systems with the parameter $A=0,$ $C=-B^{-1},$
$\left(\begin{array}[c]{cc}0&B\\\
-B^{-1}&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
D/B&1\end{array}\right)\left(\begin{array}[c]{cc}B&0\\\
0&B^{-1}\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right),$ (174)
we have
$\displaystyle\exp\left(-\frac{iB}{2D}P^{2}\right)\exp\left(\frac{i}{2}\left(QP+PQ\right)\ln
D\right)\exp\left(\frac{-i}{2DB}Q^{2}\right)$
$\displaystyle=\exp\left(\frac{-iD}{2B}Q^{2}\right)\exp\left(\frac{i}{2}\left(QP+PQ\right)\ln
B\right)\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right].$
(175)
In particular, when $A=D=0,$ $C=-B^{-1},$ from
$\left(\begin{array}[c]{cc}0&B\\\
-\frac{1}{B}&0\end{array}\right)=\left(\begin{array}[c]{cc}B&0\\\
0&\frac{1}{B}\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right),$ (176)
we have
$\displaystyle\exp\left[-\frac{B^{2}-1}{2\left(B^{2}+1\right)}a^{\dagger
2}\right]\exp\left[\left(a^{\dagger}a+\frac{1}{2}\right)\ln\left(\frac{-2Bi}{B^{2}+1}\right)\right]\exp\left[-\frac{B^{2}-1}{2\left(B^{2}+1\right)}a^{2}\right]$
$\displaystyle=\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
B\right]\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right].$
(177)
Using the following relations
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&\left(A-1\right)/C\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
C&1\end{array}\right)\left(\begin{array}[c]{cc}1&\left(D-1\right)/C\\\
0&1\end{array}\right),$ (178)
it then follows that
$F_{1}\left(A,B,C\right)=\exp\left(-\frac{i\left(A-1\right)}{2C}P^{2}\right)\exp\left(\frac{iC}{2}Q^{2}\right)\exp\left(-\frac{i\left(D-1\right)}{2C}P^{2}\right),\text{
}$ (179)
while from
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
\left(D-1\right)/B&1\end{array}\right)\left(\begin{array}[c]{cc}1&B\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
\left(A-1\right)/B&1\end{array}\right)$ (180)
we obtain
$F_{1}\left(A,B,C\right)=\exp\left(\frac{i\left(D-1\right)}{2B}Q^{2}\right)\exp\left(-\frac{iB}{2}P^{2}\right)\exp\left(\frac{i\left(A-1\right)}{2B}Q^{2}\right),\text{
}$ (181)
so we have
$\displaystyle\exp\left(\frac{i\left(D-1\right)}{2B}Q^{2}\right)\exp\left(-\frac{iB}{2}P^{2}\right)\exp\left(\frac{i\left(A-1\right)}{2B}Q^{2}\right)$
$\displaystyle=\exp\left(-\frac{i\left(A-1\right)}{2C}P^{2}\right)\exp\left(\frac{iC}{2}Q^{2}\right)\exp\left(-\frac{i\left(D-1\right)}{2C}P^{2}\right).$
(182)
In this section, based on a one-to-one correspondence between classical
Fresnel transform in phase space and quantum unitary transform in state-vector
space and the IWOP technique as well as the coherent state representation we
have found a way to directly map the classical optical transformations to the
optical operator method. We have combined the usual optical transformation
operators, such as the square phase operators, scaling operator, Fourier
transform operator and the propagation operator in free space, into a concise
and unified form. The various decompositions of Fresnel operator into the
exponential canonical operators are also obtained.
## 8 Quantum tomography and probability distribution for the Fresnel
quadrature phase
In quantum optics theory all possible linear combinations of quadratures $Q$
and $P$ of the oscillator field mode $a$ and $a^{\dagger}$ can be measured by
the homodyne measurement just by varying the phase of the local oscillator.
The average of the random outcomes of the measurement, at a given local
oscillator phase, is connected with the marginal distribution of Wigner
function (WF), thus the homodyne measurement of light field permits the
reconstruction of the WF of a quantum system by varying the phase shift
between two oscillators. In Ref. [58] Vogel and Risken pointed out that the
probability distribution for the rotated quadrature phase
$Q_{\theta}\equiv[a^{\dagger}\exp(i\theta)+a\exp(-i\theta)]/\sqrt{2},$
$\left[a,a^{\dagger}\right]=1,$which depends on only one $\theta$ angle, can
be expressed in terms of WF, and that the reverse is also true (named as
Vogel-Risken relation), i.e., one can obtain the Wigner distribution by
tomographic inversion of a set of measured probability distributions,
$P_{\theta}\left(q_{\theta}\right),$ of the quadrature amplitude. Once the
distribution $P_{\theta}\left(q_{\theta}\right)$ are obtained, one can use the
inverse Radon transformation familiar in tomographic imaging to obtain the
Wigner distribution and density matrix. The Radon transform of the WF is
closely related to the expectation values or densities formed with the
eigenstates to the rotated canonical observables. The field of problems of the
reconstruction of the density operator from such data is called quantum
tomography. (Optical tomographic imaging techniques derive two-dimensional
data from a three-dimensional object to obtain a slice image of the internal
structure and thus have the ability to peer inside the object noninvasively,
the slice image is equated with tomogram.) The theoretical development in
quantum tomography in the last decade has progressed in the direction of
determining more physical relevant parameters of the density from tomographic
data[58, 59, 60, 61, 62].
### 8.1 Relation between Fresnel transform and Radon transform of WF
In [63, 64] the Radon transform of WF which depends on two continuous
parameters is introduced, this has the advantage in conveniently associating
quantum tomography theory with squeezed coherent state theory. In this
subsection we want to derive relations between the Fresnel transform and the
Radon transform of WF in quantum optics in tomography theory.
By extending the rotated quadrature phase $Q_{\theta}$ to the Fresnel
quadrature phase
$Q_{F}\equiv\left(s^{\ast}a+ra^{\dagger}+sa^{\dagger}+r^{\ast}a\right)/\sqrt{2}=F_{1}QF_{1}^{\dagger},$
(183)
where $s$ and $r$ are related to ABCD through(85),
$s=\frac{1}{2}\left[A+D-i\left(B-C\right)\right],\;r=-\frac{1}{2}\left[A-D+i\left(B+C\right)\right],\;|s|^{2}-|r|^{2}=1,$
(184)
we shall prove that the $(D,B)$ related Radon transform of Wigner operator
$\Delta\left(q,p\right)$ is just the pure state density operator
$\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ (named as the tomographic
density operator) formed with the eigenstates belonging to the quadrature
$Q_{F}$, ( $\left|q\right\rangle_{s,r}=F_{1}\left|q\right\rangle,$ $Q$ is the
coordinate operator),
$F_{1}\left|q\right\rangle\left\langle
q\right|F_{1}^{\dagger}=\left|q\right\rangle_{s,rs,r}\left\langle
q\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(185) $D=\frac{1}{2}\left(s+s^{\ast}+r+r^{\ast}\right),\
B=\frac{1}{2i}\left(s^{\ast}-s+r^{\ast}-r\right),$ (186)
Since $F$ corresponds to classical Fresnel transform in optical diffraction
theory, so Eq. (185) indicates that the probability distribution for the
Fresnel quadrature phase is the Radon transform of WF [65].
Proof:
Firstly, from (97) we see
$F_{1}\left(s,r\right)aF_{1}^{{\dagger}}\left(s,r\right)=s^{\ast}a+ra^{\dagger},$
(187)
so from $Q=\frac{a+a^{\dagger}}{\sqrt{2}},P=i\frac{a^{\dagger}-a}{\sqrt{2}},$
indeed we have
$F_{1}QF_{1}^{\dagger}=F_{1}\frac{a+a^{\dagger}}{\sqrt{2}}F_{1}^{\dagger}=\left(s^{\ast}a+ra^{\dagger}+sa^{\dagger}+r^{\ast}a\right)/\sqrt{2}=Q_{F}.$
(188)
Secondly, we can derive the explicit form of $\left|q\right\rangle_{s,r}.$
Starting from $s^{\ast}+r^{\ast}=D+iB,$ $s^{\ast}-r^{\ast}=A-iC,$ we set up
the eigenvector equation
$Q_{F}\left|q\right\rangle_{s,r}=\left(DQ-
BP\right)\left|q\right\rangle_{s,r}=q\left|q\right\rangle_{s,r},\text{ }$
(189)
it follows
$\left|q\right\rangle_{s,r}=F_{1}\left(s,r\right)\left|q\right\rangle.$ (190)
In the coordinate and momentum representations we have
$\displaystyle\left\langle q^{\prime}\right|Q_{F}\left|q\right\rangle_{s,r}$
$\displaystyle=\left(Dq^{\prime}+iB\frac{d}{dq^{\prime}}\right)\left\langle
q^{\prime}\right|\left.q\right\rangle_{s,r}=q\left\langle
q^{\prime}\right|\left.q\right\rangle_{s,r}.$ (191) $\displaystyle\left\langle
p\right|Q_{F}\left|q\right\rangle_{s,r}$
$\displaystyle=\left(iD\frac{d}{dp}-Bp\right)\left\langle
p\right|\left.q\right\rangle_{s,r}=q\left\langle
p\right|\left.q\right\rangle_{s,r}.$ (192)
The normalizable solutions to (191) and (192) are
$\displaystyle\left\langle q^{\prime}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=c\left(q\right)\exp\left[\frac{iq^{\prime}\left(Dq^{\prime}-2q\right)}{2B}\right],$
(193) $\displaystyle\left\langle p\right|\left.q\right\rangle_{s,r}$
$\displaystyle=d\left(q\right)\exp\left[\frac{ip\left(-Bp-2q\right)}{2D}\right].$
(194)
Using the Fock representation of $\left|q\right\rangle$ and
$\left|p\right\rangle$ in Eqs.(14) and (23), we obtain
$\displaystyle\left|q\right\rangle_{s,r}$
$\displaystyle=\int_{-\infty}^{\infty}dq^{\prime}\left|q^{\prime}\right\rangle\left\langle
x^{\prime}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=\pi^{-1/4}c\left(q\right)\sqrt{\frac{2B\pi}{B-iD}}\exp\left[-\frac{q^{2}}{2B\left(B-iD\right)}+\frac{\sqrt{2}a^{\dagger}q}{D+iB}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right]\left|0\right\rangle,$ (195)
and
$\displaystyle\left|q\right\rangle_{s,r}$
$\displaystyle=\int_{-\infty}^{\infty}dp\left|p\right\rangle\left\langle
p\right|\left.q\right\rangle_{s,r}$
$\displaystyle=d\left(q\right)\pi^{-1/4}\sqrt{\frac{2\pi
D}{D+iB}}\exp\left[-\frac{q^{2}}{2D\left(D+iB\right)}+\allowbreak\frac{\sqrt{2}a^{\dagger}q}{D+iB}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right]\left|0\right\rangle.$ (196)
Comparing Eq.(195) with (196) we see
$\frac{c\left(q\right)}{d\left(q\right)}=\sqrt{\allowbreak\frac{D}{iB}}\exp\left[\frac{\allowbreak
iA}{2B}q^{2}-\frac{iCq^{2}}{2D}\right].$ (197)
On the other hand, according to the orthogonalization of
$\left|q\right\rangle_{s,r}$, ${}_{s,r}\left\langle
q^{\prime}\right.\left|q^{\prime\prime}\right\rangle_{s,r}=\delta\left(q^{\prime}-q^{\prime\prime}\right),$
we have
$\left|c\left(q\right)\right|^{2}=\frac{1}{2\pi B},\text{ \
}\left|d\left(q\right)\right|^{2}=\frac{1}{2\pi D}.$ (198)
Thus combining Eq.(197) and (198) we deduce
$c\left(q\right)=\frac{1}{\sqrt{2\pi iB}}\exp\left[\frac{\allowbreak
iA}{2B}q^{2}\right],\text{ }d\left(q\right)=\frac{1}{\sqrt{2\pi
D}}\exp\left[\frac{iCq^{2}}{2D}\right],$ (199)
and
$\left|q\right\rangle_{s,r}=\frac{\pi^{-1/4}}{\sqrt{D+iB}}\exp\left\\{-\frac{A-iC}{D+iB}\frac{q^{2}}{2}+\frac{\sqrt{2}q}{D+iB}a^{\dagger}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle,$ (200)
or
$\left|q\right\rangle_{s,r}\equiv\frac{\pi^{-1/4}}{\sqrt{s^{\ast}+\allowbreak
r^{\ast}}}\exp\left\\{-\frac{s^{\ast}-r^{\ast}}{s^{\ast}+r^{\ast}}\frac{q^{2}}{2}+\frac{\sqrt{2}q}{s^{\ast}+r^{\ast}}a^{\dagger}-\frac{s+r}{s^{\ast}+r^{\ast}}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (201)
It is easily seen that that $\left|q\right\rangle_{s,r}$ make up a complete
set (so $\left|q\right\rangle_{s,r}$ can be named as the tomography
representation),
$\int_{-\infty}^{\infty}dq\left|q\right\rangle_{s,r}{}_{s,r}\left\langle
q\right|=1.$ (202)
Then according to the Weyl quantization scheme [37]
$H\left(Q,P\right)=\int_{-\infty}^{\infty}dpdq\Delta\left(q,p\right)h\left(q,p\right),$
(203)
where $h\left(q,p\right)$ is the Weyl correspondence of $H\left(Q,P\right),$
$h\left(q,p\right)=2\pi
T_{r}\left[H\left(Q,P\right)\Delta\left(q,p\right)\right],$ (204)
$\Delta\left(q,p\right)$ is the Wigner operator [66, 67],
$\Delta\left(q,p\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}due^{ipu}\left|q+\frac{u}{2}\right\rangle\left\langle
q-\frac{u}{2}\right|,$ (205)
and using (204), (205) and (195) we know that the classical Weyl
correspondence (Weyl image) of the projection operator
$\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ is
$\displaystyle 2\pi
Tr\left[\Delta\left(q^{\prime},p^{\prime}\right)\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\right]$ $\displaystyle=\left.{}_{s,r}\left\langle
q\right|\right.\int_{-\infty}^{\infty}due^{ip^{\prime}u}\left|q^{\prime}+\frac{u}{2}\right\rangle\left\langle
q^{\prime}-\frac{u}{2}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=\frac{1}{2\pi
B}\int_{-\infty}^{\infty}du\exp\left[ip^{\prime}u+\frac{i}{B}u\left(q-Dq^{\prime}\right)\right]$
$\displaystyle=\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right],$
(206)
which means
$\left|q\right\rangle_{s,rs,r}\left\langle
q\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right).$
(207)
Combining Eqs. (187)-(190) together we complete the proof. Therefore, the
probability distribution for the Fresnel quadrature phase is the Radon
transform of WF
$|\left\langle
q\right|F_{1}^{\dagger}\left|\psi\right\rangle|^{2}=|_{s,r}\left\langle
q\right|\left.\psi\right\rangle|^{2}=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\left\langle\psi\right|\Delta\left(q^{\prime},p^{\prime}\right)\left|\psi\right\rangle,$
(208)
so we name $\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ the
tomographic density. Moreover, the tomogram of quantum state
$\left|\psi\right\rangle$ is just the squared modulus of the wave function
${}_{s,r}\left\langle q\right|\left.\psi\right\rangle,$ this new relation
between quantum tomography and optical Fresnel transform may provide
experimentalists to figure out new approach for generating tomography.
The introduction of $\left|q\right\rangle_{s,r}$ also bring convenience to
obtain the inverse of Radon transformation, using (202) we have
$e^{-igQ_{F}}=\int_{-\infty}^{\infty}dq\left|q\right\rangle_{s,r}{}_{s,r}\left\langle
q\right|e^{-igq}=\int_{-\infty}^{\infty}dqdp\Delta\left(q,p\right)e^{-ig\left(Dq-
Bp\right)}.$ (209)
Considering its right hand-side as a Fourier transformation, its reciprocal
transform is
$\displaystyle\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}dq^{\prime}\int_{-\infty}^{\infty}dg^{\prime}|g^{\prime}|\int_{0}^{\pi}d\varphi\left|q^{\prime}\right\rangle_{s,r}{}_{s,r}\left\langle
q^{\prime}\right|$
$\displaystyle\times\exp\left[-ig^{\prime}\left(\frac{q^{\prime}}{\sqrt{D^{2}+B^{2}}}-q\cos\varphi-p\sin\varphi\right)\right],$
(210)
where $g^{\prime}=g\sqrt{D^{2}+B^{2}},$
$\cos\varphi=\frac{D}{\sqrt{D^{2}+B^{2}}},$
$\sin\varphi=\frac{-B}{\sqrt{D^{2}+B^{2}}}.$So once the distribution
$|_{s,r}\left\langle q\right|\left.\psi\right\rangle|^{2}$ are obtained, one
can use the inverse Radon transformation familiar in tomographic imaging to
obtain the Wigner distribution.
By analogy, we can conclude that the $(A,C)$ related Radon transform of
$\Delta\left(q,p\right)$ is just the pure state density operator
$\left|p\right\rangle_{s,rs,r}\left\langle p\right|$ formed with the
eigenstates belonging to the conjugate quadrature of $Q_{F},$
$\displaystyle F_{1}\left|p\right\rangle\left\langle p\right|F_{1}^{\dagger}$
$\displaystyle=\left|p\right\rangle_{s,rs,r}\left\langle
p\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[p-\left(Aq^{\prime}-Cp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
$\displaystyle A$
$\displaystyle=\frac{1}{2}\left(s^{\ast}-r^{\ast}+s-r\right),\text{ \
}C=\frac{1}{2i}\left(s-r-s^{\ast}+r^{\ast}\right).$ (211)
Similarly, we find that for the momentum density,
$F_{1}\left|p\right\rangle\left\langle
p\right|F_{1}^{\dagger}=\left|p\right\rangle_{s,rs,r}\left\langle
p\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[p-\left(Ap^{\prime}-Cq^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(212)
where
$F_{1}\left|p\right\rangle=\left|p\right\rangle_{s,r}=\frac{\pi^{-1/4}}{\sqrt{A-iC}}\exp\left\\{-\frac{D+iB}{A-iC}\frac{p^{2}}{2}+\frac{\sqrt{2}ip}{A-iC}a^{\dagger}+\frac{A+iC}{A-iC}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (213)
As an application of the relation (185), recalling that the $F_{1}(r,s)$ makes
up a faithful representation of the symplectic group [r10], it then follows
from (185) that
$\displaystyle
F_{1}^{\prime}(r^{\prime},s^{\prime})F_{1}(r,s)\left|q\right\rangle\left\langle
q\right|F_{1}^{\dagger}(r,s)F_{1}^{\prime\dagger}(r^{\prime},s^{\prime})=\left|q\right\rangle_{s^{\prime\prime},r^{\prime\prime}\text{
}s^{\prime\prime},r^{\prime\prime}}\left\langle q\right|$
$\displaystyle=\int\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(\left(B^{\prime}C+DD^{\prime}\right)q^{\prime}-\left(AB^{\prime}+BD^{\prime}\right)p^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(214)
In this way a complicated Radon transform of tomography can be viewed as the
sequential operation of two Fresnel transforms. This confirms that the
continuous Radon transformation corresponds to the symplectic group
transformation [63, 64], this is an advantage of introducing the Fresnel
operator. The group property of Fresnel operators help us to analyze
complicated Radon transforms in terms of some sequential Fresnel
transformations. The new relation may provide experimentalists to figure out
new approach for realizing tomography.
### 8.2 Another new theorem to calculating the tomogram
In this subsection, we introduce a new theorem, i.e., the tomogram of a
density operator $\rho$ is equal to the marginal integration of the classical
Weyl correspondence function of $F^{\dagger}\rho F,$ where $F$ is the Fresnel
operator.
Multiplying both sides of Eq. (207) by a density matrix $\rho$ and then
performing the trace, noting the Wigner function
$W(p,q)=\mathtt{Tr}\left[\rho\Delta(p,q)\right],$ one can see
$\displaystyle
Tr\left[{\displaystyle\iint\nolimits_{-\infty}^{\infty}}dq\acute{}dp\acute{}\delta\left[q-\left(Dq\acute{}-Bp\acute{}\right)\right]\Delta\left(q\acute{},p\acute{}\right)\rho\right]$
$\displaystyle=Tr\left(\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\rho\right)=_{s,r}\left\langle
q\right|\rho\left|q\right\rangle_{s,r}=\left\langle q\right|F^{\dagger}\rho
F\left|q\right\rangle$
$\displaystyle={\displaystyle\iint\nolimits_{-\infty}^{\infty}}dq\acute{}dp\acute{}\delta\left[q-\left(Dq\acute{}-Bp\acute{}\right)\right]W(p,q).$
(215)
The right hand side of Eq. (215) is commonly defined as the tomogram of
quantum states in $(B,D)$ direction, so in our view the calculation of
tomogram in $(B,D)$ direction is ascribed to calculating
$\left\langle q\right|F^{\dagger}\rho F\left|q\right\rangle\equiv\Xi.$ (216)
This is a concise and neat formula. Similarly, the tomogram in $(A,C)$
direction is ascribed to $\left\langle p\right|F^{\dagger}\rho
F\left|p\right\rangle$.
According to the Weyl correspondence rule
$H\left(X,P\right)=\iint_{-\infty}^{\infty}dpdx\mathfrak{h}(p,x)\Delta(p,x),$
(217)
and the Weyl ordering form of $\Delta(p,q)$
$\Delta(p,q)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(218)
where the symbol$\genfrac{}{}{0.0pt}{}{:}{:}\ \genfrac{}{}{0.0pt}{}{:}{:}$
denotes Weyl ordering, the classical correspondence of a Weyl ordered operator
$\genfrac{}{}{0.0pt}{}{:}{:}\mathfrak{h}(Q,P)\genfrac{}{}{0.0pt}{}{:}{:}$ is
obtained just by replacing $Q\rightarrow q,P\rightarrow p$ in $h,$ i.e.,
$\genfrac{}{}{0.0pt}{}{:}{:}\mathfrak{h}(Q,P)\genfrac{}{}{0.0pt}{}{:}{:}=\iint_{-\infty}^{\infty}dpdq\mathfrak{h}(p,q)\Delta(p,q),$
(219)
Let the classical Weyl correspondence of $F^{\dagger}\rho F$ be $h(p,q)$
$F^{\dagger}\rho F=\iint_{-\infty}^{\infty}dpdqh(p,q)\Delta(p,q),$
then using (216) and (205) we have
$\displaystyle\Xi$ $\displaystyle=\left\langle q\right|F^{\dagger}\rho
F\left|q\right\rangle$ $\displaystyle=\left\langle
q\right|{\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\Delta\left(p,q\acute{}\right)\left|q\right\rangle$
$\displaystyle={\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\int_{-\infty}^{+\infty}\frac{dv}{2\pi}e^{ipv}\left\langle
x\right.\left.q\acute{}+\frac{v}{2}\right\rangle\left\langle
q\acute{}-\frac{v}{2}\right.\left.q\right\rangle$
$\displaystyle={\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\int_{-\infty}^{+\infty}\frac{dv}{2\pi}e^{ipv}\delta\left(q\acute{}-q+\frac{v}{2}\right)\delta\left(q\acute{}-q-\frac{v}{2}\right)$
$\displaystyle=\frac{1}{\pi}{\displaystyle\iint}dpdq\acute{}h(p,q\acute{})e^{i2p\left(q\acute{}-q\right)}\delta\left(2q\acute{}-2q\right)=\int_{-\infty}^{\infty}\frac{dp}{2\pi}h(p,q).$
(220)
Thus we reach a theorem:
The tomogram of a density operator $\rho$ is equal to the marginal integration
of the classical Weyl correspondence $h(p,q)$ of $F^{\dagger}\rho F,$ where
$F$ is the Fresnel operator, expressed by
$\mathtt{Tr}\left[\rho\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\right]=\int_{-\infty}^{\infty}\frac{dp}{2\pi}h(p,q),$ (221)
or
$\mathtt{Tr}\left[\rho\left|p\right\rangle_{s,rs,r}\left\langle
p\right|\right]=\int_{-\infty}^{\infty}\frac{dx}{2\pi}h(p,q).$ (222)
In this way the relationship between tomogram of a density operator $\rho$ and
the Fresnel transformed $\rho^{\prime}$s classical Weyl function is
established.
## 9 Two-mode GFO and Its Application
For two-dimensional optical Fresnel transforms (see (5)) in the $x-y$ plane
one may naturally think that the 2-mode GFO is just the direct product of two
independent 1-mode GFOs but with the same $(A,B,C,D)$ matrix. However, here we
present another 2-mode Fresnel operator which can not only lead to the usual
2-dimensional optical Fresnel transforms in some appropriate quantum
mechanical representations, but also provide us with some new classical
transformations (we name them entangled Fresnel transformations).
### 9.1 Two-mode GFO gained via coherent state representation
Similar in spirit to the single-mode case, we introduce the two-mode GFO
$F_{2}\left(r,s\right)$ through the following 2-mode coherent state
representation [50]
$F_{2}\left(r,s\right)=s\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\left\langle
z_{1},z_{2}\right|,$ (223)
which indicates that $F_{2}\left(r,s\right)$ is a mapping of classical
sympletic transform
$\left(z_{1},z_{2}\right)\rightarrow\left(sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right)$
in phase space. Concretely, the ket in (223) is
$\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\equiv\left|sz_{1}+rz_{2}^{\ast}\right\rangle_{1}\otimes\left|rz_{1}^{\ast}+sz_{2}\right\rangle_{2},\text{
}ss^{\ast}-rr^{\ast}=1,$ (224)
$s$ and $r$ are complex and satisfy the unimodularity condition. Using the
IWOP technique we perform the integral in (223) and obtain
$\displaystyle F_{2}\left(r,s\right)$
$\displaystyle=s\int\frac{1}{\pi^{2}}d^{2}z_{1}d^{2}z_{2}\colon\exp[-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}-rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\ast}a_{1}+z_{2}^{\ast}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}]\colon$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left[\left(\frac{1}{s^{\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\colon\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right)$
$\displaystyle=\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\exp[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln\left(s^{\ast}\right)^{-1}]\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right).$
(225)
Thus $F_{2}\left(r,s\right)$ induces the transform
$F_{2}\left(r,s\right)a_{1}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{1}-ra_{2}^{\dagger},F_{2}\left(r,s\right)a_{2}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{2}-ra_{1}^{\dagger}.$
(226)
and $F_{2}$ is actually a general 2-mode squeezing operator. Recall that (37)
implies the intrinsic relation between the EPR entangled state and the two-
mode squeezed state, which has physical implementation, i.e. in the output of
a parametric down-conversion the idler-mode and the signal-mode constitute a
two-mode squeezed state, meanwhile are entangled with each other in frequency
domain, we naturally select the entangled state representation to relate
$F_{2}\left(r,s\right)$ to two-dimensional GFT. Letting
$\left|g\right\rangle=F_{2}\left(r,s\right)\left|f\right\rangle,$ and then
projecting $\left|\psi\right\rangle$ onto the entangled state
$\left\langle\eta^{\prime}\right|$ defined by (29) and using the completeness
relation (31) of $\left|\eta\right\rangle$, we obtain
$\displaystyle g\left(\eta^{\prime}\right)$
$\displaystyle\equiv\left\langle\eta^{\prime}\right|\left.g\right\rangle=\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|f\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right.\left|f\right\rangle\equiv\int
d^{2}\eta\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)f\left(\eta\right).$
(227)
Then using the overcompleteness relation of the coherent state, and
$\displaystyle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left\\{-\frac{1}{2}(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2})\right.$
$\displaystyle\left.+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+\frac{1}{s^{\ast}}\left(z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}\right)\right\\},$
we can calculate the integral kernel
$\displaystyle\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)$
$\displaystyle=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle$
$\displaystyle=\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\left\langle\eta^{\prime}\right|\left.z_{1}^{\prime},z_{2}^{\prime}\right\rangle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right.\left|\eta\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\exp\left[-\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2}\right)-\frac{1}{2}\left(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2}\right)\right]$
$\displaystyle\times\exp\left[-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+z_{1}^{\ast}z_{2}^{\ast}+\eta
z_{1}^{\ast}+\frac{1}{s^{\ast}}\left(z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}\right)+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}+z_{1}^{\prime}z_{2}^{\prime}+\eta^{\prime\ast}z_{1}^{\prime}-\eta^{\prime}z_{2}^{\prime}-\eta^{\ast}z_{2}^{\ast}\right]$
$\displaystyle=\frac{1}{\left(-r-s+r^{\ast}+s^{\ast}\right)\pi}\exp\left[\frac{\left(-s+r^{\ast}\right)\left|\eta\right|^{2}-\left(r+s\right)\left|\eta^{\prime}\right|^{2}+\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}}{-r-s+r^{\ast}+s^{\ast}}-\frac{1}{2}\left(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2}\right)\right].$
(228)
Using the relation between $s,r$ and $\left(A,B,C,D\right)$ in Eq.(85) Eq.
(228) becomes
$\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{2iB\pi}\exp\left[\frac{i}{2B}\left(A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right)\right]\equiv\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right),$
(229)
where the superscript $M$ only means the parameters of $K_{2}^{M}$ are
$\left[A,B;C,D\right]$, and the subscript $2$ means the two-dimensional
kernel. Eq. (229) has the similar form as (12) except for its complex form.
Taking $\eta_{1}=x_{1},$ $\eta_{2}=x_{2}$ and
$\eta_{1}^{\prime}=x_{1}^{\prime}$, $\eta_{2}^{\prime}=x_{2}^{\prime}$, we
have
$\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right)=\mathcal{K}_{2}^{M}\left(x_{1}^{\prime},x_{2}^{\prime};x_{1},x_{2}\right)=\mathcal{K}_{1}^{M}\left(x_{1},x_{1}^{\prime}\right)\otimes\mathcal{K}_{1}^{M}\left(x_{2},x_{2}^{\prime}\right).$
(230)
This shows that $F_{2}\left(r,s\right)$ is really the counterpart of the
2-dimensional GFT.
If taking the matrix element of $F_{2}\left(r,s\right)$ in the
$\left|\xi\right\rangle$ representation which is conjugate to
$\left|\eta\right\rangle$, we obtain the 2-dimensional GFT in its ‘frequency
domain’, i.e.,
$\displaystyle\left\langle\xi^{\prime}\right|F_{2}\left(r,s\right)\left|\xi\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta^{\prime}d^{2}\eta}{\pi^{2}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\sigma\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
$\displaystyle=\frac{1}{8iB\pi}\int\frac{d^{2}\eta^{\prime}d^{2}\eta}{\pi^{2}}\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)$
$\displaystyle=\frac{1}{2i\left(-C\right)\pi}\exp\left[\frac{i}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\equiv\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right),$
(231)
where the superscript $N$ means that this transform kernel corresponds to the
parameter matrix $N=\left[D,-C,-B,A\right]$.
The two-mode GFO also abides by the group multiplication rule. Using the IWOP
technique and (223) we obtain
$\displaystyle F_{2}\left(r,s\right)F_{2}\left(r^{\prime},s^{\prime}\right)$
$\displaystyle=ss^{\prime}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{4}}\colon\exp\\{-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}$
$\displaystyle-
rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}-\frac{1}{2}[|z_{1}^{\prime}|^{2}+|z_{2}^{\prime}|^{2}+|s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}|^{2}+|r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}|^{2}]$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\prime\ast}a_{1}+z_{2}^{\prime\ast}a_{2}$
$\displaystyle+z_{1}^{\ast}\left(s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}\right)+z_{2}^{\ast}\left(r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}\right)-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\\}\colon$
$\displaystyle=\frac{1}{s^{\prime\prime\ast}}\exp\left(\frac{r^{\prime\prime}}{2s^{\prime\prime\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left\\{\left(\frac{1}{s^{\prime\prime\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right\\}\colon\exp\left(-\frac{r^{\prime\prime\ast}}{2s^{\prime\prime\ast}}a_{1}a_{2}\right)$
$\displaystyle=F_{2}\left(r^{\prime\prime},s^{\prime\prime}\right),$ (232)
where $\left(r^{\prime\prime},s^{\prime\prime}\right)$ are given by Eq.(106)
or (107). Therefore, (232) is a loyal representation of the multiplication
rule for ray transfer matrices in the sense of Matrix Optics.
### 9.2 Quantum Optical ABCD Law for two-mode GFO
Next we extended quantum optical ABCD Law to two-mode case. Operating with
$F_{2}(r,s)$ on two-mode number state $\left|m,n\right\rangle$ and using the
overlap between coherent state and number state, i.e.
$\left\langle z_{1},z_{2}\right.\left|m,n\right\rangle=\frac{z_{1}^{\ast
m}z_{2}^{\ast
n}}{\sqrt{m!n!}}\exp\left[-\frac{1}{2}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\right],$
(233)
and the integral formula [57]
$H_{m,n}\left(\xi,\eta\right)=(-1)^{n}e^{\xi\eta}\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}e^{-\left|z\right|^{2}+\xi z-\eta z^{\ast}},$ (234)
we can calculate
$\displaystyle F_{2}(r,s)\left|m,n\right\rangle$
$\displaystyle=s\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\left\langle
z_{1},z_{2}\right.\left|m,n\right\rangle$
$\displaystyle=\frac{s}{\sqrt{m!n!}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}z_{1}^{\ast
m}z_{2}^{\ast
n}\exp\left[-\left|s\right|^{2}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\right]$
$\displaystyle\times\exp\left[-sr^{\ast}z_{1}z_{2}-rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}\right]\left|00\right\rangle$
$\displaystyle=\frac{s}{\left|s\right|^{2m+2}\sqrt{m!n!}}\int\frac{d^{2}z_{2}}{\pi}z_{2}^{\ast
n}\left(sa_{1}^{\dagger}-sr^{\ast}z_{2}\right)^{m}\exp\left(-\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}a_{2}^{\dagger}z_{2}+\frac{ra_{1}^{\dagger}a_{2}^{\dagger}}{s^{\ast}}\right)\left|00\right\rangle$
$\displaystyle=\frac{r^{\ast n}}{s^{\ast
n+1}\sqrt{m!n!}}H_{m,n}\left[\frac{a_{1}^{\dagger}}{s^{\ast}},\frac{a_{2}^{\dagger}}{r^{\ast}}\right]\exp\left(\frac{ra_{1}^{\dagger}a_{2}^{\dagger}}{s^{\ast}}\right)\left|00\right\rangle,$
(235)
where $H_{m,n}\left(\epsilon,\varepsilon\right)$ is the two variables Hermite
polynomial [63, 64], shown in (78) and (79).
Using Eqs.(85) and (118), we recast Eq.(235) into
$\displaystyle F_{2}(r,s)\left|m,n\right\rangle$
$\displaystyle=\frac{-2/\left(C+iD\right)}{\left(q_{1}+i\right)\sqrt{m!n!}}\left(-\frac{q_{1}^{\ast}+i}{q_{1}+i}\frac{C-iD}{C+iD}\right)^{n}$
$\displaystyle\times
H_{m,n}\left[-\frac{2a_{1}^{\dagger}/\left(C+iD\right)}{q_{1}+i},\frac{2a_{2}^{\dagger}/\left(C-iD\right)}{q_{1}^{\ast}+i}\right]\exp\left(-\frac{q_{1}-i}{q_{1}+i}a_{1}^{\dagger}a_{2}^{\dagger}\right)\left|00\right\rangle.$
(236)
Noticing the multiplication rule of $F_{2}\left(r,s\right)$ in Eq.(232), which
is equivalent to
$F_{2}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{2}\left(A,B,C\right)=F_{2}\left(A^{\prime\prime},B^{\prime\prime},C^{\prime\prime}\right),$
(237)
where $\left(A^{\prime},B^{\prime},C^{\prime}\right),\left(A,B,C\right)$ and
$\left(A^{\prime\prime},B^{\prime\prime},C^{\prime\prime}\right)$ are related
to each other by Eq.(114).
Next we directly use the GFO to derive ABCD rule in quantum optics for
Gaussian beam in two-mode case. According to Eq.(235) and Eq. (237) we obtain
$\displaystyle
F_{2}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{2}\left(A,B,C\right)\left|m,n\right\rangle$
$\displaystyle=\frac{r^{\prime\prime\ast n}}{s^{\prime\prime\ast
n+1}\sqrt{m!n!}}H_{m,n}\left[\frac{a_{1}^{\dagger}}{s^{\prime\prime\ast}},\frac{a_{2}^{\dagger}}{r^{\prime\prime\ast}}\right]\exp\left[\frac{r^{\prime\prime}a_{1}^{\dagger}a_{2}^{\dagger}}{s^{\prime\prime\ast}}\right]\left|00\right\rangle,$
(238)
Similar to the way of deriving Eq. (236), we can simplify Eq. (238) as
$\displaystyle
F_{2}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{2}\left(A,B,C\right)\left|00\right\rangle$
$\displaystyle=\frac{-2/\left(C^{\prime\prime}+iD^{\prime\prime}\right)}{\left(q_{2}+i\right)\sqrt{m!n!}}\left(-\frac{q_{2}^{\ast}+i}{q_{2}+i}\frac{C^{\prime\prime}-iD^{\prime\prime}}{C^{\prime\prime}+iD^{\prime\prime}}\right)^{n}$
$\displaystyle\times
H_{m,n}\left[-\frac{2a_{1}^{\dagger}/\left(C^{\prime\prime}+iD^{\prime\prime}\right)}{q_{2}+i},\frac{2a_{2}^{\dagger}/\left(C^{\prime\prime}-iD^{\prime\prime}\right)}{q_{2}^{\ast}+i}\right]\exp\left[-\frac{q_{2}-i}{q_{2}+i}a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle,$
(239)
where the relation between $q_{2}$ and $q_{1}$ are determined by Eq.(120)
which resembles Eq.(112), this is just the new ABCD law for two-mode case in
quantum optics.
### 9.3 Optical operators derived by decomposing GFO
#### 9.3.1 GFO as quadratic combinations of canonical operators
In order to obtain the quadratic combinations of canonical operators, let
first derive an operator identity. Note
$Q_{i}=(a_{i}+a_{i}^{\dagger})/\sqrt{2},$
$P_{i}=(a_{i}-a_{i}^{\dagger})/(\sqrt{2}\mathtt{i}),$ and Eq.(30),(31) we can
prove the operator identity
$\displaystyle
e^{\frac{\lambda}{2}\left[\left(Q_{1}-Q_{2}\right)^{2}+\left(P_{1}+P_{2}\right)^{2}\right]}$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}e^{\frac{\lambda}{2}\left[\left(Q_{1}-Q_{2}\right)^{2}+\left(P_{1}+P_{2}\right)^{2}\right]}\left|\eta\right\rangle\left\langle\eta\right|$
$\displaystyle=\frac{1}{1-\lambda}\colon\exp\left[\frac{2\lambda}{1-\lambda}K_{+}\right]\colon,$
(240)
where we have set
$K_{+}\equiv\frac{1}{4}[\left(Q_{1}-Q_{2}\right)^{2}+\left(P_{1}+P_{2}\right)^{2}].$
(241)
When $B=0$, $A=1,$ $C\rightarrow C/A,$ $D=1,$ and using Eq.(225) we see that
$\displaystyle F_{2}\left(1,0,C/A\right)$
$\displaystyle=\frac{2}{2-iC/A}\colon\exp\left\\{\frac{iC/A}{2-iC/A}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}\right)\right\\}\colon$
$\displaystyle=\exp\left\\{\frac{iC}{A}K_{+}\right\\},$ (242)
which is corresponding to the square phase operator in single-mode case. In a
similar way, using (39) and (41) we can derive another operator identity
$\displaystyle
e^{\frac{\lambda}{2}\left[\left(Q_{1}+Q_{2}\right)^{2}+\left(P_{1}-P_{2}\right)^{2}\right]}$
$\displaystyle=\int\frac{d^{2}\xi}{\pi}e^{\frac{\lambda}{2}\left[\left(Q_{1}+Q_{2}\right)^{2}+\left(P_{1}-P_{2}\right)^{2}\right]}\left|\xi\right\rangle\left\langle\xi\right|$
$\displaystyle=\frac{1}{1-\lambda}\colon\exp\left[\frac{2\lambda}{1-\lambda}K_{-}\right]\colon,$
(243)
where
$K_{-}=\frac{1}{4}[\left(Q_{1}+Q_{2}\right)^{2}+\left(P_{1}-P_{2}\right)^{2}].$
(244)
It then follows from Eqs.(225) and (243)
$\displaystyle F_{2}\left(1,B/A,0\right)$
$\displaystyle=\frac{2}{2+iB/A}\colon\exp\left\\{-\frac{iB/A}{2+iB/A}\left(a_{1}^{\dagger}a_{2}^{\dagger}+a_{2}^{\dagger}a_{2}+a_{1}^{\dagger}a_{1}+a_{1}a_{2}\right)\right\\}\colon$
$\displaystyle=\frac{2}{2+iB/A}\colon\exp\left\\{-\frac{2iB/A}{2+iB/A}K_{-}\right\\}\colon$
$\displaystyle=\exp\left\\{-\frac{iB}{A}K_{-}\right\\},$ (245)
which is corresponding to Fresnel propagator in free space (single-mode case).
In particular, when $B=C=0,$ and $D=A^{-1},$ Eq. (225) becomes
$F_{2}\left(A,0,0\right)=\operatorname{sech}\lambda\colon\exp\left[-a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda+\left(\operatorname{sech}\lambda-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)+a_{1}a_{2}\tanh\lambda\right]\colon,$
(246)
where$\frac{A-A^{-1}}{A+A^{-1}}=\tanh\lambda,$ $A=e^{\lambda}.$ Eq. (246) is
just the two-mode squeezing operator,
$\displaystyle F_{2}\left(A,0,0\right)$
$\displaystyle=\exp\left[i\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\ln
A\right]\equiv\exp\left[-2K_{0}\ln A\right],$ (247) $\displaystyle K_{0}$
$\displaystyle\equiv-\frac{\mathtt{i}}{2}\left(Q_{1}P_{2}+Q_{2}P_{1}\right),$
(248)
which actually squeezes the entangled state $\left|\xi\right\rangle$ (its
conjugate state is $\left|\eta\right\rangle$),
$F_{2}\left(A,0,0\right)\left|\xi\right\rangle=\int\frac{d^{2}\xi^{\prime}}{\pi
A}\left|\xi^{\prime}/A\right\rangle\left\langle\xi^{\prime}\right|\left.\xi\right\rangle=\frac{1}{A}\left|\xi/A\right\rangle.$
(249)
Using the decomposition (132) of the matrix and combining equations (242),
(245) and (247) together, we see that
$\displaystyle F_{2}\left(A,B,C\right)$
$\displaystyle=F_{2}\left(1,0,C/A\right)F_{2}\left(A,0,0\right)F_{2}\left(1,B/A,0\right)$
$\displaystyle=\exp\left\\{\frac{iC}{A}K_{+}\right\\}\exp\left\\{-2K_{0}\ln
A\right\\}\exp\left\\{-\frac{iB}{A}K_{-}\right\\}.$ (250)
This is the two-mode quadratic canonical operator representation of
$F_{2}\left(A,B,C\right)$.
To prove Eq.(250), using (249) and (42) we see
$\displaystyle\left\langle\eta\right|F_{2}\left(A,B,C\right)\left|\xi\right\rangle$
$\displaystyle=\exp\left(\frac{iC}{2A}\left|\eta\right|^{2}-\frac{iB}{2A}\left|\xi\right|^{2}\right)\left\langle\eta\right|\int\frac{d^{2}\xi^{\prime}}{A\pi}\left|\xi^{\prime}/A\right\rangle\left\langle\xi^{\prime}\right|\left.\xi\right\rangle$
$\displaystyle=\frac{1}{A}\exp\left(\frac{iC}{2A}\left|\eta\right|^{2}-\frac{iB}{2A}\left|\xi\right|^{2}\right)\left\langle\eta\right|\left.\xi/A\right\rangle$
$\displaystyle=\frac{1}{2A}\exp\left(\frac{iC}{2A}\left|\eta\right|^{2}-\frac{iB}{2A}\left|\xi\right|^{2}\right)\exp\left[\frac{1}{2A}\left(\eta^{\ast}\xi-\eta\xi^{\ast}\right)\right].$
(251)
It then follows
$\displaystyle\left\langle\eta^{\prime}\right|F_{2}\left(A,B,C\right)\left|\eta\right\rangle$
$\displaystyle=\int_{\infty}^{\infty}\frac{d^{2}\xi}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left|\xi\right\rangle\left\langle\xi\right|\left.\eta\right\rangle$
$\displaystyle=\frac{1}{2iB}\exp\left[\frac{i}{2B}\left(A\left|\eta\right|^{2}-i\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right)\right]$
$\displaystyle\equiv\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right),$ (252)
which is just the transform kernel of a 2-dimensional GFT and the definition
given in (250) is true.
Note that the quadratic combinations in Eqs.(241), (244) and (248) of the four
canonical operators $\left(Q_{1},Q_{2};P_{1},P_{2}\right)$ obey the
commutative relations $\left[K_{+},K_{-}\right]=2K_{0},$
$\left[K_{0},K_{\pm}\right]=\pm K_{\pm},$ so $F_{2}\left(A,B,C\right)$
involves a SU(2) Lie algebra structure (this structure is also compiled by
$Q^{2}/2$, $P^{2}/2$ and $-i\left(QP+PQ\right)/2$ that have been used in
constructing $F_{1}\left(A,B,C\right)$).
#### 9.3.2 Alternate decompositions of GFO and new optical operator
identities
When $A=D=0,B=1,C=-1,$ from Eq.(225) we see
$\displaystyle F_{2}\left(0,1,-1\right)$
$\displaystyle=\exp\left[-\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln
i\right]$
$\displaystyle=\exp\left[-\mathtt{i}\frac{\pi}{2}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\right]\equiv\mathcal{F},$
(253)
which can also be named the Fourier operator, since it induces the quantum
mechanically transforms [57]
$\mathcal{F}^{\dagger}Q_{i}\mathcal{F}=P_{i},\text{
}\mathcal{F}^{\dagger}P_{i}\mathcal{F}^{\dagger}=-Q_{i}.$ (254)
it then follows that
$\mathcal{F}^{\dagger}K_{+}\mathcal{F}=K_{-}.$ (255)
On the other hand, in order to obtain the decomposition of
$F_{2}\left(A,B,C\right)$ for $A=0,$ similar to deriving Eq.(169), we have
$F_{2}\left(A,B,C\right)=\exp\left[-\frac{iB}{D}K_{-}\right]\exp\left[2K_{0}\ln
D\right]\exp\left[\frac{iC}{D}K_{+}\right],\text{ for }D\neq 0.$ (256)
While for $B\neq 0$ or $C\neq 0$, using Eqs.(170) and (171) we have another
decomposition of $F_{2}\left(A,B,C\right)$, i.e.,
$F_{2}\left(A,B,C\right)=\exp\left[\frac{iD}{B}K_{+}\right]\exp\left[-2K_{0}\ln
B\right]\mathcal{F}\exp\left[\frac{iA}{B}K_{+}\right],\text{ }B\neq 0,$ (257)
and
$F_{2}\left(A,B,C\right)=\exp\left[-\frac{iA}{C}K_{-}\right]\exp\left[-2K_{0}\ln\frac{-1}{C}\right]\mathcal{F}\exp\left[-\frac{iD}{C}K_{-}\right],C\neq
0.$ (258)
In addition, noticing Eqs.(180) and (178), we can rewrite Eqs.(257) and (258)
as follows
$F_{2}\left(A,B,C\right)=\exp\left[\frac{i}{B}\left(D-1\right)K_{+}\right]\exp\left[-iBK_{-}\right]\exp\left[\frac{i}{B}\left(A-1\right)K_{+}\right],B\neq
0,$ (259)
and
$F_{2}\left(A,B,C\right)=\exp\left[\frac{-i}{C}\left(A-1\right)K_{-}\right]\exp\left[iCK_{+}\right]\exp\left[\frac{-i}{C}\left(D-1\right)K_{-}\right],C\neq
0,$ (260)
respectively.
Next, according to some optical systems used frequently in physical optics, we
derive some new entangled optical operator identities. For a special optical
system with the parameter $A=0,$ $C=-B^{-1},$ (174) which corresponds to the
Fourier transform system, we have
$\exp\left[-\frac{iB}{D}K_{-}\right]\exp\left[2K_{0}\ln
D\right]\exp\left[-\frac{i}{BD}K_{+}\right]=\exp\left[\frac{iD}{B}K_{+}\right]\exp\left[-2K_{0}\ln
B\right]\mathcal{F}.$ (261)
In particular, when $A=D=0,$ $C=-B^{-1},$ Eq.(176) corresponding to the ideal
spectrum analyzer, we have
$\exp\left[-iBK_{-}\right]\exp\left[-\frac{i}{B}K_{+}\right]\exp\left[-iBK_{-}\right]=\exp\left[-2K_{0}\ln
B\right]\mathcal{F}.$ (262)
When $B=0,$ $D=A^{-1},$
$\left(\begin{array}[c]{cc}A&0\\\
C&A^{-1}\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
C/A&1\end{array}\right)\left(\begin{array}[c]{cc}A&0\\\
0&A^{-1}\end{array}\right),$
which corresponds to the form of image system, another operator identity is
given by
$\exp\left[-2K_{0}\ln
A\right]\exp\left[iACK_{+}\right]=\exp\left[\frac{iC}{A}K_{+}\right]\exp\left[-2K_{0}\ln
A\right].$ (263)
When $C=0,$ $A=D^{-1},$
$\left(\begin{array}[c]{cc}D^{-1}&B\\\
0&D\end{array}\right)=\left(\begin{array}[c]{cc}1&B/D\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}D^{-1}&0\\\
0&D\end{array}\right),$
which corresponds to the far foci system,
$\exp\left[\frac{iD}{B}K_{+}\right]\exp\left[-2K_{0}\ln
B\right]\mathcal{F}\exp\left[\frac{i}{BD}K_{+}\right]=\exp\left[-\frac{iB}{D}K_{-}\right]\exp\left[2K_{0}\ln
D\right].$ (264)
When $D=0,$ $C=-B^{-1},$ corresponding to the Fresnel transform system,
$\left(\begin{array}[c]{cc}A&B\\\
-B^{-1}&0\end{array}\right)=\left(\begin{array}[c]{cc}B&0\\\
0&B^{-1}\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
A/B&1\end{array}\right),$
we have
$\exp\left[-\frac{i}{AB}K_{+}\right]\exp\left[-2K_{0}\ln
A\right]\exp\left[-\frac{iB}{A}K_{-}\right]=\exp\left[-2K_{0}\ln
B\right]\mathcal{F}\exp\left[\frac{iA}{B}K_{+}\right].$ (265)
The GFO can unify those optical operators in two-mode case. Various
decompositions of the GFO into the exponential canonical operators,
corresponding to the decomposition of ray transfer matrix
$\left[A,B,C,D\right],$ are also derived. In our derivation, the entangled
state representation is of useness in our research.
### 9.4 Quantum tomography and probability distribution for the Fresnel
quadrature phase—two-mode entangled case
In section 8 we have found that under the Fresnel transformation the pure
position density $\left|q\right\rangle\left\langle q\right|$ becomes the
tomographic density $\left|q\right\rangle_{s,rs,r}\left\langle q\right|$,
which is just the Radon transform of the Wigner operator
$\Delta\left(q,p\right).$ In this section we want to generalize the above
conclusion to two-mode entangled case. Here we shall prove
$F_{2}\left|\eta\right\rangle\left\langle\eta\right|F_{2}^{\dagger}=\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\Delta\left(\sigma,\gamma\right),$
(266)
i.e., we show that $\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$
is just the Radon transform of the entangled Wigner operator
$\Delta\left(\sigma,\gamma\right).$
Similar in spirit to the single-mode case, operating $F_{2}\left(r,s\right)$
on entangled state representation $\left|\eta\right\rangle$ we see
$\displaystyle F_{2}\left(r,s\right)\left|\eta\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\exp\left[\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}+\left(\frac{1}{s^{\ast}}-1\right)\left(a_{1}^{\dagger}z_{1}+a_{2}^{\dagger}z_{2}\right)-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}\right]\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right|\left.\eta\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\exp\left[-\left|z_{1}\right|^{2}+\frac{1}{s^{\ast}}\left(a_{1}^{\dagger}-r^{\ast}z_{2}\right)z_{1}+\left(\eta+z_{2}^{\ast}\right)z_{1}^{\ast}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}\left|\eta\right|^{2}-\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}z_{2}a_{2}^{\dagger}-\eta^{\ast}z_{2}^{\ast}+\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{2}}{\pi}\exp\left[-\frac{s^{\ast}+r^{\ast}}{s^{\ast}}\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}\left(a_{2}^{\dagger}-\eta
r^{\ast}\right)z_{2}+\frac{1}{s^{\ast}}\left(a_{1}^{\dagger}-s^{\ast}\eta^{\ast}\right)z_{2}^{\ast}\right]$
$\displaystyle\times\exp\left[+\frac{\eta}{s^{\ast}}a_{1}^{\dagger}+\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}-\frac{1}{2}\left|\eta\right|^{2}\right]\left|00\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}+r^{\ast}}\exp\left\\{-\allowbreak\frac{s^{\ast}-r^{\ast}}{2\left(s^{\ast}+r^{\ast}\right)}\left|\eta\right|^{2}+\allowbreak\frac{\eta
a_{1}^{\dagger}}{s^{\ast}+r^{\ast}}\allowbreak-\allowbreak\frac{\eta^{\ast}a_{2}^{\dagger}}{s^{\ast}+r^{\ast}}+\frac{s+r}{s^{\ast}+r^{\ast}}\allowbreak
a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle\equiv\left|\eta\right\rangle_{s,r},$
(267)
or
$\left|\eta\right\rangle_{s,r}=\frac{1}{\allowbreak
D+iB}\exp\left\\{-\frac{\allowbreak A-iC}{2\left(\allowbreak
D+iB\right)}\left|\eta\right|^{2}+\frac{\eta a_{1}^{\dagger}}{\allowbreak
D+iB}-\frac{\eta^{\ast}a_{2}^{\dagger}}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak
D+iB}a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$ (268)
where we have used the integration formula
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(269)
Noticing the completeness relation and the orthogonality of
$\left|\eta\right\rangle$ we immediately derive
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=1,\text{
}_{s,r}\left\langle\eta\right|\left.\eta^{\prime}\right\rangle_{s,r}=\pi\delta\left(\eta-\eta^{\prime}\right)\delta\left(\eta^{\ast}-\eta^{\prime\ast}\right),$
(270)
a generalized entangled state representation $\left|\eta\right\rangle_{s,r}$
with the completeness relation (270). From (268) we can see that
$\displaystyle a_{1}\left|\eta\right\rangle_{s,r}$
$\displaystyle=\left(\frac{\eta}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak D+iB}a_{2}^{\dagger}\right)\left|\eta\right\rangle_{s,r},$
(271) $\displaystyle a_{2}\left|\eta\right\rangle_{s,r}$
$\displaystyle=\left(-\frac{\eta^{\ast}}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak D+iB}a_{1}^{\dagger}\right)\left|\eta\right\rangle_{s,r},$
(272)
so we have the eigen-equations for $\left|\eta\right\rangle_{s,r}$ as follows
$\displaystyle\left[D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=\sqrt{2}\eta_{1}\left|\eta\right\rangle_{s,r},\text{ }$ (273)
$\displaystyle\left[B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=\sqrt{2}\eta_{2}\left|\eta\right\rangle_{s,r},$ (274)
We can also check Eqs.(271)-(274) by another way.
#### 9.4.1 $\left|\eta\right\rangle_{s,r\text{ }s,r}\left\langle\eta\right|$
as Radon transform of the entangled Wigner operator
For two-mode correlated system, we have introduced the Wigner operator in
(45). According to the Wely correspondence rule [37]
$H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right)=\int d^{2}\gamma
d^{2}\sigma h\left(\sigma,\gamma\right)\Delta\left(\sigma,\gamma\right),$
(275)
where $h\left(\sigma,\gamma\right)$ is the Weyl correspondence of
$H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right),$ and
$h\left(\sigma,\gamma\right)=4\pi^{2}\mathtt{Tr}\left[H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right)\Delta\left(\sigma,\gamma\right)\right],$
(276)
the classical Weyl correspondence of the projection operator
$\left|\eta\right\rangle_{r,sr,s}\left\langle\eta\right|$ can be calculated as
$\displaystyle
4\pi^{2}\mathtt{Tr}\left[\left|\eta\right\rangle_{r,sr,s}\left\langle\eta\right|\Delta\left(\sigma,\gamma\right)\right]$
$\displaystyle=4\pi^{2}\int\frac{d^{2}\eta^{\prime}}{\pi^{3}}\left.{}_{r,s}\left\langle\eta\right|\left.\sigma-\eta^{\prime}\right\rangle\left\langle\sigma+\eta^{\prime}\right|\left.\eta\right\rangle_{r,s}\right.\exp(\eta^{\prime}\gamma^{\ast}-\eta^{\prime\ast}\gamma)$
$\displaystyle=4\pi^{2}\int\frac{d^{2}\eta^{\prime}}{\pi^{3}}\left\langle\eta\right|F_{2}^{\dagger}\left|\sigma-\eta^{\prime}\right\rangle\left\langle\sigma+\eta^{\prime}\right|F_{2}\left|\eta\right\rangle\exp(\eta^{\prime}\gamma^{\ast}-\eta^{\prime\ast}\gamma).$
(277)
Then using Eq.(229) we have
$4\pi^{2}\mathtt{Tr}\left[\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|\Delta\left(\sigma,\gamma\right)\right]=\pi\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right),$
(278)
which means the following Weyl correspondence
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=\pi\int d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\Delta\left(\sigma,\gamma\right),$
(279)
so the projector operator
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$ is just the Radon
transformation of $\Delta\left(\sigma,\gamma\right)$, $D$ and $B$ are the
Radon transformation parameter. Combining Eqs. (267)-(279) together we
complete the proof (266). Therefore, the quantum tomography in two-mode
entangled case is expressed as
$|_{s,r}\left\langle\eta\right|\left.\psi\right\rangle|^{2}=|\left\langle\eta\right|F^{\dagger}\left|\psi\right\rangle|^{2}=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle.$
(280)
where
$\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle$
is the Wigner function. So the probability distribution for the Fresnel
quadrature phase is the tomography (Radon transform of the two-mode Wigner
function). This new relation between quantum tomography and optical Fresnel
transform may provide experimentalists to figure out new approach for
generating tomography.
Next we turn to the “frequency” domain, that is to say, we shall prove that
the $(A,C)$ related Radon transform of entangled Wigner operator
$\Delta\left(\sigma,\gamma\right)$ is just the pure state density operator
$\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|,$ i.e.,
$F_{2}\left|\xi\right\rangle\left\langle\xi\right|F_{2}^{\dagger}=\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\Delta\left(\sigma,\gamma\right)d^{2}\sigma
d^{2}\gamma,$ (281)
where $\left|\xi\right\rangle$ is the conjugated entangled state to
$\left|\eta\right\rangle$.
By analogy with the above procedure, we obtain the 2-dimensional Fresnel
transformation in its ‘frequency domain’, i.e.,
$\displaystyle\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)$
$\displaystyle\equiv\frac{1}{\pi}\left\langle\xi^{\prime}\right|F_{2}\left(r,s\right)\left|\xi\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta
d^{2}\sigma}{\pi^{2}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
$\displaystyle=\frac{1}{8iB\pi}\int\frac{d^{2}\sigma
d^{2}\eta}{\pi^{2}}\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)\mathcal{K}_{2}^{\left(\mathtt{r},s\right)}\left(\sigma,\eta\right)$
$\displaystyle=\frac{1}{2i\left(-C\right)\pi}\exp\left[\frac{i}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right],$
(282)
where the superscript $N$ means that this transform kernel corresponds to the
parameter matrix $N=\left[D,-C,-B,A\right]$. Thus the 2D Fresnel
transformation in its ‘frequency domain’ is given by
$\Psi\left(\xi^{\prime}\right)=\int\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)d^{2}\xi.$
(283)
Operating $F_{2}\left(r,s\right)$ on $\left|\xi\right\rangle$ we also have
$\left|\xi\right\rangle_{s,r}=\frac{1}{\allowbreak\allowbreak
A-iC}\exp\left\\{-\frac{D+iB}{2\left(\allowbreak
A-iC\right)}\left|\eta\right|^{2}+\frac{\xi
a_{1}^{\dagger}}{A-iC}+\frac{\xi^{\ast}a_{2}^{\dagger}}{\allowbreak
A-iC}-\frac{\allowbreak A+iC}{\allowbreak
A-iC}a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$ (284)
or
$\left|\xi\right\rangle_{s,r}=\frac{1}{s^{\ast}-r^{\ast}}\exp\left\\{-\allowbreak\frac{s^{\ast}+r^{\ast}}{2\left(s^{\ast}-r^{\ast}\right)}\left|\xi\right|^{2}+\allowbreak\frac{\xi
a_{1}^{\dagger}}{s^{\ast}-r^{\ast}}\allowbreak+\allowbreak\frac{\xi^{\ast}a_{2}^{\dagger}}{s^{\ast}-r^{\ast}}-\frac{s-r}{s^{\ast}-r^{\ast}}\allowbreak
a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle.$ (285)
Noticing that the entangled Wigner operator in $\left\langle\xi\right|$
representation is expressed as
$\Delta\left(\sigma,\gamma\right)=\int\frac{d^{2}\xi}{\pi^{3}}\left|\gamma+\xi\right\rangle\left\langle\gamma-\xi\right|\exp(\xi^{\ast}\sigma-\sigma^{\ast}\xi),$
(286)
and using the classical correspondence of
$\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|$ which is calculated by
$\displaystyle h(\sigma,\gamma)$
$\displaystyle=4\pi^{2}\mathtt{Tr}\left[\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|\Delta\left(\sigma,\gamma\right)\right]$
$\displaystyle=4\int\frac{d^{2}\xi}{\pi}\left\langle\gamma-\xi\right|F_{2}\left|\xi\right\rangle\left\langle\xi\right|F_{2}^{{\dagger}}|\gamma+\xi\rangle\exp(\xi^{\ast}\sigma-\sigma^{\ast}\xi)$
$\displaystyle=\pi\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right),$
(287)
we obtain
$\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\Delta\left(\sigma,\gamma\right)d^{2}\sigma
d^{2}\gamma,$ (288)
so the projector operator $\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|$ is another Radon transformation of the two-mode
Wigner operator, with $A$ and $C$ being the Radon transformation parameter
(‘frequency’ domain). Therefore, the quantum tomography in
${}_{s,r}\left\langle\xi\right|$ representation is expressed as the Radon
transformation of the Wigner function
$|\left\langle\xi\right|F^{\dagger}\left|\psi\right\rangle|^{2}=|_{s,r}\left\langle\xi\right|\left.\psi\right\rangle|^{2}=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle,$
(289)
and ${}_{s,r}\left\langle\xi\right|=\left\langle\xi\right|F^{\dagger}.$
#### 9.4.2 Inverse Radon transformation
Now we consider the inverse Radon transformation. For instance, using Eq.(279)
we see the Fourier transformation of
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$ is
$\displaystyle\int
d^{2}\eta\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|\exp(-i\zeta_{1}\eta_{1}-i\zeta_{2}\eta_{2})$
$\displaystyle=\pi\int d^{2}\gamma
d^{2}\sigma\Delta\left(\sigma,\gamma\right)\exp\left[-i\zeta_{1}\left(D\sigma_{1}+B\gamma_{2}\right)-i\zeta_{2}\left(D\sigma_{2}-B\gamma_{1}\right)\right],$
(290)
the right-hand side of (290) can be regarded as a special Fourier
transformation of $\Delta\left(\sigma,\gamma\right)$, so by making its inverse
Fourier transformation, we get
$\displaystyle\Delta\left(\sigma,\gamma\right)$
$\displaystyle=\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}dr_{1}\left|r_{1}\right|\int_{-\infty}^{\infty}dr_{2}\left|r_{2}\right|\int_{0}^{\pi}d\theta_{1}d\theta_{2}$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right),$
(291)
where
$\cos\theta_{1}=\cos\theta_{2}=\frac{D}{\sqrt{B^{2}+D^{2}}},r_{1}=\zeta_{1}\sqrt{B^{2}+D^{2}},r_{2}=\zeta_{2}\sqrt{B^{2}+D^{2}}$
and
$\displaystyle K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right)$
$\displaystyle\equiv\exp\left[-ir_{1}\left(\frac{\eta_{1}}{\sqrt{B^{2}+D^{2}}}-\sigma_{1}\cos\theta_{1}-\gamma_{2}\sin\theta_{1}\right)\right]$
$\displaystyle\times\exp\left[-ir_{2}\left(\frac{\eta_{2}}{\sqrt{B^{2}+D^{2}}}-\sigma_{2}\cos\theta_{2}+\gamma_{1}\sin\theta_{2}\right)\right].$
(292)
Eq.(291) is just the inverse Radon transformation of entangled Wigner operator
in the entangled state representation. This is different from the two
independent Radon transformations’ direct product of the two independent
single-mode Wigner operators, because in (268) the
$\left|\eta\right\rangle_{s,r}$ is an entangled state. Therefore the Wigner
function of quantum state $\left|\psi\right\rangle$ can be reconstructed from
the tomographic inversion of a set of measured probability distributions
$\left|{}_{s,r}\left\langle\eta\right.\left|\psi\right\rangle\right|^{2}$,
i.e.,
$\displaystyle W_{\psi}$
$\displaystyle=\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}dr_{1}\left|r_{1}\right|\int_{-\infty}^{\infty}dr_{2}\left|r_{2}\right|\int_{0}^{\pi}d\theta_{1}d\theta_{2}$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left|{}_{s,r}\left\langle\eta\right.\left|\psi\right\rangle\right|^{2}K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right).$
(293)
Thus, based on the previous section, we have further extended the relation
connecting optical Fresnel transformation with quantum tomography to the
entangled case. The tomography representation
${}_{s,r}\left\langle\eta\right|=\left\langle\eta\right|F_{2}^{\dagger}$ is
set up, based on which the tomogram of quantum state $\left|\psi\right\rangle$
is just the squared modulus of the wave function
${}_{s,r}\left\langle\eta\right|\left.\psi\right\rangle.$ i.e. the probability
distribution for the Fresnel quadrature phase is the tomogram (Radon transform
of the Wigner function).
## 10 Fractional Fourier Transformation (FrFT) for 1-D case
The fractional Fourier transform (FrFT) has been shown to be a very useful
tool in Fourier optics and information optics. The concept of FrFT was firstly
introduced mathematically in 1980 by Namias [24] as a mathematical tool for
solving theoretical physical problems [68], but did not brought enough
attention until Mendlovic and Ozaktas [25, Ozaktas] defined the $\alpha$-th
FrFT physically, based on propagation in quadratic graded-index media (GRIN
media with medium parameters $n(r)=n_{1}-n_{2}r^{2}/2$). Since then a lot of
works have been done on its properties, optical implementations and
applications [69, 70, 71, 72].
### 10.1 Quantum version of FrFT
The FrFT of $\theta$-order is defined in a manner, i.e.,
$\mathcal{F}_{\theta}\left[f\left(x\right)\right]=\sqrt{\frac{e^{i\left(\frac{\pi}{2}-\theta\right)}}{2\pi\sin\theta}}\int_{-\infty}^{\infty}\exp\left\\{-i\frac{x^{2}+y^{2}}{2\tan\theta}+\frac{ixx^{\prime}}{\sin\theta}\right\\}f\left(x\right)dx,$
(294)
where the exponential function is an integral kernel. In order to find the
quantum correspondence of FrFT, multiplying the function
$\exp\left\\{-i\frac{x^{2}+y^{2}}{2\tan\theta}+\frac{ixx^{\prime}}{\sin\theta}\right\\}$
by the ket $\int dy\left|y\right\rangle$ and bra $\int dx\left\langle
x\right|$ from left and right, respectively, where $\left|y\right\rangle$ and
$\left|x\right\rangle$ are coordinate eigenvectors,
$X\left|x\right\rangle=x\left|x\right\rangle$, and then using (14) and the
IWOP technique to perform the integration, we obtain
$\displaystyle\int_{-\infty}^{\infty}dxdy\left|y\right\rangle\exp\left\\{-i\frac{x^{2}+y^{2}}{2\tan\theta}+\frac{ixy}{\sin\theta}\right\\}\left\langle
x\right|$ $\displaystyle=\sqrt{-2\pi i\sin\theta
e^{i\theta}}\colon\exp\left\\{\left(e^{i\theta}-1\right)a^{\dagger}a\right\\}\colon$
$\displaystyle=\sqrt{-2\pi i\sin\theta e^{i\theta}}\exp\left\\{i\theta
a^{\dagger}a\right\\},$ (295)
where we have used the operator identity in the last step of Eq.(295)
$\exp\left\\{fa^{\dagger}a\right\\}=\colon\exp\left\\{\left(e^{f}-1\right)a^{\dagger}a\right\\}\colon.$
(296)
From the orthogonal relation $\left\langle
x^{\prime}\right.\left|x\right\rangle=\delta\left(x-x^{\prime}\right),$ we
know that Eq.(295) indicates
$\sqrt{\frac{e^{i\left(\frac{\pi}{2}-\theta\right)}}{2\pi\sin\theta}}\exp\left\\{-i\frac{x^{2}+y^{2}}{2\tan\theta}+\frac{ixy}{\sin\theta}\right\\}=\left\langle
y\right|e^{i\theta a^{\dagger}a}\left|x\right\rangle,$ (297)
which implies that the integral kernel in Eq.(294) is just the matrix element
of operator $\exp\left\\{i\theta a^{\dagger}a\right\\}$ in coordinate state
($\exp\left\\{i\theta a^{\dagger}a\right\\}$ called as Fractional Fourier
Operator [73]. Therefore, if we consider $f\left(x\right)$ as $\left\langle
x\right.\left|f\right\rangle$, the wave function of quantum state
$\left|f\right\rangle$ in the coordinate representation, from Eqs. (294) and
(297) it then follows
$\mathcal{F}_{\theta}\left[f\left(x\right)\right]=\int_{-\infty}^{\infty}dx\left\langle
y\right|e^{i\theta a^{\dagger}a}\left|x\right\rangle
f\left(x\right)=\left\langle y\right|e^{i\theta
a^{\dagger}a}\left|f\right\rangle\equiv g\left(y\right),$ (298)
which suggests
$\left|g\right\rangle=e^{i\theta a^{\dagger}a}\left|f\right\rangle.$ (299)
From Eqs.(298) and (294) one can see that the FrFT in Eq.(294) corresponds
actually to the rotating operator $\left(e^{i\theta a^{\dagger}a}\right)$
transform in Eq.(298) between two quantum states, which is just the quantum
version of FrFT.
In fact, using quantum version of FrFT, one can directly derive various
properties of the FrFTs. An important feature of the FrFT is that they are
composed according to
$\mathcal{F}_{\theta^{\prime}}\mathcal{F}_{\theta}=\mathcal{F}_{\theta^{\prime}+\theta}$
(the additivity property). Without losing generality, we examine
$\mathcal{F}_{\theta+\theta^{\prime}}\left[f\left(x\right)\right]\equiv\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left\langle
y\right|e^{i\left(\theta+\theta^{\prime}\right)a^{\dagger}a}\left|x\right\rangle
f\left(x\right).$ (300)
According to the completeness relation of coordinate eigenvector,
$\int_{-\infty}^{\infty}dx^{\prime}\left|x^{\prime}\right\rangle\left\langle
x^{\prime}\right|=1,$ Eq.(300) yields
$\displaystyle\mathcal{F}_{\theta+\theta^{\prime}}\left[f\left(x\right)\right]$
$\displaystyle=\int_{-\infty}^{\infty}dx\left\langle y\right|e^{i\theta
a^{\dagger}a}e^{i\theta^{\prime}a^{\dagger}a}\left|x\right\rangle
f\left(x\right)$ $\displaystyle=\int_{-\infty}^{\infty}dx^{\prime}\left\langle
y\right|e^{i\theta
a^{\dagger}a}\left|x^{\prime}\right\rangle\int_{-\infty}^{\infty}dx\left\langle
x^{\prime}\right|e^{i\theta^{\prime}a^{\dagger}a}\left|x\right\rangle
f\left(x\right)$ $\displaystyle=\int_{-\infty}^{\infty}dx^{\prime}\left\langle
y\right|e^{i\theta
a^{\dagger}a}\left|x^{\prime}\right\rangle\mathcal{F}_{\theta}\left[f\left(x\right)\right]=\mathcal{F}_{\theta}\mathcal{F}_{\theta^{\prime}}\left[f\left(x\right)\right],$
(301)
which is just the additivity of FrFT.
In particular, when $\left|f\right\rangle$ is the number state,
$\left|f\right\rangle=\left|n\right\rangle=\frac{a^{\dagger
n}}{\sqrt{n!}}\left|0\right\rangle$, its wavefunction in coordinate
representation is
$f\left(x\right)=\left\langle
x\right.\left|n\right\rangle=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}e^{-x^{2}/2}H_{n}(x),$
(302)
the FrFT of $\left\langle x\right.\left|n\right\rangle$ is
$\mathcal{F}_{\theta}\left[\left\langle
x\right.\left|n\right\rangle\right]=\left\langle y\right|e^{i\theta
a^{\dagger}a}\left|n\right\rangle=e^{in\theta}\left\langle
y\right.\left|n\right\rangle,$ (303)
or
$\mathcal{F}_{\theta}\left[e^{-x^{2}/2}H_{n}(x)\right]=e^{in\theta}e^{-y^{2}/2}H_{n}(y),$
(304)
which indicates that the eigenfunction is Hermite-Gaussian function with the
corresponding eigenvalue being $e^{in\theta}$.
### 10.2 On the Scaled FrFT Operator
In studying various optical transformations the optical operator method is
proposed [74] as mapping of ray-transfer ABCD matrix, such that the ray
transfer through optical instruments and the diffraction can be discussed by
virtue of the commutative relations of operators and the matrix algebra. The
square phase operators, scaling operator, Fourier transform operator and the
propagation operator in free space have been proposed in the literature, two
important questions thus naturally arise: 1. what is the scaled FrFT (SFrFT)
operator which corresponds to the SFrFT’s integration kernel [24]
$\frac{1}{\sqrt{2\pi if_{e}\sin\phi}}\exp\left\\{\frac{i\left(x^{2}+x^{\prime
2}\right)}{2f_{e}\tan\phi}-\frac{ix^{\prime}x}{f_{e}\sin\phi}\right\\},$ (305)
where $f_{e}$ is standard focal length (or a scaled parameter); 2. If this
operator is found, can it be further decomposed into simpler operators and
what are their physical meaning? Since SFrFT has wide application in optical
information detection and can be implemented even by using a thick lens [76],
so our questions are worth of paying attention [75].
Figure 1: A thick lens as a kind of fractional Fourier transform device.
Let start with a thick lens (shown in Fig.1) which represents a transfer
matrix [76]
$\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)=\left(\begin{array}[c]{cc}1-\frac{\left(1-1/n\right)l}{R_{1}}&\frac{l}{n}\\\
-[\left(n-1\right)\frac{R_{1}+R_{2}}{R_{1}R_{2}}-\frac{l\left(n-1\right)^{2}}{nR_{1}R_{2}}]&1-\frac{\left(1-1/n\right)l}{R_{2}}\end{array}\right),$
(306)
where $n$ is the reflective index; $l$ is the thickness of thick lens; $R_{1}$
and $R_{2}$ denotes the curvature radius of the two surfaces of the lens,
respectively. When we choose $R_{1}=R_{2}=R,$ then Eq.(306) reduces to
$\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)=\left(\begin{array}[c]{cc}1-\frac{\left(1-1/n\right)l}{R}&\frac{l}{n}\\\
-[\left(n-1\right)\frac{2}{R}-\frac{l\left(n-1\right)^{2}}{nR^{2}}]&1-\frac{\left(1-1/n\right)l}{R}\end{array}\right).$
(307)
By defining $1-\frac{\left(1-1/n\right)l}{R}=\cos\phi,$
$\frac{l}{n}=f_{e}\sin\phi,$ and
$\frac{l}{R}=\frac{n\left(1-\cos\phi\right)}{n-1},$ $l=nf_{e}\sin\phi,$ we can
recast (307) into the simple form
$\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)=\left(\begin{array}[c]{cc}\cos\phi&f_{e}\sin\phi\\\
-\sin\phi/f_{e}&\cos\phi\end{array}\right),\text{
}\det\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)=1.$ (308)
According to (113) we immediately know that the operator of SFrFT is
$\displaystyle F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)$
$\displaystyle=\exp\left\\{\frac{i\left(f_{e}-1/f_{e}\right)\tan\phi}{2V}a^{\dagger
2}\right\\}$
$\displaystyle\times\exp\left\\{\left(a^{\dagger}a+\frac{1}{2}\right)\ln\left(\frac{2\sec\phi}{V}\right)\right\\}$
$\displaystyle\times\exp\left\\{\frac{i\left(f_{e}-1/f_{e}\right)\tan\phi}{2V}a^{2}\right\\},\text{
\ }$ (309) $\displaystyle\text{ }V$
$\displaystyle=\left[2+i\left(f_{e}+1/f_{e}\right)\tan\phi\right].$
Noting that the matrix $\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)$ can be decomposed into
$\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
-\frac{1}{f_{e}}\tan\phi&1\end{array}\right)\left(\begin{array}[c]{cc}\cos\phi&0\\\
0&\cos\phi\end{array}\right)\left(\begin{array}[c]{cc}1&f_{e}\tan\phi\\\
0&1\end{array}\right),$ (310)
according to the previous section we have
$\displaystyle F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)$
$\displaystyle=F\left(1,0,-\frac{1}{f_{e}}\tan\phi\right)F\left(\cos\phi,0,0\right)F\left(1,f_{e}\tan\phi,0\right)$
$\displaystyle=\exp\left(\frac{\tan\phi}{2if_{e}}Q^{2}\right)\exp\left\\{-\frac{i}{2}\left(QP+PQ\right)\ln\cos\phi\right\\}\exp\left(\frac{f_{e}\tan\phi}{2i}P^{2}\right),$
(311)
where $Q=\left(a+a^{\dagger}\right)/\sqrt{2},$
$P=\left(a-a^{\dagger}\right)/\left(\sqrt{2}i\right)$ and
$\exp\left(-\frac{i\tan\phi}{2f_{e}}Q^{2}\right),$
$\exp\left\\{-\frac{i}{2}\left(QP+PQ\right)\ln\cos\phi\right\\}$ and
$\exp\left(-\frac{if_{e}\tan\phi}{2}P^{2}\right)$ are the quadrature phase
operator, the squeezing operator and the free propagation operator,
respectively. On the other hand, from
$\left(\begin{array}[c]{cc}\mathcal{A}&\mathcal{B}\\\
\mathcal{C}&\mathcal{D}\end{array}\right)^{-1}=\left(\begin{array}[c]{cc}\mathcal{D}&\mathcal{-B}\\\
\mathcal{-C}&\mathcal{A}\end{array}\right),$ and (311) we see
$F_{1}^{-1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)=\exp\left(-\frac{i\mathcal{C}}{2\mathcal{D}}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln\mathcal{D}\right)\exp\left(\frac{i\mathcal{B}}{2\mathcal{D}}P^{2}\right),$
(312)
it then follows
$F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)=\exp\left(\frac{f_{e}\tan\phi}{2i}P^{2}\right)\exp\left(\frac{i}{2}\left(QP+PQ\right)\ln\cos\phi\right)\exp\left(\frac{\tan\phi}{2if_{e}}Q^{2}\right).$
(313)
Using the canonical operator form (311) or (313) of
$F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)$ we can deduce its
matrix element in the coordinate states $\left|x\right\rangle$ (its conjugate
state is $\left|p\right\rangle$)
$\left\langle
x^{\prime}\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle=\frac{1}{\sqrt{2\pi
if_{e}\sin\phi}}\exp\left\\{\left(\frac{i\left(x^{2}+x^{\prime
2}\right)}{2f_{e}\tan\phi}-\frac{ix^{\prime}x}{f_{e}\sin\phi}\right)\right\\},$
(314)
which is just the kernel of SFrFT, thus we name
$F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)$ SFrFT operator.
Noticing that the $Q^{2}/2,$ $P^{2}/2$ and $\frac{i}{4}\left(QP+PQ\right)$
construct a close SU(2) Lie algebra, we can put Eq.(313) into a more compact
form, i.e.,
$F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)=\exp\left\\{-i\frac{\phi
f_{e}}{2}\left(P^{2}+\frac{Q^{2}}{f_{e}^{2}}\right)\right\\},$ (315)
Eqs. (311), (313) and (315) are different forms of the same operator of SFrFT.
Especially, when $f_{e}=1,$
$F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\rightarrow\exp\left\\{-i\phi
a^{\dagger}a\right\\},$ which is the usual FrFT operator.
Using (314) the SFrFT of $f\left(x\right)=\left\langle
x\right|\left.f\right\rangle,$ denoted as
$\mathcal{F}_{f_{e}}^{\phi}\left[f\left(x\right)\right],$ can be expressed as
an matrix element in quantum optics context,
$\mathcal{F}_{f_{e}}^{\phi}\left[f\left(x\right)\right]=\int dx\left\langle
x^{\prime}\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle\left\langle
x\right|\left.f\right\rangle=\left\langle
x^{\prime}\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|f\right\rangle.$
(316)
The above discussions are useful since any unimodular matrix
$\left(\begin{array}[c]{cc}A&B\\\ C&D\end{array}\right)$ can be decomposed
into [77]
$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
-\mathcal{P}&1\end{array}\right)\left(\begin{array}[c]{cc}m&0\\\
0&m^{-1}\end{array}\right)\left(\begin{array}[c]{cc}\cos\phi&f_{e}\sin\phi\\\
-\sin\phi/f_{e}&\cos\phi\end{array}\right),$ (317)
where the parameters $m,\mathcal{P},\phi$ are all real,
$m^{2}=A^{2}+\frac{B^{2}}{f_{e}^{2}},\text{ }\tan\phi=\frac{B}{Af_{e}},\text{
}\mathcal{P}=-\frac{AC+DB/f_{e}^{2}}{A^{2}+\frac{B^{2}}{f_{e}^{2}}}.$ (318)
Correspondingly, the operator of FrFT is given by
$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=F_{1}\left(1,0,-\mathcal{P}\right)F_{1}\left(m,0,0\right)F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)$
$\displaystyle=\exp\left(-\frac{i}{2}\mathcal{P}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
m\right)\exp\left\\{\frac{\phi
f_{e}}{2i}\left(P^{2}+\frac{Q^{2}}{f_{e}^{2}}\right)\right\\},$ (319)
where
$F_{1}\left(1,0,-\mathcal{P}\right)=\exp\left[-\frac{i}{2}\mathcal{P}Q^{2}\right]$
is the quadratic phase operator. Thus the general Fresnel transform can always
be expressed by SFrFT as follows
$\displaystyle g\left(x^{\prime}\right)$ $\displaystyle=\int dy\left\langle
x^{\prime}\right|F_{1}\left(1,0,-\mathcal{P}\right)F_{1}\left(m,0,0\right)\left|y\right\rangle\int
dx\left\langle
y\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle
f\left(x\right)$ $\displaystyle=\sqrt{m}\int dx^{\prime\prime}dy\left\langle
x^{\prime}\right|\exp\left(-\frac{i}{2}\mathcal{P}X^{2}\right)\left|mx^{\prime\prime}\right\rangle\left\langle
x^{\prime\prime}\right.\left|y\right\rangle\int dx\left\langle
y\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle
f\left(x\right)$ $\displaystyle=\exp\left(-\frac{i}{2}\mathcal{P}x^{\prime
2}\right)\int\frac{dy}{\sqrt{m}}\delta\left(\frac{x^{\prime}}{m}-y\right)\int
dx\left\langle
y\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle
f\left(x\right)$
$\displaystyle=\frac{1}{\sqrt{m}}\exp\left(-\frac{i}{2}\mathcal{P}x^{\prime
2}\right)\int
dx\left\langle\frac{x^{\prime}}{m}\right|F_{1}\left(\mathcal{A},\mathcal{B},\mathcal{C}\right)\left|x\right\rangle
f\left(x\right)$
$\displaystyle=\frac{1}{\sqrt{m}}\exp\left(-\frac{i}{2}\mathcal{P}x^{\prime
2}\right)\mathcal{F}_{f_{e}}^{\phi}\left[f\right]\left(\frac{x^{\prime}}{m}\right).$
(320)
i.e., the output $g\left(x^{\prime}\right)$ is the SFrFT of the input
$f\left(x\right)$ plus a quadratic phase term
$\exp\left(-\frac{i}{2}\mathcal{P}x^{\prime 2}\right).$
### 10.3 An integration transformation from Chirplet to FrFT kernel
In the history of developing optics we have known that each optical setup
corresponds to an optical transformation, for example, thick lens as a
fractional Fourier transformer. In turn, once a new integration transform is
found, its experimental implementation is expected. In this subsection we
report a new integration transformation which can convert chirplet function to
FrFT kernel [78], as this new transformation is invertible and obeys Parseval
theorem, we expect it be realized by experimentalists.
The new transform we propose here is
$\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)\equiv
f\left(x,y\right),$ (321)
which differs from the usual two-fold Fourier transformation
$\iint_{-\infty}^{\infty}\frac{dxdy}{4\pi^{2}}e^{ipx+iqy}f(x,y).$ In
particular, when $h(p,q)=1,$ Eq. (321) reduces to
$\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}=\int_{-\infty}^{\infty}dq\delta\left(q-y\right)e^{-2xi\left(q-y\right)}=1,$
(322)
so $e^{2i\left(p-x\right)\left(q-y\right)}$ can be considered a basis funtion
in $p-q$ phase space, or Eq. (321) can be looked as an expansion of
$f\left(x,y\right)$ with the expansion coefficient being $h(p,q).$ We can
prove that the reciprocal transformation of (321) is
$\iint_{-\infty}^{\infty}\frac{dxdy}{\pi}e^{-2i(p-x)(q-y)}f(x,y)=h(p,q).$
(323)
In fact, substituting (321) into the left-hand side of (323) yields
$\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}h(p^{\prime},q^{\prime})\iint\frac{dxdy}{\pi}e^{2i\left[\left(p^{\prime}-x\right)\left(q^{\prime}-y\right)-\left(p-x\right)\left(q-y\right)\right]}$
$\displaystyle=\iint_{-\infty}^{\infty}dp^{\prime}dq^{\prime}h(p^{\prime},q^{\prime})e^{2i\left(p^{\prime}q^{\prime}-pq\right)}$
$\displaystyle\times\delta\left(p-p^{\prime}\right)\delta\left(q-q^{\prime}\right)\left.=h(p,q)\right..$
(324)
This transformation’s Parseval-like theorem is
$\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}|h(p,q)|^{2}$
$\displaystyle=\iint\frac{dxdy}{\pi}|f\left(x,y\right)|^{2}\iint\frac{dx^{\prime}dy^{\prime}}{\pi}e^{2i\left(x^{\prime}y^{\prime}-xy\right)}$
$\displaystyle\times\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left[\left(-y^{\prime}p-x^{\prime}q\right)+\left(py+xq\right)\right]}$
$\displaystyle=\iint\frac{dxdy}{\pi}|f\left(x,y\right)|^{2}\iint
dx^{\prime}dy^{\prime}e^{2i\left(x^{\prime}y^{\prime}-xy\right)}$
$\displaystyle\times\delta\left(x-x^{\prime}\right)\delta\left(p-p^{\prime}\right)$
$\displaystyle=\iint\frac{dxdy}{\pi}|f\left(x,y\right)|^{2}.$ (325)
Now we apply Eq. (321) to phase space transformation in quantum optics. Recall
that a signal $\psi\left(q\right)$’s Wigner transform [62, 66, 67, 79] is
$\psi\left(q\right)\rightarrow\int\frac{du}{2\pi}e^{ipu}\psi^{\ast}\left(q+\frac{u}{2}\right)\psi\left(q-\frac{u}{2}\right).$
(326)
Using Dirac’s symbol [80] to write $\psi\left(q\right)=\left\langle
q\right|\left.\psi\right\rangle,$ $\left|q\right\rangle$ is the eigenvector of
coordinate $Q$, the Wigner operator emerges from (326),
$\frac{1}{2\pi}\int_{-\infty}^{\infty}due^{-ipu}\left|q-\frac{u}{2}\right\rangle\left\langle
q+\frac{u}{2}\right|=\Delta\left(p,q\right),\text{ }\hbar=1.$ (327)
If $h\left(q,p\right)$ is quantized as the operator $\hat{H}\left(P,Q\right)$
through the Weyl-Wigner correspondence [37]
$H\left(P,Q\right)=\iint_{-\infty}^{\infty}dpdq\Delta\left(p,q\right)h\left(q,p\right),$
(328)
then
$h\left(q,p\right)=\int_{-\infty}^{\infty}due^{-ipu}\left\langle
q+\frac{u}{2}\right|\hat{H}\left(Q,P\right)\left|q-\frac{u}{2}\right\rangle,$
(329)
this in the literature is named the Weyl transform, $h\left(q,p\right)$ is the
Weyl classical correspondence of the operator $\hat{H}\left(Q,P\right)$.
Substituting (329) into (321) we have
$\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)$
$\displaystyle=\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}\int_{-\infty}^{\infty}due^{-ipu}$
$\displaystyle\times\left\langle
q+\frac{u}{2}\right|\hat{H}\left(Q,P\right)\left|q-\frac{u}{2}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dq\int_{-\infty}^{\infty}du\left\langle
q+\frac{u}{2}\right|\hat{H}\left(Q,P\right)\left|q-\frac{u}{2}\right\rangle$
$\displaystyle\times\delta\left(q-y-\frac{u}{2}\right)e^{-2ix\left(q-y\right)}$
$\displaystyle=\int_{-\infty}^{\infty}due^{-ixu}\left\langle
y+u\right|\hat{H}\left(Q,P\right)\left|y\right\rangle.$ (330)
Using $\left\langle y+u\right|=\left\langle u\right|e^{iPy}$ and
$(\sqrt{2\pi})^{-1}e^{-ixu}=\left\langle p_{=x}\right|\left.u\right\rangle,$
where $\left\langle p\right|$ is the momentum eigenvector, and
$\displaystyle\int_{-\infty}^{\infty}due^{-ixu}\left\langle y+u\right|$
$\displaystyle=\int_{-\infty}^{\infty}due^{-ixu}\left\langle u\right|e^{iPy}$
$\displaystyle=\sqrt{2\pi}\int_{-\infty}^{\infty}du\left\langle
p_{=x}\right|\left.u\right\rangle\left\langle u\right|e^{iPy}$
$\displaystyle=\sqrt{2\pi}\left\langle p_{=x}\right|e^{ixy},$ (331)
then Eq. (330) becomes
$\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)=\sqrt{2\pi}\left\langle
p_{=x}\right|\hat{H}\left(Q,P\right)\left|y\right\rangle e^{ixy},$ (332)
thus through the new integration transformation a new relationship between a
phase space function $h(p,q)$ and its Weyl-Wigner correspondence operator
$\hat{H}\left(Q,P\right)$ is revealed. The inverse of (332), according to
(323), is
$\iint_{-\infty}^{\infty}\frac{dxdy}{\sqrt{\pi/2}}e^{-2i\left(p-x\right)\left(q-y\right)}\left\langle
p_{=x}\right|\hat{H}\left(Q,P\right)\left|y\right\rangle e^{ixy}=h(p,q).$
(333)
For example, when $\hat{H}\left(Q,P\right)=e^{f(P^{2}+Q^{2}-1)/2},$ its
classical correspondence is
$e^{f\left(P^{2}+Q^{2}-1\right)/2}\rightarrow
h(p,q)=\frac{2}{e^{f}+1}\exp\left\\{2\frac{e^{f}-1}{e^{f}+1}\left(p^{2}+q^{2}\right)\right\\}.$
(334)
Substituting (334) into (332) we have
$\displaystyle\frac{2}{e^{f}+1}\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}\exp\left\\{2\frac{e^{f}-1}{e^{f}+1}\left(p^{2}+q^{2}\right)\right\\}$
$\displaystyle=\sqrt{2\pi}\left\langle
p_{=x}\right|e^{f\left(P^{2}+Q^{2}-1\right)/2}\left|y\right\rangle e^{ixy}.$
(335)
Using the Gaussian integration formula
$\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}e^{-\lambda\left(p^{2}+q^{2}\right)}$
$\displaystyle=\frac{1}{\sqrt{\lambda^{2}+1}}\exp\left\\{\frac{-\lambda\left(x^{2}+y^{2}\right)}{\lambda^{2}+1}+\frac{2i\lambda^{2}}{\lambda^{2}+1}xy\right\\},$
(336)
in particular, when
$\lambda=-i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right),$ with
$\frac{-\lambda}{\lambda^{2}+1}=\frac{i}{2\tan\alpha},$
$\frac{2\lambda^{2}}{\lambda^{2}+1}=1-\frac{1}{\sin\alpha},$ Eq. (336) becomes
$\displaystyle\frac{2}{ie^{-i\alpha}+1}\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}$
$\displaystyle\times\exp\left\\{i\left(p^{2}+q^{2}\right)\tan(\frac{\pi}{4}-\frac{\alpha}{2})\right\\}$
$\displaystyle=\frac{1}{\sqrt{ie^{-i\alpha}\sin\alpha}}\exp\left\\{\frac{i\left(x^{2}+y^{2}\right)}{2\tan\alpha}-\frac{ixy}{\sin\alpha}\right\\}e^{ixy},$
(337)
where
$\exp\\{i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\left(p^{2}+q^{2}\right)\\}$
represents an infinite long chirplet function. Comparing (337) with (335) we
see $ie^{-i\alpha}=e^{f},$ $f=i\left(\frac{\pi}{2}-\alpha\right),$ it then
follows
$\displaystyle\left\langle
p_{=x}\right|e^{i(\frac{\pi}{2}-\alpha)\left(P^{2}+Q^{2}-1\right)/2}\left|y\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2\pi
ie^{-i\alpha}\sin\alpha}}\exp\left\\{\frac{i\left(x^{2}+y^{2}\right)}{2\tan\alpha}-\frac{ixy}{\sin\alpha}\right\\},$
(338)
where the right-hand side of (338) is just the FrFT kernel. Therefore the new
integration transformation (321) can convert spherical wave to FrFT kernel. We
expect this transformation could be implemented by experimentalists.
Moreover, this transformation can also serve for solving some operator
ordering problems. We notice
$\displaystyle\frac{1}{\pi}\exp[2i\left(p-x\right)\left(q-y\right)]$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dv}{2\pi}\delta\left(q-y-\frac{v}{2}\right)\exp\left\\{i\left(p-x\right)v\right\\},$
(339)
so the transformation (321) is equivalent to
$\displaystyle h(p,q)$
$\displaystyle\rightarrow\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)$
$\displaystyle=\iint_{-\infty}^{\infty}dpdq\int_{-\infty}^{\infty}\frac{dv}{2\pi}\delta\left(q-y-\frac{v}{2}\right)e^{i\left(p-x\right)v}h(p,q)$
$\displaystyle=\iint_{-\infty}^{\infty}\frac{dpdq}{2\pi}h(p+x,y+\frac{q}{2})e^{ipq}.$
(340)
For example, using (327) and (339) we have
$\displaystyle\Delta(p,q)$
$\displaystyle\rightarrow\iint_{-\infty}^{\infty}\frac{dpdq}{2\pi}\Delta(p+x,y+\frac{q}{2})e^{ipq}$
$\displaystyle=\iint_{-\infty}^{\infty}\frac{dpdq}{4\pi^{2}}\int_{-\infty}^{\infty}due^{-i\left(p+x\right)u}$
$\displaystyle\times\left|y+\frac{q}{2}-\frac{u}{2}\right\rangle\left\langle
y+\frac{q}{2}+\frac{u}{2}\right|e^{ipq}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{2\pi}\int_{-\infty}^{\infty}due^{-ixu}\delta\left(q-u\right)$
$\displaystyle\times\left|y+\frac{q}{2}-\frac{u}{2}\right\rangle\left\langle
y+\frac{q}{2}+\frac{u}{2}\right|$
$\displaystyle=\int_{-\infty}^{\infty}\frac{du}{2\pi}e^{-ixu}\left|y\right\rangle\left\langle
y+u\right|=\left|y\right\rangle\left\langle
y\right|\int_{-\infty}^{\infty}\frac{du}{2\pi}e^{iu\left(P-u\right)}$
$\displaystyle=\delta\left(y-Q\right)\delta\left(x-P\right),$ (341)
so
$\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(q-Q\right)\delta\left(p-P\right),$
(342)
thus this new transformation can convert the Wigner operator to
$\delta\left(q-Q\right)\delta\left(p-P\right).$ Similarly, we have
$\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(p-P\right)\delta\left(q-Q\right).$
Then for the Wigner function of a density operator $\rho$,
$W_{\psi}(p,q)\equiv\mathtt{Tr}\left[\rho\Delta(p,q)\right],$ we have
$\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}\mathtt{Tr}\left[\rho\Delta(p^{\prime},q^{\prime})\right]e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$
$\displaystyle=\mathtt{Tr}\left[\rho\delta\left(q-Q\right)\delta\left(p-P\right)\right]$
$\displaystyle=\int\frac{dudv}{4\pi^{2}}\mathtt{Tr}\left[\rho
e^{i\left(q-Q\right)u}e^{i\left(p-P\right)v}\right],$ (343)
we may define $\mathtt{Tr}\left[\rho
e^{i\left(q-Q\right)u}e^{i\left(p-P\right)v}\right]$ as the $Q-P$
characteristic function. Similarly,
$\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}\mathtt{Tr}\left[\rho\Delta(p^{\prime},q^{\prime})\right]e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$
$\displaystyle=\mathtt{Tr}\left[\rho\delta\left(p-P\right)\delta\left(q-Q\right)\right]$
$\displaystyle=\int\frac{dudv}{4\pi^{2}}\mathtt{Tr}\left[\rho
e^{i\left(p-P\right)v}e^{i\left(q-Q\right)u}\right]$ (344)
we name $\mathtt{Tr}\left[\rho
e^{i\left(p-P\right)v}e^{i\left(q-Q\right)u}\right]$ as the $P-Q$
characteristic function.
## 11 Complex Fractional Fourier Transformation
In this section, we extend 1-D FrFT to the complex fractional Fourier
transformation (CFrFT).
### 11.1 Quantum version of CFrFT
According to Ref. [81], based on the entangled state $\left|\eta\right\rangle$
in two-mode Fock space and its orthonormal property, we can take the matrix
element of
$\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]$
in the entangled state $\left|\eta\right\rangle,$
$\mathcal{K}^{F}\left(\eta^{\prime},\eta\right)=\left\langle\eta^{\prime}\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\left|\eta\right\rangle,$
(345)
as the integral transform kernel of CFrFT,
$\mathcal{F}_{\alpha}\left[f\right]\left(\eta\right)=\int\frac{d^{2}\eta}{\pi}\mathcal{K}^{F}\left(\eta^{\prime},\eta\right)f\left(\eta\right).$
(346)
Using the normally ordered expansion of $e^{\lambda
a_{1}^{\dagger}a_{1}}=\colon\exp\left[\left(e^{\lambda}-1\right)a_{1}^{\dagger}a_{1}\right]\colon$
and the completeness relation of the coherent state representation,
$\left|z_{i}\right\rangle=\exp\left\\{-\frac{1}{2}\left|z_{i}\right|^{2}+z_{i}a_{i}^{\dagger}\right\\}\left|0\right\rangle_{i},$
we calculate that $\mathcal{K}^{F}\left(\eta^{\prime},\eta\right)$ is
$\displaystyle\mathcal{K}^{F}\left(\eta^{\prime},\eta\right)$
$\displaystyle=\left\langle\eta^{\prime}\right|\frac{d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{2}}\left|z_{1}^{\prime},z_{2}^{\prime}\right\rangle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|\colon\exp\left[\left(e^{-i\alpha}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\colon$
$\displaystyle\times\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right|\left.\eta\right\rangle$
$\displaystyle=\frac{e^{i(\alpha-\frac{\pi}{2})}}{2\sin\alpha}\exp\left[\frac{i(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2})}{2\tan\alpha}-\frac{i\left(\eta^{\prime\ast}\eta+\eta^{\ast}\eta^{\prime}\right)}{2\sin\alpha}\right],$
(347)
which is just the integral kernel of the CFrFT in [82]. Thus we see that the
matrix element of
$\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]$
between two entangled state representations $\left|\eta\right\rangle$ and
$\left|\eta^{\prime}\right\rangle$ corresponds to CFrFT. This is a new route
from quantum optical transform to classical CFrFT transform. Let
$\eta=x_{2}+iy_{2},$ $\eta^{\prime}=x_{1}+iy_{1},$ (346) becomes
$\displaystyle\mathcal{F}_{\alpha}\left[f\right]\left(x_{2},y_{2}\right)$
$\displaystyle=\frac{e^{i(\alpha-\frac{\pi}{2})}}{2\sin\alpha}\exp\left[\frac{i\left(x_{2}^{2}+y_{2}^{2}\right)}{2\tan\alpha}\right]$
$\displaystyle\times\int\frac{dx_{1}dy_{1}}{\pi}\exp\left[\frac{i\left(x_{1}^{2}+y_{1}^{2}\right)}{2\tan\alpha}-i\frac{\left(x_{1}x_{2}+y_{1}y_{2}\right)}{\sin\alpha}\right]f\left(x_{1},y_{1}\right).$
(348)
In fact, letting
$f\left(\eta\right)=\left\langle\eta\right|\left.f\right\rangle,$ then using
Eqs.(31) and (347) we have
$\displaystyle\left\langle\eta^{\prime}\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\left|f\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\left|\eta\right\rangle\left\langle\eta\right|\left.f\right\rangle=\int\frac{d^{2}\eta}{\pi}\mathcal{K}^{F}\left(\eta^{\prime},\eta\right)f\left(\eta\right)$
$\displaystyle=\frac{e^{i(\alpha-\frac{\pi}{2})}}{2\sin\alpha}\int\frac{d^{2}\eta}{\pi}\exp\left[\frac{i(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2})}{2\tan\alpha}-\frac{i\left(\eta^{\prime\ast}\eta+\eta^{\ast}\eta^{\prime}\right)}{2\sin\alpha}\right]f\left(\eta\right).$
(349)
Thus the quantum mechanical version of CFrFT is given by
$\mathcal{F}_{\alpha}\left[f\right]\left(\eta^{\prime}\right)\equiv\left\langle\eta^{\prime}\right|\exp\left[-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\left|f\right\rangle.$
(350)
The standard complex Fourier transform is $\mathcal{F}_{\pi/2}.$
$\mathcal{F}_{0}$ is the identity operator.
### 11.2 Additivity property and eigenmodes of CFrFT
We will show later that this CFrFT can help us to reveal some new property
which has been overlooked in the formulation of the direct product of two real
FrFTs [83]. The definition (360) is of course required to satisfy the basic
postulate that
$\mathcal{F}_{\alpha}\mathcal{F}_{\beta}\left[f\right]\left(\eta^{\prime}\right)=\mathcal{F}_{\alpha+\beta}\left[f\left(\eta\right)\right]$
(the additivity property). For this purpose, using Eq.(360) and Eq.(31) we see
$\displaystyle\mathcal{F}_{\alpha+\beta}\left[f\left(\eta\right)\right]$
$\displaystyle\equiv\left\langle\eta^{\prime}\right|e^{-i(\alpha+\beta)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|f\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d^{2}\eta^{\prime\prime}}{\pi}\left\langle\eta^{\prime}\right|e^{-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|\eta^{\prime\prime}\right\rangle$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime\prime}\right|e^{-i\beta\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|\eta\right\rangle
f\left(\eta\right)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d^{2}\eta^{\prime\prime}}{\pi}\left\langle\eta^{\prime}\right|e^{-i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|\eta^{\prime\prime}\right\rangle\mathcal{F}_{\beta}\left[f\left(\eta\right)\right]$
$\displaystyle=\mathcal{F}_{\alpha}\mathcal{F}_{\beta}\left[f\left(\eta\right)\right].$
(351)
This derivation is clear and concise by employing the
$\left|\eta\right\rangle$ representation and quantum mechanical version of
CFrFT.
On the other hand, the formula (350) can help us to derive CFrFT of some wave
functions easily. For example, when $\left|f\right\rangle$ is a two-mode
number state
$\left|m,n\right\rangle=a_{1}^{{\dagger}m}a_{2}^{{\dagger}n}/\sqrt{m!n!}\left|00\right\rangle$,
then the CFrFT of the wave function
$\left\langle\eta\right|\left.m,n\right\rangle$ is
$\displaystyle\mathcal{F}_{\alpha}\left[\left\langle\eta\right|\left.m,n\right\rangle\right]$
$\displaystyle=\left\langle\eta^{\prime}\right|e^{i(\alpha+\beta)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|m,n\right\rangle$
$\displaystyle=e^{i(\alpha+\beta)\left(m+n\right)}\left\langle\eta^{\prime}\right|\left.m,n\right\rangle.$
(352)
To calculate $\left\langle\eta^{\prime}\right|\left.m,n\right\rangle$, let us
recall the definition of two-variable Hermite polynomial
$H_{m,n}\left(\xi,\xi^{\ast}\right)$ (LABEL:4.20,4.21), we can expand
$\left\langle\eta^{\prime}\right|$ as
$\left\langle\eta^{\prime}\right|=\left\langle
00\right|\sum_{m,n=0}^{\infty}i^{m+n}\frac{a_{1}^{m}a_{2}^{n}}{m!n!}H_{m,n}\left(-i\eta^{\prime\ast},i\eta^{\prime}\right)e^{-\left|\eta^{\prime}\right|^{2}/2},$
(353)
thus
$\left\langle\eta^{\prime}\right|\left.m,n\right\rangle=\frac{i^{m+n}}{\sqrt{m!n!}}H_{m,n}\left(-i\eta^{\prime\ast},i\eta^{\prime}\right)e^{-\left|\eta^{\prime}\right|^{2}/2}.$
(354)
As a result of (354) we see that equation (352) becomes
$\mathcal{F}_{\alpha}\left[H_{m,n}\left(-i\eta^{\ast},i\eta\right)e^{-\left|\eta\right|^{2}/2}\right]=e^{i(\alpha+\beta)\left(m+n\right)}H_{m,n}\left(-i\eta^{\prime\ast},i\eta^{\prime}\right)e^{-\left|\eta^{\prime}\right|^{2}/2}.$
(355)
If we consider the operation $\mathcal{F}_{\alpha}$ as an operator, one can
say that the eigenfunction of $\mathcal{F}_{\alpha}$ (the eigenmodes of CFrFT)
is the two-variable Hermite polynomials $H_{m,n}$ with the eigenvalue being
$e^{i(\alpha+\beta)\left(m+n\right)}$. This is a new property of CFrFT. Since
the function space spanned by $H_{m,n}\left(\eta,\eta^{\ast}\right)$ is
complete,
$\int\frac{d^{2}\eta}{\pi}e^{-|\eta|^{2}}H_{m,n}\left(\eta,\eta^{\ast}\right)\left[H_{m,n}\left(\eta,\eta^{\ast}\right)\right]^{\ast}=\sqrt{m!n!m^{\prime}!n^{\prime}!}\delta_{m,m^{\prime}}\delta_{n,n^{\prime}},$
(356)
and
$\sum_{m,n=0}^{\infty}\frac{1}{m!n!}H_{m,n}\left(\eta,\eta^{\ast}\right)\left[H_{m,n}\left(\eta^{\prime},\eta^{\prime\ast}\right)\right]^{\ast}e^{-\left|\eta\right|^{2}}=\pi\delta\left(\eta-\eta^{\prime}\right)\delta\left(\eta^{\ast}-\eta^{\prime\ast}\right),$
(357)
one can confirms that the eigenmodes of CFrFT form an orthogonal and complete
basis set [84]. Note that the two variable Hermite polynomial
$H_{m,n}\left(\eta,\eta^{\ast}\right)$ is not the direct product of two
independent ordinary Hermite polynomials, so CFrFT differs from the direct
product of two FrFTs.
### 11.3 From Chirplet to CFrFT kernel
In this subsection, by developing Eq. (321) to more general case which can be
further related to the transformation between two mutually conjugate entangled
state representations $\left|\xi\right\rangle$ and $\left|\eta\right\rangle$,
we shall propose a new integration transformation in $\xi-\eta$ phase space
(see Eq. (358) below) and its inverse transformation. We find that Eq. (358)
also possesses some well-behaved transformation properties and can be used to
obtain the CFrFT kernel from a chirplet [85].
#### 11.3.1 New complex integration transformation
Corresponding to the structure of phase space spanned by
$\left|\xi\right\rangle$ and $\left|\eta\right\rangle$ and enlightened by Eq.
(321), we propose a new complex integration transformation in $\xi-\eta$ phase
space
$\int\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}\mathcal{F}(\eta,\xi)\equiv
D\left(\nu,\mu\right).$ (358)
When $\mathcal{F}(\xi,\eta)=1,$ (358) becomes
$\displaystyle\int\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}$
$\displaystyle=\int
d^{2}\xi\delta\left(\xi-\mu\right)\delta\left(\xi^{\ast}-\mu^{\ast}\right)e^{\nu\left(\xi^{\ast}-\mu^{\ast}\right)-\nu^{\ast}\left(\xi-\mu\right)}=1,$
(359)
so
$e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}$
can be considered a basis function in $\xi-\eta$ phase space, or Eq. (358) can
be looked as an expansion of $D\left(\nu,\mu\right)$ in terms of
$e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)},$
with the expansion coefficient being $\mathcal{F}(\eta,\xi).$
We can prove that the inverse transform of (358) is
$\int\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}e^{\left(\xi^{\ast}-\mu^{\ast}\right)\left(\eta-\nu\right)-\left(\eta^{\ast}-\nu^{\ast}\right)\left(\xi-\mu\right)}D\left(\nu,\mu\right)\equiv\mathcal{F}(\eta,\xi).$
(360)
In fact, substituting (358) into the left-hand side of (360) yields
$\displaystyle\int\frac{d^{2}\xi^{\prime}d^{2}\eta^{\prime}}{\pi^{2}}\mathcal{F}(\eta^{\prime},\xi^{\prime})\int\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}$ $\displaystyle\times
e^{\left(\xi^{\prime}-\mu\right)\left(\eta^{\prime\ast}-\nu^{\ast}\right)-\left(\eta^{\prime}-\nu\right)\left(\xi^{\prime\ast}-\mu^{\ast}\right)+\left(\xi^{\ast}-\mu^{\ast}\right)\left(\eta-\nu\right)-\left(\eta^{\ast}-\nu^{\ast}\right)\left(\xi-\mu\right)}$
$\displaystyle=\int\frac{d^{2}\xi^{\prime}d^{2}\eta^{\prime}}{\pi^{2}}\mathcal{F}(\eta^{\prime},\xi^{\prime})e^{\left(\xi^{\prime}\eta^{\prime\ast}-\eta^{\prime}\xi^{\prime\ast}+\xi^{\ast}\eta-\eta^{\ast}\xi\right)}$
$\displaystyle\times\iint\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}e^{\left(\eta^{\ast}-\eta^{\prime\ast}\right)\mu+\left(\eta^{\prime}-\eta\right)\mu^{\ast}}e^{\left(\xi^{\prime\ast}-\xi^{\ast}\right)\nu+\left(\xi-\xi^{\prime}\right)\nu^{\ast}}$
$\displaystyle=\int
d^{2}\xi^{\prime}d^{2}\eta^{\prime}\mathcal{F}(\eta^{\prime},\xi^{\prime})e^{\left(\xi^{\prime}\eta^{\prime\ast}-\eta^{\prime}\xi^{\prime\ast}+\xi^{\ast}\eta-\eta^{\ast}\xi\right)}$
$\displaystyle\times\delta^{\left(2\right)}\left(\eta^{\prime}-\eta\right)\delta^{\left(2\right)}\delta\left(\xi-\xi^{\prime}\right)\left.=\mathcal{F}(\eta,\xi)\right..$
(361)
This Parseval-like theorem for this transformation can also be demonstrated,
$\displaystyle\int\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}|\mathcal{F}(\eta,\xi)|^{2}$
$\displaystyle=\int\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}|D\left(\nu,\mu\right)|^{2}\iint\frac{d^{2}\mu^{\prime}d^{2}\nu^{\prime}}{\pi^{2}}$
$\displaystyle\times\exp\left[\left(\mu^{\ast}\nu-\nu^{\ast}\mu\right)+\left(\mu^{\prime}\nu^{\prime\ast}-\nu^{\prime}\mu^{\prime\ast}\right)\right]$
$\displaystyle\times\int\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}\exp\left[\left(\mu^{\prime\ast}-\mu^{\ast}\right)\eta+\left(\mu-\mu^{\prime}\right)\eta^{\ast}\right]$
$\displaystyle\times\exp\left[(\nu^{\ast}-\nu^{\prime\ast})\xi+(\nu^{\prime}-\nu)\xi^{\ast}\right]$
$\displaystyle=\int\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}|D\left(\nu,\mu\right)|^{2}\iint
d^{2}\mu^{\prime}d^{2}\nu^{\prime}$
$\displaystyle\times\exp\left[\left(\mu^{\ast}\nu-\nu^{\ast}\mu\right)+\left(\mu^{\prime}\nu^{\prime\ast}-\nu^{\prime}\mu^{\prime\ast}\right)\right]$
$\displaystyle\times\delta^{\left(2\right)}\left(\mu-\mu^{\prime}\right)\delta^{\left(2\right)}\delta(\nu^{\prime}-\nu)$
$\displaystyle=\int\frac{d^{2}\mu
d^{2}\nu}{\pi^{2}}|D\left(\nu,\mu\right)|^{2}.$ (362)
#### 11.3.2 Complex integration transformation and complex Weyl
transformation
In Ref. [86] for correlated two-body systems, we have successfully established
the so-called entangled Wigner operator, expressed in the entangled state
$\left\langle\eta\right|$ representation as (45),
$\Delta\left(\sigma,\gamma\right)\rightarrow\Delta(\eta,\xi)=\int\frac{d^{2}\sigma}{\pi^{3}}\left|\eta-\sigma\right\rangle\left\langle\eta+\sigma\right|e^{\sigma\xi^{\ast}-\sigma^{\ast}\xi},$
(363)
the advantage of introducing $\Delta(\eta,\xi)$ can be seen in Ref. [87]. The
corresponding Wigner function for a density matrix $\rho$ is
$W_{\rho}(\eta,\xi)=\int\frac{d^{2}\eta}{\pi^{3}}\left\langle\eta+\sigma\right|\rho\left|\eta-\sigma\right\rangle
e^{\sigma\xi^{\ast}-\sigma^{\ast}\xi}.$ (364)
If $F(\eta,\xi)$ is quantized as the operator
$F\left(Q_{1},Q_{2},P_{1},P_{2}\right)$ through the Weyl-Wigner correspondence
$F\left(Q_{1},Q_{2},P_{1},P_{2}\right)=\int d^{2}\eta
d^{2}\xi\mathcal{F}(\eta,\xi)\Delta(\eta,\xi),$ (365)
then using (364) we see
$\displaystyle\mathcal{F}(\eta,\xi)$
$\displaystyle=4\pi^{2}\mathtt{Tr}\left[F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\Delta(\eta,\xi)\right]$
$\displaystyle=4\int\frac{d^{2}\sigma}{\pi}e^{\sigma\xi^{\ast}-\sigma^{\ast}\xi}\left\langle\eta+\sigma\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\eta-\sigma\right\rangle,$
(366)
which is named as the complex Weyl transform, and $\mathcal{F}(\eta,\xi)$ is
the Weyl classical correspondence of $F\left(Q_{1},Q_{2},P_{1},P_{2}\right)$.
Substituting (366) into (358) we get
$\displaystyle\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}\mathcal{F}(\eta,\xi)$
$\displaystyle=\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}$
$\displaystyle\times
4\int\frac{d^{2}\sigma}{\pi}e^{\sigma\xi^{\ast}-\sigma^{\ast}\xi}\left\langle\eta+\sigma\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\eta-\sigma\right\rangle$
$\displaystyle=4\int\frac{d^{2}\sigma
d^{2}\eta}{\pi}e^{-\mu\left(\eta^{\ast}-\nu^{\ast}\right)+\mu^{\ast}\left(\eta-\nu\right)}\delta\left(\eta^{\ast}-\nu^{\ast}-\sigma^{\ast}\right)\delta\left(\eta-\nu-\sigma\right)$
$\displaystyle\times\left\langle\eta+\sigma\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\eta-\sigma\right\rangle$
$\displaystyle=4\int\frac{d^{2}\sigma}{\pi}e^{\mu^{\ast}\sigma-\mu\sigma^{\ast}}\left\langle\nu+2\sigma\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\nu\right\rangle.$
(367)
Using (30), we have
$\displaystyle\left\langle\nu+2\sigma\right|$ $\displaystyle=\left\langle
2\sigma\right|\exp\left\\{\frac{i}{\sqrt{2}}\left[\nu_{1}\left(P_{1}-P_{2}\right)-\nu_{2}\left(Q_{1}+Q_{2}\right)\right]\right\\},$
(368) $\displaystyle\nu$ $\displaystyle=\nu_{1}+i\nu_{2}.$
As a result of (368) and
$\frac{1}{2}e^{\mu^{\ast}\sigma-\mu\sigma^{\ast}}=\left\langle\xi_{=\mu}\right|\left.2\sigma\right\rangle,$
we see
$\displaystyle 4\int d^{2}\sigma
e^{\mu^{\ast}\sigma-\mu\sigma^{\ast}}\left\langle\nu+2\sigma\right|$
$\displaystyle=8\int
d^{2}\sigma\left\langle\xi_{=\mu}\right|\left.2\sigma\right\rangle\left\langle
2\sigma\right|\exp\\{\frac{i}{\sqrt{2}}\left[\nu_{1}\left(P_{1}-P_{2}\right)-\nu_{2}\left(Q_{1}+Q_{2}\right)\right]\\}$
$\displaystyle=2\pi\left\langle\xi_{=\mu}\right|\exp\\{\frac{i}{\sqrt{2}}\left[\nu_{1}\left(P_{1}-P_{2}\right)-\nu_{2}\left(Q_{1}+Q_{2}\right)\right]\\}$
$\displaystyle=2\pi\left\langle\xi_{=\mu}\right|e^{i\left(\mu_{2}\nu_{1}-\mu_{1}\nu_{2}\right)}.$
(369)
Using (369), we convert Eq. (367) as
$\displaystyle\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}F(\eta,\xi)$
$\displaystyle=2\pi\left\langle\xi_{=\mu}\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\nu\right\rangle
e^{i\left(\nu_{1}\mu_{2}-\nu_{2}\mu_{1}\right)}.$ (370)
The inverse of (370), according to (360), is
$\displaystyle\mathcal{F}(\eta,\xi)$ $\displaystyle=\iint\frac{2d^{2}\mu
d^{2}\nu}{\pi}e^{\left(\xi^{\ast}-\mu^{\ast}\right)\left(\eta-\nu\right)-\left(\eta^{\ast}-\nu^{\ast}\right)\left(\xi-\mu\right)}$
$\displaystyle\times\left\langle\xi_{=\mu}\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\nu\right\rangle
e^{i\left(\nu_{1}\mu_{2}-\nu_{2}\mu_{1}\right)}.$ (371)
Thus through the new integration transformation, a new relationship between a
phase space function $\mathcal{F}(\eta,\xi)$ and its Weyl-Wigner
correspondence operator $F\left(Q_{1},Q_{2},P_{1},P_{2}\right)$ is revealed.
For example, from the following Weyl-Wigner correspondence
$\frac{4}{\left(e^{f}+1\right)^{2}}\exp\left[\frac{e^{f}-1}{e^{f}+1}(\left|\eta\right|^{2}+\left|\xi\right|^{2})\right]\rightarrow\exp\\{f[K_{+}+K_{-}-1]\\},$
(372)
($K_{+}$ and $K_{-}$ are defined in Eqs.(241) and (244)) and (371) we have
$\displaystyle\frac{4}{\left(e^{f}+1\right)^{2}}\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}$
$\displaystyle\times\exp\left[\frac{e^{f}-1}{e^{f}+1}(\left|\eta\right|^{2}+\left|\xi\right|^{2})\right]$
$\displaystyle=2\pi\left\langle\xi_{=\mu}\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\nu\right\rangle
e^{i\left(\nu_{1}\mu_{2}-\nu_{2}\mu_{1}\right)}.$ (373)
Using the Gaussian integration formula
$\displaystyle\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}e^{-\lambda(\left|\eta\right|^{2}+\left|\xi\right|^{2})}$
$\displaystyle=\frac{1}{1+\lambda^{2}}\exp\left[-\frac{\lambda(\left|\mu\right|^{2}+\left|\nu\right|^{2})}{1+\lambda^{2}}+\frac{\lambda^{2}\left(\mu\nu^{\ast}-\mu^{\ast}\nu\right)}{1+\lambda^{2}}\right],$
(374)
in particular, when
$\lambda=-i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right),$ with
$\frac{-\lambda}{\lambda^{2}+1}=\frac{i}{2\tan\alpha},$
$\frac{\lambda^{2}}{\lambda^{2}+1}=\frac{1}{2}-\frac{1}{2\sin\alpha},$Eq.
(374) becomes
$\displaystyle\iint\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{\left(\xi-\mu\right)\left(\eta^{\ast}-\nu^{\ast}\right)-\left(\eta-\nu\right)\left(\xi^{\ast}-\mu^{\ast}\right)}\exp\left[i\tan(\frac{\pi}{4}-\frac{\alpha}{2})(\left|\eta\right|^{2}+\left|\xi\right|^{2})\right]$
$\displaystyle=\frac{e^{i\alpha}}{i\sin\alpha}\exp\left[\frac{i(\left|\mu\right|^{2}+\left|\nu\right|^{2})}{2\tan\alpha}-\frac{\mu\nu^{\ast}-\mu^{\ast}\nu}{2\sin\alpha}+i\mu_{2}\nu_{1}-i\mu_{1}\nu_{2}\right],$
(375)
where
$\exp[i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)(\left|\eta\right|^{2}+\left|\xi\right|^{2})]$
represents an infinite long chirplet function. By taking
$f=i(\frac{\pi}{2}-\alpha)$ in (373), such that $ie^{-i\alpha}=e^{f},$ and
comparing with (375) we obtain
$\displaystyle\left\langle\xi_{=\mu}\right|F\left(Q_{1},Q_{2},P_{1},P_{2}\right)\left|\nu\right\rangle$
$\displaystyle=\frac{-ie^{i\alpha}}{2\pi\sin\alpha}\exp\left[\frac{i(\left|\mu\right|^{2}+\left|\nu\right|^{2})}{2\tan\alpha}-\frac{\mu\nu^{\ast}-\mu^{\ast}\nu}{2\sin\alpha}\right],$
(376)
where the right-hand side of (376) is just the CFrFT kernel whose properties
can be seen in Ref. [87]. (One may compare the forms (347) and (376) to see
their slight difference. For the relation between them we refer to Ref.[85,
87]). Dragoman has shown that the kernel of the CFrFT can be classically
produced with rotated astigmatic optical systems that mimic the quantum
entanglement. Therefore the new integration transformation (358) can convert
spherical wave to CFrFT kernel. We expect this transformation could be
implemented by experimentalists.
### 11.4 Squeezing for the generalized scaled FrFT
In some practical applications it is necessary to introduce input and output
scale parameters [17, 18] into FrFT, i.e., scaled FrFT. The reason lies in
that two facts: (1) the scaled FrFT may be more useful and convenient for
optical information processing due to the scale parameters (free parameters)
introduced into FrFT; (2) it can be reduced to the conventional FrFT under a
given condition. In this subsection, by establishing the relation between the
optical scaled FrFT and quantum mechanical squeezing-rotating operator
transform in one-mode case, we employ the IWOP technique and the bipartite
entangled state representation of two-mode squeezing operator to extend the
scaled FrFT to more general cases, such as scaled complex FrFT and entangled
scaled FrFT. The properties of scaled FrFTs can be seen more clearly from the
viewpoint of representation transform in quantum mechanics.
#### 11.4.1 Quantum correspondence of the scaled FrFT
The scaled FrFT [76] of $\alpha$-order is defined in a manner such that the
usual FrFT is its special case, i.e.,
$\mathcal{F}_{\alpha}\left[f\left(x\right)\right]=\sqrt{\frac{e^{i\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\mu\nu\sin\alpha}}\int_{-\infty}^{\infty}\exp\left\\{-i\frac{x^{2}/\mu^{2}+y^{2}/\nu^{2}}{2\tan\alpha}+\frac{ixx^{\prime}}{\mu\nu\sin\alpha}\right\\}f\left(x\right)dx,$
(377)
where the exponential function is an integral kernel. In a similar way to
deriving the quantum correspondence of FrFT in (297), and using the natural
repression of single-mode squeezing operator $S_{1}$ in coordinate
representation [88],
$S_{1}\left(\mu\right)=\frac{1}{\sqrt{\mu}}\int_{-\infty}^{\infty}dx\left|\frac{x}{\mu}\right\rangle\left\langle
x\right|,$ (378)
we have
$\displaystyle\exp\left\\{-i\frac{x^{2}/\mu^{2}+y^{2}/\nu^{2}}{2\tan\alpha}+\frac{ixx^{\prime}}{\mu\nu\sin\alpha}\right\\}$
$\displaystyle=\sqrt{-2\pi i\mu\nu e^{i\alpha}\sin\alpha}\left\langle
y\right|S_{1}^{\dagger}\left(\nu\right)\exp\left\\{i\alpha
a^{\dagger}a\right\\}S_{1}\left(\mu\right)\left|x\right\rangle,$ (379)
which implies that the integral kernel in Eq.(377) is just the matrix element
of operator $S_{1}^{\dagger}\left(\nu\right)\exp\left\\{i\theta
a^{\dagger}a\right\\}S_{1}\left(\mu\right)$ in coordinate states. From
Eq.(379) it then follows that
$\displaystyle\mathcal{F}_{\alpha}\left[f\left(x\right)\right]$
$\displaystyle=\int_{-\infty}^{\infty}dx\left\langle
y\right|S_{1}^{\dagger}\left(\nu\right)e^{i\alpha
a^{\dagger}a}S_{1}\left(\mu\right)\left|x\right\rangle f\left(x\right)$
$\displaystyle=\left\langle y\right|S_{1}^{\dagger}\left(\nu\right)e^{i\alpha
a^{\dagger}a}S_{1}\left(\mu\right)\left|f\right\rangle\equiv g\left(y\right),$
(380)
which suggests
$\left|g\right\rangle=S_{1}^{\dagger}\left(\nu\right)e^{i\alpha
a^{\dagger}a}S_{1}\left(\mu\right)\left|f\right\rangle.$ (381)
From Eqs.(380) and (377) one can see that the scaled FrFT in Eq.(380)
corresponds actually to the squeezing-rotating operator
$\left(S_{1}^{\dagger}\left(\nu\right)e^{i\theta
a^{\dagger}a}S_{1}\left(\mu\right)\right)$ transform in Eq.(380) between two
quantum states.
#### 11.4.2 The Scaled CFrFT
On the basis of quantum mechanical version of one-mode scaled FrFT, we
generalize it to two-mode case, i.e., we can introduce the integral
$\displaystyle\mathcal{F}_{\alpha}^{C}\left[f\left(\eta\right)\right]$
$\displaystyle\equiv\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|f\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|\eta\right\rangle
f\left(\eta\right),$ (382)
where $f\left(\eta\right)=$ $\left\langle\eta\right.\left|f\right\rangle$.
Using the natural expression of the two-mode squeezing operator $S_{2}$
(LABEL:3.16), and noticing that
$\left\langle\eta^{\prime}\right|e^{i\theta\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}\left|\eta\right\rangle$
is just the integral kernel of CFrFT (347), we can reform (382) as
$\displaystyle\mathcal{F}_{\alpha}^{C}\left[f\left(\eta\right)\right]$
$\displaystyle=\frac{e^{i\left(\frac{\pi}{2}-\alpha\right)}}{2\mu\nu\sin\alpha}\int\frac{d^{2}\eta}{\pi}f\left(\eta\right)$
$\displaystyle\times\exp\left\\{-\frac{i(\left|\eta^{\prime}\right|^{2}/\nu^{2}+\allowbreak\left|\eta\right|^{2}/\mu^{2})}{2\tan\alpha}+\frac{i\left(\eta^{\prime}{}^{\ast}\allowbreak\eta+\eta^{\ast}\allowbreak\eta^{\prime}\right)}{2\mu\nu\sin\alpha}\right\\}.$
(383)
It is obvious that Eq.(383) is just a generalized CFrFT with squeezing
parameters, we name it the scaled CFrFT. Thus we link a two-mode squeezing-
rotating operator transform to the scaled CFrFT of complex functions.
#### 11.4.3 Entangled scaled FrFT
On the other hand, recall that the entangled state $\left|\eta\right\rangle$
can be Schmidt-decomposed as [89]
$\left|\eta\right\rangle=e^{-i\eta_{1}\eta_{2}}\int_{-\infty}^{\infty}dx\left|x\right\rangle_{1}\otimes\left|x-\sqrt{2}\eta_{1}\right\rangle_{2}e^{i\sqrt{2}x\eta_{2}},$
(384)
we see that
$\displaystyle\left\langle
x_{1}^{\prime},x_{2}^{\prime}\right|\left.\eta^{\prime}\right\rangle$
$\displaystyle=e^{-i\eta_{1}^{\prime}\eta_{2}^{\prime}}\delta\left(\sqrt{2}\eta_{1}^{\prime}+x_{2}^{\prime}-x_{1}^{\prime}\right)e^{i\sqrt{2}x_{1}^{\prime}\eta_{2}^{\prime}},$
$\displaystyle\left\langle\eta\right|\left.x_{1},x_{2}\right\rangle$
$\displaystyle=e^{i\eta_{1}\eta_{2}}\delta\left(\sqrt{2}\eta_{1}+x_{2}-x_{1}\right)e^{-i\sqrt{2}x_{1}\eta_{2}}.$
(385)
Using Eq.(31) we have
$\displaystyle K\left(x_{1}^{\prime},x_{2}^{\prime},x_{1},x_{2}\right)$
$\displaystyle\equiv\left\langle
x_{1}^{\prime},x_{2}^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|x_{1},x_{2}\right\rangle.$
$\displaystyle=\int\frac{d^{2}\eta d^{2}\eta^{\prime}}{\pi^{2}}\left\langle
x_{1}^{\prime},x_{2}^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|\eta\right\rangle\left\langle\eta\right.\left|x_{1},x_{2}\right\rangle,$
(386)
where
$\left|x_{1},x_{2}\right\rangle=\left|x_{1}\right\rangle\otimes\left|x_{2}\right\rangle.$
On substituting Eqs. (384) and (385) into Eq.(386), we can derive
$\displaystyle K\left(x_{1}^{\prime},x_{2}^{\prime},x_{1},x_{2}\right)$
$\displaystyle=\left\\{\sqrt{\frac{e^{i\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\sin\alpha}}\exp\left[-i\frac{\lambda_{\nu}^{\prime
2}+\lambda_{\mu}^{2}}{2\tan\alpha}+\frac{i\lambda_{\mu}\lambda_{\nu}^{\prime}}{\sin\alpha}\right]\right\\}$
$\displaystyle\times\left\\{\sqrt{\frac{e^{i\left(\frac{\pi}{2}-\alpha\right)}}{2\pi\sin\alpha}}\exp\left[-i\frac{\kappa_{\nu}^{\prime
2}+\kappa_{\mu}^{2}}{2\tan\alpha}+\frac{i\kappa_{\mu}\kappa_{\nu}^{\prime}}{\sin\alpha}\right]\right\\},$
(387)
where $\lambda_{\mu}=\frac{x_{1}-x_{2}}{\sqrt{2}\mu},$
$\lambda_{\nu}^{\prime}=\frac{x_{1}^{\prime}-x_{2}^{\prime}}{\sqrt{2}\nu};\kappa_{\mu}=\frac{\mu\left(x_{2}+x_{1}\right)}{\sqrt{2}},\kappa_{\nu}^{\prime}=\frac{\nu\left(x_{1}^{\prime}+x_{2}^{\prime}\right)}{\sqrt{2}}.$
From Eq.(387) one can see that a new 2-dimensional (2D) scaled FrFT can be
composed of one 1D scaled FFT in its space domain and the other in its
”frequency” domain, while the transform variables being the combination of two
coordinates as shown in Eq.(LABEL:13.44), so Eq.(387) is quite different from
the direct product two 1D scaled FrFTs that are both in ‘space domain‘ are
indicated in Eq.(380). Note that the new 2D scaled FFT is still characterized
by only 3-parameter. Therefore, for any function
$f\left(x_{1},x_{2}\right)=\left\langle
x_{1},x_{2}\right.\left|f\right\rangle$ we can define an entangled scaled
FrFT, i.e.,
$\displaystyle\mathcal{F}_{\alpha}^{E}\left[f\left(x_{1},x_{2}\right)\right]$
$\displaystyle=\int_{-\infty}^{\infty}K\left(x_{1}^{\prime},x_{2}^{\prime},x_{1},x_{2}\right)f\left(x_{1},x_{2}\right)dx_{1}dx_{2}$
$\displaystyle=\left\langle
x_{1}^{\prime},x_{2}^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|f\right\rangle.$
(388)
Next we examine the properties of these scaled FrFTs in the quantum optics
context. Without losing generality, for the additivity property, we consider
the scaled CFrFT,
$\mathcal{F}_{\alpha+\beta}^{C}\left[f\left(\eta\right)\right]\equiv\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i(\alpha+\beta)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|\eta\right\rangle
f\left(\eta\right).$ (389)
Inserting the completeness relation of $\left|\eta\right\rangle$ into Eq.(389)
yields
$\displaystyle\mathcal{F}_{\alpha+\beta}^{C}\left[f\left(\eta\right)\right]$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\tau\right)S_{2}^{\dagger}\left(\tau\right)e^{i\beta\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|\eta\right\rangle
f\left(\eta\right)$
$\displaystyle=\int\frac{d^{2}\eta^{\prime\prime}}{\pi}\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu_{=\tau}^{\prime}\right)\left|\eta^{\prime\prime}\right\rangle$
$\displaystyle\times\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime\prime}\right|S_{2}^{\dagger}\left(\nu_{=\tau}^{\prime}\right)e^{i\beta\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\mu\right)\left|\eta\right\rangle
f\left(\eta\right)$
$\displaystyle=\int\frac{d^{2}\eta^{\prime\prime}}{\pi}\left\langle\eta^{\prime}\right|S_{2}^{\dagger}\left(\nu\right)e^{i\alpha\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)}S_{2}\left(\tau\right)\left|\eta_{=\tau}^{\prime\prime}\right\rangle\mathcal{F}_{\beta}\left[f\left(\eta\right)\right]=\mathcal{F}_{\alpha}\mathcal{F}_{\beta}\left[f\left(\eta\right)\right],$
(390)
which is just the additivity property. It should be pointed out that the
condition of additive operator for the scaled FrFTs is that the parameter
$\nu^{\prime}$ of the prior cascade should be equal to the parameter
$\mu^{\prime}$ of the next one, i.e., $\mu^{\prime}=\nu^{\prime}.$ For other
scaled FrFTs, the properties can also be discussed in the similar way
(according to their quantum versions).
To this end, we should emphasize that different scaled FrFTs correspond to
different quantum mechanical squeezing operators or representations. That is
to say, it is possible that some other scaled FrFT can be presented by using
different quantum mechanical squeezing operators or representations.
## 12 Adaption of Collins diffraction formula and CFrFT
The connection between the Fresnel diffraction in free space and the FrFT had
been bridged by Pellat-Finet [90] who found that FrFTs are adapted to the
mathematical expression of Fresnel diffraction, just as the standard Fourier
transform is adapted to Fraunhofer diffraction. In previous sections, a new
formulation of the CFrFT and the Collins diffraction formula are respectively
derived in the context of representation transform of quantum optics. In this
section we inquire if the adaption problem of Collins diffraction formula to
the CFrFT can also be tackled in the context of quantum optics. We shall treat
this topic with the use of two-mode (3 parameters) squeezing operator and in
the entangled state representation of continuous variables, in so doing the
quantum mechanical version of associated theory of classical diffraction and
classical CFrFT is obtained, which connects classical optics and quantum
optics in this aspect.
Figure 2: The Fresnel diffraction through ABCD optical system.
For Gaussian beam, the $ABCD$ rule is equally derived via optical diffraction
integral theory—the Collins integral formula. As shown in Fig.2, if
$f\left(\eta\right)$ represents the input field amplitude at point $\eta$ on
$S_{1}$, and $g\left(\eta^{\prime}\right)$ denotes the diffraction field
amplitude at point $\eta^{\prime}$ on $S_{2},$ then Collins formula in complex
form takes the form (LABEL:11.5,11.7). Next we shall examine adaption of the
Collins formula to the CFrFT by virtue of the entangled state representation
in quantum optics [83].
### 12.1 Adaption of the Collins formula to CFrFT
Using the completeness relation of $\left|\eta\right\rangle,$ we can further
put Eq.(227) into
$g\left(\eta^{\prime}\right)=\left\langle\eta^{\prime}\right|U_{2}\left(r,s\right)\left|f\right\rangle=\left\langle\eta^{\prime}\right|U_{2}\left(r,s\right)\mu_{1}^{2}\int\frac{d^{2}\eta}{\pi}\left|\mu_{1}\eta\right\rangle\left\langle\mu_{1}\eta\right.\left|f\right\rangle,$
(391)
and taking $\eta^{\prime}=\sqrt{\frac{B}{D}}\frac{\sigma}{K},$
$\mu_{1}=\sqrt{\frac{B}{A}}/L$ as well as writing
$g\left(\eta^{\prime}\right)\rightarrow\left\langle\sqrt{\frac{B}{D}}\frac{\sigma}{K}\right|\left.g\right\rangle\equiv
G\left(\sigma\right),\text{ }f\left(\mu_{1}\eta\right)\equiv
F\left(\eta\right),$ (392)
where $K$ and $L$ are two constants to be determined later, then according to
Eqs. (227) and (391) we have
$\displaystyle G\left(\sigma\right)$
$\displaystyle=\mu_{1}^{2}\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|U_{2}\left(r,s\right)\left|\mu_{1}\eta\right\rangle
F\left(\eta\right)$
$\displaystyle=\frac{1}{2iAL^{2}}\exp\left[\frac{i\left|\sigma\right|^{2}}{2K^{2}}\right]\int\frac{d^{2}\eta}{\pi}\exp\left\\{\frac{i\left|\eta\right|^{2}}{2L^{2}}-\frac{i\left(\sigma^{\ast}\eta+\sigma\eta^{\ast}\right)}{2LK\sqrt{AD}}\right\\}F\left(\eta\right).$
(393)
Comparing Eq.(393) with Eq.(229) leads us to choose
$L^{2}=\tan\alpha,\ K=\sqrt{\sin 2\alpha/\left(2AD\right)}.$ (394)
Then Eq.(393) becomes
$\displaystyle G\left(\sigma\right)$
$\displaystyle=\frac{\cos\alpha}{i2A\sin\alpha}\exp\left[i\frac{AD-\cos^{2}\alpha}{\sin
2\alpha}\left|\sigma\right|^{2}\right]$
$\displaystyle\times\int\frac{d^{2}\eta}{\pi}\exp\left\\{\frac{i\left(\left|\eta\right|^{2}+\left|\sigma\right|^{2}\right)}{2\tan\alpha}-\frac{i\left(\sigma^{\ast}\eta+\sigma\eta^{\ast}\right)}{2\sin\alpha}\right\\}F\left(\eta\right)$
$\displaystyle=\frac{\cos\alpha}{A}e^{-i\alpha}\exp\left[i\frac{AD-\cos^{2}\alpha}{\sin
2\alpha}\left|\sigma\right|^{2}\right]\mathcal{F}_{\alpha}\left[F\right]\left(\sigma\right),$
(395)
so Eq. (395) is a standard CFrFT up to a quadratic phase term
$\exp\left[i\frac{AD-\cos^{2}\alpha}{\sin
2\alpha}\left|\sigma\right|^{2}\right]$, according to Eq.(394) and
$\sqrt{\frac{B}{D}}\frac{\sigma}{K}=\eta^{\prime}$, it can also be written as
$\exp\left[i\frac{AD-\cos^{2}\alpha}{\sin
2\alpha}\left|\sigma\right|^{2}\right]=\exp\left[\frac{i}{R}\left|\eta^{\prime}\right|^{2}\right],$
(396)
which represents a quadratic approximation to a sphere wave diverging from a
luminous point at distance
$R=\frac{2AB}{AD-\cos^{2}\alpha}$ (397)
from $S_{2}.$ Let $S$ be the sphere tangent to $S_{2}$ with radius $R$ (see
Fig.2). A point on $S$ is located by its projection on $S_{2}$, this means
that coordinates on $S_{2}$ can also be used as coordinates on $S$. Therefore,
the quadratic phase term can be compensated if the output field is observed on
$S$ but $S_{2}$. Then, after considering the phase compensation, the field
transforms from $S_{1}$ to $S$ is
$G_{S}\left(\sigma\right)=\frac{\cos\alpha}{A}e^{-i\alpha}\mathcal{F}_{\alpha}\left[F\right]\left(\sigma\right),$
(398)
In this way, the field amplitude on $S$ is the perfect $\alpha-th$ FFT-C of
the field amplitude on $S_{1}$.
### 12.2 Adaption of the additivity property of CFrFT to the Collins formula
for two successive Fresnel diffractions
The most important property of FrFT is that $\mathcal{F}_{\alpha}$ obeys the
additivity rule, i.e., two successive FrFT of order $\alpha$ and $\beta$ makes
up the FFT of order $\alpha+\beta$. For the CFrFT, its additivity property is
proven in Eq.(351). For Collins diffraction from $S_{1}$ to $S^{\prime}$ (see
Fig.1), the additivity means that the diffraction pattern observed on
$S^{\prime}$($\bar{\eta}$) (the sphere tangent to $S_{3}$ with radius
$R^{\prime}$) and associated with $\mathcal{F}_{\alpha+\beta}$ should be the
result of a first diffraction phenomenon (associated with
$\mathcal{F}_{\alpha})$ on $S$ (with $\eta^{\prime}$), followed by a second
diffraction phenomenon (associated with $\mathcal{F}_{\beta})$ from $S$ to
$S^{\prime}$. This is a necessary consequence of the Huygens principle. Next
we prove that such is indeed the case.
Firstly, let us consider the field transform from $S_{1}$ (with $\eta$) to
$S^{\prime}$ (see Fig.2) described by the ray transfer matrix
[$A^{\prime},B^{\prime},C^{\prime},D^{\prime}$]. Similar to deriving Eq.(397),
after the squeezing transform and the phase compensation,
$R^{\prime}=\frac{2A^{\prime}B^{\prime}}{A^{\prime}D^{\prime}-\cos^{2}\alpha^{\prime}}\rightarrow\exp\left(\frac{i}{R^{\prime}}\left|\bar{\eta}\right|^{2}\right),$
(399)
thus we can obtain the expression of CFrFT for Collins diffraction from
$S_{1}$ to $S^{\prime}$ (not $S_{3}$),
$G_{S^{\prime}}\left(\sigma^{\prime}\right)=\frac{\cos\alpha^{\prime}}{A^{\prime}}e^{-i\alpha^{\prime}}\mathcal{F}_{\alpha^{\prime}}\left[f\left(\mu_{1}^{\prime}\eta\right)\right]\left(\sigma^{\prime}\right),$
(400)
where
$\bar{\eta}=\sqrt{\frac{B^{\prime}}{D^{\prime}}}\frac{\sigma^{\prime}}{K^{\prime}}\
$and $\mu_{1}^{\prime}=\sqrt{\frac{B^{\prime}}{A^{\prime}}}/L^{\prime},$
$L^{\prime 2}=\tan\alpha^{\prime},\ K^{\prime}=\sqrt{\sin
2\alpha^{\prime}/\left(2A^{\prime}D^{\prime}\right)}.$ (401)
Eq.(400) is the same in form as Eq.(398) but with primed variables. Using
Eqs.(399) and (227) one can prove that the transform from $S_{1}$ to
$S^{\prime}$ is (see Eqs. (227), (395)-(397))
$g_{S^{\prime}}\left(\bar{\eta}\right)=\exp\left(-\frac{i}{R^{\prime}}\left|\bar{\eta}\right|^{2}\right)\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime},s^{\prime}\right)\left|f\right\rangle\equiv
G_{S^{\prime}}\left(\sigma^{\prime}\right).$ (402)
In Eq.(402) we have taken the phase compensation term (399) into account.
Secondly, let us consider the second diffraction from $S$ to $S_{3}$
determined by the ray transfer matrix
[$A^{\prime\prime},B^{\prime\prime},C^{\prime\prime},D^{\prime\prime}$]. For
this purpose, using the group multiplication rule of $F_{2}\left(r,s\right)$,
we can decompose the diffraction from $S_{1}$ to $S^{\prime}$ into two parts:
one is described as the matrix $[A,B,C,D]$ from plane $S_{1}$ (with $\eta$) to
$S_{2}\left(S\right)$ (with $\eta^{\prime}$), the other is
$[A^{\prime\prime},B^{\prime\prime},C^{\prime\prime},D^{\prime\prime}]$ from
plane $S_{2}$ to $S_{3}\left(S^{\prime}\right)$ (with $\bar{\eta}$), then the
total matrix from $S_{1}$ to $S_{3}$ is
$\left(\begin{array}[c]{cc}A^{\prime}&B^{\prime}\\\
C^{\prime}&D^{\prime}\end{array}\right)=\left(\begin{array}[c]{cc}A^{\prime\prime}&B^{\prime\prime}\\\
C^{\prime\prime}&D^{\prime\prime}\end{array}\right)\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right).$ (403)
Using Eq.(31) and the group multiplication rule of
$F_{2}\left(r^{\prime},s^{\prime}\right)$, we can further put Eq.(402) into
another form
$\displaystyle G_{S^{\prime}}\left(\sigma^{\prime}\right)$
$\displaystyle=\exp\left[-\frac{i}{R^{\prime}}\left|\bar{\eta}\right|^{2}\right]\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime\prime},s^{\prime\prime}\right)U_{2}\left(r,s\right)\left|f\right\rangle$
$\displaystyle=\exp\left[-\frac{i}{R^{\prime}}\left|\bar{\eta}\right|^{2}\right]\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime\prime},s^{\prime\prime}\right)\mu_{2}^{\prime
2}\int\frac{d^{2}\sigma}{\pi}\left|\mu_{2}^{\prime}\sigma\right\rangle\left\langle\mu_{2}^{\prime}\sigma\right|U_{2}\left(r,s\right)\mu_{1}^{\prime
2}\int\frac{d^{2}\eta}{\pi}\left|\mu_{1}^{\prime}\eta\right\rangle\left\langle\mu_{1}^{\prime}\eta\right.\left|f\right\rangle$
$\displaystyle=\mu_{2}^{\prime
2}\int\frac{d^{2}\sigma}{\pi}\exp\left(-\frac{i\left|\bar{\eta}\right|^{2}}{R^{\prime}}\right)\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime\prime},s^{\prime\prime}\right)\left|\mu_{2}^{\prime}\sigma\right\rangle\left[\mu_{1}^{\prime
2}\int\frac{d^{2}\eta}{\pi}\left\langle\mu_{2}^{\prime}\sigma\right|U_{2}\left(r,s\right)\left|\mu_{1}^{\prime}\eta\right\rangle
F\left(\eta\right)\right]$
$\displaystyle=\frac{B}{DK^{2}}\int\frac{d^{2}\sigma}{\pi}\exp\left(-\frac{i\left|\bar{\eta}\right|^{2}}{R^{\prime}}\right)\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime\prime},s^{\prime\prime}\right)\left|\mu_{2}^{\prime}\sigma\right\rangle
G\left(\sigma\right),$ (404)
where $\mu_{2}^{\prime}=\sqrt{\frac{B}{D}}\frac{1}{K}$ and we have made a
reasonable assumption that $\mu_{1}^{\prime}=\mu_{1}$(so
$f\left(\mu_{1}^{\prime}\eta\right)=F\left(\eta\right)$), which means that
there are same scaled variants for the input field amplitudes on $S_{1}$ of
the diffractions from $S_{1}$ to $S$ and from $S_{1}$ to $S^{\prime}$.
In order to examine the second diffraction domain from $S$ to $S^{\prime}$
(not $S_{3}$), we need to translate the output field amplitude
$G\left(\sigma\right)$ observed on plane $S_{2}$ to the field amplitude
observed on sphere plane $S,$ i.e., putting $G\left(\sigma\right)$ into
$G_{S}\left(\sigma\right)$ (see Eq.(398)) by taking the phase compensation
(see Eq.(393)) into account. Thus the field transform from $S$ to $S^{\prime}$
is ($G_{S}\left(\sigma\right)\rightarrow
G_{S^{\prime}}\left(\sigma^{\prime}\right)$)
$\displaystyle G_{S^{\prime}}\left(\sigma^{\prime}\right)$
$\displaystyle=\frac{B}{DK^{2}}\int\frac{d^{2}\sigma}{\pi}\exp\left(\frac{i}{R}\frac{B\left|\sigma\right|^{2}}{DK^{2}}-\frac{i\left|\bar{\eta}\right|^{2}}{R^{\prime}}\right)\left\langle\bar{\eta}\right|U_{2}\left(r^{\prime\prime},s^{\prime\prime}\right)\left|\mu_{2}^{\prime}\sigma\right\rangle
G_{S}\left(\sigma\right)$
$\displaystyle=\frac{B}{DK^{2}}\frac{1}{2iB^{\prime\prime}}\int\frac{d^{2}\sigma}{\pi}\exp\left\\{\frac{iB^{\prime}}{K^{\prime
2}D^{\prime}}\left(\frac{D^{\prime\prime}}{2B^{\prime\prime}}-\frac{1}{R^{\prime}}\right)\left|\sigma^{\prime}\right|^{2}\right.$
$\displaystyle\left.+\frac{iB\left|\sigma\right|^{2}}{DK^{2}}\left(\frac{A^{\prime\prime}}{2B^{\prime\prime}}+\frac{1}{R}\right)-\frac{i\left(\sigma\sigma^{\prime\ast}+\sigma^{\prime}\sigma^{\ast}\right)}{2B^{\prime\prime}K^{\prime}K\sqrt{\frac{DD^{\prime}}{BB^{\prime}}}}\right\\}G_{S}\left(\sigma\right).$
(405)
Comparing Eq.(405) with Eq.(229) leads us to choose
$\sin\beta=B^{\prime\prime}K^{\prime}K\sqrt{\frac{DD^{\prime}}{BB^{\prime}}}=\frac{B^{\prime\prime}}{2}\sqrt{\frac{\sin
2\alpha^{\prime}\sin 2\alpha}{A^{\prime}ABB^{\prime}}},$ (406)
and noticing that $\mu_{1}^{\prime}=\mu_{1}$ yields
$\frac{B^{\prime}}{A^{\prime}\tan\alpha^{\prime}}=\frac{B}{A\tan\alpha},$ thus
we have
$\text{
}A^{\prime}=\frac{B^{\prime\prime}}{B}\frac{\cos\alpha^{\prime}\sin\alpha}{\sin\beta},\text{
}A=\frac{B^{\prime\prime}}{B^{\prime}}\frac{\sin\alpha^{\prime}\cos\alpha}{\sin\beta}.$
(407)
Combining Eqs.(391) and (407) it then follows (letting
$\alpha^{\prime}=\alpha+\beta$)
$\frac{B}{DK^{2}}\frac{1}{2iB^{\prime\prime}}=\frac{AB}{\sin
2\alpha}\frac{1}{iB^{\prime\prime}}=\frac{A}{iA^{\prime}\cos\alpha}\frac{\cos\alpha^{\prime}}{2\sin\beta},$
(408)
and
$\displaystyle\frac{B^{\prime}}{K^{\prime
2}D^{\prime}}\left(\frac{D^{\prime\prime}}{2B^{\prime\prime}}-\frac{1}{R^{\prime}}\right)$
$\displaystyle=\frac{1}{2}\cot\beta,$
$\displaystyle\frac{B}{DK^{2}}\left(\frac{A^{\prime\prime}}{2B^{\prime\prime}}+\frac{1}{R}\right)$
$\displaystyle=\frac{1}{2}\cot\beta,$ (409)
where we have used Eqs.(397), (399) and (403).
Substitution of Eqs.(398), (406), (408) and (409) into Eq.(405) yields
$\displaystyle G_{S^{\prime}}\left(\sigma^{\prime}\right)$
$\displaystyle=\frac{A}{iA^{\prime}\cos\alpha}\frac{\cos\alpha^{\prime}}{2\sin\beta}\int\frac{d^{2}\sigma}{\pi}\exp\left\\{\frac{i\left(\left|\sigma^{\prime}\right|^{2}+\left|\sigma\right|^{2}\right)}{2\tan\beta}-\frac{i\left(\sigma\sigma^{\prime\ast}+\sigma^{\prime}\sigma^{\ast}\right)}{2\sin\beta}\right\\}G_{S}\left(\sigma\right)$
$\displaystyle=\frac{\cos\alpha^{\prime}e^{-i\alpha}}{iA^{\prime}}\frac{1}{2\sin\beta}\int\frac{d^{2}\sigma}{\pi}\exp\left\\{\frac{i\left(\left|\sigma^{\prime}\right|^{2}+\left|\sigma\right|^{2}\right)}{2\tan\beta}-\frac{i\left(\sigma\sigma^{\prime\ast}+\sigma^{\prime}\sigma^{\ast}\right)}{2\sin\beta}\right\\}\mathcal{F}_{\alpha}\left[F\right]\left(\sigma\right)$
$\displaystyle=\frac{\cos\alpha^{\prime}}{A^{\prime}}e^{-i\left(\alpha+\beta\right)}\mathcal{F}_{\beta}\mathcal{F}_{\alpha}\left[F\right]\left(\sigma^{\prime}\right).$
(410)
The first equation of Eq.(410) indicates that it is just a CFrFT of
$G_{S}\left(\sigma\right)$ from $S$ to $S^{\prime}.$ Comparing Eq.(410) with
Eq.(400), we see
$\mathcal{F}_{\beta}\mathcal{F}_{\alpha}\left[F\right]\left(\sigma^{\prime}\right)=\mathcal{F}_{\alpha+\beta}\left[F\right]\left(\sigma^{\prime}\right).$
(411)
Thus we complete the study of adaption of CFrFT to the mathematical
representation of Collins diffraction formula in quantum optics context.
## 13 The Fractional Radon transform
Optical tomographic imaging techniques derive two-dimensional data from a
three-dimensional object to obtain a slice image of the internal structure and
thus have the ability to peer inside the object noninvasively. The
mathematical method which complete this task is the Radon transformation.
Similarly, one can use the inverse Radon transformation to obtain the Wigner
distribution by tomographic inversion of a set of measured probability
distributions of the quadrature amplitude [91, 92]. Based on the Radon
transform [93] and the FrFT we can introduce the conception of fractional
Radon transformation (FRT) which combines both of them in a reasonable way. We
notice the well-known fact that the usual Radon transform of a function
$f\left(\vec{r}\right)$ can be proceeded in two successive steps, the first
step is an $n-$dimensional ordinary Fourier transform, i.e. performing a usual
FT of $f\left(\vec{r}\right)$ in $n$-dimensional $\vec{k}$ space,
$F\left(\vec{k}\right)=F\left(t\hat{e}\right)=\int
f\left(\vec{r}\right)e^{-2\pi i\vec{k}\cdot\vec{r}}d\vec{r},$ (412)
where $\vec{k}=t\hat{e},$ $\hat{e}$ is a unit vector, $t$ is a real number.
Its inverse is
$f\left(\vec{r}\right)=\int F\left(\vec{k}\right)e^{2\pi
i\vec{k}\cdot\vec{r}}d\vec{k}.$ (413)
Letting $s=t\lambda$ and rewriting (412) as
$F\left(t\hat{e}\right)=\int_{-\infty}^{\infty}ds\int
d\vec{r}f\left(\vec{r}\right)e^{-2\pi
is}\delta\left(s-\vec{k}\cdot\vec{r}\right)=\int_{-\infty}^{\infty}d\lambda
e^{-2\pi it\lambda}\int
f\left(\vec{r}\right)\delta\left(\lambda-\hat{e}\cdot\vec{r}\right)d\vec{r},$
(414)
one can see that the integration over $d\vec{r}$ has been defined as a Radon
transform of $f\left(\vec{r}\right)$, denoted as
$\int
f\left(\vec{r}\right)\delta\left(\lambda-\hat{e}\cdot\vec{r}\right)d\vec{r}=f_{R}\left(\lambda,\hat{e}\right).$
(415)
So $F\left(t\hat{e}\right)$ can be considered as a $1-$dimensional Fourier
transform of $f_{R}\left(\lambda,\hat{e}\right),$
$F\left(t\hat{e}\right)=\int_{-\infty}^{\infty}d\lambda e^{-2\pi
it\lambda}f_{R}\left(\lambda,\hat{e}\right).$ (416)
Its inverse transform is
$f_{R}\left(\lambda,\hat{e}\right)=\int_{-\infty}^{\infty}F\left(t\hat{e}\right)e^{2\pi
it\lambda}dt,$ (417)
this ordinary $1-$dimensional Fourier transform is considered as the second
step. Combining result of (412) and (417) we have
$f_{R}\left(\lambda,\hat{e}\right)=\int_{-\infty}^{\infty}\int
f\left(\vec{r}\right)e^{-2\pi it\hat{e}\cdot\vec{r}}e^{2\pi
it\lambda}d\vec{r}dt.$ (418)
i. e. two usual FTs make up a Radon transform, The inverse of (418) is
$\int_{-\infty}^{\infty}\int f_{R}\left(\lambda,\hat{e}\right)e^{2\pi
i\vec{k}\cdot\vec{r}}e^{-2\pi
it\lambda}d\vec{k}d\lambda=f\left(\vec{r}\right).$ (419)
By analogy with these procedures we can make two successively FRFTs to realize
the new fractional Radon transformation [FRT] . The $n$-dimensional FrFT of
$f\left(\vec{r}\right)$ is defined as
$\mathfrak{F}_{\alpha,\vec{k}}\left[f\right]=\left(C_{\alpha}\right)^{n}\int\exp\left(\frac{i\left(\vec{r}^{2}+\vec{k}^{2}\right)}{2\tan\alpha}-\frac{i\vec{k}\cdot\vec{r}}{\sin\alpha}\right)f\left(\vec{r}\right)d\vec{r}\equiv
F_{\alpha}\left(t\hat{e}\right),\text{ \ }\vec{k}=t\hat{e}.$ (420)
where $\alpha$ is named as the order of FrFT,
$C_{\alpha}=\left[\frac{e^{i\alpha}}{2\pi i\sin\alpha}\right]^{1/2}.$
Firstly, we perform an $1-$dimensional inverse fractional Fourier transform
for $F_{\alpha}\left(t\hat{e}\right)\ $in $t$-space,
$\displaystyle f_{R,\alpha}\left(\lambda,\hat{e}\right)$
$\displaystyle=\left[C_{\alpha}\right]^{1-n}\mathfrak{F}_{-\alpha,t}\left[F_{\alpha}\left(t\hat{e}\right)\right]$
$\displaystyle=\left[C_{\alpha}\right]^{1-n}C_{-\alpha}\int_{-\infty}^{\infty}\exp\left(-\frac{i\left(\lambda^{2}+t^{2}\right)}{2\tan\alpha}+\frac{i\lambda
t}{\sin\alpha}\right)F_{\alpha}\left(t\hat{e}\right)dt,$ (421)
$\left[C_{\alpha}\right]^{1-n}$ was introduced for later’s convenience. Then
substituting (420) into (421) we have
$\displaystyle f_{R,\alpha}\left(\lambda,\hat{e}\right)$
$\displaystyle=\left[C_{\alpha}\right]^{1-n}C_{-\alpha}\left(C_{\alpha}\right)^{n}\int\exp\left(-\frac{i\left(t^{2}+\lambda^{2}\right)}{2\tan\alpha}+\frac{i\lambda
t}{\sin\alpha}+\frac{i\left(\vec{r}^{2}+t^{2}\right)}{2\tan\alpha}-\frac{it\hat{e}\cdot\vec{r}}{\sin\alpha}\right)f\left(\vec{r}\right)d\vec{r}dt$
$\displaystyle=\int\exp\left(\frac{i\left(\vec{r}^{2}-\lambda^{2}\right)}{2\tan\alpha}\right)\delta\left(\lambda-\hat{e}\cdot\vec{r}\right)f\left(\vec{r}\right)d\vec{r},$
(422)
which completes the $n$-dimensional fractional Radon transformation.
Especially, when $\alpha=\pi/2,$ (422) reduces to the usual Radon transform
(415). Now we examine if the additive property of FrFT is consistent with
(422). According to the additive property of FrFT
$\mathfrak{F}_{\alpha}\mathfrak{F}_{\alpha}=\mathfrak{F}_{\alpha+\beta},$ and
(420) we see
$\displaystyle F_{\alpha+\beta}\left(\vec{k}=t\hat{e}\right)$
$\displaystyle=\mathfrak{F}_{\beta,\vec{k}}\mathfrak{F}_{\alpha,\vec{\xi}}\left[f\right]$
$\displaystyle=\left(C_{\beta}\right)^{n}\left(C_{\alpha}\right)^{n}\int\int\exp\left(\frac{i\left(\vec{r}^{2}+\vec{\xi}^{2}\right)}{2\tan\alpha}+\frac{i\left(\vec{\xi}^{2}+\vec{k}^{2}\right)}{2\tan\beta}\right)$
$\displaystyle\times\exp\left(-\frac{i\vec{\xi}\cdot\vec{r}}{\sin\alpha}-\frac{i\vec{k}\cdot\vec{\xi}}{\sin\beta}\right)f\left(\vec{r}\right)d\vec{r}d\vec{\xi}$
$\displaystyle=\mathfrak{F}_{\beta+\alpha,\vec{k}}\left[f\right].$ (423)
The corresponding one-dimensional inverse FrFT should be
$\displaystyle\left[\frac{e^{i\left(\alpha+\beta\right)}}{2\pi
i\sin\left(\alpha+\beta\right)}\right]^{\left(1-n\right)/2}C_{-\beta}C_{-\alpha}\int_{-\infty}^{\infty}\exp\left(-\frac{i\left(\lambda^{2}+\mu^{2}\right)}{2\tan\beta}-\frac{i\left(t^{2}+\mu^{2}\right)}{2\tan\alpha}\right)$
$\displaystyle\times\exp\left(+\frac{i\lambda\mu}{\sin\beta}+\frac{i\mu
t}{\sin\alpha}\right)F_{\alpha+\beta}\left(t\hat{e}\right)dtd\mu$
$\displaystyle=\left[\frac{e^{i\left(\alpha+\beta\right)}}{2\pi
i\sin\left(\alpha+\beta\right)}\right]^{\left(1-n\right)/2}C_{-\left(\alpha+\beta\right)}\int_{-\infty}^{\infty}F_{\alpha+\beta}\left(t\hat{e}\right)\exp\left(-\frac{i\left(t^{2}+\lambda^{2}\right)}{2\tan\left(\alpha+\beta\right)}+\frac{i\lambda
t}{\sin\left(\alpha+\beta\right)}\right)dt$
$\displaystyle=\int\exp\left(\frac{i\left(\vec{r}^{2}-\lambda^{2}\right)}{2\tan\left(\alpha+\beta\right)}\right)\delta\left(\lambda-\hat{e}\cdot\vec{r}\right)f\left(\vec{r}\right)d\vec{r}=f_{R,\alpha+\beta}\left(\lambda,\hat{e}\right),$
(424)
which coincides with (422). From (422) and (424), we can confirm that the
transform kernel of $\alpha$th FrFT is
$\exp\left(\frac{i\left(\vec{r}^{2}-\lambda^{2}\right)}{2\tan\alpha}\right)\delta\left(\lambda-\hat{e}\cdot\vec{r}\right).$
(425)
For example, one can calculate the fractional Radon transform of the $n-$mode
Wigner operator to obtain some new quantum mechanical representations. Finally
we give the inversion of the fractional Radon transformation, From (422) we
have
$\frac{1}{\left(2\pi\sin^{2}\alpha\right)^{n/2}}\int\int
f_{R,\alpha}\left(\lambda,\hat{e}\right)\exp\left(\frac{i\left(\lambda^{2}-\vec{r}^{2}\right)}{2\tan\alpha}-\frac{i\lambda
t}{\sin\alpha}+\frac{it\hat{e}\cdot\vec{r}}{\sin\alpha}\right)d\vec{k}d\lambda=f\left(\vec{r}\right),$
(426)
which is an extension of (419).
In summary, based on the Radon transform and fractional Fourier transform we
have naturally introduced the $n$-dimensional FRFT, in Ref. Zalevsky and
Mendlovic [94] also defined 2-dimensional FRFT, but in different approach. We
have identified the transform kernel for FrFT. The generalization to complex
fractional Radon transformation is also possible [95].
## 14 Wavelet transformation and the IWOP technique
In recent years wavelet transforms [96, 97] have been developed which can
overcome some shortcomings of the classical Fourier analysis and therefore has
been widely used in Fourier optics and information science since 1980s. Here
we present a quantum optical version of classical wavelet transform (WT) by
virtue of the IWOP technique.
### 14.1 Quantum optical version of classical WTs
A wavelet has its energy concentrated in time to give a tool for the analysis
of transient, nonstationary, or time-varying phenomena. (It is a wavelet
because it is localized and it resembles a wave because it oscillates.)
Mathematically, wavelets are defined by starting with a function $\psi$ of the
real variable $x$, named a mother wavelet which is required to decrease
rapidly to zero as $|x|$ tends to infinity,
$\int_{-\infty}^{\infty}\psi\left(x\right)dx=0,$ (427)
A more general requirement for a mother wavelet is to demanded
$\psi\left(x\right)$ to have vanishing moments
$\int_{-\infty}^{\infty}x^{l}\psi\left(x\right)dx=0,$ $l=0,1,2...,L.$ (A
greater degree of smoothness than continuity also leads to vanishing moments
for the mother wavelet). The theory of wavelets is concerned with the
representation of a function in terms of a two-parameter family of dilates and
translates of a fixed function. The mother wavelet $\psi$ generates the other
wavelets of the family $\psi_{\left(\mu,s\right)}$, ($\mu$ is scaling
parameter, $s$ is a translation parameter, $s\in\mathrm{R),}$ the dilated-
translated function is defined as
$\psi_{\left(\mu,s\right)}\left(x\right)=\frac{1}{\sqrt{\left|\mu\right|}}\psi\left(\frac{x-s}{\mu}\right),$
(428)
while the wavelet integral transform of a signal function $f\left(x\right)\in
L^{2}\left(\mathrm{R}\right)$ by $\psi$ is defined by
$W_{\psi}f\left(\mu,s\right)=\frac{1}{\sqrt{\left|\mu\right|}}\int_{-\infty}^{\infty}f\left(x\right)\psi^{\ast}\left(\frac{x-s}{\mu}\right)dx.$
(429)
We can express (429) as
$W_{\psi}f\left(\mu,s\right)=\left\langle\psi\right|U\left(\mu,s\right)\left|f\right\rangle.$
(430)
where $\left\langle\psi\right|$ is the state vector corresponding to the given
mother wavelet, $\left|f\right\rangle$ is the state to be transformed, and
$U\left(\mu,s\right)\equiv\frac{1}{\sqrt{\left|\mu\right|}}\int_{-\infty}^{\infty}\left|\frac{x-s}{\mu}\right\rangle\left\langle
x\right|dx$ (431)
is the squeezing-translating operator [96, 98, 99], $\left\langle x\right|$ is
the eigenvector of coordinate operator. In order to combine the wavelet
transform with quantum states transform more tightly, using the IWOP technique
we can directly perform the integral in (431)
($Q=(a+a^{{\dagger}})/\sqrt{2},\mu>0$)
$\displaystyle U\left(\mu,s\right)$
$\displaystyle=\frac{1}{\sqrt{\pi\mu}}\int_{-\infty}^{\infty}dx\colon\exp\left[-\frac{\mu^{2}+1}{2\mu^{2}}x^{2}+\frac{xs}{\mu^{2}}+\sqrt{2}\frac{x-s}{\mu}a^{\dagger}+\sqrt{2}xa-\frac{s^{2}}{2\mu^{2}}-Q^{2}\right]\colon$
$\displaystyle=\sqrt{\frac{2\mu}{1+\mu^{2}}}\colon\exp\left[\frac{1}{2\left(1+\mu^{2}\right)}\left(\frac{s}{\mu}+\sqrt{2}a^{\dagger}+\sqrt{2}\mu
a\right)^{2}-\sqrt{2}\frac{s}{\mu}a^{\dagger}-\frac{s^{2}}{2\mu^{2}}-Q^{2}\right]\colon.$
(432)
This is the explicitly normal product form. Let $\mu=e^{\lambda}$,
$\operatorname{sech}\lambda=\frac{2\mu}{1+\mu^{2}},$
$\tanh\lambda=\frac{\mu^{2}-1}{\mu^{2}+1},$ using the operator identity
$e^{ga^{\dagger}a}=\colon\exp\left[\left(e^{g}-1\right)a^{\dagger}a\right]\colon,$
Eq. (432) becomes
$\displaystyle U\left(\mu,s\right)$
$\displaystyle=\exp\left[\frac{-s^{2}}{2\left(1+\mu^{2}\right)}-\frac{a^{\dagger
2}}{2}\tanh\lambda-\frac{a^{\dagger}s}{\sqrt{2}}\operatorname{sech}\lambda\right]$
$\displaystyle\times\exp\left[\left(a^{\dagger}a+\frac{1}{2}\right)\ln\operatorname{sech}\lambda\right]$
$\displaystyle\times\exp\left[\frac{a^{2}}{2}\tanh\lambda+\frac{sa}{\sqrt{2}}\operatorname{sech}\lambda\right].$
(433)
In particular, when $s=0$, it reduces to the well-known squeezing operator,
$U\left(\mu,0\right)=\frac{1}{\sqrt{\mu}}\int_{-\infty}^{\infty}\left|\frac{x}{\mu}\right\rangle\left\langle
x\right|dx=\exp[\frac{\lambda}{2}\left(a^{2}-a^{\dagger 2}\right).$ (434)
For a review of the squeezed state theory we refer to [9].
### 14.2 The condition of mother wavelet in the context of quantum optics
Now we analyze the condition (427) for mother wavelet from the point of view
of quantum optics. Due to
$\int_{-\infty}^{\infty}\left|x\right\rangle dx=\left|p=0\right\rangle,$ (435)
where $\left|p\right\rangle$ is the momentum eigenstate, we can recast the
condition into quantum mechanics as
$\int_{-\infty}^{\infty}\psi\left(x\right)dx=0\rightarrow\left\langle
p=0\right|\left.\psi\right\rangle=0,$ (436)
which indicates that the probability of a measurement of
$\left|\psi\right\rangle$ by the projection operator
$\left|p\right\rangle\left\langle p\right|$ with value $p=0$ is zero. Without
loss of generality, we suppose
$\left|\psi\right\rangle_{M}=G\left(a^{\dagger}\right)\left|0\right\rangle=\sum_{n=0}^{\infty}g_{n}a^{\dagger
n}\left|0\right\rangle,$ (437)
where $g_{n}$ are such chosen as to letting $\left|\psi\right\rangle$ obeying
the condition (427). Using the coherent states’ overcompleteness relation we
have
$\displaystyle\left\langle p=0\right|\left.\psi\right\rangle$
$\displaystyle=\left\langle
p=0\right|\int\frac{d^{2}z}{\pi}\left|z\right\rangle\left\langle
z\right|\sum\limits_{n}g_{n}a^{\dagger n}\left|0\right\rangle$
$\displaystyle=\sum\limits_{n}g_{n}\int\frac{d^{2}z}{\pi}e^{-|z|^{2}}z^{\ast
n}\sum\limits_{m}\frac{\left(\frac{z^{2}}{2}\right)^{m}}{m!}$
$\displaystyle=\sum\limits_{m}\sum\limits_{n}\frac{1}{m!2^{m}}g_{n}\delta_{n,2m}n!=\sum\limits_{n}g_{2n}=0.$
(438)
Eq.(438) provides a general formalism to find the qualified wavelets. For
example, assuming $g_{2n}=0$ for $n>3$, so the coefficients of the survived
terms should satisfy
$g_{0}+g_{2}+3g_{4}+15g_{6}=0,$ (439)
and $\left|\psi\right\rangle$ becomes
$\left|\psi\right\rangle=\left(g_{0}+g_{2}a^{\dagger 2}+g_{4}a^{\dagger
4}+g_{6}a^{\dagger 6}\right)\left|0\right\rangle.$ (440)
Projecting it onto the coordinate representation, we get the qualified
wavelets
$\displaystyle\psi\left(x\right)$
$\displaystyle=\pi^{-1/4}e^{-x^{2}/2}\left[g_{0}+g_{2}\left(2x^{2}-1\right)+g_{4}\left(4x^{4}-12x^{2}+3\right)\right.$
$\displaystyle\left.+g_{6}\left(8x^{6}-60x^{4}+90x^{2}-15\right)\right],$
(441)
where we have used $\left\langle
x\right.\left|n\right\rangle=\left(2^{n}n!\sqrt{\pi}\right)^{-1/2}H_{n}\left(x\right)e^{-x^{2}/2}$,
and $H_{n}\left(x\right)$ is the Hermite polynomials. Now we take some
examples.
Case 1: in (438) by taking $g_{0}=\frac{1}{2},$ $g_{2}=-\frac{1}{2},$
$g_{2n}=0$ (otherwise), we have
$\left|\psi\right\rangle_{M}=\frac{1}{2}\left(1-a^{\dagger
2}\right)\left|0\right\rangle,$ (442)
it then follows
$\psi_{M}\left(x\right)\equiv\frac{1}{2}\left\langle
x\right|\left(1-a^{\dagger
2}\right)\left|0\right\rangle=\frac{1}{2}\left\langle
x\right|\left(\left|0\right\rangle-\sqrt{2}\left|2\right\rangle\right)=\pi^{-1/4}e^{-x^{2}/2}\left(1-x^{2}\right),$
(443)
which is just the Maxican hat wavelet, satisfying the condition
$\int_{-\infty}^{\infty}e^{-x^{2}/2}\left(1-x^{2}\right)dx=0.$Hence
$\frac{1}{2}\left(1-a^{\dagger 2}\right)\left|0\right\rangle$ is the state
vector corresponding to the Maxican hat mother wavelet (see Fig. 3). Once the
state vector $\left\langle\psi\right|$ corresponding to mother wavelet is
known, for any state $\left|f\right\rangle$ the matrix element
${}_{M}\left\langle\psi\right|U\left(\mu,s\right)\left|f\right\rangle$ is just
the wavelet transform of $f(x)$ with respect to $\left\langle\psi\right|.$
Figure 3: Traditional Mexican hat wavelet. Figure 4: Generalized Mexican hat
wavelet $\psi_{2}\left(x\right)$when $g_{0}=-2$, $g_{2}=-1$, $g_{4}=1$and
$g_{6}=0$. Figure 5: Generalized Mexican hat wavelet when $g_{0}=-1$,
$g_{2}=-2$, $g_{4}=1$and $g_{6}=0$.
Case 2: when $g_{0}=-2$, $g_{2}=-1$, $g_{4}=1$ and $g_{6}=0$, from (441) we
obtain (see Fig. 4)
$\psi_{2}\left(x\right)=2\pi^{-1/4}e^{-x^{2}/2}\left(2x^{4}-7x^{2}+1\right),$
(444)
which also satisfies $\int_{-\infty}^{\infty}\psi_{2}\left(x\right)dx=0$. Note
that when $g_{0}=-1$, $g_{2}=-2$, $g_{4}=1$ and $g_{6}=0$, we obtain a
slightly different wavelet (see Fig. 5). Therefore, as long as the parameters
$g_{2n}$ conforms to condition (439), we can adjust their values to control
the shape of the wavelets.
Case 3: when $g_{0}=1$, $g_{2}=2$, $g_{4}=4$ and $g_{6}=-1$, we get (see Fig.
6)
$\psi_{3}\left(x\right)=\pi^{-1/4}e^{-x^{2}/2}\left(-8x^{6}+76x^{4}-134x^{2}+26\right),$
(445)
and $\int_{-\infty}^{\infty}\psi_{3}\left(x\right)dx=0$. From these figures we
observe that the number of the nodes of the curves at the $x$-axis is equal to
the highest power of the wavelet functions.
Figure 6: Generalized Mexican hat wavelet $\psi_{3}\left(x\right)$when
$g_{0}=1$, $g_{2}=2$, $g_{4}=4$and $g_{6}=-1$.
To further reveal the properties of the newly found wavelets, we compare the
wavelet transform computed with the well-known Mexican hat wavelet
$\psi_{1}\left(x\right)$ and that with our new wavelet
$\psi_{2}\left(x\right)$. Concretely, we map a simple cosine signal $\cos\pi
x$ by performing the wavelet transforms with
$\psi_{i}\left[T\left(x-X\right)\right]$, $i=1,2$, into a two-dimensional
space $\left(X,T\right)$, where $X$ denotes the location of a wavelet and $a$
its size. The resulting wavelet transforms by $\psi_{1}\left(x\right)$
(=$\psi_{M}\left(x\right)$) and $\psi_{2}\left(x\right)$ are
$\displaystyle\Omega_{1}\left(X,T\right)$
$\displaystyle=\frac{2}{\sqrt{3}}\int_{-\infty}^{\infty}dx\psi_{1}\left[T\left(x-X\right)\right]\cos\pi
x,$ (446) $\displaystyle\Omega_{2}\left(X,T\right)$
$\displaystyle=\frac{1}{\sqrt{30}}\int_{-\infty}^{\infty}dx\psi_{2}\left[T\left(x-X\right)\right]\cos\pi
x,$ (447)
where $2/\sqrt{3}$ and $1/\sqrt{30}$ are the normalization factors for
$\psi_{1}$ and $\psi_{2}$ respectively, the wavelet integral
$\Omega_{i}\left(X,T\right)$ are also called wavelet coefficient which
measures the variation of cos$\pi x$ in a neighborhood of $X,$ whose size is
proportional to $1/T$. The contour line representation of
$\Omega_{1}\left(X,T\right)$ and $\Omega_{2}\left(X,T\right)$ are depicted in
Fig. 7 and Fig. 8, respectively, where the transverse axis is $X$-axis (time
axis), while the longitudinal axis ($T$-axis) is the frequency axis.
Figure 7: Contour line representation of $\Omega_{1}\left(X,a\right).$ Figure
8: Contour line representation of $\Omega_{2}\left(X,a\right).$
It is remarkable that although two overall shapes of the two contour lines
look similar, there exist two notable differences between these two figures :
1) Along $T$-axis $\Omega_{1}\left(X,T\right)$ has one maximum, while
$\Omega_{2}\left(X,T\right)$ has one main maximum and one subsidiary maximum
(“two islands”), so when $\psi_{2}$ scales its size people have one more
chance to identify the frequency information of the cosine wave than using
$\psi_{M}$. Interesting enough, the “two islands” of
$\Omega_{2}\left(X,T\right)$ in Fig. 8 can be imagined as if they were
produced while the figure of $\Omega_{1}\left(X,T\right)$ deforms into two
sub-structures along $a$-axis. 2) Near the maximum of
$\Omega_{2}\left(X,T\right)$ the density of the contour lines along $a$-axis
is higher than that of $\Omega_{1}\left(X,T\right)$, which indicates that the
new wavelet $\psi_{2}$ is more sensitive in detecting frequency information of
the signal at this point. Therefore, $\psi_{2}\left(x\right)$ may be superior
to $\psi_{M}\left(x\right)$ in analyzing some signals. Finally, we mention
that there exist some remarkable qualitative similarities between the mother
wavelets presented in Figs. 3 through 6 and some of the amplitude envelopes of
higher order laser spatial modes and spatial supermodes of phase locked diode
laser arrays [100, 101, 102], which are due to spatial coherence.
### 14.3 Quantum mechanical version of Parseval theorem for WT
In this subsection, we shall prove that the Parseval theorem of 1D WT [98, 99,
103]:
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dsW_{\psi}f_{1}\left(\mu,s\right)W_{\psi}^{\ast}f_{2}\left(\mu,s\right)=2C_{\psi}\int
f_{1}\left(x\right)f_{2}^{\ast}\left(x\right)dx,$ (448)
where $\psi\left(x\right)$ is a mother wavelet whose Fourier transform is
$\psi\left(p\right),$
$C_{\psi}=2\pi\int_{0}^{\infty}\frac{\left|\psi\left(p\right)\right|^{2}}{p}dp<\infty$.
In the context of quantum mechanics, according to Eq.(430) we see that the
quantum mechanical version of Parseval theorem should be
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}ds\left\langle\psi\right|U\left(\mu,s\right)\left|f_{1}\right\rangle\left\langle
f_{2}\right|U^{\dagger}\left(\mu,s\right)\left|\psi\right\rangle=2C_{\psi}\left\langle
f_{2}\right.\left|f_{1}\right\rangle.$ (449)
and since $\psi\left(x\right)=\left\langle x\right|\left.\psi\right\rangle$,
so $\psi\left(p\right)$ involved in $C_{\psi}$ is $\left\langle
p\right|\left.\psi\right\rangle,$ $\left\langle p\right|$ is the momentum
eigenvector
$\psi\left(p\right)=\left\langle
p\right|\left.\psi\right\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx\psi\left(x\right)e^{-ipx}.$
(450)
Eq.(449) indicates that once the state vector $\left\langle\psi\right|$
corresponding to mother wavelet is known, for any two states
$\left|f_{1}\right\rangle$ and $\left|f_{2}\right\rangle$, their overlap up to
the factor $C_{\psi}$ (determined by Eq.(463)) is just their corresponding
overlap of WTs in the ($\mu,s)$ parametric space.
Proof of Equation (449): In order to show Eq.(449), we calculate
$\displaystyle U^{\dagger}\left(\mu,s\right)\left|p\right\rangle$
$\displaystyle=\frac{1}{\sqrt{\left|\mu\right|}}\int_{-\infty}^{\infty}dx\left|x\right\rangle\left\langle\frac{x-s}{\mu}\right|\left.p\right\rangle$
$\displaystyle=\frac{e^{-i\frac{ps}{\mu}}}{\sqrt{2\pi\left|\mu\right|}}\int_{-\infty}^{\infty}dx\left|x\right\rangle
e^{i\frac{p}{\mu}x}$
$\displaystyle=\frac{1}{\sqrt{\left|\mu\right|}}e^{-i\frac{ps}{\mu}}\left|\frac{p}{\mu}\right\rangle,$
(451)
which leads to
$\int_{-\infty}^{\infty}dsU^{\dagger}\left(\mu,s\right)\left|p^{\prime}\right\rangle\left\langle
p\right|U\left(\mu,s\right)=2\pi\delta\left(p-p^{\prime}\right)\left|\frac{p^{\prime}}{\mu}\right\rangle\left\langle\frac{p}{\mu}\right|,$
(452)
where we have used the formula
$\int_{-\infty}^{\infty}\frac{dx}{2\pi}e^{ix\left(p-p^{\prime}\right)}=\delta\left(p-p^{\prime}\right).$
(453)
Inserting the completeness relation
$\int_{-\infty}^{\infty}dp\left|p\right\rangle\left\langle p\right|=1$ into
the left side of Eq.(448) and then using Eq.(452) we have
L.H.S of Eq.(448)
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dsdpdp^{\prime}\psi^{\ast}\left(p\right)\psi\left(p^{\prime}\right)\left\langle
f_{2}\right|U^{\dagger}\left(\mu,s\right)\left|p^{\prime}\right\rangle\left\langle
p\right|U\left(\mu,s\right)\left|f_{1}\right\rangle$
$\displaystyle=2\pi\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dp\psi^{\ast}\left(p\right)\psi\left(p\right)\left\langle
f_{2}\right.\left|\frac{p}{\mu}\right\rangle\left\langle\frac{p}{\mu}\right|\left.f_{1}\right\rangle$
$\displaystyle\equiv I_{1}+I_{2},$ (454)
where
$\displaystyle I_{1}$
$\displaystyle=2\pi\int_{0}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dp\psi^{\ast}\left(p\right)\psi\left(p\right)\left\langle
f_{2}\right.\left|\frac{p}{\mu}\right\rangle\left\langle\frac{p}{\mu}\right|\left.f_{1}\right\rangle$
$\displaystyle=2\pi\int_{-\infty}^{\infty}dp\left[\int_{0}^{\infty}\left|\psi\left(\mu
p\right)\right|^{2}\frac{d\mu}{\mu}\right]\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle,$ (455)
and
$\displaystyle I_{2}$
$\displaystyle=2\pi\int_{-\infty}^{0}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dp\left|\psi\left(p\right)\right|^{2}\left\langle
f_{2}\right.\left|\frac{p}{\mu}\right\rangle\left\langle\frac{p}{\mu}\right|\left.f_{1}\right\rangle$
$\displaystyle=2\pi\int_{-\infty}^{\infty}dp\left[\int_{0}^{\infty}\left|\psi\left(-\mu
p\right)\right|^{2}\frac{d\mu}{\mu}\right]\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle.$ (456)
Further, we can put Eqs.(455) and (456) into the following forms,
$\displaystyle I_{1}$
$\displaystyle=2\pi\int_{0}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dp\psi^{\ast}\left(p\right)\psi\left(p\right)\left\langle
f_{2}\right.\left|\frac{p}{\mu}\right\rangle\left\langle\frac{p}{\mu}\right|\left.f_{1}\right\rangle$
$\displaystyle=C_{\psi}\int_{0}^{\infty}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle+2\pi\int_{-\infty}^{0}dp\left[\int_{0}^{\infty}\left|\psi\left(\mu
p\right)\right|^{2}\frac{d\mu p}{\mu p}\right]\left\langle
f_{2}\right.\left|p\right\rangle\left\langle p\right|\left.f_{1}\right\rangle$
$\displaystyle=C_{\psi}\int_{0}^{\infty}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle+2\pi\int_{-\infty}^{0}dp\left[\int_{0}^{-\infty}\left|\psi\left(p^{\prime}\right)\right|^{2}\frac{dp^{\prime}}{p^{\prime}}\right]\left\langle
f_{2}\right.\left|p\right\rangle\left\langle p\right|\left.f_{1}\right\rangle$
$\displaystyle=C_{\psi}\int_{0}^{\infty}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle+C_{\psi}^{\prime}\int_{-\infty}^{0}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle p\right|\left.f_{1}\right\rangle$
(457)
and
$I_{2}=C_{\psi}^{\prime}\int_{0}^{\infty}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle+C_{\psi}\int_{-\infty}^{0}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle,$ (458)
where
$\displaystyle C_{\psi}$
$\displaystyle=2\pi\int_{0}^{\infty}\left|\psi\left(\mu
p\right)\right|^{2}\frac{d\mu}{\mu}=2\pi\int_{0}^{\infty}\left|\psi\left(p\right)\right|^{2}\frac{dp}{p},$
$\displaystyle C_{\psi}^{\prime}$
$\displaystyle=2\pi\int_{0}^{-\infty}\left|\psi\left(p^{\prime}\right)\right|^{2}\frac{dp^{\prime}}{p^{\prime}}=2\pi\int_{0}^{\infty}\left|\psi\left(-p\right)\right|^{2}\frac{dp}{p},$
(459)
thus when the definite integration satisfies the admissible condition, i.e.,
$\int_{0}^{\infty}\left|\psi\left(p\right)\right|^{2}\frac{dp}{p}=\int_{0}^{\infty}\left|\psi\left(-p\right)\right|^{2}\frac{dp}{p},$
(460)
which leads to
$2\pi\int_{-\infty}^{\infty}\left|\psi\left(p\right)\right|^{2}\frac{dp}{\left|p\right|}=2C_{\psi}.$
(461)
Eq. (454) can be transformed to
$\text{L.H.S of
Eq.(\ref{15.16})}=2C_{\psi}\int_{-\infty}^{\infty}dp\left\langle
f_{2}\right.\left|p\right\rangle\left\langle
p\right|\left.f_{1}\right\rangle=\text{R.H.S of Eq.(\ref{15.16}).}$ (462)
where
$C_{\psi}\equiv
2\pi\int_{0}^{\infty}\frac{\left|\psi\left(p\right)\right|^{2}}{p}dp<\infty,\text{
}$ (463)
thus the theorem is proved. Especially, when $\left|f_{1}\right\rangle=$
$\left|f_{2}\right\rangle,$ Eq.(449) becomes
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}ds|\left\langle\psi\right|U\left(\mu,s\right)\left|f_{1}\right\rangle|^{2}=2C_{\psi}\left\langle
f_{1}\right.\left|f_{1}\right\rangle,$ (464)
which is named isometry of energy.
### 14.4 Inversion formula of WT
Now we can directly derive the inversion formula of WT, i.e. we take
$\left\langle f_{2}\right|=\left\langle x\right|$ in Eq.(449), then using
Eq.(430) we see that Eq.(449) reduces to
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dsW_{\psi}f_{1}\left(\mu,s\right)\left\langle
x\right|U^{\dagger}\left(\mu,s\right)\left|\psi\right\rangle=2C_{\psi}\left\langle
x\right.\left|f_{1}\right\rangle.$ (465)
Due to Eq.(431) we have
$\left\langle
x\right|U^{\dagger}\left(\mu,s\right)=\frac{1}{\sqrt{\left|\mu\right|}}\left\langle
x\right|\int_{-\infty}^{\infty}dx^{\prime}\left|x^{\prime}\right\rangle\left\langle\frac{x^{\prime}-s}{\mu}\right|=\frac{1}{\sqrt{\left|\mu\right|}}\left\langle\frac{x-s}{\mu}\right|.$
(466)
It then follows
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}dsW_{\psi}f_{1}\left(\mu,s\right)\frac{1}{\sqrt{\left|\mu\right|}}\left\langle\frac{x-s}{\mu}\right|\left.\psi\right\rangle=2C_{\psi}\left\langle
x\right.\left|f_{1}\right\rangle,$ (467)
which means
$f_{1}\left(x\right)=\frac{1}{2C_{\psi}}\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}\sqrt{\left|\mu\right|}}\int_{-\infty}^{\infty}ds\psi\left(\frac{x-s}{\mu}\right)W_{\psi}f_{1}\left(\mu,s\right),$
(468)
this is the inversion formula of WT.
### 14.5 New orthogonal property of mother wavelet in parameter space
Form the Parserval theorem (448) of WT in quantum mechanics we can derive some
new property of mother wavelet [104]. Taking
$\left|f_{1}\right\rangle=\left|x\right\rangle$,
$\left|f_{2}\right\rangle=\left|x^{\prime}\right\rangle$ in (448) one can see
that
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}\left|\mu\right|}\int_{-\infty}^{\infty}ds\psi\left(\frac{x-s}{\mu}\right)\psi^{\ast}\left(\frac{x^{\prime}-s}{\mu}\right)=2C_{\psi}\delta\left(x-x^{\prime}\right),$
(469)
which is a new orthogonal property of mother wavelet in parameter space
spanned by $\left(\mu,s\right)$. In a similar way, we take
$\left|f_{1}\right\rangle=\left|f_{2}\right\rangle=\left|n\right\rangle,$ a
number state, since $\left\langle n\right.\left|n\right\rangle=1,$ then we
have
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}ds|\left\langle\psi\right|U\left(\mu,s\right)\left|n\right\rangle|^{2}=2C_{\psi},$
(470)
or take
$\left|f_{1}\right\rangle=\left|f_{2}\right\rangle=\left|z\right\rangle,$
$\left|z\right\rangle=\exp\left(-\left|z\right|^{2}/2+za^{\dagger}\right)\left|0\right\rangle$
is the coherent state, then
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}ds|\left\langle\psi\right|U\left(\mu,s\right)\left|z\right\rangle|^{2}=2C_{\psi}.$
(471)
This indicates that $C_{\psi}$ is $\left|f_{1}\right\rangle$-independent,
which coincides with the expression in (463). Next, we consider a special
example. When the mother wavelet is the Mexican hat (443), we have
$\psi_{M}\left(p\right)\equiv\left\langle
p\right.\left|\psi_{M}\right\rangle=\frac{1}{2}\left(\left\langle
p\right.\left|0\right\rangle-\sqrt{2}\left\langle
p\right.\left|2\right\rangle\right)=\pi^{-1/4}\allowbreak
p^{2}e^{-\frac{1}{2}p^{2}}.$ (472)
where
$\left\langle
p\right.\left|n\right\rangle=\frac{\left(-i\right)^{n}}{\sqrt{2^{n}n!\sqrt{\pi}}}e^{-p^{2}/2}H_{n}\left(p\right).$
(473)
Here $H_{n}\left(p\right)$ is the single-variable Hermit polynomial [105].
Substituting Eq.(472) into Eq.(463) we have
$C_{\psi}\equiv
2\pi\int_{0}^{\infty}\frac{\left|\psi_{M}\left(p\right)\right|^{2}}{p}dp=\sqrt{\pi}.$
(474)
Thus, for the Mexican hat wavelet (443), we see
$\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}\left|\mu\right|}\int_{-\infty}^{\infty}ds\psi_{M}\left(\frac{x-s}{\mu}\right)\psi_{M}^{\ast}\left(\frac{x^{\prime}-s}{\mu}\right)=2\sqrt{\pi}\delta\left(x-x^{\prime}\right).$
(475)
Eq.(475) can be checked as follows. Using Eq.(443) and noticing that
$\psi_{M}\left(x\right)=\psi_{M}\left(-x\right)$, we can put the left hand
side of Eq.(475) into
$\displaystyle\mathtt{L.H.S.of}\text{ }(\text{\ref{15.43}})$
$\displaystyle=2\int_{0}^{\infty}du\int_{-\infty}^{\infty}ds\psi_{M}\left(ux-s\right)\psi_{M}^{\ast}\left(ux^{\prime}-s\right)$
$\displaystyle=\left\\{\begin{array}[c]{cc}0,&x\neq x^{\prime}\\\
\frac{3}{2}\int_{0}^{\infty}du\rightarrow\infty,&x=x^{\prime}\end{array}\right.=\mathtt{R.H.S.of}\text{
}(\ref{15.43}).$ (478)
where we have used the integration formulas
$\int_{-\infty}^{\infty}\left(1-s^{2}\right)^{2}\exp\left(-s^{2}\right)ds=\allowbreak\frac{3}{4}\sqrt{\pi},$
(479)
and
$\displaystyle\int_{-\infty}^{\infty}\left(1-s^{2}\right)\left[1-\left(s-b\right)^{2}\right]e^{-s^{2}/2-\left(b-s\right)^{2}/2}ds$
$\displaystyle=\frac{\sqrt{\pi}}{16}e^{-\frac{b^{2}}{4}}\left[12+b^{2}\left(b^{2}-12\right)\right].$
(480)
Next, we examine if the Morlet wavelet obey the formalism (469). The the
Morlet wavelet is defined as [106, 107, 108]
$\psi_{mor}\left(x\right)=\pi^{-1/4}\left(e^{ifx}-e^{-f^{2}/2}\right)e^{-x^{2}/2}.$
(481)
Substituting (481) into the left hand side of (469) yields
$\displaystyle I$
$\displaystyle\equiv\int_{-\infty}^{\infty}\frac{d\mu}{\mu^{2}\left|\mu\right|}\int_{-\infty}^{\infty}ds\psi_{mor}\left(\frac{x-s}{\mu}\right)\psi_{mor}^{\ast}\left(\frac{x^{\prime}-s}{\mu}\right)$
$\displaystyle=\left\\{\begin{array}[c]{cc}0,&x\neq x^{\prime}\\\
2\left(1+e^{-f^{2}}-2e^{-3f^{2}/4}\right)\int_{0}^{\infty}\frac{d\mu}{\mu^{2}}\rightarrow\infty,&x=x^{\prime}\end{array}\right..$
(484)
Thus the Morlet wavelet satisfies Eq.(469).
### 14.6 WT and Wigner-Husimi Distribution Function
Phase space technique has been proved very useful in various branches of
physics. Distribution functions in phase space have been a major topic in
studying quantum mechanics and quantum statistics. Among various phase space
distributions the Wigner function $F_{w}\left(q,p\right)$ [66, 67] is the most
popularly used, since its two marginal distributions lead to measuring
probability density in coordinate space and momentum space, respectively. But
the Wigner distribution function itself is not a probability distribution due
to being both positive and negative. In spite of its some attractive formal
properties, it needs to be improved. To overcome this inconvenience, the
Husimi distribution function $F_{h}\left(q^{\prime},p^{\prime}\right)$ is
introduced [109], which is defined in a manner that guarantees it to be
nonnegative. Its definition is smoothing out the Wigner function by averaging
over a “coarse graining” function,
$F_{h}\left(q,p,\kappa\right)=\int\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}F_{w}\left(q^{\prime},p^{\prime}\right)\exp\left[-\kappa\left(q^{\prime}-q\right)^{2}-\frac{\left(p^{\prime}-p\right)^{2}}{\kappa}\right],$
(485)
where $\kappa>0$ is the Gaussian spatial width parameter, which is free to be
chosen and which determines the relative resolution in $p$-space versus
$q$-space.
In the following, we shall employ the optical wavelet transformation to study
the Husimi distribution function, this is to say, we shall show that the
Husimi distribution function of a quantum state $\left|\psi\right\rangle$ can
be obtained by making a WT of the Gaussian function $e^{-x^{2}/2},$ i.e.,
$\left\langle\psi\right|\Delta_{h}\left(q,p,\kappa\right)\left|\psi\right\rangle=\frac{e^{-\frac{p^{2}}{\kappa}}}{\sqrt{\pi\kappa}}\left|\int_{-\infty}^{\infty}dx\psi^{\ast}\left(\frac{x-s}{\mu}\right)e^{-x^{2}/2}\right|^{2},$
(486)
where
$s=\frac{-1}{\sqrt{\kappa}}\left(\kappa q+ip\right),\text{
}\mu=\sqrt{\kappa},$ (487)
and $\left\langle\psi\right|\Delta_{h}\left(q,p\right)\left|\psi\right\rangle$
is the Husimi distribution function as well as
$\Delta_{h}\left(q,p,\kappa\right)$ is the Husimi operator,
$\Delta_{h}\left(q,p,\kappa\right)=\frac{2\sqrt{\kappa}}{1+\kappa}\colon\exp\left\\{\frac{-\kappa\left(q-Q\right)^{2}}{1+\kappa}-\frac{\left(p-P\right)^{2}}{1+\kappa}\right\\}\colon,$
(488)
here $\colon\colon$ denotes normal ordering; $Q$ and $P$ are the coordinate
and the momentum operator.
Proof of Eq.(486). According to Eqs.(429) and (430), when
$\left|f\right\rangle$ is the vacuum state $\left|0\right\rangle$,
$e^{-x^{2}/2}=\pi^{1/4}\left\langle x\right.\left|0\right\rangle$, we see that
$\pi^{-1/4}\int_{-\infty}^{\infty}\frac{dx}{\sqrt{\mu}}\psi^{\ast}\left(\frac{x-s}{\mu}\right)e^{-x^{2}/2}dx=\left\langle\psi\right|U\left(\mu,s\right)\left|0\right\rangle.$
(489)
From Eq.(433) it then follows that
$U\left(\mu,s\right)\left|0\right\rangle=\operatorname{sech}^{1/2}\lambda\exp\left[\frac{-s^{2}}{2\left(1+\mu^{2}\right)}-\frac{a^{\dagger}s}{\sqrt{2}}\operatorname{sech}\lambda-\frac{a^{\dagger
2}}{2}\tanh\lambda\right]\left|0\right\rangle.$ (490)
Substituting Eq.(487) and $\tanh\lambda=\frac{\kappa-1}{\kappa+1},$
$\cosh\lambda=\frac{1+\kappa}{2\sqrt{\kappa}}$ into Eq.(490) yields
$\displaystyle
e^{-\frac{p^{2}}{2\kappa}+\frac{ipq}{\kappa+1}}U\left(\mu=\sqrt{\kappa},s=-\sqrt{\kappa}q-ip/\sqrt{\kappa}\right)\left|0\right\rangle$
$\displaystyle=\left(\frac{2\sqrt{\kappa}}{1+\kappa}\right)^{1/2}\exp\left\\{\frac{-\kappa
q^{2}}{2\left(1+\kappa\right)}-\frac{p^{2}}{2\left(1+\kappa\right)}\right.$
$\displaystyle\left.+\frac{\sqrt{2}a^{\dagger}}{1+\kappa}\left(\kappa
q+ip\right)+\frac{1-\kappa}{2\left(1+\kappa\right)}a^{\dagger
2}\right\\}\left|0\right\rangle\left.\equiv\left|p,q\right\rangle_{\kappa}\right.,$
(491)
then the WT of Eq.(489) can be further expressed as
$e^{-\frac{p^{2}}{2\kappa}+\frac{ipq}{\kappa+1}}\int_{-\infty}^{\infty}\frac{dx}{\left(\kappa\pi\right)^{1/4}}\psi^{\ast}\left(\frac{x-s}{\mu}\right)e^{-x^{2}/2}=\left\langle\psi\right.\left|p,q\right\rangle_{\kappa}.$
(492)
Using normally ordered form of the vacuum state projector
$\left|0\right\rangle\left\langle 0\right|=\colon e^{-a^{\dagger}a}\colon,$and
the IWOP method as well as Eq.(LABEL:15.55) we have
$\left|p,q\right\rangle_{\kappa\kappa}\left\langle
p,q\right|=\frac{2\sqrt{\kappa}}{1+\kappa}\colon\exp\left[\frac{-\kappa\left(q-Q\right)^{2}}{1+\kappa}-\frac{\left(p-P\right)^{2}}{1+\kappa}\right]\colon=\Delta_{h}\left(q,p,\kappa\right).$
(493)
Now we explain why $\Delta_{h}\left(q,p,\kappa\right)$ is the Husimi operator.
Using the formula for converting an operator $A$ into its Weyl ordering form
[110]
$\displaystyle A$
$\displaystyle=2\int\frac{d^{2}\beta}{\pi}\left\langle-\beta\right|A\left|\beta\right\rangle\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)\\}\genfrac{}{}{0.0pt}{}{:}{:},$
(494) $\displaystyle d^{2}\beta$ $\displaystyle=d\beta_{1}d\beta_{2},\text{
}\beta=\beta_{1}+i\beta_{2},$
where the symbol $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$
denotes the Weyl ordering, $\left|\beta\right\rangle$ is the usual coherent
state, substituting Eq.(493) into Eq.(494) and performing the integration by
virtue of the technique of integration within a Weyl ordered product of
operators, we obtain
$\left|p,q\right\rangle_{\kappa\kappa}\left\langle
p,q\right|=2\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[-\kappa\left(q-Q\right)^{2}-\frac{\left(p-P\right)^{2}}{\kappa}\right]\genfrac{}{}{0.0pt}{}{:}{:}.$
(495)
This is the Weyl ordering form of
$\left|p,q\right\rangle_{\kappa\kappa}\left\langle p,q\right|.$ Then according
to Weyl quantization scheme [37] we know the classical corresponding function
of a Weyl ordered operator is obtained by just replacing $Q\rightarrow
q^{\prime},P\rightarrow p^{\prime},$
$\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[-\kappa\left(q-Q\right)^{2}-\frac{\left(p-P\right)^{2}}{\kappa}\right]\genfrac{}{}{0.0pt}{}{:}{:}\rightarrow\exp\left[-\kappa\left(q-q^{\prime}\right)^{2}-\frac{\left(p-p^{\prime}\right)^{2}}{\kappa}\right],$
(496)
and in this case the Weyl rule is expressed as
$\displaystyle\left|p,q\right\rangle_{\kappa\kappa}\left\langle p,q\right|$
$\displaystyle=2\int
dq^{\prime}dp^{\prime}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q^{\prime}-Q\right)\delta\left(p^{\prime}-P\right)\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[-\kappa\left(q-q^{\prime}\right)^{2}-\frac{\left(p-p^{\prime}\right)^{2}}{\kappa}\right]$
$\displaystyle=2\int
dq^{\prime}dp^{\prime}\Delta_{w}\left(q^{\prime},p^{\prime}\right)\exp\left[-\kappa\left(q^{\prime}-q\right)^{2}-\frac{\left(p^{\prime}-p\right)^{2}}{\kappa}\right],$
(497)
where at the last step we used the Weyl ordering form of the Wigner operator
$\Delta_{w}\left(q,p\right)$ [111]
$\Delta_{w}\left(q,p\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:}.$
(498)
In reference to Eq.(485) in which the relation between the Husimi function and
the WF is shown, we know that the right-hand side of Eq. (497) should be just
the Husimi operator, i.e.
$\displaystyle\left|p,q\right\rangle_{\kappa\kappa}\left\langle p,q\right|$
$\displaystyle=2\int
dq^{\prime}dp^{\prime}\Delta_{w}\left(q^{\prime},p^{\prime}\right)\exp\left[-\kappa\left(q^{\prime}-q\right)^{2}-\frac{\left(p^{\prime}-p\right)^{2}}{\kappa}\right]$
$\displaystyle=\Delta_{h}\left(q,p,\kappa\right),$ (499)
thus Eq. (486) is proved by combining Eqs.(499) and (492).
Thus the optical WT can be used to study the Husimi distribution function in
quantum optics phase space theory [112].
## 15 Complex Wavelet transformation in entangled state representations
We now turn to 2-dimensional complex wavelet transform (CWT) [113].
### 15.1 CWT and the condition of Mother Wavelet
Since wavelet family involves squeezing transform, we recall that the two-mode
squeezing operator has a natural representation in the entangled state
representation (ESR),
$\exp[\lambda\left(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}\right)]=\frac{1}{\mu}\int_{-\infty}^{\infty}\left|\frac{\eta}{\mu}\right\rangle\left\langle\eta\right|dx,$
$\mu=e^{\lambda}$, thus we are naturally led to studying 2-dimensional CWT in
ESR. Using ESR we can derive some new results more conveniently than using the
direct-product of two single-particle coordinate eigenstates. To be concrete,
we impose the condition on qualified mother wavelets also in
$\left|\eta\right\rangle$ representation,
$\int_{-\infty}^{\infty}\frac{d^{2}\eta}{2\pi}\psi\left(\eta\right)=0,$ (500)
where $\psi\left(\eta\right)=$
$\left\langle\eta\right|\left.\psi\right\rangle.$ Thus we see
$\int_{-\infty}^{\infty}\frac{d^{2}\eta}{2\pi}\left|\eta\right\rangle=\exp\\{-a_{1}^{\dagger}a_{2}^{\dagger}\\}\left|00\right\rangle=\left|\xi=0\right\rangle,$
(501)
and the condition (500) becomes
$\left\langle\xi=0\right|\left.\psi\right\rangle=0.$ (502)
Without loss of generality, assuming
$\left|\psi\right\rangle=\sum_{n,m=0}^{\infty}K_{n,m}a_{1}^{\dagger
n}a_{2}^{\dagger m}\left|00\right\rangle,$ (503)
then using the two-mode coherent $\left|z_{1}z_{2}\right\rangle$ state we can
write (502) as
$\displaystyle\left\langle\xi=0\right|\left.\psi\right\rangle$
$\displaystyle=\left\langle\xi=0\right|\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|z_{1}z_{2}\right\rangle\left\langle
z_{1}z_{2}\right|\sum_{n,m=0}^{\infty}K_{n,m}a_{1}^{\dagger n}a_{2}^{\dagger
m}\left|00\right\rangle$
$\displaystyle=\sum_{n,m=0}^{\infty}K_{n,m}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}z_{1}^{\ast
n}z_{2}^{\ast m}\exp\left[-|z_{1}|^{2}-|z_{2}|^{2}-z_{1}z_{2}\right]$
$\displaystyle=\sum_{n,m=0}^{\infty}K_{n,m}\int\frac{d^{2}z_{2}}{\pi}\exp\left[-|z_{2}|^{2}\right]z_{2}^{n}z_{2}^{\ast
m}\left(-1\right)^{n}$
$\displaystyle=\sum_{n=0}^{\infty}n!K_{n,n}\left(-1\right)^{n}=0,$ (504)
this is the constraint on the coefficient $K_{n,n}$ in (504), i.e., the
admissibility condition for $\left|\psi\right\rangle$. Thus Eq. (503) is in
the form:
$\left|\psi\right\rangle=\sum_{n=0}^{\infty}n!K_{n,n}\left|n,n\right\rangle.$
(505)
To derive the qualified mother wavelet $\psi\left(\eta\right)=$
$\left\langle\eta\right|\left.\psi\right\rangle$ from
$\left|\psi\right\rangle$, noticing Eq.(353) and (505) we have
$\displaystyle\psi\left(\eta\right)$
$\displaystyle=e^{-\left|\eta\right|^{2}/2}\sum_{n=0}^{\infty}K_{n,n}H_{n,n}\left(\eta^{\ast},\eta\right)\left(-1\right)^{n}$
$\displaystyle=e^{-\left|\eta\right|^{2}/2}\sum_{n=0}^{\infty}n!K_{n,n}L_{n}\left(\left|\eta\right|^{2}\right),$
(506)
where $L_{n}\left(x\right)$ is the Laguerre polynomial. In this case, we may
name the wavelet in Eq. (506) as the Laguerre–Gaussian mother wavelets,
analogous to the name of Laguerre–Gaussian modes in optical propagation. For
example:
(1) When taking $K_{0,0}=\frac{1}{2},$ $K_{1,1}=\frac{1}{2},$ $K_{n,n}=0$ for
$n\geqslant 2,$ so we see
$\left|\psi\right\rangle_{1}=\frac{1}{2}\left(1+a_{1}^{\dagger}a_{2}^{\dagger}\right)\left|00\right\rangle,$
(507)
which differs from the direct-product state $\left(1-a_{1}^{\dagger
2}\right)\left|0\right\rangle_{1}\otimes\left(1-a_{2}^{\dagger
2}\right)\left|0\right\rangle_{2}$. It then follows from Eq. (506) that
$\psi_{1}\left(\eta\right)\equiv\frac{1}{2}\left\langle\eta\right|\left(\left|00\right\rangle+\left|11\right\rangle\right)=e^{-\frac{1}{2}\left|\eta\right|^{2}}\\{1-\frac{1}{2}\left|\eta\right|^{2}\\},$
(508)
which differs from
$e^{-\left(x^{2}+y^{2}\right)/2}(1-x^{2})\left(1-y^{2}\right)$, the direct
product of two 1D Mexican hat wavelets (see also the difference between Figs.
9 and 10).
Figure 9: The Laguree-Gaussian mother wavelet $\psi_{1}\left(\eta\right)$.
Figure 10: 2D Mexican hat mother wavelet (Hermite Gaussian mother wavelet).
(2) when $K_{0,0}=1$, $K_{1,1}=3$, $K_{2,2}=1$, $K_{n,n}=0$ for $n\geqslant
3,$ we have (see Fig. 11)
$\psi_{2}\left(\eta\right)\equiv\left(6-7\left|\eta\right|^{2}+\left|\eta\right|^{4}\right)e^{-\frac{1}{2}\left|\eta\right|^{2}}.$
(509)
Figure 11: Laguerre-Gaussian mother wavelet $\psi_{2}\left(\eta\right).$
(3) when $K_{0,0}=1$, $K_{1,1}=1$, $K_{2,2}=3$, $K_{3,3}=3$, $K_{n,n}=0$ for
$n\geqslant 4,$ the mother wavelet $\psi_{3}\left(\eta\right)$ (see Fig. 12)
reads
$\psi_{3}\left(\eta\right)=\left(14-31\left|\eta\right|^{2}+12\left|\eta\right|^{4}-\left|\eta\right|^{6}\right)e^{-\frac{1}{2}\left|\eta\right|^{2}}.$
(510)
Figure 12: Laguerre-Gaussian mother wavelet $\psi_{3}\left(\eta\right).$
From the figures we can see that as long as the coefficients $K_{n,n}$ satisfy
condition (506), we can construct arbitrary complex mother wavelet by adding
or reducing the number of coefficients, or by adjusting the value of them. And
since only $K_{n,m}$ ($m=n$) survive in all the coefficients, the mother
wavelets obtained are all circularly symmetric on the complex plane.
Moreover, the CWT of a signal function $F\left(\eta\right)$ by $\Psi$ is
defined by
$W_{\psi}F\left(\mu,\kappa\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}F\left(\eta\right)\psi^{\ast}\left(\frac{\eta-\kappa}{\mu}\right).$
(511)
Using the $\left\langle\eta\right|$ representation we can treat it from the
quantum mechanically,
$W_{\psi}F\left(\mu,\kappa\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left\langle\psi\right|\left.\frac{\eta-\kappa}{\mu}\right\rangle\left\langle\eta\right|\left.F\right\rangle=\left\langle\Psi\right|U_{2}\left(\mu,\kappa\right)\left|F\right\rangle,$
(512)
where
$U_{2}\left(\mu,\kappa\right)\equiv\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta-\kappa}{\mu}\right\rangle\left\langle\eta\right|,\;\mu=e^{\lambda},$
(513)
is the two-mode squeezing-displacing operator. Using the IWOP technique we can
calculate its normally ordered form,
$\displaystyle U_{2}\left(\mu,\kappa\right)$
$\displaystyle=\operatorname{sech}\lambda\colon\exp\\{\left(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}\right)\tanh\lambda+\left(\operatorname{sech}\lambda-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)$
$\displaystyle+\frac{1}{2}\left(\sigma^{\ast}a_{2}^{\dagger}-\sigma
a_{1}^{\dagger}\right)\operatorname{sech}\lambda+\frac{1}{1+\mu^{2}}\left(\kappa^{\ast}a_{1}-\kappa
a_{2}-\frac{1}{2}\left|\kappa\right|^{2}\right)\\}\colon.$ (514)
When $\kappa=0,$ it reduces to the usual normally ordered two-mode squeezing
operator. Once the state vector ${}_{M}\left\langle\Psi\right|$ corresponding
to mother wavelet is known, for any state $\left|F\right\rangle$ the matrix
element
${}_{M}\left\langle\Psi\right|U_{2}\left(\mu,\kappa\right)\left|F\right\rangle$
is just the wavelet transform of $F(\eta)$ with respect to
${}_{M}\left\langle\Psi\right|.$ Therefore, various quantum optical field
states can then be analyzed by their wavelet transforms.
### 15.2 Parseval Theorem in CWT
In order to complete the CWT theory, we must ask if the corresponding Parseval
theorem exists [114]. This is important since the inversion formula of CWT may
appear as a lemma of this theorem. Noting that CWT involves two-mode squeezing
transform, so the corresponding Parseval theorem differs from that of the
direct-product of two 1D wavelet transforms, too.
Next let us prove the Parseval theorem for CWT,
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}W_{\psi}g_{1}\left(\mu,\kappa\right)W_{\psi}^{\ast}g_{2}\left(\mu,\kappa\right)=C_{\psi}^{\prime}\int\frac{d^{2}\eta}{\pi}g_{2}^{\ast}\left(\eta\right)g_{1}\left(\eta\right),$
(515)
where $\kappa=\kappa_{1}+i\kappa_{2},$ and
$C_{\psi}^{\prime}=4\int_{0}^{\infty}\frac{d\left|\xi\right|}{\left|\xi\right|}\left|\psi\left(\xi\right)\right|^{2}.$
(516)
$\psi\left(\xi\right)$ is the complex Fourier transform of
$\psi\left(\eta\right),\psi\left(\xi\right)=\left\langle\xi\right|\left.\psi\right\rangle=\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left\langle\xi\right|\left.\eta\right\rangle\left\langle\eta\right|\left.\psi\right\rangle$.
According to (512) and (513) the quantum mechanical version of Parseval
theorem should be
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\left\langle
g_{2}\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\psi\right\rangle=C_{\psi}^{\prime}\left\langle
g_{2}\right.\left|g_{1}\right\rangle.$ (517)
Eq.(517) indicates that once the state vector $\left\langle\psi\right|$
corresponding to mother wavelet is known, for any two states
$\left|g_{1}\right\rangle$ and $\left|g_{2}\right\rangle$, their overlap up to
the factor $C_{\psi}$ (determined by (516)) is just their corresponding
overlap of CWTs in the ($\mu,\kappa$) parametric space.
Next we prove Eq.(515) or (517). In the same procedure as the proof of
Eq.(449). We start with calculating
$U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi\right\rangle.$ Using (42) and
(513), we have
$U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi\right\rangle=\frac{1}{\mu}\left|\frac{\xi}{\mu}\right\rangle
e^{\frac{i}{\mu}\left(\xi_{1}\kappa_{2}-\xi_{2}\kappa_{1}\right)},$ (518)
it then follows
$\displaystyle\int\frac{d^{2}\kappa}{\pi}U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi^{\prime}\right\rangle\left\langle\xi\right|U_{2}\left(\mu,\kappa\right)$
$\displaystyle=\frac{1}{\mu^{2}}\int\frac{d^{2}\kappa}{\pi}e^{\frac{i}{\mu}\left[\left(\xi_{1}^{\prime}-\xi_{1}\right)\kappa_{2}+\left(\xi_{2}-\xi_{2}^{\prime}\right)\kappa_{1}\right]}\left|\frac{\xi^{\prime}}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right|$
$\displaystyle=4\pi\left|\frac{\xi}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right|\delta\left(\xi_{1}^{\prime}-\xi_{1}\right)\delta\left(\xi_{2}-\xi_{2}^{\prime}\right).$
(519)
Using the completeness of $\left|\xi\right\rangle$ and (519) the left-hand
side (LHS) of (517) can be reformed as
LHS of Eq.(517)
$\displaystyle=\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa
d^{2}\xi
d^{2}\xi^{\prime}}{\pi^{3}}\left\langle\psi\right|\left.\xi\right\rangle$
$\displaystyle\times\left\langle\xi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\left\langle
g_{2}\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi^{\prime}\right\rangle\left\langle\xi^{\prime}\right|\left.\psi\right\rangle$
$\displaystyle=4\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\xi}{\pi}\left|\psi\left(\xi\right)\right|^{2}\left\langle
g_{2}\right.\left|\frac{\xi}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right.\left|g_{1}\right\rangle$
$\displaystyle=\int\frac{d^{2}\xi}{\pi}\left\\{4\int_{0}^{\infty}\frac{d\mu}{\mu}\left|\psi\left(\mu\xi\right)\right|^{2}\right\\}\left\langle
g_{2}\right.\left|\xi\right\rangle\left\langle\xi\right.\left|g_{1}\right\rangle,$
(520)
where the integration value in $\\{..\\}$ is actually $\xi-$independent.
Noting that the mother wavelet $\psi\left(\eta\right)$ in Eq.(506) is just the
function of $\left|\eta\right|,$ so $\psi\left(\xi\right)$ is also the
function of $\left|\xi\right|.$ In fact, using Eqs.(506) and (42), we have
$\psi\left(\xi\right)=e^{-1/2\left|\xi\right|^{2}}\sum_{n=0}^{\infty}K_{n,n}H_{n,n}\left(\left|\xi\right|,\left|\xi\right|\right),$
(521)
where we have used the integral formula
$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(522)
So we can rewrite (520) as
$\text{LHS of
(\ref{16.18})}=C_{\psi}^{\prime}\int\frac{d^{2}\xi}{\pi}\left\langle
g_{2}\right.\left|\xi\right\rangle\left\langle\xi\right.\left|g_{1}\right\rangle=C_{\psi}^{\prime}\left\langle
g_{2}\right.\left|g_{1}\right\rangle,$ (523)
where
$C_{\psi}^{\prime}=4\int_{0}^{\infty}\frac{d\mu}{\mu}\left|\psi\left(\mu\xi\right)\right|^{2}=4\int_{0}^{\infty}\frac{d\left|\xi\right|}{\left|\xi\right|}\left|\psi\left(\xi\right)\right|^{2}.$
(524)
Then we have completed the proof of the Parseval theorem for CWT in (517).
Here, we should emphasize that (517) is not only different from the product of
two 1D WTs, but also different from the usual WT in 2D.
When $\left|g_{2}\right\rangle=\left|\eta\right\rangle,$ by using (513) we see
$\left\langle\eta\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\psi\right\rangle=\frac{1}{\mu}\psi\left(\frac{\eta-\kappa}{\mu}\right),$
then substituting it into (517) yields
$g_{1}\left(\eta\right)=\frac{1}{C_{\psi}^{\prime}}\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi\mu}W_{\psi}g_{1}\left(\mu,\kappa\right)\psi\left(\frac{\eta-\kappa}{\mu}\right),$
(525)
which is just the inverse transform of the CWT. Especially, when
$\left|g_{1}\right\rangle=$ $\left|g_{2}\right\rangle,$ Eq. (517) reduces to
$\displaystyle\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|W_{\psi}g_{1}\left(\mu,\kappa\right)\right|^{2}$
$\displaystyle=C_{\psi}^{\prime}\int\frac{d^{2}\eta}{\pi}\left|g_{1}\left(\eta\right)\right|^{2},$
$\displaystyle\text{or
}\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\right|^{2}$
$\displaystyle=C_{\psi}^{\prime}\left\langle
g_{1}\right.\left|g_{1}\right\rangle,$ (526)
which is named isometry of energy.
### 15.3 Orthogonal property of mother wavelet in parameter space
On the other hand, when $\left|g_{1}\right\rangle=\left|\eta\right\rangle,$
$\left|g_{2}\right\rangle=\left|\eta^{\prime}\right\rangle$, Eq.(517) becomes
$\frac{1}{C_{\psi}^{\prime}}\int_{0}^{\infty}\frac{d\mu}{\mu^{5}}\int\frac{d^{2}\kappa}{\pi}\psi\left(\frac{\eta^{\prime}-\kappa}{\mu}\right)\psi^{\ast}\left(\frac{\eta-\kappa}{\mu}\right)=\pi\delta^{(2)}\left(\eta-\eta^{\prime}\right),$
(527)
which is a new orthogonal property of mother wavelet in parameter space
spanned by $\left(\mu,\kappa\right)$. In a similar way, we take
$\left|g_{1}\right\rangle=\left|g_{2}\right\rangle=\left|m,n\right\rangle,$ a
two-mode number state, since $\left\langle
m,n\right.\left|m,n\right\rangle=1,$ then we have
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|m,n\right\rangle\right|^{2}=C_{\psi}^{\prime},$
(528)
or take
$\left|g_{1}\right\rangle=\left|g_{2}\right\rangle=\left|z_{1},z_{2}\right\rangle,$
$\left|z\right\rangle=\exp\left(-\left|z\right|^{2}/2+za^{\dagger}\right)\left|0\right\rangle$
is the coherent state, then
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|z_{1},z_{2}\right\rangle\right|^{2}=C_{\psi}^{\prime}.$
(529)
Next we examine a special example. When the mother wavelet is
$\psi_{1}\left(\eta\right)$ in (508), using (42) we have
$\psi\left(\xi\right)=\frac{1}{2}\left|\xi\right|^{2}e^{-\frac{1}{2}\left|\xi\right|^{2}},$
which leads to
$C_{\psi}^{\prime}=\int_{0}^{\infty}\left|\xi\right|^{3}e^{-\left|\xi\right|^{2}}d\left|\xi\right|=\frac{1}{2}.$
Thus for $\psi_{1}\left(\eta\right)$, we see
$2\int_{0}^{\infty}\frac{d\mu}{\mu^{5}}\int\frac{d^{2}\kappa}{\pi}\psi_{1}\left(\frac{\eta^{\prime}-\kappa}{\mu}\right)\psi_{1}^{\ast}\left(\frac{\eta-\kappa}{\mu}\right)=\pi\delta^{(2)}\left(\eta-\eta^{\prime}\right).$
(530)
Eq. (530) can be checked as follows. Using (508) and the integral formula
$\displaystyle\int_{0}^{\infty}u\left(1-\frac{ux^{2}}{2}\right)\left(1-\frac{uy^{2}}{2}\right)e^{-u\frac{x^{2}+y^{2}}{2}}du$
$\displaystyle=-\frac{4(x^{4}-4x^{2}y^{2}+y^{4})}{(x^{2}+y^{2})^{4}},\text{
}\operatorname{Re}\left(x^{2}+y^{2}\right)>0,$ (531)
we can put the left-hand side (LHS) of (530) into
$\text{LHS of
(\ref{16.31})}=-\int\frac{d^{2}\kappa}{\pi}\frac{4(x^{4}-4x^{2}y^{2}+y^{4})}{(x^{2}+y^{2})^{4}},$
(532)
where $x^{2}=\left|\eta^{\prime}-\kappa\right|^{2},$
$y^{2}=\left|\eta-\kappa\right|^{2}.$
When $\eta^{\prime}=\eta,$ $x^{2}=y^{2},$
$\text{LHS of
(\ref{16.31})}=\int\allowbreak\frac{d^{2}\kappa}{2\pi\left|\kappa-\eta\right|^{4}}=\int_{0}^{\infty}\allowbreak\int_{0}^{2\pi}\frac{drd\theta}{2\pi
r^{3}}\rightarrow\infty.$ (533)
On the other hand, when $\eta\neq\eta^{\prime}$ and noticing that
$\displaystyle x^{2}$
$\displaystyle=\left(\eta_{1}^{\prime}-\kappa_{1}\right)^{2}+\left(\eta_{2}^{\prime}-\kappa_{2}\right)^{2},$
$\displaystyle y^{2}$
$\displaystyle=\left(\eta_{1}-\kappa_{1}\right)^{2}+\left(\eta_{2}-\kappa_{2}\right)^{2},$
(534)
which leads to
$\mathtt{d}x^{2}\mathtt{d}y^{2}=4\left|J\right|\mathtt{d}\kappa_{1}\mathtt{d}\kappa_{2}$,
where
$J\left(x,y\right)=\left|\begin{array}[c]{cc}\kappa_{1}-\eta_{1}^{\prime}&\kappa_{2}-\eta_{2}^{\prime}\\\
\kappa_{1}-\eta_{1}&\kappa_{2}-\eta_{2}\end{array}\right|$. As a result of
(534), (532) reduces to
$\text{LHS of
(\ref{16.31})}=-4\int_{-\infty}^{\infty}\frac{dxdy}{\pi}\frac{xy(x^{4}-4x^{2}y^{2}+y^{4})}{\left|J\right|(x^{2}+y^{2})^{4}}=0\text{,}$
(535)
where we have noticed that $J\left(x,y\right)$ is the funtion of
$\left(x^{2},y^{2}\right).$ Thus we have
$\text{LHS of
(\ref{16.31})}=\left\\{\begin{array}[c]{cc}\infty,&\eta=\eta^{\prime},\\\
0,&\eta\neq\eta^{\prime}.\end{array}\right.=\text{RHS of (\ref{16.31}).}$
(536)
### 15.4 CWT and Entangled Husimi distribution
Recalling that in Ref.[115], the so-called entangled Husimi operator
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ has been introduced, which is
endowed with definite physical meaning, and it is found that the two-mode
squeezed coherent state $\left|\sigma,\gamma\right\rangle_{\kappa}$
representation of $\Delta_{h}\left(\sigma,\gamma,\kappa\right),$
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)=$
$\left|\sigma,\gamma,\kappa\right\rangle\left\langle\sigma,\gamma,\kappa\right|$.
The entangled Husimi operator $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$
and the entangled Husimi distribution $F_{h}\left(\sigma,\gamma,\kappa\right)$
of quantum state $\left|\psi\right\rangle$ are given by
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(537)
and
$F_{h}\left(\sigma,\gamma,\kappa\right)=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}F_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(538)
respectively, where
$F_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)=\left\langle\psi\right|\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\left|\psi\right\rangle$
is two-mode Wigner function, with
$\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)$ being the two-mode
Wigner operator. Thus we are naturally led to studying the entangled Husimi
distribution function from the viewpoint of wavelet transformation. In this
subsection, we shall extend the relation between wavelet transformation and
Wigner-Husimi distribution function to the entangled case, that is to say, we
employ the CWT to investigate the entangled Husimi distribution function
(EHDF) by bridging the relation between CWT and EHDF. This is a convenient
approach for calculating various entangled Husimi distribution functions of
miscellaneous two-mode quantum states.
#### 15.4.1 CWT and its quantum mechanical version
In Ref.[113], the CWT has been proposed, i.e., the CWT of a complex signal
function $g\left(\eta\right)$ by $\psi$ is defined by
$W_{\psi}g\left(\mu,z\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}g\left(\eta\right)\psi^{\ast}\left(\frac{\eta-z}{\mu}\right),$
(539)
whose admissibility condition for mother wavelets,
$\int\frac{d^{2}\eta}{2\pi}\psi\left(\eta\right)=0,$ is examined in the
entangled state representations $\left\langle\eta\right|$ and a family of new
mother wavelets (named the Laguerre–Gaussian wavelets) are found to match the
CWT [113]. In fact, by introducing the bipartite entangled state
representation $\left\langle\eta=\eta_{1}+\mathtt{i}\eta_{2}\right|,$we can
treat (538) quantum mechanically,
$W_{\psi}g\left(\mu,z\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left\langle\psi\right|\left.\frac{\eta-z}{\mu}\right\rangle\left\langle\eta\right|\left.g\right\rangle=\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|g\right\rangle,$
(540)
where $z=z_{1}+iz_{2}\in C,$ $0<\mu\in R,$
$g\left(\eta\right)\equiv\left\langle\eta\right|\left.g\right\rangle\ $and
$\psi\left(\eta\right)=\left\langle\eta\right|\left.\psi\right\rangle$ are the
wavefunction of state vector $\left|g\right\rangle$ and the mother wavelet
state vector $\left|\psi\right\rangle$ in $\left\langle\eta\right|$
representation, respectively, and
$U_{2}\left(\mu,z\right)\equiv\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta-z}{\mu}\right\rangle\left\langle\eta\right|,\;\mu=e^{\lambda},$
(541)
is the two-mode squeezing-displacing operator. Noticing that the two-mode
squeezing operator has its natural expression in $\left\langle\eta\right|$
representation (36), which is different from the direct product of two single-
mode squeezing (dilation) operators, and the two-mode squeezed state is
simultaneously an entangled state, thus we can put Eq.(541) into the following
form,
$U_{2}\left(\mu,z\right)=S_{2}\left(\mu\right)\mathfrak{D}\left(z\right),$
(542)
where $\mathfrak{D}\left(z\right)$ is a two-mode displacement operator,
$\mathfrak{D}\left(z\right)\left|\eta\right\rangle=\left|\eta-z\right\rangle$
and
$\displaystyle\mathfrak{D}\left(z\right)$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left|\eta-z\right\rangle\left\langle\eta\right|$
$\displaystyle=\exp\left[iz_{1}\frac{P_{1}-P_{2}}{\sqrt{2}}-iz_{2}\frac{Q_{1}+Q_{2}}{\sqrt{2}}\right]$
$\displaystyle=D_{1}\left(-z/2\right)D_{2}\left(z^{\ast}/2\right).$ (543)
It the follows the quantum mechanical version of CWT is
$W_{\psi}g\left(\mu,\zeta\right)=\left\langle\psi\right|S_{2}\left(\mu\right)\mathfrak{D}\left(z\right)\left|g\right\rangle=\left\langle\psi\right|S_{2}\left(\mu\right)D_{1}\left(-z/2\right)D_{2}\left(z^{\ast}/2\right)\left|g\right\rangle.$
(544)
Eq.(544) indicates that the CWT can be put into a matrix element in the
$\left\langle\eta\right|$ representation of the two-mode displacing and the
two-mode squeezing operators in Eq.(LABEL:e10) between the mother wavelet
state vector $\left|\psi\right\rangle$ and the state vector
$\left|g\right\rangle$ to be transformed. Once the state vector
$\left\langle\psi\right|$ as mother wavelet is chosen, for any state
$\left|g\right\rangle$ the matrix element
$\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|g\right\rangle$ is just
the wavelet transform of $g(\eta)$ with respect to $\left\langle\psi\right|.$
Therefore, various quantum optical field states can then be analyzed by their
wavelet transforms.
#### 15.4.2 Relation between CWT and EHDF
In the following we shall show that the EHDF of a quantum state
$\left|\psi\right\rangle$ can be obtained by making a complex wavelet
transform of the Gaussian function $e^{-\left|\eta\right|^{2}/2},$ i.e.,
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle=e^{-\frac{1}{\kappa}\left|\gamma\right|^{2}}\left|\int\frac{d^{2}\eta}{\sqrt{\kappa}\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-z}{\sqrt{\kappa}}\right)\right|^{2},$
(545)
where $\mu=e^{\lambda}=\sqrt{\kappa},$ $z=z_{1}+iz_{2},$ and
$\displaystyle z_{1}$
$\displaystyle=\frac{\cosh\lambda}{1+\kappa}\left[\gamma^{\ast}-\gamma-\kappa\left(\sigma^{\ast}+\sigma\right)\right],$
(546) $\displaystyle z_{2}$
$\displaystyle=\frac{i\cosh\lambda}{1+\kappa}\left[\gamma+\gamma^{\ast}+\kappa\left(\sigma-\sigma^{\ast}\right)\right],$
(547)
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ is named the entangled Husimi
operator by us,
$\displaystyle\Delta_{h}\left(\sigma,\gamma,\kappa\right)$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle-\left.\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon.$
(548)
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle$
is the Husimi distribution function.
Proof of Eq.(545).
When the state to be transformed is
$\left|g\right\rangle=\left|00\right\rangle$ (the two-mode vacuum state), by
noticing that
$\left\langle\eta\right.\left|00\right\rangle=e^{-\left|\eta\right|^{2}/2},$
we can express Eq.(540) as
$\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-z}{\mu}\right)=\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|00\right\rangle.$
(549)
To combine the CWTs with transforms of quantum states more tightly and
clearly, using the IWOP technique we can directly perform the integral in
Eq.(541) [116]
$\displaystyle U_{2}\left(\mu,z\right)$
$\displaystyle=\operatorname{sech}\lambda\exp\left[-\frac{1}{2\left(1+\mu^{2}\right)}\left|z\right|^{2}+a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda+\frac{1}{2}\left(z^{\ast}a_{2}^{\dagger}-za_{1}^{\dagger}\right)\operatorname{sech}\lambda\right]$
$\displaystyle\times\exp\left[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\ln\operatorname{sech}\lambda\right]\exp\left(\frac{z^{\ast}a_{1}-za_{2}}{1+\mu^{2}}-a_{1}a_{2}\tanh\lambda\right).$
(550)
where we have set $\mu=e^{\lambda}$,
$\operatorname{sech}\lambda=\frac{2\mu}{1+\mu^{2}}$,
$\tanh\lambda=\frac{\mu^{2}-1}{\mu^{2}+1}$, and we have used the operator
identity
$e^{ga^{\dagger}a}=\colon\exp\left[\left(e^{g}-1\right)a^{\dagger}a\right]\colon$.
In particular, when $z=0,$ $U_{2}\left(\mu,z=0\right)$ becomes to the usual
normally ordered two-mode squeezing operator $S_{2}\left(\mu\right)$. From
Eq.(550) it then follows that
$\displaystyle U_{2}\left(\mu,z\right)\left|00\right\rangle$
$\displaystyle=\operatorname{sech}\lambda\exp\left\\{-\frac{\left(z_{1}-iz_{2}\right)\left(z_{1}+iz_{2}\right)}{2\left(1+\mu^{2}\right)}+a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda\right.$
$\displaystyle\left.+\frac{1}{2}\left[\left(z_{1}-iz_{2}\right)a_{2}^{\dagger}-\left(z_{1}+iz_{2}\right)a_{1}^{\dagger}\right]\operatorname{sech}\lambda\right\\}\left|00\right\rangle.$
(551)
Substituting Eqs.(546), (547) and $\tanh\lambda=\frac{\kappa-1}{\kappa+1},$
$\cosh\lambda=\frac{1+\kappa}{2\sqrt{\kappa}}$ into Eq.(551) yields
$\displaystyle
e^{-\frac{1}{2\kappa}\left|\gamma\right|^{2}-\frac{\sigma\gamma^{\ast}-\gamma\sigma^{\ast}}{2\left(\kappa+1\right)}}U_{2}\left(\mu,z_{1},z_{2}\right)\left|00\right\rangle$
$\displaystyle=\frac{2\sqrt{\kappa}}{1+\kappa}\exp\left\\{-\frac{\left|\gamma\right|^{2}+\kappa\left|\sigma\right|^{2}}{2\left(\kappa+1\right)}+\frac{\kappa\sigma+\gamma}{1+\kappa}a_{1}^{\dagger}+\frac{\gamma^{\ast}-\kappa\sigma^{\ast}}{1+\kappa}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\frac{\kappa-1}{\kappa+1}\right\\}\allowbreak\left|00\right\rangle\left.\equiv\right.\left|\sigma,\gamma\right\rangle_{\kappa},$
(552)
then the CWT of Eq.(549) can be further expressed as
$e^{-\frac{1}{2\kappa}\left|\gamma\right|^{2}-\frac{\sigma\gamma^{\ast}-\gamma\sigma^{\ast}}{2\left(\kappa+1\right)}}\int\frac{d^{2}\eta}{\mu\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-
z_{1}-iz_{2}}{\mu}\right)=\left\langle\psi\right.\left|\sigma,\gamma\right\rangle_{\kappa}.$
(553)
Using normally ordered form of the vacuum state projector
$\left|00\right\rangle\left\langle 00\right|=\colon
e^{-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}}\colon,$ and the IWOP method as
well as Eq.(552) we have
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left[-\frac{\left|\gamma\right|^{2}+\kappa\left|\sigma\right|^{2}}{\kappa+1}+\frac{\kappa\sigma+\gamma}{1+\kappa}a_{1}^{\dagger}+\frac{\gamma^{\ast}-\kappa\sigma^{\ast}}{1+\kappa}a_{2}^{\dagger}\right.$
$\displaystyle\left.+\frac{\kappa\sigma^{\ast}+\gamma^{\ast}}{1+\kappa}a_{1}+\frac{\gamma-\kappa\sigma}{1+\kappa}a_{2}+\frac{\kappa-1}{\kappa+1}\left(a_{1}^{\dagger}a_{2}^{\dagger}+a_{1}a_{2}\right)-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right]\colon$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle-\left.\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon\left.=\right.\Delta_{h}\left(\sigma,\gamma,\kappa\right).$
(554)
Now we explain why $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ is the
entangled Husimi operator. Using the formula for converting an operator $A$
into its Weyl ordering form [37]
$A=4\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left\langle-\alpha,-\beta\right|A\left|\alpha,\beta\right\rangle\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{2\left(\alpha^{\ast}a_{1}-a_{1}^{\dagger}\alpha+\beta^{\ast}a_{2}-a_{2}^{\dagger}\beta+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\\}\genfrac{}{}{0.0pt}{}{:}{:},$
(555)
where the symbol $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$
denotes the Weyl ordering, $\left|\beta\right\rangle$ is the usual coherent
state, substituting Eq.(554) into Eq.(555) and performing the integration by
virtue of the technique of integration within a Weyl ordered product of
operators, we obtain
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=\frac{16\kappa}{\left(1+\kappa\right)^{2}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left\langle-\alpha,-\beta\right|\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle\left.-\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon\left|\alpha,\beta\right\rangle$
$\displaystyle\times\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{2\left(\alpha^{\ast}a_{1}-a_{1}^{\dagger}\alpha+\beta^{\ast}a_{2}-a_{2}^{\dagger}\beta+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\\}\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=4\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\frac{1}{\kappa}\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\genfrac{}{}{0.0pt}{}{:}{:},$
(556)
where we have used the integral formula
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(557)
Eq.(556) is the Weyl ordering form of
$\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|.$
Then according to Weyl quantization scheme we know the Weyl ordering form of
two-mode Wigner operator is given by
$\Delta_{w}\left(\sigma,\gamma\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\delta\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)\delta\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\delta\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(558)
thus the classical corresponding function of a Weyl ordered operator is
obtained by just replacing
$a_{1}-a_{2}^{{\dagger}}\rightarrow\sigma^{\prime},a_{1}+a_{2}^{{\dagger}}\rightarrow\gamma^{\prime},$
i.e.,
$\displaystyle
4\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\frac{1}{\kappa}\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle\rightarrow
4\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(559)
and in this case the Weyl rule is expressed as
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\delta\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)\delta\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)$
$\displaystyle\times\delta\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}$
$\displaystyle=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}.$
(560)
In reference to Eq.(538) in which the relation between the entangled Husimi
function and the two-mode Wigner function is shown, we know that the right-
hand side of Eq. (560) should be just the entangled Husimi operator, i.e.
$\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}=\Delta_{h}\left(\sigma,\gamma,\kappa\right),$
(561)
thus Eq. (545) is proved by combining Eqs.(561) and (553).
Thus we have further extended the relation between wavelet transformation and
Wigner-Husimi distribution function to the entangled case. That is to say, we
prove that the entangled Husimi distribution function of a two-mode quantum
state $\left|\psi\right\rangle$ is just the modulus square of the complex
wavelet transform of $e^{-\left|\eta\right|^{2}/2}$ with
$\psi\left(\eta\right)$ being the mother wavelet up to a Gaussian function,
i.e.,
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle=e^{-\frac{1}{\kappa}\left|\gamma\right|^{2}}\left|\int\frac{d^{2}\eta}{\sqrt{\kappa}\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\left(\eta-z\right)/\sqrt{\kappa}\right)\right|^{2}$.
Thus is a convenient approach for calculating various entangled Husimi
distribution functions of miscellaneous quantum states.
## 16 Symplectic Wavelet transformation (SWT)
In this section we shall generalize the usual wavelet transform to symplectic
wavelet transformation (SWT) by using the coherent state representation [117].
### 16.1 Single-mode SWT
First we are motivated to generalize the usual wavelet transform, which
concerns about dilation, to optical Fresnel transform (we will explain this in
detail in section below), i.e. we shall use the symplectic-
transformed—translated versions of the mother wavelet
$\psi_{r,s;\kappa}\left(z\right)=\sqrt{s^{\ast}}\psi\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]$
(562)
as a weighting function to synthesize the original complex signal
$f\left(z\right)$,
$\displaystyle W_{\psi}f\left(r,s;\kappa\right)$
$\displaystyle=\int\frac{d^{2}z}{\pi}f\left(z\right)\psi_{r,s;\kappa}^{\ast}\left(z\right),\text{
}$ (563) $\displaystyle d^{2}z$ $\displaystyle=dxdy,\text{ }z=x+iy,$
this is named the symplectic-transformed—translated wavelet transform. One can
see that the mother wavelet $\psi$ generates the other wavelets of the family
$\psi^{\ast}\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]$
through a translating transform followed by a symplectic transform, ($r,s$ are
the symplectic transform parameter, $|s|^{2}-|r|^{2}=1,$ $\kappa$ is a
translation parameter, $s$, $r$ and $\kappa\in\mathrm{C}$), this can be seen
more clearly by writing the second transform in matrix form
$\left(\begin{array}[c]{c}z-\kappa\\\
z^{\ast}-\kappa^{\ast}\end{array}\right)\rightarrow
M\left(\begin{array}[c]{c}z-\kappa\\\
z^{\ast}-\kappa^{\ast}\end{array}\right),\text{
}M\equiv\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right),$ (564)
where $M$ is a symplectic matrix satisfies $M^{T}JM=J$,
$J=\left(\begin{array}[c]{cc}0&I\\\ -I&0\end{array}\right)$.
Symplectic matrices in Hamiltonian dynamics correspond to canonical
transformations and keep the Poisson bracket invariant, while in matrix optics
they represent ray transfer matrices of optical instruments, such as lenses
and fibers.
#### 16.1.1 Properties of symplectic-transformed—translated WT
It is straightforward to evaluate this transform and its reciprocal transform
when $f\left(z\right)$ is the complex Fourier exponentials,
$f\left(z\right)=\exp\left(z\beta^{\ast}-z^{\ast}\beta\right),$ (note that
$z\beta^{\ast}-z^{\ast}\beta$ is pure imaginary):
$\displaystyle W_{\psi}f$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)\psi^{\ast}\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\exp\left[\left(z+\kappa\right)\beta^{\ast}-\left(z^{\ast}+\kappa^{\ast}\right)\beta\right]\psi^{\ast}\left(sz-
rz^{\ast}\right)$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z^{\prime}}{\pi}\exp\left[\left(s^{\ast}z^{\prime}+rz^{\prime\ast}+\kappa\right)\beta^{\ast}-\left(sz^{\prime\ast}+r^{\ast}z^{\prime}+\kappa^{\ast}\right)\beta\right]\psi^{\ast}\left(z^{\prime}\right)$
$\displaystyle=\exp\left[\kappa\beta^{\ast}-\kappa^{\ast}\beta\right]\sqrt{s}\int\frac{d^{2}z^{\prime}}{\pi}\exp\left[z^{\prime}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)-z^{\prime\ast}\left(s\beta-r\beta^{\ast}\right)\right]\psi^{\ast}\left(z^{\prime}\right)$
$\displaystyle=\sqrt{s}\exp\left[\kappa\beta^{\ast}-\kappa^{\ast}\beta\right]\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right),$
(565)
where $\Phi$ is the complex Fourier transform of $\psi^{\ast}.$ Then we form
the adjoint operation
$\displaystyle W_{\psi}^{\ast}\left(W_{\psi}f\right)\left(z\right)$
$\displaystyle=\sqrt{s^{\ast}}\int\frac{d^{2}\kappa}{\pi}\left(W_{\psi}f\right)\left(r,s;\kappa\right)\psi\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]$
$\displaystyle=\left|s\right|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)\int\frac{d^{2}\kappa}{\pi}\exp\left[\left(\kappa+z\right)\beta^{\ast}-\left(\kappa^{\ast}+z^{\ast}\right)\beta\right]\psi\left[-s\kappa+r\kappa^{\ast}\right]$
$\displaystyle=\left|s\right|\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)$
$\displaystyle\times\int\frac{d^{2}\kappa^{\prime}}{\pi}\exp\left[-\kappa^{\prime}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)+\kappa^{\prime\ast}\left(s\beta-r\beta^{\ast}\right)\right]\psi\left(\kappa^{\prime}\right)$
$\displaystyle=\left|s\right|\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)\right|^{2},$
(566)
from which we have
$\int\frac{W_{\psi}^{\ast}\left(W_{\psi}f\right)\left(z\right)d^{2}s}{\left|s\right|^{2}}=\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)\int
d^{2}s\frac{\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)\right|^{2}}{\left|s\right|}$
(567)
so we get the inversion formula
$f\left(z\right)=\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)=\frac{\int
d^{2}sW_{\psi}^{\ast}\left(W_{\psi}f\right)\left(z\right)/\left|s\right|^{2}}{\int
d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)\right|^{2}/\left|s\right|}.$
(568)
Eq.(568) leads us to impose the normalization
$\int
d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)\right|^{2}/\left|s\right|=1,$
(569)
in order to get the wavelet representation
$f\left(z\right)=\int
d^{2}sW_{\psi}^{\ast}\left(W_{\psi}f\right)\left(z\right)/\left|s\right|^{2}.$
(570)
Then we can have a form of Parseval’s theorem for this new wavelet transform:
Preposition: For any $f$ and $f^{\prime}$ we have
$\int\int
W_{\psi}f\left(r,s;\kappa\right)W_{\psi}f^{\prime\ast}\left(r,s;\kappa\right)\frac{d^{2}\kappa
d^{2}s}{\left|s\right|^{2}}=\int\frac{d^{2}z}{2\pi}f\left(z\right)f^{\prime\ast}\left(z\right).$
(571)
Proof: Let us assume $F(\beta)$ and $F^{\prime}(\beta)$ are the complex
Fourier transform of $f\left(z\right)$ and $f^{\prime}\left(z\right)$
respectively,
$F(\beta)=\int\frac{d^{2}z}{2\pi}f\left(z\right)\exp\left(z\beta^{\ast}-z^{\ast}\beta\right)$
(572)
recall the convolution theorem defined on complex Fourier transform,
$\displaystyle\int
d^{2}zf\left(\alpha-z,\alpha^{\ast}-z^{\ast}\right)f^{\prime}\left(z\right)$
$\displaystyle=\int
d^{2}z\int\frac{d^{2}\beta}{2\pi}F(\beta)e^{\left(\alpha^{\ast}-z^{\ast}\right)\beta-\left(\alpha-z\right)\beta^{\ast}}\int\frac{d^{2}\beta^{\prime}}{2\pi}F^{\prime}(\beta^{\prime})\exp\left(z^{\ast}\beta^{\prime}-z\beta^{\prime\ast}\right)$
$\displaystyle=\int\int d^{2}\beta
d^{2}\beta^{\prime}F^{\prime}(\beta^{\prime})F(\beta)e^{\alpha^{\ast}\beta-\alpha\beta^{\ast}}\delta\left(\beta-\beta^{\prime}\right)\delta\left(\beta^{\ast}-\beta^{\prime\ast}\right)$
$\displaystyle=\int d^{2}\beta
F(\beta)F^{\prime}(\beta)e^{\alpha^{\ast}\beta-\alpha\beta^{\ast}}$ (573)
so from (573) and (562), (563) we see that
$W_{\psi}f\left(r,s;\kappa\right)=\int\frac{d^{2}z}{\pi}f\left(z\right)\psi_{r,s;\kappa}^{\ast}\left(z\right)$
can be considered as a convolution in the form
$\displaystyle\int
d^{2}zf\left(z\right)\psi^{\ast}\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]$
$\displaystyle=\int d^{2}\beta
F(\beta)\Phi^{\ast}\left(s\beta-r\beta^{\ast}\right)\exp\left(\kappa\beta^{\ast}-\kappa^{\ast}\beta\right)$
(574)
It then follows from (573) that
$\displaystyle\int
W_{\psi}f\left(r,s;\kappa\right)W_{\psi}f^{\prime\ast}\left(r,s;\kappa\right)d^{2}\kappa$
$\displaystyle=\left|s\right|\int\frac{d^{2}\beta
d^{2}\beta^{\prime}}{2\pi}F(\beta)\Phi^{\ast}\left(s\beta-r\beta^{\ast}\right)$
$\displaystyle\times
F^{\prime\ast}(\beta^{\prime})\Phi^{\prime}\left(s\beta^{\prime}-r\beta^{\prime\ast}\right)\delta\left(\beta-\beta^{\prime}\right)\delta\left(\beta^{\ast}-\beta^{\prime\ast}\right)$
$\displaystyle=\left|s\right|\int\frac{d^{2}\beta}{2\pi}F(\beta)F^{\prime\ast}(\beta)\left|\Phi\left(s\beta-r\beta^{\ast}\right)\right|^{2},$
(575)
Therefore, using (569) we see that the further integration yields
$\displaystyle\int\frac{d^{2}s}{\left|s\right|^{2}}\int
W_{\psi}f\left(r,s;\kappa\right)W_{\psi}f^{\prime\ast}\left(r,s;\kappa\right)d^{2}\kappa$
$\displaystyle=\int\frac{d^{2}\beta}{2\pi}F(\beta)F^{\prime\ast}(\beta)\int\frac{d^{2}s}{\left|s\right|}\left|\Phi\left(s\beta-r\beta^{\ast}\right)\right|^{2}$
$\displaystyle=\int\frac{d^{2}\beta}{2\pi}F(\beta)F^{\prime\ast}(\beta)=\int\frac{d^{2}z}{2\pi}f\left(z\right)f^{\prime\ast}\left(z\right),$
(576)
which completes the proof.
Theorem: From the Proposition (571) we have
$\int\int
W_{\psi}f\left(r,s;\kappa\right)\psi_{r,s;\kappa}\left(z\right)\frac{d^{2}\kappa
d^{2}s}{\left|s\right|^{2}}=f\left(z\right),$ (577)
that is, there exists an inversion formula for arbitrary function
$f\left(z\right)$. In fact, in Eq. (563) when we take
$f\left(z\right)=\delta\left(z-z^{\prime}\right),$ then
$W_{\psi}f\left(r,s;\kappa\right)=\int\frac{d^{2}z}{\pi}f\left(z\right)\psi_{r,s;\kappa}^{\ast}\left(z\right)=\psi_{r,s;\kappa}^{\ast}\left(z^{\prime}\right).$
(578)
Substituting (578) into (576) we obtain (577).
#### 16.1.2 Relation between $W_{\psi}f\left(r,s;\kappa\right)$ and optical
Fresnel transform
Now we explain why the idea of $W_{\psi}f\left(r,s;\kappa\right)$ is
originated from the optical Fresnel transform. We can visualize the
symplectic-transformed—translated wavelet transform in the context of quantum
mechanics, letting $f\left(z\right)\equiv\left\langle
z\right|\left.f\right\rangle$, $\left\langle z\right|$ is the coherent state,
$\left|z\right\rangle=\exp\left[za^{\dagger}-z^{\ast}a\right]\equiv\left|\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle$, $|0\rangle$ is the vacuum state in
Fock space, then Eq. (562) can be expressed as
$\displaystyle W_{\psi}f\left(r,s;\kappa\right)$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\psi^{\ast}\left[s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)\right]f\left(z\right)$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\left\langle\psi\right|\left.\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{c}z-\kappa\\\
z^{\ast}-\kappa^{\ast}\end{array}\right)\right\rangle\left\langle
z\right|\left.f\right\rangle$ (583)
$\displaystyle=\left\langle\psi\right|F_{1}\left(r,s,\kappa\right)\left|f\right\rangle,$
(584)
where $F^{\left(r,s,\kappa\right)}$ is defined as
$F_{1}\left(r,s,\kappa\right)=\sqrt{s}\int\frac{d^{2}z}{\pi}\left|sz-
rz^{\ast}\right\rangle\left\langle z+\kappa\right|,\;\;$ (585)
and $\left|sz-
rz^{\ast}\right\rangle\equiv\left|\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle.$To know the explicit form of
$F_{1}\left(r,s,\kappa\right)$, we employ the normal ordering of the vacuum
projector $\left|0\right\rangle\left\langle
0\right|=:\exp\left(-a^{\dagger}a\right):$ and the IWOP technique to perform
the integration in (585), which leads to
$\displaystyle F_{1}\left(r,s,\kappa\right)$
$\displaystyle=\exp(\frac{1}{4}\left|\kappa\right|^{2}+\frac{r}{8s}\kappa^{\ast
2}+\frac{r^{\ast}\kappa^{2}}{8s^{\ast}})\exp\left(-\frac{r}{2s^{\ast}}a^{\dagger
2}-\frac{\kappa}{2s^{\ast}}\left(\left|s\right|^{2}+\left|r\right|^{2}\right)a^{\dagger}\right)$
$\displaystyle\exp\left[\left(a^{\dagger}a+\frac{1}{2}\right)\ln\frac{1}{s^{\ast}}\right]\exp\left(\frac{r^{\ast}}{2s^{\ast}}a^{2}-\frac{1}{2s^{\ast}}\left(s^{\ast}\kappa^{\ast}+r^{\ast}\kappa\right)a\right).$
(586)
The transformation matrix element of $F_{1}\left(r,s,\kappa=0\right)$ in the
coordinate representation $\left|x\right\rangle$ is just the kernel of optical
diffraction integration (101) (Fresnel transform), this explains our
motivation to introduce $W_{\psi}f\left(r,s;\kappa\right)$. In particular,
when $r^{\ast}=r\equiv\sinh\lambda$, $s=s^{\ast}\equiv\cosh\lambda,$
$F_{1}\left(r,s,\kappa=0\right)$ reduces to it the well-known single-mode
squeezing operator $\exp[\frac{\lambda}{2}\left(a^{2}-a^{\dagger 2}\right)]$
which corresponds to dilation in the usual WT.
### 16.2 Entangled SWT
In the above subsection, the mother wavelet is gained through a translating
transform followed by a symplectic transform. This motivation arises from the
consideration that symplectic transforms are more general than the dilated
transform, and are useful in Fresnel transform of Fourier optics, e.g. ray
transfer matrices of optical instruments, such as lenses and fibers in matrix
optics, while in quantum optics symplectic transforms correspond to single-
mode Fresnel operator (or generalized SU(1,1) squeezing operator).
Recalling that in section 9 we have introduced the 2-mode entangled Fresnel
operator which is a mapping of classical mixed transformation
$\left(z,z^{\prime}\right)\rightarrow\left(sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right)$
in 2-mode coherent state $\left|z,z^{\prime}\right\rangle$ representation onto
quantum operator $F_{2}\left(r,s\right)$, thus we are naturally led to develop
the SWT in (563) to the so-called entangled SWT (ESWT) [118] for signals
$g\left(z,z^{\prime}\right)$ defined in two complex planes,
$W_{\phi}g\left(r,s;k,k^{\prime}\right)={\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}g\left(z,z^{\prime}\right)\phi_{r,s;k,k^{\prime}}^{\ast}\left(z,z^{\prime}\right),$
(587)
here
$\phi_{r,s;k.k^{\prime}}\left(z,z^{\prime}\right)\equiv
s^{\ast}\phi\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right],$
(588)
is used as a weighting function to synthesize the signal
$g\left(z,z^{\prime}\right)$ regarding to two complex planes. One can see that
the mother wavelet $\phi$ generates the family
$\phi^{\ast}\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right]$
through a translating transform followed by an entangled symplectic transform.
We emphasize that this transform mixes the two complex planes, which is
different from the tensor product of two independent transforms
$\left(z,z^{\prime}\right)\rightarrow\left[s\left(z-k\right)-r\left(z^{\ast}-k^{\ast}\right),s\left(z^{\prime}-k^{\prime}\right)-r\left(z^{\prime\ast}-k^{\prime\ast}\right)\right]$
given by (584). The new symplectic transform can be seen more clearly by
writing it in matrix form:
$\left(\begin{array}[c]{c}z-k\\\ z^{\ast}-k^{\ast}\\\ z^{\prime}-k^{\prime}\\\
z^{\prime\ast}-k^{{}^{\prime}\ast}\end{array}\right)\longrightarrow\mathcal{M}\left(\begin{array}[c]{c}z-k\\\
z^{\ast}-k^{\ast}\\\ z^{\prime}-k^{\prime}\\\
z^{\prime\ast}-k^{{}^{\prime}\ast}\end{array}\right)\text{,
}\mathcal{M}=\left[\begin{array}[c]{cccc}s&0&0&r\\\ 0&s^{\ast}&r^{\ast}&0\\\
0&r&s&0\\\ r^{\ast}&0&0&s^{\ast}\end{array}\right]$ (589)
where $\mathcal{M}$ is symplectic satisfying
$\mathcal{M}^{T}\mathcal{J}\mathcal{M}=\mathcal{J}$,
$\mathcal{J}=\left[\begin{array}[c]{cc}0&\mathcal{I}\\\
-\mathcal{I}&0\end{array}\right]$ , $\mathcal{I}$ is the $2\times 2$ unit
matrix.
For Eq. (587) being qualified as a new wavelet transform we must prove that it
possesses fundamental properties of the usual wavelet transforms, such as the
admissibility condition, the Parseval’s theorem and the inversion formula.
When $g\left(z,z^{\prime}\right)$ is the complex Fourier exponential,
$g_{1}\left(z,z^{\prime}\right)=\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right),$
(590)
according to (587)-(588) we evaluate its ESWT
$\displaystyle W_{\phi}g_{1}$
$\displaystyle={\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right)\phi_{r,s;k.k^{\prime}}^{\ast}\left(z,z^{\prime}\right)$
$\displaystyle=s{\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\phi^{\ast}\left[sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right]$
$\displaystyle\times\exp\left[\left(z+k\right)\beta^{\ast}-\left(z^{\ast}+k^{\ast}\right)\beta+\left(z^{\prime}+k^{\prime}\right)\gamma^{\ast}-\left(z^{\prime\ast}+k^{\prime\ast}\right)\gamma\right].$
(591)
Making the integration variables transform $sz+rz^{\prime\ast}\rightarrow w,$
$sz^{\prime}+rz^{\ast}\rightarrow w^{\prime},$ Eq. (591) becomes
$\displaystyle W_{\phi}g_{1}$
$\displaystyle=s\exp\left(k\beta^{\ast}-k^{\ast}\beta+k^{\prime}\gamma^{\ast}-k^{\prime\ast}\gamma\right){\displaystyle\iint}\frac{d^{2}wd^{2}w^{\prime}}{\pi^{2}}\phi^{\ast}\left(w,w^{\prime}\right)$
$\displaystyle\times\exp\left[w\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma\right)-w^{\ast}\left(s\beta+r\gamma^{\ast}\right)+w^{\prime}\left(s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)-w^{\prime\ast}\left(s\gamma+r\beta^{\ast}\right)\right],$
(592)
the last integration is just the complex Fourier transform (CFT) of
$\phi^{\ast},$ denoting it as $\Phi^{\ast},$ we have
$W_{\phi}g_{1}=s\exp\left(k\beta^{\ast}-k^{\ast}\beta+k^{\prime}\gamma^{\ast}-k^{\prime\ast}\gamma\right)\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right).$ (593)
Then we form the adjoint operation of (593),
$\displaystyle
W_{\phi}^{\ast}\left(W_{\phi}g_{1}\right)\left(z,z^{\prime}\right)$
$\displaystyle=s^{\ast}{\displaystyle\iint}\frac{d^{2}kd^{2}k^{\prime}}{\pi^{2}}\\{\left(W_{\phi}g\right)\left(r,s;k,k^{\prime}\right)\\}\phi\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right]$
$\displaystyle=\left|s\right|^{2}\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right){\displaystyle\iint}\frac{d^{2}kd^{2}k^{\prime}}{\pi^{2}}\exp\left(k\beta^{\ast}-k^{\ast}\beta+k^{\prime}\gamma^{\ast}-k^{\prime\ast}\gamma\right)$
$\displaystyle\times\phi\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right]$
$\displaystyle=\left|s\right|^{2}\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right){\displaystyle\iint}\frac{d^{2}kd^{2}k^{\prime}}{\pi^{2}}\phi\left[-sk-
rk^{\prime\ast},-sk^{\prime}-rk^{\ast}\right]$
$\displaystyle\times\exp\left[\left(k+z\right)\beta^{\ast}-\left(k^{\ast}+z^{\ast}\right)\beta+\left(k^{\prime}+z^{\prime}\right)\gamma^{\ast}-\left(k^{\prime\ast}+z^{\prime\ast}\right)\gamma\right]$
$\displaystyle=\left|s\right|^{2}\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right){\displaystyle\iint}\frac{d^{2}vd^{2}v^{\prime}}{\pi^{2}}\phi\left(v,v^{\prime}\right)$
$\displaystyle\times\exp\left[-v\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma\right)+v^{\ast}\left(s\beta+r\gamma^{\ast}\right)-v^{\prime}\left(s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)+v^{\prime\ast}\left(s\gamma+r\beta^{\ast}\right)\right],$
(594)
where the integration in the last line is just the CFT of $\phi$ (comparing
with (592)), thus (594) leads to
$W_{\phi}^{\ast}\left(W_{\phi}g_{1}\right)\left(z,z^{\prime}\right)=\left|s\right|^{2}\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right)\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}.$ (595)
From Eq. (595) we have
$\displaystyle\int
d^{2}sW_{\phi}^{\ast}\left(W_{\phi}g_{1}\right)\left(z,z^{\prime}\right)/\left|s\right|^{4}$
$\displaystyle=\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right)$
$\displaystyle\times\int
d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}/\left|s\right|^{2},$
(596)
which together with (590) lead to
$g_{1}\left(z,z^{\prime}\right)=\frac{\int
d^{2}sW_{\phi}^{\ast}\left(W_{\phi}g_{1}\right)\left(z,z^{\prime}\right)/\left|s\right|^{4}}{\int
d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}/\left|s\right|^{2}}.$
(597)
Eq. (597) implies that we should impose the normalization
$\int d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}/\left|s\right|^{2}=1,$
(598)
such that the reproducing process exists
$g_{1}\left(z,z^{\prime}\right)=\int
d^{2}sW_{\phi}^{\ast}\left(W_{\phi}g_{1}\right)\left(z,z^{\prime}\right)/\left|s\right|^{4}.$
(599)
(598) may be named the generalized admissibility condition. Now we can have
the corresponding _Parseval theorem_ : For any $g$ and $g^{\prime}$ we have
${\displaystyle\iiint}W_{\phi}g\left(r,s;k,k^{\prime}\right)W_{\phi}^{\ast}g^{\prime}\left(r,s;k,k^{\prime}\right)\frac{d^{2}kd^{2}k^{\prime}d^{2}s}{\left|s\right|^{4}}={\displaystyle\iint}d^{2}zd^{2}z^{\prime}g\left(z,z^{\prime}\right)g^{\prime\ast}\left(z,z^{\prime}\right).$
(600)
_Proof:_ Assuming $F\left(\beta,\gamma\right)$ and
$F^{\prime}\left(\beta,\gamma\right)$ be CFT of $g\left(z,z^{\prime}\right)$
and $g^{\prime}\left(z,z^{\prime}\right)$, respectively,
$F\left(\beta,\gamma\right)={\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}g\left(z,z^{\prime}\right)\exp\left(z\beta^{\ast}-z^{\ast}\beta+z^{\prime}\gamma^{\ast}-z^{\prime\ast}\gamma\right),$
(601)
recalling the corresponding convolution theorem
$\displaystyle{\displaystyle\iint}d^{2}zd^{2}z^{\prime}g\left(\alpha-z,\alpha^{\ast}-z^{\ast};\alpha^{\prime}-z^{\prime},\alpha^{\prime\ast}-z^{\prime\ast}\right)g^{\prime}\left(z,z^{\prime}\right)$
$\displaystyle={\displaystyle\iint}d^{2}\beta d^{2}\gamma
F\left(\beta,\gamma\right)F^{\prime}\left(\beta,\gamma\right)\exp\left(\alpha^{\ast}\beta-\alpha\beta^{\ast}+\alpha^{\prime\ast}\gamma-\alpha^{\prime}\gamma^{\ast}\right),$
(602)
so from Eqs. (587) and (601)-(602) we see that
$W_{\phi}g\left(r,s;k,k^{\prime}\right)={\displaystyle\iint}\frac{d^{2}zd^{2}z\prime}{\pi^{2}}g\left(z,z^{\prime}\right)\phi_{r,s;k,k^{\prime}}^{\ast}\left(z,z^{\prime}\right)$
can be considered as a convolution in the form (noting that the CFT of
$\phi^{\ast}$ is $\Phi^{\ast},$ see (592)-(593))
$\displaystyle{\displaystyle\iint}d^{2}zd^{2}z^{\prime}g\left(z,z^{\prime}\right)\phi^{\ast}\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right]$
$\displaystyle={\displaystyle\iint}d^{2}\beta d^{2}\gamma
F\left(\beta,\gamma\right)\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\exp\left(k\beta^{\ast}-k^{\ast}\beta+k^{\prime}\gamma^{\ast}-k^{\prime\ast}\gamma\right).$
(603)
Using Eq. (603) we calculate
$\displaystyle{\displaystyle\iint}W_{\phi}g\left(r,s;k,k^{\prime}\right)W_{\phi}^{\ast}g^{\prime}\left(r,s;k,k^{\prime}\right)d^{2}kd^{2}k^{\prime}$
$\displaystyle=\left|s\right|^{2}{\displaystyle\iiiint}d^{2}\beta d^{2}\gamma
d^{2}\beta^{\prime}d^{2}\gamma^{\prime}F\left(\beta,\gamma\right)\Phi^{\ast}\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)$ $\displaystyle\times
F^{\prime^{\ast}}\left(\beta^{\prime},\gamma^{\prime}\right)\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\delta\left(\beta-\beta^{\prime}\right)\delta\left(\beta^{\ast}-\beta^{\prime\ast}\right)\delta\left(\gamma-\gamma^{\prime}\right)\delta\left(\gamma^{\ast}-\gamma^{\prime\ast}\right)$
$\displaystyle=\left|s\right|^{2}{\displaystyle\iint}d^{2}\beta d^{2}\gamma
F\left(\beta,\gamma\right)F^{\prime\ast}\left(\beta,\gamma\right)\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}.$ (604)
As a consequence of (598) and (604) the further integration yields
$\displaystyle\int\frac{d^{2}s}{\left|s\right|^{4}}{\displaystyle\iint}W_{\phi}g\left(r,s;k,k^{\prime}\right)W_{\phi}^{\ast}g^{\prime}\left(r,s;k,k^{\prime}\right)d^{2}kd^{2}k^{\prime}$
$\displaystyle={\displaystyle\iint}d^{2}\beta d^{2}\gamma
F\left(\beta,\gamma\right)F^{\prime\ast}\left(\beta,\gamma\right)\int
d^{2}s\left|\Phi\left(s^{\ast}\beta^{\ast}+r^{\ast}\gamma,\text{
}s^{\ast}\gamma^{\ast}+r^{\ast}\beta\right)\right|^{2}/\left|s\right|^{2}$
$\displaystyle={\displaystyle\iint}d^{2}\beta d^{2}\gamma
F\left(\beta,\gamma\right)F^{\prime\ast}\left(\beta,\gamma\right)={\displaystyle\iint}d^{2}zd^{2}z^{\prime}g\left(z,z^{\prime}\right)g^{\prime\ast}\left(z,z^{\prime}\right),$
(605)
which completes the proof.
_Inversion Formula_ :__ From Eq. (600) we have
$g\left(z,z^{\prime}\right)={\displaystyle\iiint}W_{\phi}g\left(r,s;k,k^{\prime}\right)\phi_{r,s;k,k^{\prime}}\left(z,z^{\prime}\right)\frac{d^{2}kd^{2}k^{\prime}d^{2}s}{\pi^{2}\left|s\right|^{4}},$
(606)
that is, there exists an inversion formula for $g\left(z,z^{\prime}\right)$
which represents the original signal $g\left(z,z^{\prime}\right)$ as a
superposition of wavelet functions $\phi_{r,s;k,k^{\prime}},$ with the value
of entangled wavelet transform $W_{\phi}g\left(r,s;k,k^{\prime}\right)$
serving as coefficients. In fact, in Eq. (587) when we take
$g\left(z,z^{\prime}\right)=\delta\left(z-u\right)\delta\left(z^{\ast}-u^{\ast}\right)\delta\left(z^{\prime}-u^{\prime}\right)\delta\left(z^{\prime\ast}-u^{\prime\ast}\right),$
(607)
then
$W_{\phi}g\left(r,s;k,k^{\prime}\right)=\frac{1}{\pi^{2}}\phi_{r,s;k,k^{\prime}}^{\ast}\left(u,u^{\prime}\right).$
(608)
Substituting (607)-(608) into (605), we obtain (606). We can visualize the
ESWT in the context of quantum mechanics, letting
$g\left(z,z^{\prime}\right)=\left\langle
z,z^{\prime}\right|\left.g\right\rangle\ $and using Eqs. (588)-(589), Eq.
(587) is expressed as
$\displaystyle W_{\phi}g\left(r,s;k,k^{\prime}\right)$
$\displaystyle=s{\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\phi^{\ast}\left[s\left(z-k\right)+r\left(z^{\prime\ast}-k^{\prime\ast}\right),s\left(z^{\prime}-k^{\prime}\right)+r\left(z^{\ast}-k^{\ast}\right)\right]g\left(z,z^{\prime}\right)$
$\displaystyle=s{\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left\langle\phi\right.\left|\mathcal{M}\left(\begin{array}[c]{c}z-k\\\
z^{\ast}-k^{\ast}\\\ z^{\prime}-k^{\prime}\\\
z^{\prime\ast}-k^{\prime\ast}\end{array}\right)\right\rangle\left\langle
z,z^{\prime}\right|\left.g\right\rangle=\left\langle\phi\right|F_{2}\left(r,s;k,k^{\prime}\right)\left|g\right\rangle$
(613)
where $F_{2}\left(r,s;k,k^{\prime}\right)$ is defined as
$\begin{array}[c]{c}F_{2}\left(r,s;k,k^{\prime}\right)=s{\displaystyle\iint}\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left|sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right\rangle\left\langle
z+k,z^{\prime}+k^{\prime}\right|,\\\
\left|sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right\rangle=\left|sz+rz^{\prime\ast}\right\rangle_{1}\otimes\left|sz^{\prime}+rz^{\ast}\right\rangle_{2}.\end{array}$
(614)
When $k=0$ and $k^{\prime}=0$, $F_{2}\left(r,s;k=k^{\prime}=0\right)$ is just
the 2-mode Fresnel operator.
Thus, we have extended the SWT of signals in one complex plane to ESWT of
signals defined in two complex planes, the latter is not the tensor product of
two independent SWTs, this generalization is inevitable, since it resembles
the extending from the single-mode squeezing transform (or Fresnel operator)
to the two-mode squeezing transform (or entangled Fresnel operator) in quantum
optics.
### 16.3 Symplectic-dilation mixed WT
Next we shall introduce a new kind of WT, i.e., symplectic-dilation mixed WT
[119]. Recalling that in Ref. [120] we have constructed a new entangled-
coherent state representation (ECSR) $\left|\alpha,x\right\rangle$,
$\displaystyle\left|\alpha,x\right\rangle$
$\displaystyle=\exp\left[-\frac{1}{2}x^{2}-\frac{1}{4}\left|\alpha\right|^{2}+(x+\frac{\alpha}{2})a_{1}^{\dagger}\right.$
$\displaystyle\left.+(x-\frac{\alpha}{2})a_{2}^{\dagger}-\frac{1}{4}(a_{1}^{\dagger}+a_{2}^{\dagger})^{2}\right]\left|00\right\rangle,$
(615)
which is the common eigenvector of the operator $\left(X_{1}+X_{2}\right)/2$
and $a_{1}-a_{2},$ i.e.,
$\left(a_{1}-a_{2}\right)\left|\alpha,x\right\rangle=\alpha\left|\alpha,x\right\rangle$
and
$\frac{1}{2}(X_{1}+X_{2})\left|\alpha,x\right\rangle=\frac{1}{\sqrt{2}}x\left|\alpha,x\right\rangle,$
where $X_{i}=\frac{1}{\sqrt{2}}(a_{i}+a_{i}^{\dagger})$ is the coordinate
operator, ($i=1,2)$. $\left|\alpha,x\right\rangle$ is complete,
$\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}\left|\alpha,x\right\rangle\left\langle\alpha,x\right|=1,$
(616)
and exhibits partly non-orthogonal property (for $\alpha)$ and orthonormal
property (for $x),$
$\displaystyle\left\langle\alpha^{\prime},x^{\prime}\right.\left|\alpha,x\right\rangle$
$\displaystyle=\sqrt{\pi}\exp\left[-\frac{1}{4}(\left|\alpha\right|^{2}+\left|\alpha^{\prime}\right|^{2})+\frac{1}{2}\alpha\alpha^{\prime\ast}\right]\delta\left(x^{\prime}-x\right),$
(617)
so $\left|\alpha,x\right\rangle$ possess behavior of both the coherent state
and the entangled state. An interesting question is: Can we introduce a new
kind of continuous WT for which the $\left|\alpha,x\right\rangle$
representation underlies? The answer is affirmative. Our motivation of this
issue comes from the mixed lens-Fresnel transform in classical optics [121]
(see (287) below).
By synthesizing (429) and (563) and in reference to (616) we propose the mixed
WT for $g\left(\alpha,x\right)$ ($\alpha=\alpha_{1}+\mathtt{i}\alpha_{2}$):
$W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)\equiv\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}g\left(\alpha,x\right)\psi_{s,r,\kappa;\mathrm{a},b}^{\ast}\left(\alpha,x\right).$
(618)
where $\mathtt{d}^{2}\alpha=\mathtt{d}\alpha_{1}\mathtt{d}\alpha_{2},$ the
family of mother wavelet $\psi$ involves both the the symplectic transform of
$\alpha$ and the dilation-transform of $x$,
$\psi_{s,r,\kappa;\mathrm{a},b}\left(\alpha,x\right)=\sqrt{\frac{s^{\ast}}{\left|\mathrm{a}\right|}}\psi\left[s\left(\alpha-\kappa\right)-r\left(\alpha^{\ast}-\kappa^{\ast}\right),\frac{x-b}{\mathrm{a}}\right].$
(619)
Letting $g\left(\alpha,x\right)\equiv$
$\left\langle\alpha,x\right|\left.g\right\rangle,$ then (618) can be expressed
as quantum mechanical version
$W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)=\left\langle\psi\right|U\left(s,r,\kappa;\mathrm{a},b\right)\left|g\right\rangle,$
(620)
where $U\left(s,r,\kappa;\mathrm{a},b\right)$ is defined as
$\displaystyle U\left(s,r,\kappa;\mathrm{a},b\right)$
$\displaystyle=\sqrt{\frac{s}{\left|\mathrm{a}\right|}}\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}$
$\displaystyle\times\left|s\alpha-r\alpha^{\ast},\frac{x-b}{\mathrm{a}}\right\rangle\left\langle\alpha+\kappa,x\right|.$
(621)
$U\left(s,r,\kappa=0;\mathrm{a},b=0\right)$ is just the generalized squeezing
operator, which causes a lens-Fresnel mixed transform.
For Eq. (618) being qualified as a new WT we must prove that it possesses
fundamental properties of the usual WTs, such as the admissibility condition,
the Parseval theorem and the inversion formula. It is straightforward to
evaluate the transform (618) and its reciprocal transform when
$g\left(\alpha,x\right)$ is the exponential
$g_{1}\left(\alpha,x\right)=\exp\left(\alpha^{\ast}\beta-\alpha\beta^{\ast}-\mathtt{i}px\right),$
$\displaystyle W_{\psi}g_{1}$
$\displaystyle=\sqrt{\frac{s}{\left|\mathrm{a}\right|}}e^{\kappa^{\ast}\beta-\kappa\beta^{\ast}-\mathtt{i}pb}\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}$
$\displaystyle\times\psi^{\ast}(s\alpha-r\alpha^{\ast},\frac{x}{\mathrm{a}})e^{\alpha^{\ast}\beta-\alpha\beta^{\ast}-\mathtt{i}px}.$
(622)
Making the integration variables transform $s\alpha-r\alpha^{\ast}\rightarrow
w,\frac{x}{\mathrm{a}}\rightarrow x^{\prime},$ leading to
$\mathtt{d}^{2}\alpha\rightarrow\mathtt{d}^{2}w$ and
$\int_{-\infty}^{\infty}\mathtt{d}x\rightarrow\left|\mathrm{a}\right|\int_{-\infty}^{\infty}\mathtt{d}x^{\prime}$,
(622) becomes
$W_{\psi}g_{1}=\sqrt{s\left|\mathrm{a}\right|}\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)e^{\kappa^{\ast}\beta-\kappa\beta^{\ast}-\mathtt{i}pb},$
(623)
where $\Phi^{\ast}$ is just the Fourier transform of $\psi^{\ast},$
$\displaystyle\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\mathtt{d}x^{\prime}}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}w}{2\pi}\psi^{\ast}\left(w,x^{\prime}\right)$
$\displaystyle\times
e^{w^{\ast}\left(s\beta-r\beta^{\ast}\right)-w\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta\right)-\mathtt{i}\mathrm{a}px^{\prime}}.$
(624)
Then we perform the adjoint WT of (618), using (619) and (623) we see
$\displaystyle W_{\psi}^{\ast}\left(W_{\psi}g_{1}\right)\left(\alpha,x\right)$
$\displaystyle=\sqrt{\frac{s^{\ast}}{\left|\mathrm{a}\right|}}\int_{-\infty}^{\infty}\frac{\mathtt{d}b}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\kappa}{2\pi}W_{\psi}g_{1}$
$\displaystyle\times\psi\left[s\left(\alpha-\kappa\right)-r\left(\alpha^{\ast}-\kappa^{\ast}\right),\frac{x-b}{\mathrm{a}}\right]$
$\displaystyle=\left|s\right|\left|\mathrm{a}\right|g_{1}\left(\alpha,x\right)\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\int_{-\infty}^{\infty}\frac{\mathtt{d}b^{\prime}}{\sqrt{\pi}}$
$\displaystyle\times\int\frac{\mathtt{d}^{2}\kappa^{\prime}}{2\pi}e^{\kappa^{\prime}\beta^{\ast}-\kappa^{\prime\ast}\beta+\mathtt{i}\mathrm{a}pb^{\prime}}\psi\left(s\kappa^{\prime}-r\kappa^{\prime\ast},b^{\prime}\right)$
$\displaystyle=\left|s\right|\left|\mathrm{a}\right|g_{1}\left(\alpha,x\right)\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\right|^{2}.$ (625)
From Eq. (625) we obtain
$\displaystyle\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\mathrm{a}^{2}}\int\frac{\mathtt{d}^{2}s}{\left|s\right|^{2}}W_{\psi}^{\ast}\left(W_{\psi}g_{1}\right)\left(\alpha,x\right)$
$\displaystyle=g_{1}\left(\alpha,x\right)\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\left|\mathrm{a}\right|}\int\frac{\mathtt{d}^{2}s}{\left|s\right|}\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\right|^{2},$ (626)
which leads to
$g_{1}\left(\alpha,x\right)=\frac{\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\mathrm{a}^{2}}\int\frac{\mathtt{d}^{2}s}{\left|s\right|^{2}}W_{\psi}^{\ast}\left(W_{\psi}g_{1}\right)\left(\alpha,x\right)}{\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\left|\mathrm{a}\right|}\int\frac{\mathtt{d}^{2}s}{\left|s\right|}\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\right|^{2}}.$ (627)
Eq. (627) implies that we should impose the normalization
$\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\left|\mathrm{a}\right|}\int\frac{\mathtt{d}^{2}s}{\left|s\right|}\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\right|^{2}=1,$ (628)
such that the reproducing process exists
$g_{1}\left(\alpha,x\right)=\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\mathrm{a}^{2}}\int\frac{\mathtt{d}^{2}s}{\left|s\right|^{2}}W_{\psi}^{\ast}\left(W_{\psi}g_{1}\right)\left(\alpha,x\right).$
(629)
(628) may be named the generalized admissibility condition. Now we can have
the corresponding _Parseval theorem_ : For any $g$ and $g^{\prime}$ we have
$\displaystyle\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a\mathtt{d}}b}{\mathrm{a}^{2}}\int\frac{\mathtt{d}^{2}\kappa\mathtt{d}^{2}s}{\left|s\right|^{2}}W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)W_{\psi}^{\ast}g^{\prime}\left(s,r,\kappa;\mathrm{a},b\right)$
$\displaystyle=\int_{-\infty}^{\infty}\mathtt{d}x\int\mathtt{d}^{2}\alpha
g\left(\alpha,x\right)g^{\prime\ast}\left(\alpha,x\right).$ (630)
_Proof:_ Assuming $F\left(\beta,p\right)$ and $F^{\prime}\left(\beta,p\right)$
be the Fourier transforms of $g\left(\alpha,x\right)$ and
$g^{\prime}\left(\alpha,x\right)$, respectively,
$F\left(\beta,p\right)=\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{2\pi}}\int\frac{\mathtt{d}^{2}\alpha}{\pi}g\left(\alpha,x\right)e^{\alpha\beta^{\ast}-\alpha^{\ast}\beta+\mathtt{i}px},$
(631)
In order to prove (630), we first calculate
$W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)$. In similar to deriving
Eq.(221), using (LABEL:14), (217) and the inversion formula of (631) we have
$\displaystyle W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)$
$\displaystyle=\sqrt{\frac{s}{\left|\mathrm{a}\right|}}\int_{-\infty}^{\infty}\frac{\mathtt{d}p}{\sqrt{2\pi}}\int\frac{\mathtt{d}^{2}\beta}{\pi}F\left(\beta,p\right)\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}$
$\displaystyle\times
e^{\alpha^{\ast}\beta-\alpha\beta^{\ast}-\mathtt{i}px}\psi^{\ast}\left[s\left(\alpha-\kappa\right)-r\left(\alpha^{\ast}-\kappa^{\ast}\right),\frac{x-b}{\mathrm{a}}\right]$
$\displaystyle=\sqrt{s\left|\mathrm{a}\right|}\int_{-\infty}^{\infty}\frac{\mathtt{d}p}{\sqrt{2\pi}}\int\frac{\mathtt{d}^{2}\beta}{\pi}F\left(\beta,p\right)$
$\displaystyle\times\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)e^{\kappa^{\ast}\beta-\kappa\beta^{\ast}-\mathtt{i}pb}.$
(632)
It then follows
$\displaystyle\int_{-\infty}^{\infty}\mathtt{d}b\int\mathtt{d}^{2}\kappa
W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)W_{\psi}^{\ast}g^{\prime}\left(s,r,\kappa;\mathrm{a},b\right)$
$\displaystyle=\left|\mathrm{a}s\right|\int_{-\infty}^{\infty}\mathtt{d}p\mathtt{d}p^{\prime}\int\mathtt{d}^{2}\beta\mathtt{d}^{2}\beta^{\prime}F\left(\beta,p\right)F^{\prime\ast}\left(\beta^{\prime},p^{\prime}\right)$
$\displaystyle\times\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\text{
}\mathrm{a}p\right)\Phi\left(s^{\ast}\beta^{\prime\ast}-r^{\ast}\beta^{\prime},\mathrm{a}p^{\prime}\right)$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{\mathtt{d}b}{2\pi}\int\frac{\mathtt{d}^{2}\kappa}{\pi^{2}}e^{\kappa^{\ast}\left(\beta-\beta^{\prime}\right)-\kappa\left(\beta^{\ast}-\beta^{\prime\ast}\right)+\mathtt{i}\left(p^{\prime}-p\right)b}$
$\displaystyle=\left|\mathrm{a}s\right|\int_{-\infty}^{\infty}\mathtt{d}p\int\mathtt{d}^{2}\beta
F\left(\beta,p\right)F^{\prime\ast}\left(\beta,p\right)\left|\Phi\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\mathrm{a}p\right)\right|^{2}.$
(633)
Substituting (633) into the left-hand side (LHS) of (630) and using (628) we
see
LHS of (271)
$\displaystyle=\int_{-\infty}^{\infty}\mathtt{d}p\int\mathtt{d}^{2}\beta
F\left(\beta,p\right)F^{\prime\ast}\left(\beta,p\right)$
$\displaystyle\times\int_{-\infty}^{\infty}\frac{\mathtt{d}\mathrm{a}}{\left|\mathrm{a}\right|}\int\frac{\mathtt{d}^{2}s}{\left|s\right|}\left|\Phi^{\ast}\left(s^{\ast}\beta^{\ast}-r^{\ast}\beta,\mathrm{a}p\right)\right|^{2}$
$\displaystyle=\int_{-\infty}^{\infty}\mathtt{d}p\int\mathtt{d}^{2}\beta
F\left(\beta,p\right)F^{\prime\ast}\left(\beta,p\right).$ (634)
Thus we complete the proof of Eq.(630).
_Inversion Formula_ :__ From Eq. (630) we have
$g\left(\alpha,x\right)=\int_{-\infty}^{\infty}\frac{\mathtt{d\mathrm{a}d}b}{\sqrt{\pi}\mathrm{a}^{2}}\int\frac{\mathtt{d}^{2}\kappa\mathtt{d}^{2}s}{2\pi\left|s\right|^{2}}W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)\psi_{s,r,\kappa;\mathrm{a},b}\left(\alpha,x\right),$
(635)
that is the inversion formula for the original signal $g\left(\alpha,x\right)$
expressed by a superposition of wavelet functions
$\psi_{s,r,\kappa;\mathrm{a},b}\left(\alpha,x\right),$ with the value of
continuous WT $W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)$ serving as
coefficients. In fact, in Eq. (618) when we take
$g\left(\alpha,x\right)=\delta\left(\alpha-\alpha^{\prime}\right)\delta\left(\alpha^{\ast}-\alpha^{\prime\ast}\right)\delta\left(x-x^{\prime}\right),$
then
$W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)=\frac{1}{2\pi\sqrt{\pi}}\psi_{s,r,\kappa;\mathrm{a},b}^{\ast}\left(\alpha^{\prime},x^{\prime}\right).$
(636)
Substituting (636) into (634) yields (635).
We can visualize the new WT $W_{\psi}g\left(s,r,\kappa;\mathrm{a},b\right)$ in
the context of quantum optics. Noticing that the generalized squeezing
operator $U\left(s,r,\kappa=0;\mathrm{a},b=0\right)$ in (621) is an image of
the combined mapping of the classical real dilation transform $x\rightarrow$
$x/\mathrm{a}$ ($\mathrm{a}>0$) and the classical complex symplectic transform
$\left(\alpha,\alpha^{\ast}\right)\rightarrow\left(s\alpha-r\alpha^{\ast},s^{\ast}\alpha^{\ast}-r^{\ast}\alpha\right)$
in $\left|\alpha,x\right\rangle$ representation, one can use the technique of
integration within normal product of operators to perform the integration in
(621) to derive its explicit form (see Eq. (15) in Ref. [r12a]). The transform
matrix element of $U\left(s,r,\kappa=0;\mathrm{a},b=0\right)$ in the entangled
state representation $\left|\eta\right\rangle$ is
$\displaystyle\left\langle\eta\right|U\left(s,r,\kappa=0;\mathrm{a},b=0\right)\left|\eta^{\prime}\right\rangle$
$\displaystyle=\sqrt{\frac{s}{\mathrm{a}}}\int_{-\infty}^{\infty}\frac{\mathtt{d}x}{\sqrt{\pi}}\int\frac{\mathtt{d}^{2}\alpha}{2\pi}\left\langle\eta\right.\left|s\alpha-r\alpha^{\ast},\frac{x}{\mathrm{a}}\right\rangle\left\langle\alpha,x\right|\left.\eta^{\prime}\right\rangle.$
(637)
In Fock space $\left|\eta=\eta_{1}+\mathtt{i}\eta_{2}\right\rangle$ is two-
mode EPR entangled state in (29).
Then using (615) and (29), we obtain
$\left\langle\eta\right.\left|\alpha,x\right\rangle=\frac{1}{\sqrt{2}}\exp\left[-\frac{\alpha^{2}+\left|\alpha\right|^{2}}{4}-\frac{1}{2}\eta_{1}^{2}+\eta_{1}\alpha-\mathtt{i}\eta_{2}x\right].$
(638)
Substituting (638) into (637) and using (85), we obtain
$\displaystyle\left\langle\eta\right|U\left(s,r,\kappa=0;\mathrm{a},b=0\right)\left|\eta^{\prime}\right\rangle$
$\displaystyle=\frac{\pi}{\sqrt{\mathrm{a}}}\delta\left(\eta_{2}^{\prime}-\eta_{2}/\mathrm{a}\right)\frac{1}{\sqrt{2\mathtt{i}\pi
B}}$
$\displaystyle\times\exp\left[\frac{\mathtt{i}}{2B}\left(A\eta_{1}^{\prime
2}-2\eta_{1}\eta_{1}^{\prime}+D\eta_{1}^{2}\right)\right].$ (639)
which is just the kernel of a mixed lens$-$Fresnel transform, i.e., the
variable $\eta_{1}$ of the object experiences a generalized Fresnel transform,
while $\eta_{2}$ undergoes a lens transformation. Thus, based on
$\left|\alpha,x\right\rangle$ we have introduced SDWT which involves both the
real variable dilation-transform and complex variable symplectic transform,
corresponding to the lens-Fresnel mixed transform in classical optics.
## 17 Fresnel-Hadamard combinatorial transformation
In the theoretical study of quantum computer, of great importance is the
Hadamard transform. This operation is $n$ Hadamard gates acting in parallel on
$n$ qubits. The Hadamard transform produces an equal superposition of all
computational basis states. From the point of view of Deutsch-Jozsa quantum
algorithm, the Hadamard transform is an example of the $N=2^{n}$ quantum
Fourier transform, which can be expressed as [122]
$\left|j\right\rangle=\frac{1}{\sqrt{2^{n}}}{\displaystyle\sum\limits_{k=0}^{2^{n}-1}}e^{2\pi
ijk/2^{n}}\left|k\right\rangle.$ (640)
Now the continuous Hadamard transform, used to go from the coordinate basis
$\left|x\right\rangle$ to the momentum basis, is defined as[123]
$\mathfrak{F}\left|x\right\rangle=\frac{1}{\sqrt{\pi}\sigma}\int_{-\infty}^{\infty}dy\exp\left(2ixy/\sigma^{2}\right)\left|y\right\rangle,$
(641)
where $\sigma$ is the scale length. $\mathfrak{F}$ is named Hadamard operator.
Using the completeness of
$\int_{-\infty}^{\infty}dx\left|x\right\rangle\left\langle x\right|=1,$ we
have
$\mathfrak{F}=\frac{1}{\sqrt{\pi}\sigma}{\displaystyle\iint_{-\infty}^{\infty}}dxdy\exp\left(2ixy/\sigma^{2}\right)\left|y\right\rangle\left\langle
x\right|.$ (642)
The above two transforms (Fresnel transform and Hadmard transform) are
independent of each other, an interesting question thus naturally arises: can
we combine the two transforms together? To put it in another way, can we
construct a combinatorial operator which play the role of both Fresnel
transform and Hadmard transform for two independent optical modes? The answer
is affirmative, in this section we try to construct so-called Fresnel-Hadmard
combinatorial transform.
### 17.1 The Hadamard-Fresnel combinatorial operator
Based on the coherent-entangled representation $\left|\alpha,x\right\rangle$,
and enlightened by Eq. (96) and (641) we now construct the following ket-bra
integration [124]
$U=\frac{\sqrt{s}}{\sqrt{\pi}\sigma}\int\frac{d^{2}\alpha}{\pi}{\displaystyle\iint}dxdy\exp\left(2ixy/\sigma^{2}\right)\left|s\alpha-r\alpha^{\ast},y\right\rangle\left\langle\alpha,x\right|,$
(643)
we name $U$ the Hadamard-Fresnel combinatorial operator.
Substituting Eq.(641) into Eq.(643), and using the two-mode vacuum projector’s
normally ordered form $\left|00\right\rangle\left\langle
00\right|=\colon\exp\left[-a_{1}^{+}a_{1}-a_{2}^{+}a_{2}\right]\colon$ as well
as the IWOP technique we get
$U=\frac{\sqrt{s}}{\sqrt{\pi}\sigma}\colon\int\frac{d^{2}z}{\pi}A\left(z,z^{\ast}\right){\displaystyle\iint}dxdyB(x,y)e^{C}\colon,$
(644)
where
$C\equiv-\frac{\left(a_{1}^{+}+a_{2}^{+}\right)^{2}+\left(a_{1}+a_{2}\right)^{2}}{4}-a_{1}^{+}a_{1}-a_{2}^{+}a_{2},$
$B(x,y)\equiv\exp\left[-\frac{y^{2}+x^{2}}{2}+y\left(a_{1}^{+}+a_{2}^{+}\right)+x\left(a_{1}+a_{2}\right)+\frac{2ixy}{\sigma^{2}}\right],$
and
$A\left(z,z^{\ast}\right)\equiv\exp\left[-\frac{\left|sz-
rz^{\ast}\right|^{2}+\left|z\right|^{2}}{4}+\frac{sz-
rz^{\ast}}{2}\left(a_{1}^{\dagger}-a_{2}^{\dagger}\right)+\frac{z^{\ast}\left(a_{1}-a_{2}\right)}{2}\right],$
they are all within the normal ordering symbol $::$ . Now performing the
integration over $dxdy$ within $::$ and remembering that all creation
operators are commute with all annihilation operators (the essence of the IWOP
technique) so that they can be considered c-number during the integration, we
can finally obtain
$\displaystyle U$
$\displaystyle=\frac{1}{\sqrt{s^{\ast}}}\frac{4\sqrt{\pi}\sigma}{\sqrt{\sigma^{4}+4}}\colon\exp\left\\{-\frac{1}{2}\frac{r}{s^{\ast}}\left(\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}\right)^{2}+\frac{\sigma^{4}-4}{2\left(\sigma^{4}+4\right)}\left(\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\right)^{2}\right.$
$\displaystyle+\left(\frac{1}{s^{\ast}}-1\right)\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}\frac{a_{1}-a_{2}}{\sqrt{2}}+\left(\frac{4i\sigma^{2}}{\sigma^{4}+4}-1\right)\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\frac{a_{1}+a_{2}}{\sqrt{2}}$
$\displaystyle+\left.\frac{1}{2}\frac{r^{\ast}}{s^{\ast}}\left(\frac{a_{1}-a_{2}}{\sqrt{2}}\right)^{2}+\frac{\sigma^{4}-4}{2\left(\sigma^{4}+4\right)}\left(\frac{a_{1}+a_{2}}{\sqrt{2}}\right)^{2}\right\\}\colon,$
(645)
which is the normally ordered form of Hadamard-Fresnel combinatorial operator.
### 17.2 The properties of Hadamard-Fresnel operator
Note
$\left[\frac{a_{1}-a_{2}}{\sqrt{2}},\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\right]=0,$
(646)
and
$\left[\frac{a_{1}-a_{2}}{\sqrt{2}},\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}\right]=1,\text{
\
}\left[\frac{a_{1}+a_{2}}{\sqrt{2}},\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\right]=1,$
(647)
$\frac{a_{1}-a_{2}}{\sqrt{2}}$ can be considered a mode independent of another
mode $\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}},$ thus we have the
operator identity
$\text{exp}\left[f\left(a_{1}^{{\dagger}}\pm
a_{2}^{{\dagger}}\right)\left(a_{1}\pm
a_{2}\right)\right]=\colon\exp[\frac{1}{2}\left(e^{2f}-1\right)\left(a_{1}^{{\dagger}}\pm
a_{2}^{{\dagger}}\right)\left(a_{1}\pm a_{2}\right)]\colon.$ (648)
Using (648) we can rewrite Eq.(647) as
$U=U_{2}U_{1}=U_{1}U_{2},$ (649)
where
$\displaystyle U_{1}$
$\displaystyle=\frac{4\sqrt{\pi}\sigma}{\sqrt{\sigma^{4}+4}}\exp\left[\frac{\sigma^{4}-4}{2\left(\sigma^{4}+4\right)}\left(\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\right)^{2}\right]$
$\displaystyle\exp\left[\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\frac{a_{1}+a_{2}}{\sqrt{2}}\ln\frac{4i\sigma^{2}}{\left(\sigma^{4}+4\right)}\right]\exp\left[\frac{\sigma^{4}-4}{2\left(\sigma^{4}+4\right)}\left(\frac{a_{1}+a_{2}}{\sqrt{2}}\right)^{2}\right]$
(650)
and
$\displaystyle U_{2}$
$\displaystyle=\exp\left[-\frac{r}{2s^{\ast}}\left(\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}\right)^{2}\right]\exp\left[\left(\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}\frac{a_{1}-a_{2}}{\sqrt{2}}+\frac{1}{2}\right)\ln\frac{1}{s^{\ast}}\right]$
$\displaystyle\exp\left[\frac{r^{\ast}}{2s^{\ast}}\left(\frac{a_{1}-a_{2}}{\sqrt{2}}\right)^{2}\right],$
(651)
while $U_{2}$ is the Fresnel operator for mode $\frac{a_{1}-a_{2}}{\sqrt{2}},$
$U_{1}$ is named the Hadamard operator for mode
$\frac{a_{1}+a_{2}}{\sqrt{2}}.$
It then follows
$\displaystyle U\frac{a_{1}-a_{2}}{\sqrt{2}}U^{-1}$
$\displaystyle=U_{2}\frac{a_{1}-a_{2}}{\sqrt{2}}U_{2}^{-1}=s^{\ast}\frac{a_{1}-a_{2}}{\sqrt{2}}+r\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}},$
$\displaystyle U\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}U^{-1}$
$\displaystyle=U_{2}\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}}U_{2}^{-1}=r^{\ast}\frac{a_{1}-a_{2}}{\sqrt{2}}+s\frac{a_{1}^{{\dagger}}-a_{2}^{{\dagger}}}{\sqrt{2}},$
(652)
from which we see the Hadamard-Fresnel combinatorial operator can play the
role of Fresnel transformation for $\frac{a_{1}-a_{2}}{\sqrt{2}}.$ Physically,
$\frac{a_{1}-a_{2}}{\sqrt{2}}$ and $\frac{a_{1}+a_{2}}{\sqrt{2}}$ can be two
output fields of a beamsplitter.
In a similar way, we have
$\displaystyle U\frac{a_{1}+a_{2}}{\sqrt{2}}U^{-1}$
$\displaystyle=U_{1}\frac{a_{1}+a_{2}}{\sqrt{2}}U_{1}^{-1}=\frac{1}{4i\sigma^{2}}\left[\left(\sigma^{4}+4\right)\frac{a_{1}+a_{2}}{\sqrt{2}}-\left(\sigma^{4}-4\right)\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}\right],$
$\displaystyle U\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}U^{-1}$
$\displaystyle=U_{1}\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}U_{1}^{-1}=\frac{1}{4i\sigma^{2}}\left[-\left(\sigma^{4}+4\right)\frac{a_{1}^{{\dagger}}+a_{2}^{{\dagger}}}{\sqrt{2}}+\left(\sigma^{4}-4\right)\frac{a_{1}+a_{2}}{\sqrt{2}}\right]$
(653)
which for the quadrature
$X_{i}=\left(a_{i}+a_{i}^{{\dagger}}\right)/\sqrt{2},$
$P_{i}=\left(a_{i}-a_{i}^{{\dagger}}\right)/\sqrt{2}i,$ $\left(i=1,2\right),$
leads to
$U\frac{X_{1}+X_{2}}{2}U^{-1}=\frac{\sigma^{2}}{4}\left(P_{1}+P_{2}\right),\text{
\ \
}U\left(P_{1}+P_{2}\right)U^{-1}=-\frac{4}{\sigma^{2}}\frac{X_{1}+X_{2}}{2},$
(654)
from which we see that the Hadamard-Fresnel combinatorial operator also plays
the role of exchanging the total momentum—average position followed by a
squeezing transform, with the squeezing parameter being
$\frac{\sigma^{2}}{4}.$
The mutual transform in (654) can be realized by
$\displaystyle
e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}\left(X_{1}+X_{2}\right)e^{-i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}$
$\displaystyle=P_{1}+P_{2},\text{ }$ (655) $\displaystyle\text{\
}e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}\left(P_{1}+P_{2}\right)e^{-i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}$
$\displaystyle=-\left(X_{1}+X_{2}\right)\text{\ }$ (656)
while the two-mode squeezing operator is
$S_{2}=\exp\left[\ln\frac{2}{\sigma^{2}}\left(a_{1}^{{\dagger}}a_{2}^{{\dagger}}-a_{1}a_{2}\right)\right],$
therefore
$U_{1}=S_{2}^{-1}e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}.$
(657)
From Eq.(649) and Eq.(657), we see that the Hadamard-Fresnel combinatorial
operator can be decomposed as
$U=U_{2}S_{2}^{-1}e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}=S_{2}^{-1}e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)}U_{2}.$
(658)
It can be also seen that $U$ is unitary, $U^{+}U=UU^{+}=1$.
In this section, we have introduced the Fresnel-Hadamard combinatorial
operator by virtue of the IWOP technique. This unitary operator plays the role
of both Fresnel transformation for mode $\frac{a_{1}-a_{2}}{\sqrt{2}}$ and
Hadamard transformation for mode $\frac{a_{1}+a_{2}}{\sqrt{2}},$ respectively,
and the two transformations are combinatorial. We have shown that the two
transformations are concisely expressed in the coherent-entangled state
representation as a projective operator in integration form. We also found
that the Fresnel-Hadamard operator can be decomposed as
$U_{2}S_{2}^{-1}e^{i\frac{\pi}{2}\left(a_{1}^{{\dagger}}a_{1}+a_{2}^{{\dagger}}a_{2}\right)},$
a Fresnel operator $U_{2}$, a two-mode squeezing operator $S_{2}^{-1}$ and the
total momentum-average position exchanging operator. Physically,
$\frac{a_{1}-a_{2}}{\sqrt{2}}$ and $\frac{a_{1}+a_{2}}{\sqrt{2}}$ can be two
output fields of a beamsplitter. If an optical device can be designed for
Fresnel-Hadamard combinatorial transform, then it can be directly applied to
these two output fields of the beamsplitter.
In summary, although quantum optics and classical optics are so different, no
matter in the mathematical tools they employed or in a conceptual view
(quantum optics concerning the wave-particle duality of optical field with an
emphasis on its nonclassical properties, whereas classical optics works on the
distribution aqnd propagation of the light waves), that it is a new
exploration to link them systematically. However, In this review, via the
route of developing Dirac’s symbolic method we have revealed some links
between them by mapping classical symplectic transformation in the coherent
state representation onto quantum unitary operators (GFO), throughout our
discussion the IWOP technique is indispensable for the derivation. We have
resorted to the quantum optical interpretation of various classical optical
transformations by adopting quantum optics concepts such as the coherent
states, squeezed states, and entangled states, etc. Remarkably, we have
endowed complex fractional Fourier transform, Hankel transform with quantum
optical representation-transform interpretation. Our formalism, starting from
quantum optics theory, not only provides quantum mechanical account of various
classical optical transformations, but also have found their way back to some
new classical transformations, e.g. entangled Fresnel transform, Fresnel-
wavelet transform, etc, which may have realistic optical interpretation in the
future. As Dirac predicted, functions that have been applied in classical
optical problems may be translated in an operator language in quantum
mechanics, and vice-versa. We expect that the content of this work may play
some role in quantum states engineering, i.e., optical field states’
preparation and design.
Once the correspondence in this respect between the two distinct fields is
established, the power of Dirac’s symbolic method can be fully displayed to
solve some new problems in classical optics, e.g., to find new eigen-modes of
some optical transforms; to extend the research region of classical optics
theoretically by introducing new transforms (for example, the entangled
Fresnel transforms), which may bring attention of experimentalists, who may
get new ideas to implement these new classical optical transformations.
Acknowledgement: This work supported by the National Natural Science
Foundation of China, Grant No. 10775097 and 10874174, and a grant from the Key
Programs Foundation of Ministry of Education of China (No. 210115), and the
Research Foundation of the Education Department of Jiangxi Province of China
(grant no. GJJ10097).
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|
arxiv-papers
| 2010-10-03T04:21:13 |
2024-09-04T02:49:13.352069
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-yi Fan and Li-yun Hu",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1010.0377"
}
|
1010.0417
|
VISUAL-HINT BOUNDARY TO SEGMENT ALGORITHM FOR IMAGE SEGMENTATION Yu Su and
Margaret H. Dunham Department of Computer Science and Engineering Southern
Methodist University Dallas, Texas 75275–0122
###### Abstract
Image segmentation has been a very active research topic in image analysis
area. Currently, most of the image segmentation algorithms are designed based
on the idea that images are partitioned into a set of regions preserving
homogeneous intra-regions and inhomogeneous inter-regions. However, human
visual intuition does not always follow this pattern. A new image segmentation
method named Visual-Hint Boundary to Segment (VHBS) is introduced, which is
more consistent with human perceptions. VHBS abides by two visual hint rules
based on human perceptions: (i) the global scale boundaries tend to be the
real boundaries of the objects; (ii) two adjacent regions with quite different
colors or textures tend to result in the real boundaries between them. It has
been demonstrated by experiments that, compared with traditional image
segmentation method, VHBS has better performance and also preserves higher
computational efficiency.
## 1 Introduction
Image segmentation is a vast topic in image analysis. In this chapter, we
present a low-level image segmentation method, which has been proposed to
segment images in a way that agrees with human perceptions. In recent years,
Most of the image segmentation algorithms are designed based on an idea that
partitions the images into a set of regions preserving homogeneous intra-
regions and inhomogeneous inter-regions. By this idea, these methods segment
images in classification or clustering manner. However, human visual intuition
does not always follow this manner. Our goal of this research is to define a
low-level image segmentation algorithm which is consistent with human visual
perceptions.
The proposed new image segmentation method is called Visual-hint Boundary to
Segment (VHBS). VHBS abides by two visual hint rules based on human
perceptions: (i) the global scale boundaries tend to be the real boundaries of
the objects; (ii) two adjacent regions with quite different colors or textures
tend to result the real boundaries between them. Compared with other
unsupervised segmentation methods, the outputs of VHBS are more consistent to
the human perceptions. Beside, reducing complexity is another objective of
VHBS since high performance segmentation methods usually are computationally
intensive. Therefore, chaos and non-chaos concepts are introduced in VHBS to
prevent algorithm going down to details of pixel level. It guarantees that
segmentation process stays at a coarse level and keeps the computational
efficiency.
VHBS is composed by two phases, Entropy-driven Hybrid Segmentation (EDHS) and
Hierarchical Probability Segmentation (HPS.) VHBS starts from EDHS, which
produces a set of initial segments by combining local regions and boundaries.
These local regions and boundaries are generated by a top-down decomposition
process and initial segments are formed by a bottom-up composition process.
The top-down decomposition process recursively decomposes the given images
into small partitions by setting a stopping condition for each branch of
decomposition. We set an entropy measurement as the stopping condition since
smaller entropy of local partitions implies lower disorder in the local
partitions. To preserve the computational efficiency, we set up a size
threshold of the partitions to prevent the decomposition going down to pixel
level. Based on this threshold, local partitions are grouped into two types,
chaos if the size of a partition is less than the threshold and non-chaos
otherwise. Local regions and boundaries are computed in local partitions. Each
local region is described by a vector called feature description and the local
boundaries are weighted by the probabilities. To calculate the probabilities,
we design two scale filter, $f_{1}$ and $f_{2}$, which are based on the two
visual hints (i) and (ii) respectively. The boundaries between two adjacent
regions are weighted by the product of $f_{1}$ and $f_{2}$. A bottom-up
composition process is followed and combines these local regions and
boundaries to form a set of the initial segments, $S=(s_{0},\ldots,s_{n})$.
The second phase of VHBS is Hierarchical Probability Segmentation (HPS,) which
constructs a Probability Binary Tree (PBT) based on these initial segments
$S$. PBT presents the hierarchy segments based on boundary probabilities
between these initial segments, which forms the leaves of PBT. The root
represents the original images and the intern nodes of PBT are the segments
combined by their children. Links are labeled by the boundary probabilities.
PBT can be visualized in number of segments or even provides the local
details. The difference compared with the methods based on MST such as [20,
26, 21] is that these methods generate the tree structure based on the
similarities between pixels. Whereas, our method generate the tree structure
based on probabilities between regions. It makes the algorithm insensitive to
the noise and it greatly reduces the computational complexity. A similar
approach is proposed by [1]. Compared with this approach, VHBS is more
efficient since VHBS prevents the decomposition process going down to pixel
level by setting a chaos threshold. The novel aspects of VHBS include:
1. 1.
Visual-Hint: Algorithm abides by two visual hint rules which force the outputs
of VHBS are more consistent to human perceptions;
2. 2.
feature Detection: VHBS outputs a set of feature descriptors, which describe
the features for each segment;
3. 3.
computational Efficiency: VHBS has high computational efficiency since the
algorithm does not go down to pixel level;
4. 4.
hybrid algorithm: VHBS combines edge-, region-, cluster- and graph-based
techniques.
## 2 Relative work
Image segmentation is one of major research areas in image analysis and is
already explored for many years. Regularly, segmentation methods partition the
given images into a set of segments, where a segment contains a set of pixels
that preserve high similarity within a segment and maximize differences
between different segments. Some examples of classical image segmentation
algorithms are k-means clustering [31], histogram threshold, region growth and
watershed transformation. These methods are efficient and easy to understand
but with obvious weaknesses which become barriers for applications. These
weaknesses include sensitivity of image noise and textures; improperly merging
two disjoint areas with the similar gray values by histogram threshold
methods; improper initial condition resulting in incorrect outputs and tending
to produce excessive over-segmentation by watershed transformation methods
[27]. All these examples demonstrate that image segmentation is an extensive
and difficult research area. In recent years, numerous image segmentation
methods have been proposed and greatly overcome those weaknesses. Commonly,
segmentation algorithms fall into one or more than one of the following
categories: edge-based, region-based, cluster-based and graph-based.
The idea of edge-based segmentation methods is straightforward. Contours and
segments highly correlate each other. The closed contours give the segments
and segments automatically produce the boundaries. Edge-based segmentation
methods rely on contours located in images and then these contours produce the
boundaries of the segments. Therefore, much research has focused on contour
detection. The classical approaches to detect the edges are to look for the
discontinuities of brightness such as in Canny Edge Detection [6]. [36]
demonstrates that these approaches by looking for discontinuities of
brightness are inadequate models for locating the boundaries in natural
images. One reason is that texture is a common phenomenon in natural images
and these textures produce some unwanted edges. Another reason is that to
locate segments in an image by edges requests closed boundaries. These
approaches usually provide incontinuous contours, which is inadequate for
locating the segments. In recent years, many high performance contour
detections have been proposed such as [42, 36, 32, 55, 33]. One category of
contour detections is locating the boundary of an image by measuring the local
features. To improve the edge detection performance on the natural images,
some approaches consider one or combine more descriptors for each pixel in
several feature channels over a different scales and orientations to locate
boundaries. [36] proposes a learning schema approach that considers
brightness, color and texture features at each local position and uses a
classifier to combine these local features. Based on the research of [36, 33]
combines the spectral component to form the so called globalized probability
of boundary to improve the accuracy of the boundary detection. There are many
boundary detection and segmentation methods which are oriented energy approach
[37]. Examples of these approaches are [51, 40]. To achieve high accuracy,
these approaches usually combine more than one local feature channels. The
computational complexity becomes a bottleneck if the application requests high
computational efficiency. Of course there are many other proposed boundary
detection and segmentation algorithms based on rich texture analysis such as
[4, 28, 22]. However, highly accurate image contour and segment detection
methods are computationally intensive [1].
To locate the segments from the contours, the contours must form closed
regions. An example of such research is [45] by bridging the disconnecting
contours or contours tracking to locate the regions. Another recent research
is [1], which can be divided into two phases: (i) Oriented Watershed Transform
(OWT) produces a set of initial regions from a contour detection. Paper
selects gPb proposed by [33] as the contour detection algorithm since this
contour detector gives high accuracy by the benchmark of BSDB [35]; (ii)
Ultrametric Contour Map (UCM) constructs the hierarchical segments. A tree is
generated by a greedy graph-based region merging algorithm, where the leaves
are those initial regions and the root is the entire images. The segmentation
algorithm proposed by [1] has high accuracy segmentation performance. But the
disadvantage is obvious. gPb is a expensive contour detection and gPb provides
fine gradient initial regions. It can be proved that the time complexity of
constructing hierarchical segments over such a fine gradient is also
computationally intensive. Other examples of recent contour-segment researches
are [22, 23].
Typically, a region-based algorithm is combined with clustering techniques to
assemble the sub-regions into final segments and numerous methods fall into
these two schemas such as [48, 30, 10, 12, 11, 24, 13]. The common used
techniques include region growth, split-merge, supervised and unsupervised
classification. [24] proposes a region growth algorithm, called random walk
segmentation, which is a multi-label, user interactive image segmentation.
This algorithm starts from a small number of seeds with user-predefined
labels. Random walk algorithm can determine the probabilities by assuming a
random walker starting at each unlabeled pixel that will first reach one of
these user-predefined seeds. By this assumption, pixels are assigned to the
label which is the greatest probability based on a random walker. Mean shift
[13] is a well known density estimation cluster algorithm and been widely used
for image segmentation and object tracking. Based on the domain probability
distribution, the algorithm iteratively climbs the gradient to locate the
nearest peak. [13] demonstrates that mean shift provides good segmentation
results and is suitable for real data analysis. But the quadratic
computational complexity of the algorithm is a disadvantage and the choice of
moving window size is not trivial.
In recent years, much research has been built based on graph theoretic
techniques. It has been demonstrated by [44, 20, 16, 26, 19, 18, 15, 21, 52,
5, 39, 50, 8, 34, 17] that these approaches support image segmentation as
well. As pointed out by [26], graph-based segmentation could be roughly
divided into two groups. One is tree-structure segmentation and another is
graph-cut segmentation. Assuming a 2D image as space $P$, both of these two
approaches view $P$ as the collection of a set of subgraphs
$<p_{1},\ldots,p_{n}>$, where each $p_{i}$ is an undivided partition,
$P=\bigcup p_{i}$ and $\phi=p_{i}\bigcap p_{j}$ for all $1\leq i\neq j\leq n$.
Commonly, $p_{i}$ denotes a pixel of the images. Tree-structure [20, 16, 26,
19, 18, 15, 21] expresses the split-merge process in a hierarchical manner.
The links between parents and children indicate the including relationship and
the nodes of the tree denote the pieces of subgraphs. Graph-cut [52, 5, 39,
50, 8, 34, 17] views each element of $<p_{1},\ldots,p_{n}>$ as a vertex and
the edges of the graph are defined by the similarities between these adjacent
vertices. This process forms a weighted undirected graph $G=(V,E)$ and relies
the graph cutting to process the graph partition.
A common tree-structure approach is minimum spanning tree (MST) [41]. [20, 21]
propose an algorithm based on MST. That is using the local variation of
intensities to locate the proper granularity of the segments based on the so
called Kruskal’s minimum spanning tree (KMST). Another recent example of tree-
structure approach is [16]. The purpose of this approach is to find the
semantic coherence regions based on $\epsilon$-neighbor coherence segmentation
criterion by a so called connected coherence tree algorithm (CCTA). Rather
than generating tree based on the pixel similarities, [1] generate a tree
structure based on the region similarities. Tree structure based on region
similarities should provide better computational complexity than the structure
based on the pixel similarities since $|V|$ is greatly reduced by replacing
pixels by regions.
Graph-cut approaches are also called spectral clustering. The basic idea of
the graph-cut approach is partitioning $G=(V,E)$ into disjoint subsets by
removing the edges linking subsets. Among these approaches, the normalized cut
(Ncut) [44] is widely used. Ncut proposed a minimization cut criterion which
measures the cut cost as a fraction of the total edge connection to all the
nodes in the graph. This paper demonstrates that minimizing the normalized cut
criterion is equivalent to solving a generalized eigenvector system. Other
recent examples of graph-cut approaches are [52, 5]. Graph-cut approaches have
been proved to be NP-complete problems and the computational complexity is
expensive. These disadvantages become the main barriers for graph-cut methods.
## 3 Entropy-driven Hybrid Segmentation
Entropy-driven Hybrid Segmentation, EDHS, begins with a top-down decomposition
from the original input images and finishes with a bottom-up composition. Top-
down decomposition quarterly partitions a given image and correspondly
produces a quadtree based on a stopping condition. EDHS uses an edge detector,
such as Canny Detector [6], to locate the boundaries between the local regions
in the leaves. These boundaries are weighted by the probabilities computed
based on the two visual hint rules.
Bottom-up composition recursively combines the local regions when the two
adjacent local regions share a boundary with zero probability. This process
forms the initial segments, $S=(s_{0},\ldots,s_{n})$, and a set of
probabilities, $C_{b}=\\{c_{i,j}\\}$, which describes the weights of the
boundaries between each pair of the adjacent initial segments, where index $i$
and $j$ imply two initial segments $s_{i}$ and $s_{j}$, which share a boundary
valued by a real number $c_{i,j}\in[0,1]$, $0\leq i\neq j\leq n$. For each
initial segment $s_{i}$, a feature vector, $fd_{i}=<v_{1},\ldots,v_{m}>$, is
generated to describe this segment. The feature descriptor, such as the $CF$
in BIRCH [47], summarizes the important features of each area (cluster.)
Although the specific values used in feature descriptor may vary, in this
chapter we assume $<r,g,b>$, where $r$, $g$ and $b$ are mean values of color
channels of red, green and blue.
### 3.1 Top-down decomposition
Decomposition mechanism is a wildly used technique in hierarchical image
segmentation [2, 38]. Our decomposition process recursively decomposes the
images into four quadrants. The decomposition process is presented by an
unbalance quadtree. The root represents the original image and nodes represent
the four partitions. A stopping condition is assigned for each branch of
decomposition. Partition process is stopped when the desired stopping
condition is reached. Figure 1 demonstrates an example of the data structures.
We summarize the top-down decomposition as follows:
1. i
Partitioning the images into small pieces reduces the information presented in
local images, which helps VHBS conquer the sub-problems at the local position;
2. ii
Decomposition provides the relative scale descriptor for the scale filter to
calculate the probabilities of the boundaries. We will discuss the relative
scale descriptor in section 3.2.1;
3. iii
Divide and conquer schema potentially supports the parallel implementation,
which could greatly improve the computational efficiency.
To describe the decomposition, the dyadic rectangle [29] is introduced. A
dyadic rectangle is the family of regions
$I_{s,t}=\\{[i2^{s},(i+1)2^{s}-1]\times[j2^{t},(j+1)2^{t}-1],0\leq i\leq
n/2^{s}-1,0\leq j\leq m/2^{t}-1\\}$, for $0\leq s\leq\log n,0\leq t\leq\log
m$. The dyadic rectangle of $I$ has some nice properties. Every dyadic
rectangle is contained in exactly one “parent” dyadic rectangle, where the
“child” dyadic rectangles are located inside of a “parent” dyadic rectangle.
The area of “paren” is always an integer power of two times of “chil” dyadic
rectangle. Mapping the images into Cartesian plane, dyadic rectangle provides
a model to uniformly decompose images into sub-images recursively.
Given an $n\times m$ $I$, assuming that $n$ and $m$ are the power of $2$, the
set of dyadic rectangles at levels $(0,0)$ through $(\log n,\log m)$ form a
complete quadtree, whose root is the level $(\log n,\log m)$ dyadic rectangle
$[0,n-1]\times[0,m-1]$. Each dyadic rectangle $I_{s,t}$ with level $1\leq
s\leq\log n,1\leq t\leq\log m$ has four children that are dyadic rectangles at
levels $s-1$ and $t-1$, which are four quadrants of the $I_{s,t}$. Suppose
$I_{s,t}=[i2^{s},(i+1)2^{s}-1]\times[j2^{t},(j+1)2^{t}-1]$, for $0\leq i\leq
n2^{-s},0\leq j\leq m2^{-t}$, then, the first quadrant of $I_{s,t}$ is
$[(2i+1)n2^{-(s+1)},(2i+2)n2^{-(s+1)}-1]\times[2jm2^{-(t+1)},(2j+1)m2^{-(t+1)}-1]$;
the second quadrant is
$[2in2^{-(s+1)},(2i+1)n2^{-(s+1)}-1]\times[2jm2^{-(t+1)},(2j+1)m2^{-(t+1)}-1]$;
the third quadrant is
$[2in2^{-(s+1)},(2i+1)n2^{-(s+1)}-1]\times[(2j+1)m2^{-(t+1)},(2j+2)m2^{-(t+1)}-1]$
and the fourth quadrant is
$[(2i+1)n2^{-(s+1)},(2i+2)n2^{-(s+1)}-1]\times[(2j+1)m2^{-(t+1)},(2j+2)m2^{-(t+1)}-1]$.
Figure 1: Top-down decomposition and the quadtree structure
#### 3.1.1 Stopping Condition
In information theory, entropy is a measure of the uncertainty associated with
a random variable [14]. We choose entropy [43] as the stopping condition for
the top-down decomposition since entropy provides a measurement of disorder of
a data set. Let $\zeta$ denote the stopping condition for each branch of the
quadtree. If $\zeta$ holds, then EDHS stops the partition process of this
branch. By decreasing the size of images, the decomposition reduces the
information presented in the local positions. Follows give the concept of
segment set. Based on this concept, we define the entropy of images and
K-Color Rule.
###### Definition 1 (Segment set:).
Given a partition $P$ of the interval $[a,b],a=j_{0}\leq j_{1}\leq\ldots\leq
j_{n}=b$, where $a$ and $b$ are minimum and maximum of feature values of a
given image $I$, it gives a segment set $X=\\{x_{1},\ldots,x_{k}\\}$, where
$x_{i}$ is a set of pixels that all the pixels in $x_{i}$ form a connected
region and all the feature values of the pixels are located in interval
$[j_{h},j_{h+1}],0\leq h\leq n$. $I=\bigcup x_{i}$ and $\phi=x_{i}\bigcap
x_{j}$ for all $0\leq i\neq j\leq k$.
###### Definition 2 (Entropy :).
Given a segment set $X=\\{x_{1},\ldots,x_{k}\\}$ based on a partition $P$,
then entropy of $I$ to the base $b$ is
$H_{k}(X)=-\sum_{i=1}^{k}p(x_{i})\log_{b}p(x_{i})$ (1)
where $p$ denote the probability mass function of segment set
$X=\\{x_{i}\\},1\leq i\leq k$. To make the analysis simple, assume the
logarithm base is $e$. This gives
$H_{k}(X)=-\sum_{i=1}^{k}p(x_{i})\ln p(x_{i})$ (2)
###### Theorem 1.
Let $k$ be the number of segments of an image, supremum of $H_{k}(X)$ is
strictly increasing function with respect to $k$ and the supermum of
$H_{k}(X)$ is $\ln(k)$.
###### Proof.
Let $H_{k}(p_{1},\ldots,p_{k})$ denote the entropy of an image with segment
set $X=\\{x_{1},x_{2},\ldots,x_{k}\\}$ and $P(X=x_{i})=p_{i}$. To show
supremum of $H_{k}(X)$ is strictly increasing function respect to $k$, we need
to show that $\sup H_{k}(p_{1},\ldots,p_{k})\leq\sup
H_{k+1}(p^{\prime}_{1},\ldots,p^{\prime}_{k=1})$ for any $k\in Z^{+}$.
By [14] Theorem 2.6.4, we have
$\displaystyle H_{k}(p_{1},\ldots,p_{k})\leq
H_{k}(\frac{1}{k},\ldots,\frac{1}{k})\text{ and
}H_{k+1}(p^{\prime}_{1},\ldots,p^{\prime}_{k+1})\leq
H_{k+1}(\frac{1}{k+1},\ldots,\frac{1}{k+1})$ $\displaystyle
H_{k}(\frac{1}{k},\ldots,\frac{1}{k})=-\sum_{i=1}^{k}\frac{1}{k}\ln{1}{k}=\ln(k)$
$\displaystyle
H_{k+1}(\frac{1}{k+1},\ldots,\frac{1}{k+1})=-\sum_{i=1}^{k+1}\frac{1}{k+1}\ln{1}{k+1}=\ln(k+1)$
Then we have $\displaystyle
H_{k+1}(\frac{1}{k+1},\ldots,\frac{1}{k+1})=\ln(k+1)>\ln(k)=H_{k}(\frac{1}{k},\ldots,\frac{1}{k})$
∎
###### Definition 3 (K - Color Rule:).
Using different colors for different segments in segment set
$X=\\{x_{1},\ldots,x_{k}\\}$, if the image holds no more than $k$ segments,
which means image can be covered by $k$ colors, we say condition ‘K - Color
Rule’ (K-CR) is true; else, K-CR is false.
Assume an $m\times n$ image $I$. If $I$ is a one color (1-CR) image, then it
is a zero entropy image by Theorem 1. Consider another case. Assume the image
is too complicated that none of the segments holds more than one pixel. This
case gives the maximum entropy, $\ln(mn)$ ($k$ yields $mn$.) Then, the range
of $H_{k}(X)$ for a $m\times n$ $I$ is $[0,\ln(mn)]$. The larger the entropy
is, the more information is contained in the images.
Based on this observation of $H_{k}(X)$, we choose image entropy as the
stopping condition for the top-down decomposition because $H_{k}(X)$ is highly
related to the number of segments. $k$ denotes the number of segments and the
range of $k$ is $[1,mn]$. If a proper value of $k$ is chosen for a given
image, then $\zeta$ yields to $H_{k}(X)\leq\ln(k)$ by Theorem 1. That is, for
a certain branch of the quadtree, decomposition approach partitions the given
images until the local entropy is no larger than $\ln(k)$.
#### 3.1.2 K as An Algorithm Tuning Parameter
The value of $k$ impacts the depth of decomposition. A small value of $k$
results a deep quadtree because $\ln(k)$ is small. Small leaves do not contain
too much information, which results few boundaries within the leaves. Thus $k$
is a key issue since it decides the weights of the edge- and
region/clustering-based segmentation techniques used in EDHS. In other words,
$k$ is a measurement that indicates the degree to which each technique is
used. Figure 2 demonstrates that $k$ can be viewed as a sliding block ranging
from $1$ to $mn$. If $k$ is close to $1$, EDHS is closer to a region/cluster-
based segmentation method since few boundaries are detected in the leaves. The
weight of edge-based technique increases as long as the value of $k$ becomes
large.
Figure 2: Sliding block $k$ and entropy measurement $\ln(k)$
Suppose $k=1$, then the stopping condition $\zeta$ yields
$H_{k}(X)\leq\ln(1)=0$. To meet this stopping condition, the decomposition
process goes down to the pixel level if the neighbor pixels are inhomogeneous.
Then, EDHS is a pure region/cluster-based segmentation since there is no
necessary to detect the boundaries for the one color images.
Suppose $k=mn$, then the stopping condition $\zeta$ yields
$H_{k}(X)\leq\ln(mn)$ . By Theorem 1, no decomposition approach is processed
since $\zeta$ holds for an $m\times n$ image by Theorem 1. Then, EDHS is a
pure edge-based segmentation since no decomposition approach is employed. EDHS
just runs an edge detector locating the boundaries to form the local regions.
For an $m\times n$ image, the possible values of $k$ range from $1$ to $mn$.
Are all these integers from $1$ to $mn$ valid for $k$? The answer is no. Let
us take a close look at the cases when $k=1$ and $k=2$.
Points, lines and regions are three essential elements in a two-dimensional
image plane. We are looking for a value of $k$ which can efficiently recognize
the lines and regions (we treat a point as noise if this point is
inhomogeneous with its neighbors.) Keep in mind that the aim of decomposition
is to reduce the disorder. It suggests that $k$ should be a small integer.
When $k=1$, as discussed above, it forces the leaves to be one color. EDHS
yields a pure region/cluster-based segmentation. Previous algorithms of this
type have proved to be computationally expensive [46].
Consider $k=2$. The decomposition approach continually partitions the image
until the local entropy is less than $\ln(2)$, which tends to force the leaves
holding no more than two colors. Assuming a line passing through the sub-
images, to recognize this line, one of the local regions of the leaves needs
to be this line or part of a line. It makes the size of leaves quite small and
forces the decomposition process to go down to pixel level. Under this
circumstance, the time and space complexities are quite expensive. Another
fatal drawback is that the small size of leaves makes the EDHS sensitive to
noise. Even a pixel, which is inhomogeneous with its neighbors, could cause
invalid recognition around this pixel area.
$k\geq 3$ is a good choice because it can efficiently recognize the lines and
regions in 2D plane. An example is shown in Figure 3 (a). Top-down
decomposition goes down to pixel level to locate the curve if set $k<3$. But
for $k\geq 3$, no decomposition is needed since the entropy of Figure 3 (a)
must be less than $\ln(3)$ by Theorem 1. It suggests that EDHS is stable and
reliable when $k\geq 3$.
Figure 3: Lines in images
There is an extreme (the worst) case that we need to consider. This case is
shown in Figure 3 (b). Multiple lines pass through one point, say $p$. Define
a closed ball $B(p,\epsilon)$, where $p$ is the center and $\epsilon>0$ is the
radius of the ball. No matter how small $\epsilon$ is, $B(p,\epsilon)$
contains $2N_{l}+1$ segments divided by $N_{l}$ lines. In other words,
partitioning does not help reduce the number of segments around the area
$B(p,\epsilon)$. Therefore, decomposing images into small pieces does not
decrease entropy inside $B(p,\epsilon)$. To handle this case, we introduce
‘chaos’ leaves.
As shown in Figure 3 (b), entropy does not decrease inside $B(p,\epsilon)$
along with decomposition. To solve this problem, we introduce a threshold 1,
which is the smallest size of the leaves. If the size of partition is less
than $l$, the top-down decomposition does not continue even though the desired
$\zeta$ has not been reached. If this case happens, we call these leaves
chaos.
#### 3.1.3 Approximate Image Entropy
To calculate the entropy defined by Definition 2, we should know the
probability distribution $Pr(X=x_{i})=p_{i}$, where $X=(x_{1},\ldots,x_{k})$
is a set of segments of $I$. In most cases, we have no prior knowledge of the
distribution of the segments for the given images. In other words, we are not
able to directly compute the image entropy defined by Definition 2. Definition
4 gives an alternative calculation called approximate image entropy, which
does not require any prior knowledge of the distribution of the segments but
provides an approximate entropy value.
###### Definition 4 (approximate entropy:).
Given a partition $P$ of the interval $[a,b]$, $a=j_{0}\leq
j_{1}\leq\ldots\leq j_{n}=b$, where $a$ and $b$ are minimum and maximum of
feature values of a given image $I$, then approximate entropy $H(V)$ is
defined as
$H(V)=-\sum_{i=1}^{n}p_{i}\log_{b}p_{i}$ (3)
where $p$ denotes the probability mass function of the feature value set
$V=\\{v_{1},\ldots,v_{n}\\}$. Each $v_{i}$ denotes a collection of pixels
whose feature values are located in $[j_{i-1},j_{i}]$ and $p_{i}=Pr(V=v_{i})$,
$1\leq i\leq n$. After setting the logarithm base as $e$, $H(V)$ yields
$H(V)=-\sum_{i=1}^{n}p_{i}\ln p_{i}$ (4)
###### Theorem 2.
Given an $I$ and a partition $P$, $H(V)$ is less or equal then $H(X)$.
###### Proof.
Given an $I$ and a partition $P=[j_{0},j_{1},\ldots,j_{n}]$, where
$a=j_{0}\leq j_{1}\leq\ldots\leq j_{n}=b$, $a$ and $b$ are minimum and maximum
of pixel feature values of $I$. Let $k$ is the number of segments defined in
Definition 2. By the Definition of 2, $k$ must be greater or equal to $n$.
There are two cases need to be considered. One is $k=n$ and another is $k>n$.
Case I: if $k=n$, by the Definition 2 and Definition 4, $p(x_{i})=p(v_{i})$,
which induces $H(X)=H(V)$.
Case II: if $k>n$, it implies that there must exist at least two segments
which locate at the same partition interval. Without loss of generality,
assuming $H(X)$ and $H(V)$ are defined over the partition $P$, where $H(X)$
with two segments $x_{i}$ and $x_{j}$, $1\leq i\neq j\leq k$, both feature
values of $x_{i}$ and $x_{j}$ locate in the interval $[j_{h-1},j_{h}]$, $1\leq
h\leq n$. If we can prove $H(X)>H(V)$, then the theorem can be proved by
repeating following proof arbitrary times.
By Definition 2 and 4 respectively,
$\displaystyle H(X)=-(p(x_{1})\ln p(x_{1})+\ldots+p(x_{h^{\prime}})\ln
p(x_{h^{\prime}})+p(x_{h^{\prime\prime}})\ln
p(x_{h^{\prime\prime}})+\ldots+p(x_{n})\ln p(x_{n}))$ $\displaystyle
H(V)=-(p(x_{1})\ln p(x_{1})+\ldots+p(x_{h})\ln p(x_{h})+\ldots+p(x_{n})\ln
p(x_{n}))$ $\displaystyle\text{where
}p(x_{h})=p(x_{h^{\prime}})+p(x_{h^{\prime\prime}}),0<p(x_{h})<1\text{ and
}0<p(x_{h^{\prime}}),p(x_{h^{\prime\prime}})<p(x_{h})$ $\displaystyle\text{To
prove }H(X)>H(V)\text{, we consider }H(X)-H(V).$ $\displaystyle
H(X)-H(V)=p(x_{h})\ln p(x_{h})-(p(x_{h^{\prime}})\ln
p(x_{h^{\prime}})+p(x_{h^{\prime\prime}})\ln p(x_{h^{\prime\prime}}))$
$\displaystyle\text{Let }p(x_{h})=y,p(x_{h^{\prime}})=x\text{, therefore
}p(x_{h^{\prime\prime}})=y-x\text{, where }0<x<y<1$ $\displaystyle
H(X)-H(V)=y\ln y-(x\ln x+(y-x)\ln(y-x))$ $\displaystyle=\ln y^{y}-(\ln
x^{x}+\ln(y-x)^{(y-x)})$ $\displaystyle=\ln y^{y}-\ln x^{x}(y-x)^{(y-x)}$
$\displaystyle\text{Let }f(x,y)=y^{y}-x^{x}(y-x)^{(y-x)}$
$\displaystyle\text{If we can prove }f(x,y)>0\text{ for all }0<x<y<1$
$\displaystyle\text{then }H(X)-H(V)=\ln y^{y}-\ln x^{x}(y-x)^{(}y-x)>0$
$\displaystyle\text{ since }\ln(x)\text{ is an increasing function.}$
$\displaystyle f(x,y)=y^{y}-x^{x}(y-x)^{(y-x)}$
$\displaystyle=y^{y}-x^{x}\frac{(y-x)^{y}}{(y-x)^{x}}$
$\displaystyle=y^{y}-(\frac{x}{y-x})^{x}(y-x)^{y}$
$\displaystyle=y^{y}(1-(\frac{x}{y-x})^{x}(\frac{y-x}{y})^{y})$
$\displaystyle\text{Let }\alpha=\frac{x}{y-x}\text{,
then}f(x,y)=y^{y}(1-\alpha^{x}(\frac{1}{1+\alpha})^{y})=y^{y}(1-\frac{\alpha^{x}}{(1+\alpha)^{y}})$
$\displaystyle\text{Notice that }\alpha^{x}<(1+\alpha)^{y}\text{ since
}\alpha\text{ is a positive number and }0<x<y<1$ $\displaystyle\text{Then,
}1-\frac{\alpha^{x}}{(1+\alpha)^{y}}\text{ is positive. It implies
}f(x,y)>0\text{ because }y^{y}>0$
∎
Assume an $I$ and $k$-CR is true. It implies that $H_{k}(X)\leq\ln(k)$ by
Theorem 1. By Theorem 2, $H(V)$ must be equal or less than $H_{k}(X)$.
Therefore, It induces the a logical chain, truth of $k$-CR $\Rightarrow
H_{k}(X)\leq\ln(k)\Rightarrow H(V)\leq\ln(k)$. Both $H_{k}(X)\leq\ln(k)$ and
$H(V)\leq\ln(k)$ are necessary but not sufficient conditions for the truth of
$k$-CR.
#### 3.1.4 Noise Segments
The term noise of an image usually refers to unwanted color or texture
information. We do not count a small drop of ink on an A4 paper as a valid
segment. In most circumstances, it would be considered as noise. How to
distinguish those valid and invalid segments is an important issue.
###### Definition 5 (Dominant and Noise segments:).
Let $p$ denote the probability mass function of segment set
$X=\\{x_{1},\ldots,x_{k}\\}$ in $I$. Given a threshold $t_{noise}\in[0,1]$,
$x_{i}$ is a noise segment if $Pr(X=x_{i})=p_{i}\leq t_{noise}$ and
$\sum_{p_{i}\leq t_{noise}}p_{i}$ is greatly less than
$\sum_{p_{i}>t_{noise}}p_{i}$ , $1\leq i\leq k$. Other segments are called
dominant segments.
If the segments are small enough and the total area of those segments occupies
a small portion of a given image, we call those segments noise segments. The
first requirement of noise segment is understandable because the noise
segments should be small. The reason of defining the second condition is to
avoid the cases that the images are totally composed by small pieces.
The value of $k$ of K-CR in Definition 3 refers the number of dominant
segments. By Theorem 1 and 2, the supremum of $H(V)$ for this given image is
no longer $\ln(k)$. The noise supremum of $H(V)$ should be slightly larger
than $ln(k)$. Assuming the noise redundancy be $\epsilon$, then redundancy
stopping condition, $\zeta_{r}$, yields $H(V)\leq\ln(k)+\epsilon$.
Consider dividing segments into two groups, noise and dominant segments. By
Definition 4, $H(V)$ yields as follows:
$H(V)=-\sum_{x_{i}\in X}p(x_{i})\ln
p(x_{i})=-\sum_{p(x_{i})>t_{noise}}p(x_{i})\ln p(x_{i})-\sum_{p(x_{i})\leq
t_{noise}}p(x_{i})\ln p(x_{i})$
Given an image $I$, let $a$ be the total portion of dominant segments. Then
$a=\sum_{p(x_{i})>t_{noise}}p(x_{i})$ and the rest area,
$1-a=\sum_{p(x_{i})\leq t_{noise}}p(x_{i})$, is the portion of noise segments.
By Definition 5, $a>>(1-a)$. Let $k$ and $k^{\prime}$ be the number of
dominant and noise segments respectively. After applying Theorem 2.6.4 [14],
we get the noise supermum of $H(V)$ as follows.
$H(V)=-\sum_{p(x_{i})>t_{noise}}p(x_{i})\ln p(x_{i})-\sum_{p(x_{i})\leq
t_{noise}}p(x_{i})\ln
p(x_{i})\leq-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))$
The noise redundancy
$\epsilon=-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))-\ln(k)$. The
redundancy stopping condition, $\zeta_{r}$, yields
$H(V)\leq-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))$.
Following gives an example to compute the noise redundancy. Suppose a 3-CR
application $(k=3)$, setting $a=0.98$ and $k^{\prime}=3$. The redundancy
stopping condition for 3-CR yields
$-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))=1.1968$, which is
slight greatly than $ln(k)=1.0986$. Noise redundancy
$\epsilon=-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))-\ln(k)=0.0981$.
We summarize the top-down decomposition by Algorithm 1 and demonstrate some
examples of the top-down decomposition in Figure 4 by varying different $k$
values, where $l=3$, $a=0.998$ and $k^{\prime}=3$.
1em boxed
input : $I$: An image
output : $T_{qud}$: A decomposition quadtree
if _size of $I$ $<l$_ then
// current $I$ is chaos
Create a chaos leaf for $I$ and generate a feature descriptor for $I$.
else
if _$H(V) >-(a\ln(\frac{a}{k})+(1-a)\ln(\frac{1-a}{k^{\prime}}))$_ then
Partition $I$ into four partitions: $quad1$, $quad2$, $quad3$, $quad4$;
Append $quad1$, $quad2$, $quad3$ and $quad4$ as children of $I$ in the
$T_{qud}$;
Topdowndecomposition($quad1$);
Topdowndecomposition($quad2$);
Topdowndecomposition($quad3$);
Topdowndecomposition($quad4$);
else
Locate the local regions by detecting the boundaries within $I$;
Create a non-chaos leaf and generate a feature descriptor for each local
region;
Algorithm 1 Topdowndecomposition
1em
Figure 4: Quarter decomposition by different stopping conditions
### 3.2 Bottom-up Composition
Bottom-up composition stands at a kernel position of VHBS since this process
combines the local regions at the leaves of the quadtree to form the initial
segment set $S=(s_{0},\ldots,s_{n})$. It also calculates the probabilities of
the boundaries between these initial segments. At the same time, bottom-up
composition process generates the feature descriptors for each initial segment
by combining the local region feature descriptors.
The probabilities of the boundaries between these initial segments are
computed by two filters, which are designed based on the two visual hint rules
(i) and (ii) separately. The first one called scale filter, $f_{1}$, abides by
rule (i). The probabilities are measured by the length of the boundaries.
Longer boundaries result higher probabilities. The second one called
similarity filter, $f_{2}$, abides by rule (ii). The probabilities of the
boundaries are measured by the differences of two adjacent regions. Larger
different features of two adjacent regions result higher probability
boundaries. The finial weights of the boundaries are the trade-off of two
filters by taking the products of these two filters. If the probability of the
boundary between two local regions is zero, these two local regions are
combined together.
#### 3.2.1 Scale Filter
Scale filter is defined based on the visual hint (i): the global scale
boundaries tend to be the real boundaries of the objects. It suggests that
these boundaries caused by the local texture are not likely to be the
boundaries of our interesting objects because the objects with large size are
more likely to be our interesting objects. To measure the relative length of
each boundary, we use the sizes of the decomposition partitions in which the
objects are fully located. These local scale boundaries are not likely to
extend to a number of partitions since the length of these boundaries are
short. By this observation, we define the scale filter $f_{1}$ based on the
sizes of the partitions.
Scale filter $f_{1}$ is a function which calculates the confidence of the
boundary based on the scale observations. The input parameter of $f_{1}$ is
the relative scale descriptor $s$, which is ratio between the sizes of the
local partitions and the original images. The relative scale descriptor $s$ is
the measurement of the relative scale of the boundaries. We assume the sizes
of the images are the length of the longer sides. An example of calculating
scale descriptor $s$ is shown in Figure 5. The boundaries inside the marked
partition have the relative scale descriptor $s$, which is defined as:
$s=z/L$ (5)
where $z$ is the size of a partition and $L$ is the size of a original image.
Figure 5: The relative scale descriptor
Based on visual hint (i), $f_{1}(s)$ must be a strictly increasing function on
domain $[0,1]$ and the range of $f_{1}(s)$ locates in the interval $[0,1]$. If
s is small, it suggests that the confidence of the boundary should be small
since the boundaries are just located in a small area. If $s$ is close to one,
it suggests that the boundaries have high confidence. The gradient of
$f_{1}(s)$ is decreasing. This is because human perception is not linearly
dependent on the relative scale descriptor. For the same difference, human
perception is more sensitive when both of them are short rather than both of
them are long. Therefore we define $f_{1}(s)$ as follows:
$f_{1}(s)=\frac{2}{1+e^{-\beta_{1}s}}-1$ (6)
where $\beta_{1}$ is the scale damping coefficient and $s$ is the relative
scale descriptor. Figure 6 gives the $f_{1}(s)$ with different scale damping
coefficients.
Figure 6: Scale filter $f_{1}(s)$ with different scale damping coefficients.
#### 3.2.2 Similarity Filter
Compared these regions with similar colors or textures, human perception is
more impressed by regions with quite different features. Visual hint (ii)
suggests that two adjacent regions with different colors or textures tend to
produce the high confidence of boundaries between them. Based on this
observation, we defined a similarity filter $f_{2}$ to filter out the boundary
signals which pass through similar regions.
Similarity filter $f_{2}$ is a function which calculates the confidence of the
boundaries based on the similarity measurements between two adjacent regions.
Examples of similarity measurements are Dice, Jaccard, Cosine, Overlap. The
similarity measurement is a real number in interval $[0,1]$. The higher the
value, the more similar two regions are. $Similarity=1$ implies that they are
absolutely the same and zero means they are totally different. Based on the
visual hint (ii), small similarity measurement should result in high
confidence boundary. Let $x$ denote the similarity measurement of two adjacent
regions. Then $f_{2}(x)$ is a strictly decreasing function over domain $[0,1]$
and the range of $f_{2}(x)$ is inside $[0,1]$. Human perception is not linear
relationship with the similarity measures. Human perception is sensitive to
the regions when these regions have obvious different colors or textures. For
example, similarity measure $0.1$ and $0.2$ are not a big difference for human
visual because both of them are obviously different. This fact suggests that
the gradient of $f_{2}(x)$ is decreasing over the domain $[0,1]$. Then we
define $f_{2}(x)$ as follows to satisfy the requirements above.
$f_{2}(x)=(\frac{2}{1+e^{-\beta_{2}(1-x)}}-1)(\frac{1+e^{-\beta_{2}}}{1-e^{-\beta_{2}}})$
(7)
where $\beta_{2}$ is the similarity damping coefficient and $x$ is the
similarity measurement between two adjacent regions defined as
$x=similar(fd_{1},fd_{2})\in[0,1]$. $fd_{1}$ and $fd_{2}$ are feature
descriptors of the local regions. Figure 7 gives the curves of $f_{2}(x)$ with
respect to different similarity damping coefficients.
Figure 7: Similarity filter $f_{2}(s)$ with different similarity damping
coefficients.
We implemented VHBS by using $f_{2}(x)$ given by equation 6. We found that the
boundary signals are over-damped by the similarity filter $f_{2}(x)$. The
algorithm assigns low confidence values for boundaries with global scales that
preserved similar feature descriptors. Human visual is also sensitive to these
sorts of boundaries. To avoid these cases, we redesign the similarity filter
by considering the relative scale descriptor as well. Similarity filter is
redefined as $f_{2}(x,s)$, which outputs high confidence weights when either
parameter $s$ or $x$ is close to one. We also introduce a threshold similarity
$t$. If the similarity measurement is higher than this threshold, algorithm
sets the confidence as $0$, which means that there are no boundaries between
these two regions if human visual cannot tell the difference of these two
adjacent regions.
$f_{2}(x,s)=\left\\{\begin{array}[]{l
l}(\frac{2}{1+e^{-y(1-x)}}-1)(\frac{1+e^{-y}}{1-e^{-y}})&\quad\text{if
$x<t$}\\\ \text{ where
}y=\alpha(\frac{2}{1+e^{-\beta_{2}s}}-1)(\frac{1+e^{-\beta_{2}}}{1-e^{-\beta_{2}}})\\\
0&\quad\text{if $x\geq t$}\\\ \end{array}\right.$ (8)
where $\beta_{2}$ is the similarity damping coefficient, $\alpha$ is amplitude
modulation and $x$ is the similarity measurement between two feature
descriptors. Figure 8 demonstrates $f_{2}(x,s)$ when $\beta_{2}=10$ and
$\alpha=20$.
Figure 8: Similarity filter $f_{2}(x,s)$ with $\beta_{2}=10$ and $\alpha=20$.
#### 3.2.3 Partition Combination
The bottom-up composition starts from the very bottom leaves. Composition
process iteratively combines partitions from the next lower layer and this
process continues until reaching the root of the quadtree.
For the leaves of the quadtree, each boundary is marked by a confidence value,
$cnf$, which is given by formula $f_{1}(s)\cdot f_{2}(x,s)$, where $f_{1}(s)$
and $f_{2}(x,s)$ are scale and similarity filters defined by equations 6 and 8
respectively. The relative scale descriptor, $s$, is computed by equation 5
and $x$ is the similarity of two adjacent local segments. Figure 9
demonstrates the function $cnf=f_{1}(s)\cdot f_{2}(x,s)$ where with
$\beta_{1}=8$, $\beta_{2}=10$ and $\alpha=20$.
Figure 9: $f_{1}(s)\cdot f_{2}(x,s)$ with $\beta_{1}=8$, $\beta_{2}=10$ and
$\alpha=20$.
For these non-chaos leaves, the contours which form the closed areas and the
borders of the leaves form the boundaries of local regions. For these chaos
leaves, the boundaries only refer to the leaf borders. Each running time of
bottom-up combination, four leaves are combined together to the next lower
layer. Figure 10 shows that there are several possible cases during combining
four leaves together.
1. i
No interconnection happens during the combination;
2. ii
A new segment is formed by connecting several local regions which locate in
different leaves;
3. iii
The boundaries of leaves happen to be the boundaries of segments.
Figure 10: Three cases of partition combination.
For the case (i) such as region A shown in Figure 10, $cnf$ is calculated when
it is in leaf $1$ and there is no necessary to recalculate $cnf$ during the
combination. But for the case (ii) such as region B, region B is connected by
four local regions. Each $cnf$ is calculated separately. But the $cnf$ region
B needs to be recalculated after four local regions combined together since
the new combined segment is located in a large partition and the relative
scale descriptor is increased. Besides, a feature descriptor for region B is
generated based on feature descriptors for each local region. The third case
is that the boundaries of partitions happen to be the boundaries of the
regions. One example is region C shown in Figure 10. During the combination
process, algorithm also calculates $cnf$ of the leaf boundaries. After
combination hits the root of the quadtree, the process generates the initial
segment set $S=(s_{0},\ldots,s_{n})$, the boundary confidence set
$C_{b}=\\{c_{i,j}\\}$, where $c_{i,j}$ indicates the boundary probability
between segments $s_{i}$ and $s_{j}$, $0\leq i\neq j\leq n$ and the feature
descriptor set $FD=(fd_{0},\ldots,fd_{n})$. Follows is the Algorithm 2 of
iterative bottom-up composition.
1em boxed
input : $T_{qud}$: A decomposition quadtree
$id$: root of $T_{qud}$
output : $S=(s_{0},\ldots,s_{n})$: the initial segment set
$C_{b}=\\{c_{i,j}\\}$: the boundary confidence set
$FD=(fd_{0},\ldots,fd_{n})$: the feature descriptor set
node$\leftarrow$read a node of $T_{qud}$ by $id$;
if _node has children_ then
Read the four children of node as, $id1$,$id2$,$id3$,$id4$;
quad1$\leftarrow$Bottomupcomposition($id1$);
quad2$\leftarrow$Bottomupcomposition($id2$);
quad3$\leftarrow$Bottomupcomposition($id3$);
quad4$\leftarrow$Bottomupcomposition($id4$);
//Combine partitions. Recompute $cnf=f_{1}(s)\cdot f_{2}(x,s)$ if needed
img$\leftarrow$combine(quad1,quad2,quad3,quad4);
else
//This node is a leaf of $T_{qud}$
img$\leftarrow$computelocalcnf(node); //$cnf=f_{1}(s)\cdot f_{2}(x,s)$
return img;
Algorithm 2 Bottomupcomposition
1em
## 4 Hierarchical Probability Segmentation
Hierarchical segmentation is a widely used technique for image segmentation.
Regular hierarchical segmentation is modeled in layer-built structures.
Compared with the regular hierarchical structures, Hierarchical Probability
Segmentation, HPS, presents the hierarchical segmentation by a Probability
Binary Tree (PBT,) where the links are weighted by the confidence values,
$cnf\in[0,1]$. The root represents an image. Nodes represent segments and the
children of a node are the sub-segments of this node. Initial segments
$S=(s_{0},\ldots,s_{n})$ compose the leaves of PBT. Since PBT is generated in
greedy manner, higher level nodes always have higher probabilities than the
lower level nodes. One can visualize the PBT in arbitrary number of segments.
Of course, this number is less than the number of the initial segments.
### 4.1 Probability Binary Tree (PBT)
###### Definition 6 (Probability Binary Tree (PBT):).
Let $n_{0}$ denote the root of a PBT and $n_{0}$ represents the original
images $I$. Nodes of PBT denote the segments and links represent the
relationship of inclusion. Assume nodes $n_{i}$, $n_{i1}$, $n_{i2}$ and links
$l_{i1}$, $l_{i2}$, where $n_{i1}$ and $n_{i2}$ are children of $n_{i}$ linked
by $l_{i1}$ and $l_{i2}$ respectively. $n_{i}$, $n_{i1}$, $n_{i2}$, $l_{i1}$,
$l_{i2}$ preserve the following properties:
1. i
Let $DS=\\{(n_{i1}^{1},n_{i2}^{1}),(n_{i1}^{2},n_{i2}^{2}),\ldots\\}$ denote
the set of all the possible pairs of sub-segments of $n_{i}$. Function
$g(n_{i1}^{j},n_{i2}^{j})$ gives the $cnf$ of segments $n_{i1}^{j}$ and
$n_{i2}^{j}$. Assume $(n_{i1},n_{i2})=\arg\max g(DS)$ and $l_{i1}$. Therefore
$l_{i2}$ are weighted by $g(n_{i1},n_{i2})$;
2. ii
for any element of $DS$, $(n_{i1}^{j},n_{i2}^{j})$, $n_{i}=n_{i1}^{j}\cup
n_{i2}^{j}$ and $\phi=n_{i1}^{j}\cap n_{i2}^{j}$.
Definition 6 recursively gives the definition of PBT, which has the following
properties:
1. i
Every PBT node (except root) is contained in exactly one parent node;
2. ii
every PBT node (except the leaves) is spanned by two child nodes;
3. iii
a number of pairs of nodes span $n_{i}$. These pairs are candidates to be the
children of $n_{i}$ and each pair is labeled with the probability of these two
nodes,
$\\{cnf_{0}=g(<n_{i1}^{0},n_{i2}^{0}>),cnf_{1}=g(<n_{i1}^{1},n_{i2}^{1}>),\ldots\\}$.
PBT chooses the pairs $<n_{i1}^{j},n_{i2}^{j}>$, which have the highest
$cnf_{j}$ to span the node $n_{i}$. The links $l_{i1}$ and $l_{i2}$ are
weighted by $cnf_{j}$;
4. iv
assume a node $n_{i}$ with a link $l_{i}$ pointed in from its parent and the
links $l_{i1}$, $l_{i2}$ pointing out to its children. Weights of $l_{i1}$,
$l_{i2}$ must be no larger than the weight of $l_{i}$;
5. v
if two nodes (segments) overlap, one of them must be a child of the other.
By presenting the segmentation in PBT, the images are recursively partitioned
in two segments with the highest probability among all the possible pair of
segments. Figure 11 gives an example of the PBT.
Figure 11: Probability Binary Tree.
Let $C_{b}=\\{c_{i,j}\\}$ denote the set of boundary probabilities.
$c_{i,j}\in C_{b}$ represents the boundary probability between initial
segments $s_{i}$ and $s_{j}$, where $c_{i,j}\in[0,1]$, $0\leq i\neq j\leq n$.
HPS constructs PBT in bottom-up manner, which means leaves are first created
and the root is the last node created. To generate a PBT in greedy schema,
$C_{b}$ is sorted in ascending order. Let $T_{pbt}$ be a PBT and
$C^{\prime}_{b}$ be the ascending order sequence of $C_{b}$. Algorithm 3
describes to generate a $T_{pbt}$.
1em boxed
input : $C^{\prime}_{b}$: sorted $C_{b}$ in ascending order
output : $T_{pbt}$: a probability binary tree, PBT
while _$C^{\prime}_{b}$ is not empty_ do
$c_{ij}\leftarrow$ read the first element of $C^{\prime}_{b}$ and remove it
from $C^{\prime}_{b}$;
$i\leftarrow$ read the index of segment $s_{i}$ from $c_{ij}$;
$j\leftarrow$ read the index of segment $s_{j}$ from $c_{ij}$;
if _$s_{i}$ is not exist in $T_{pbt}$_ then
Create a node $s_{i}$ in $T_{pbt}$;
else
$s_{i}\leftarrow$ read the $s_{i}$ from $T_{pbt}$;
if _$s_{j}$ is not exist in $T_{pbt}$_ then
Create a node $s_{j}$ in $T_{pbt}$;
else
$s_{j}\leftarrow$ read the $s_{j}$ from $T_{pbt}$;
Create a new node $n_{new}$, which is the parent of $s_{i}$ and $s_{j}$;
Create the links from $n_{new}$ to $s_{i}$ and from $n_{new}$ to $s_{i}$
weighted by $c_{ij}$;
Replace index $i$ and $j$ in current $C^{\prime}_{b}$ by the index of
$n_{new}$ and remove the duplicate elements in $C^{\prime}_{b}$;
Algorithm 3 Generating a probability binary tree.
1em
### 4.2 Visualization of the Segmentation
Generally, there are two ways to visualize the segments. One is threshold-
based visualization and another is number-based visualization. As discussed in
previous section, the root of the PBT represents the original images. For the
other nodes, the more shallow the positions are, the more coarse-gradient the
segments are. For example, visualizing image in segments $a$ and $b$ shown in
Figure 11 is combining initial segments $1$, $2$ and $3$ together to form
segment $a$ and combining initial segments $4$ and $5$ together to form
segment $b$.
Suppose a threshold, $t_{visual}\in[0,1]$, is selected for visualization. By
the properties of the PBT, the weights of the links are the probabilities of
the segments. The weighs are decreasing as long as the depths are increasing.
Given a $t_{visual}$, threshold-based visualization only displays the segments
whose link weights are greater than the given $t_{visual}$.
The number-based visualization displays a certain number of segments. Let
$n_{visual}$ denote the number of visualization segments. The implementation
of number-based visualization is trivial. Algorithm sorts the nodes in
descending order with respect to the link weights and picks the first
$n_{visual}$ number of nodes to display.
## 5 Algorithm Complexity Analysis
Since the algorithm is divided into two stages, EDHS and HPS, we discuss the
computational complexity of them separately.
### 5.1 EDHS Computational Complexity
Assume the depth of the quadtree generated by top-down decomposition is $d$
and the depth of the root is zero. The maximum $d$ is
$(\min(\lfloor\log_{2}n\rfloor,\lfloor\log_{2}m\rfloor)-\lceil\log_{2}l\rceil)$,
where $n\times m$ is the size of original images and $l$ is the chaos
threshold. Depending on the different images and the chosen stopping condition
$\zeta$, decomposition process generates an unbalance quadtree with depth of
$d$. To analysis the complexity of the decomposition, we assume the worst
cases that the images are fully decomposed. It implies that depth of all the
leaves is
$d=(\min(\lfloor\log_{2}n\rfloor,\lfloor\log_{2}m\rfloor)-\lceil\log_{2}l\rceil)$.
At the $i$th level of the quadtree, there are $4^{i}$ numbers of nodes and the
size of each node is $\frac{mn}{4^{i}}$. Then the running time of computing
the stopping condition $\zeta$ of the ith depth is $4^{i}\frac{mn}{4^{i}}=mn$
and the total running time of decomposition is $dmn$. Commonly, the time
complexity of an edge detector is $O(mn)$ such as Canny Edge Detection [6].
Plus the time complexity of generating the feature descriptors $O(mn)$. The
running time of top-down decomposition is $(d+2)mn$, which gives the time
complexity of top-down decomposition $O(mn)$.
The combination process starts from the leaves to calculate the boundary
confidence by $cnf=f_{1}(s)\cdot f_{2}(x,s)$, which gives the running time
$\frac{m}{2^{d}}\frac{n}{2^{d}}$ for each leaf (leaf size is
$\frac{m}{2^{d}}\frac{n}{2^{d}}$) and total running time is
$4^{d}\frac{mn}{4^{d}}=mn$ since there are $4^{d}$ numbers of leaves totally.
At the $i$th level of the quadtree, the composition process combines the four
quadrants into one, which gives the running time
$m+n+\frac{m}{2^{i}}\frac{n}{2^{i}}$ and total running time of the $i$th depth
is $4^{i}(m+n+\frac{m}{2^{i}}\frac{n}{2^{i}})=mn+4^{i}(m+n)$. Then the total
running time of bottom-up composition is $(d+1)mn+\frac{4^{d}-1}{3}(m+n)$. It
can be proved when $d$ is large enough, term of $\frac{(4^{d}-1)(m+n)}{3}$
dominates the running time. It gives the time complexity $O(4^{d}(m+n))$. Then
the time complexity of the EDHS is $O(4^{d}(m+n))$, where $d$ is the depth of
the quadtree and mn is the size of the input images.
### 5.2 HPS Computational Complexity
To make the analysis simple, we assume maximum $d$ is $log_{2}n$ (under the
worst situation.) It suggests that the maximum number leaves is
$4^{log_{2}n}=n^{2}$. This implies the worst running complexity case is that
one pixel is one local segment. At pixel level, PHS considers four pixel
neighbors as adjacent local segments. They are left, right, upper and below
pixels. Then the maximum size of $C_{b}$ is $2mn$. PHS first sorts $C_{b}$ to
$C^{\prime}_{b}$, which gives the time complexity $O(mn\log(mn))$. Considering
the algorithm to generate the $T_{pbt}$, the total running time is
$\sum_{i=1}^{2mn}(2mn-i)=2(mn)^{2}-mn$. This gives time complexity
$O((mn)^{2})$. Compare with the complexity of EDHS, generating $T_{pbt}$ is
more expensive part. In real applications, the number of elements in $C_{b}$
is far less than $2mn$ since the algorithm does not go down to pixel level by
setting the size threshold of the partitions. The running time also highly
depends on the input images because the number of local segments function
depends on the complexity of the image itself. For the test running
experiments of image set [35], the average number of elements of $C_{b}$ is
around $3000$ or less. Then the practical time complexity of algorithm is far
less than $O((mn)^{2})$.
## 6 Experimental Results
In this section, we demonstrate experiments of VHBS by tuning different
parameters and evaluate the results by comparing with outputs of Normalized
Cut [44] and KMST [20, 21] based on the same test set of Berkeley Segmentation
dataset (BSDB) [35].
### 6.1 Weighted Boundary Evaluations
As discussed in previous section, the visualization of boundary probability
set $C_{b}$ results in the weighted boundaries of the initial segments
$S=(s_{0},\ldots,s_{n})$. Some examples are demonstrated in Figure 10. We use
the benchmark provided by BSDB, which uses Precision-Recall curves [49] to
evaluate the boundary detection. Precision is defined as the number of true
positives over the number of elements retrieved by the detection, and Recall
is defined as the number of elements retrieved by detection divided by the
total number of existing relevant elements. Precision measure can be viewed as
the correctness and Recall measure can be viewed as the completeness.
Since the purpose of our algorithm is to locate the segments, algorithm only
marks the contours that form the boundaries of the regions. Incontinuities or
unclosed contours are ignored. These ignored contours include three portions.
One is the contours within the chaos leaves. The second is the contours in
non-chaos leaves that are Incontinuities or unclosed. These boundaries between
two regions with similarity measure higher than the threshold $t$ (in equation
8) are also ignored. These suggest that Precision-Recall curves mainly located
at the left side of the PR graphs.
The first set of experiments is based on the observation of different values
of $k$. We discussed the value of $k$ in section 3.1.2. Recall one of the
purposes of decomposition is to reduce the information contained in leaves.
From this point of view, small integer is preferred for $k$ since small values
of $k$ result more strict stopping condition. On the other hand, we need to
avoid too many chaos leaves since chaos means that the algorithm has failed to
recognize these regions. According to the arguments above, the ideal value of
$k$ should be as small as possible but large enough to lower the number of
chaos leaves. We call this value optimization point. Obviously, different
images have different optimization points. Figure 12 shows the boundary
detection Precision-Recall based on $k$ over the BSDB test set. As shown in
Figure, algorithm has better Precision-Recall performances as long as $k$
increasing since larger values of $k$ give less number of chaos leaves. But
after some points, there are no improvements because numbers of chaos leaves
are zero.
Figure 12: Boundary evaluation based on $k$.
Based the observation of $k$, we used a program to automatically locate the
optimization points for each image. The program is trivial. It pre-generates
the decomposition quadtree and selects the value of $k$ which is the smallest
integer in range $[3,mn]$ but holding zero number of chaos leaves. Then the
rest experiments are generated based on the optimization points for each
image.
We also demonstrate the experiments based on damping coefficient $\beta_{1}$
of scale filter ($f_{1}$ 6) in Figure 13 (ii), the experiments based on
damping coefficient $\beta_{2}$ of similarity filter ($f_{2}$ 8) in Figure 13
(iii), the experiments based on amplitude modulation $\alpha$ of similarity
filter ($f_{2}$ 8) in Figure 13 (i) and the experiments based on similarity
threshold $t$ of similarity filter ($f_{2}$ 8) in Figure 13 (iv). As
demonstrated, different values of these parameters slightly shift the
Precision-Recall curves. Based on these experiments, we choose $\alpha=1$,
$\beta_{1}=8$, $\beta_{2}=3$ and $t=0.994$ for the remaining experiments.
Figure 14 provides some examples of boundary detections.
Figure 13: Boundary evaluation based on $\alpha$, $\beta_{1}$, $\beta_{2}$ and
$t$. Figure 14: Examples of the weighted boundaries.
### 6.2 Visualization of the Segments
We demonstrate some examples of the natural images based on different number
of segments in Figure 15. The visualizations of the segments are generated
based on top-down manner of PBT. Since PBT is constructed based on the
probabilities of the segments, the segments located at the higher level of PBT
suggest higher possibilities of the segments. If visualization PBT in small
number of segments, it gives the general outline of the images since only the
segments with high possibilities are visualized. The more details are provided
when visualization goes down to the depth of PBT.
Figure 15: Examples of the segmentation results by VHBS.
### 6.3 Evaluation Experiments by Tuning Parameters
Rather than giving the subjective running experiments of the algorithm, the
segmentation results need to be compared with other existed methods
quantitatively. One usually used segmentation evaluation is supervised
evaluation [7], which compares the results of segmentation against the
manually-segmented reference images. The disadvantage of such methods is that
the quality of segmentation is inherently subjective. Then, there is a
requirement to evaluate the image segmentations unsubjectively. Such methods
are called unsupervised evaluation [54] methods such as [53, 3, 9].
To evaluate the results of segmentation quantitatively and objectively, [25]
proposed four criteria: (i) the characteristics should be homogeneous within
the segments; (ii) the characteristics should be quite different between the
adjacent segments; (iii) shape of the segments should be simple and without
holes inside; (iv) boundaries between segments need to be smooth and
continues, not ragged. For our experiments, we choose $Q(I)$ [3], $H_{r}(I)$,
$H_{l}(I)$ and $E$ [53] as evaluators. $Q(I)$ and $H_{r}(I)$ measure the
intra-region uniformity, which is described as the criteria (i). $H_{l}(I)$
measures the inter-region uniformity, which is the criteria (ii). $E$ is
defined as $E=H_{r}(I)+H_{l}(I)$, which combine the criteria (i) and criteria
(ii). Table 1 gives the details of the evaluators used in this chapter. The
disadvantage of unsupervised evaluations is that these evaluation criteria
might not appropriate for the natural images since the perception of human
segmentation is based on the semantic understanding. It might result different
conclusions by human perception [54].
Table 1: The unsupervised evaluators used in this chapter. Evaluator | Description | Formula
---|---|---
$Q(I)$ [3] | Intra-region evaluator based on color error | $Q(I)=\frac{\sqrt{R}}{1000N\cdot M}\sum_{i=1}^{R}[\frac{e_{i}^{2}}{1+\log S_{i}}+(\frac{R(S_{i})}{S_{i}})^{2}]$
$H_{r}(I)$ [53] | | $H_{r}(I)=\sum_{i=1}^{R}(\frac{S_{i}}{S_{I}})H(X_{i})$
| Intra-region evaluator based on entropy | $H(X_{i})=-\sum_{m\in V_{i}^{\mu}}\frac{L_{i}(m)}{S_{i}}\log(\frac{L_{i}(m)}{S_{i}})$
$H_{l}(I)$ [53] | Inter-region evaluator based on entropy | $H_{l}(I)=-\sum_{i=1}^{R}\frac{S_{i}}{S_{I}}\log(\frac{S_{i}}{S_{I}})$
$E$ [53] | Composite evaluator | $E=H_{r}(I)+H_{l}(I)$
$I$: the segmented image
$NM$: the size of the image
$R$: the number of regions in the segmented image
$S_{i}$: the area of pixels of the ith segment
$S_{I}$: the area of pixels of the image $I$
$R(S_{i})$: the number of segments having the area of pixels equal to $S_{i}$
$e_{i}$: the color error in RGB space defined as
$e_{i}^{2}=\sum_{\gamma\in\\{r,g,b\\}}\sum_{p\in
X_{i}}(C_{\gamma}(p)-\overline{C_{\gamma}}(X_{i}))^{2}$
where $C_{\gamma}(p)$ is the value of component $\gamma$ of pixel $p$ and
$\overline{C_{\gamma}}(X_{i})=\frac{\sum_{p\in X_{i}}C_{\gamma}(p)}{S_{i}}$
$V_{i}^{\mu}$: the set of all possible values associated with feature $\mu$ in
segment $i$
$L_{i}(m)$: the number of pixels in $i$th segment that have a value of $m$ for
feature $\mu$
Figure 16 gives the experimental evaluations based on damping coefficient
$\beta_{1}$ of scale filter $f_{1}$ 6, damping coefficient $\beta_{2}$ ,
amplitude modulation $\alpha$ and similarity threshold $t$ of similarity
filter $f_{2}$ 8.
Figure 16: Experimental evaluations based on $\beta_{1}$, $\beta_{2}$,
$\alpha$ and $t$.
Figure 16 demonstrates that the evaluations of $Q(I)$, $H_{r}(I)$, $H_{l}(I)$
and $E$ are not varied much by different damping coefficients $\beta_{1}$,
$\beta_{2}$ and amplitude modulation $\alpha$. It suggests that $Q(I)$,
$H_{r}(I)$, $H_{l}(I)$ and $E$ are not sensitive to parameters of $\beta_{1}$,
$\beta_{2}$ and $\alpha$. Figure 16 (iv) presents the same information as
Figure 13 (iv) demonstrated that similarity threshold $t=1$ obviously has
worse performance of $Q(I)$ than other similarity threshold. This is the
reason we do not set $t=1$ in the experiments.
Figure 17 shows the evaluations of $Q(I)$, $H_{r}(I)$, $H_{l}(I)$ and $E$
based on parameter of $k$, where $N_{nc}$ and $N_{c}$ are the number of non-
chaos leaves and chaos leaves. The information shown in Figure 17 (i)
demonstrates that both the numbers of chaos and non-chaos leaves are
decreasing as long as $k$ increasing. Notice that the number of non-chaos
leaves is slightly increasing at very left side of the graph is because that
there are too many areas transferring from chaos to non-chaos. Based on Figure
17 (i), we can make a judge that the average optimization point locates around
$k=25$ for the whole test set of BSDB since $N_{c}$ is close to zero when $k$
is about $25$. The evaluation $Q(I)$ shown in Figure 17 (ii) strongly supports
this estimation. Obviously, the optimization points are different when images
are different.
Figure 17: Experimental evaluations based on $k$.
### 6.4 Comparison with Other Segmentation Methods
In this section, we compare VHBS with Ncut [44] and KMST [20, 21] over the
entire test set of Berkeley Segmentation Dataset and Benchmark [35]. The
source codes of Ncut and KMST are got from the author’s websites. To fairly
compare these three algorithms, we tune the parameters to output the same
number of segments of each image in the test set. Figure 18 provides the
evaluations of $Q(I)$, $H_{r}(I)$, $H_{l}(I)$ and $E$ based on number of
segments.
Figure 18: VHBS compares with Ncut [44] and KMST [20, 21].
Figure 18 demonstrates that our algorithm gives the best performance of $Q(I)$
and $H_{r}(I)$ almost over all the number of segments and Normalized Cut has
the best performance of $H_{l}(I)$. Let us take a close look at the evaluator
$H_{l}(I)$. As [53] points out, $H_{l}(I)$ favors very few large segments and
many small segments. In other words, Segmentation with very few large segments
and many small segments gives high evaluation $H_{l}(I)$.
It is expected that VHBS performs poorly over the evaluator $H_{l}(I)$ because
$H_{l}(I)$ contradicts to the mechanism of scale filter $f_{1}$, where scale
filter favors large areas. Scale filter $f_{1}$ 6 makes VHBS prefer to select
large area segments, which is more consistent with human visual perception.
Meanwhile, similarity filter $f_{2}$ 8 favors the segments preserving high
uniformity within the segments because $f_{2}$ marks the boundaries with high
weights when two adjacent regions are quite different. Based on the discussion
above, it is not hard to understand that VHBS performs well $Q(I)$ and
$H_{r}(I)$, but not $H_{l}(I)$ and $E$ since $H_{l}(I)$ and $E$ is penalized
by number of small area segments.
## 7 Conclusion
Our contribution lies in proposing a new low-level image segmentation
algorithm, VHBS, which obeys two visual hint rules. Unlike most unsupervised
segmentation methods, which are based on the clustering techniques, VHBS is
based on the human perceptions since $f_{1}$ and $f_{2}$ are designed based on
two visual hint rules and somehow contradict clustering ideas. The evaluations
of experiments demonstrate that VHBS has high performance over the natural
images. VHBS still preserves high efficiency because VHBS does not go down to
the pixel level by setting the entropy and chaos thresholds as the stopping
condition of the image decompositions. Rather than outputting the segments of
the given images, at the same time, VHBS also provides the feature
descriptors, which are statistic summarization for each segment. To improve
the performance, one of our future works is to construct the algorithm in
learning schema to get the optimized parameters by a learning process.
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|
arxiv-papers
| 2010-10-03T15:27:56 |
2024-09-04T02:49:13.401893
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Su and Margaret H. Dunham",
"submitter": "Yu Su",
"url": "https://arxiv.org/abs/1010.0417"
}
|
1010.0422
|
Convolutional Matching Pursuit and Dictionary Training
Arthur Szlam, Koray Kavukcuoglu, and Yann LeCun
## 1 Introduction
One of the most succesful recent signal processing paradigms has been the
sparse coding/dictionary design model [8, 4]. In this model, we try to
represent a given $d\times n$ data matrix $X$ of $n$ points in
$\mathbb{R}^{d}$ written as columns via a solution to the problem
$\\{W_{*},Z_{*}\\}=\\{W_{*}(K,X,q),Z_{*}(K,X,q)\\}$
$=arg\min\limits_{Z\in\mathbb{R}^{K\times n},W\in\mathbb{R}^{d\times
K}}\sum_{k}||Wz_{k}-x_{k}||^{2},\,\,||z_{k}||_{0}\leq q,$ (1.1)
or its $Z$ coordinate convexification
$\\{\tilde{W}_{*},\tilde{Z}_{*}\\}=\\{\tilde{W}_{*}(K,X,\lambda),\tilde{Z}_{*}(K,X,\lambda)\\}$
$=arg\min\limits_{Z\in\mathbb{R}^{K\times n},W\in\mathbb{R}^{d\times
K}}\sum_{k}||Wz_{k}-x_{k}||^{2}+\lambda||z_{k}||_{1}.$ (1.2)
Here, $\\{W,Z\\}$ are the dictionary and the coefficients, respectively, and
$z_{k}$ is the $k$th column of $Z$. $K$, $q$, and $\lambda$ are user selected
parameters controlling the power of the model.
More recently, many models with additional structure have been proposed. For
example, in [9, 2], the dictionary elements are arranged in groups and the
sparsity is on the group level. In [3, 5, 7], the dictionaries are constructed
to be translation invariant. In the former work, the dictionary is constructed
via a non-negative matrix factorization. In the latter two works, the
construction is a convolutional analogue of 1.2 or an $l^{p}$ variant, with
$0<p<1$. In this short note we work with greedy algorithms for solving the
convolutional analogues of 1.1. Specifically, we demonstrate that sparse
coding by matching pursuit and dictionary learning via K-SVD [1] can be used
in the translation invariant setting.
## 2 Matching Pursuit
Matching pursuit [6] is a greedy algorithm for the solution of the sparse
coding problem
$\min_{z}||Wz-x||^{2},$ $||z||_{0}\leq q,$
where the $d\times k$ matrix $W$ is the dictionary, the $k\times 1$ $z$ is the
code, and $x$ is an $d\times 1$ data vector.
1. 1.
Set $e=x$, and $z$ the $k$-dimensional zero vector.
2. 2.
Find $j={\rm arg}\max\limits_{i}||W_{i}^{T}e||_{2}^{2}$.
3. 3.
Set $a=W_{j}^{T}x$.
4. 4.
Set $e\leftarrow e-aW_{j}$, and $z_{j}=z_{j}+a$.
5. 5.
Repeat for $q$ steps
Note that with a bit of bookkeeping, it is only necessary to multiply $W$
against $x$ once, instead of $q$ times. This at a cost of an extra $O(K^{2})$
storage: set $e_{r}$ and $a_{r}$ be $e$ and $a$ from the $r$th step above.
Then:
$W^{T}e_{0}=W^{T}x;$ $W^{T}e_{1}=W^{T}x-a_{0}W^{T}W_{j_{0}},$
and so on. If the Gram matrix for $W$ is stored, this is just a lookup.
### 2.1 Convolutional MP
We consider the special case
$\min_{z}||\sum_{j=1}^{k}w_{j}*z_{j}-x||^{2},$ $||\overline{z}||_{0}\leq q,$
where each $w_{j}$ is a filter, and $\overline{z}$ is all of the responses.
Note that the Gram matrix of the “Toeplitz” dictionary consisting of all the
shifts of the $w_{j}$ is usually too big to be used as a lookup table.
However, because of the symmetries of the convolution, it is also unnecessary;
we only need store a $4*h_{f}\times w_{f}\times k^{2}$ array of inner
products, where $h_{f}$ and $w_{f}$ are the dimensions of the filters.
With this additional storage, to run $q$ basis pursuit steps with $k$ filters
on an $h\times w$ image costs the computation of one application of the filter
bank plus $O(kqhw)$ operations.
## 3 Learning the filters
Given a set of $x$, we can learn the filters and the codes simultaneously.
Several methods are available. A simple one is to alternate between updating
the codes and updating the filters, as in K-SVD [1]:
1. 1.
Initialize $k$ $h_{f}\times w_{f}$ filters $\\{w_{1},...,w_{k}\\}$.
2. 2.
Solve for $z$ as above.
3. 3.
For each filter $w_{j}$,
* •
find all locations in all the data images where $w_{j}$ is activated
* •
extract the $h_{f}\times w_{f}$ patch $E_{p}$ from the reconstruction via $z$
at each activated point $p$.
* •
remove the contribution of $w_{j}$ from each $E_{p}$ (i.e. $E_{p}\leftarrow
E_{p}-c_{(p,j)}w_{j}$, where $c_{(p,j)}$ was the activation determined by
$z$).
* •
update $w_{j}\leftarrow\text{PCA}(E_{p})$
4. 4.
Repeat from step 2 until fixed number of iterations.
We note that the forward subproblem (finding $Z$ with $W$ fixed) is not
convex, and so the alternation is not guaranteed to decrease the energy or to
converge to even a local minimum. However, in practice, on image and audio
data, this method generates good filters.
## 4 Some experiments
We train filters on three data sets: the AT&T face database, the motorcycles
from a Caltech database, and the VOC PASCAL database. For all the images in
all our experiments, we perform an additive contrast normalization: each image
$x$ is transformed into $x^{\prime}=x-x*b$, where $b$ is a $5\times 5$
averaging box filter. This is very nearly transforming
$x^{\prime}=\nabla^{2}x$, that is, using the discrete Laplacian of the image
instead of the image. Using the Laplacian would correspond to using the energy
$\sum_{x}||\nabla\left(\sum_{j}w_{j}*z_{j}-x\right)||^{2},$
that is, the energy sees the difference between gradients, not intensities.
### 4.1 Faces
The AT&T face database, available at
http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html is a set of
400 images of 40 individuals. The faces are centered in each image. We resize
each image to $64\times 64$ and contrast normalize. We train $8$ $16\times 16$
filters. After training the filters we find the feature maps of each image in
the database, obtaining a new set of 400 $8$ channel images. We take the
elementwise absolute value of each of the 8 channel images, and then average
pool over $8\times 8$ blocks. We then train a new 16 element dictionary on the
subsampled images. In figure 1 we display the first level filters, and the
second level filters up to shifts of size 8 and sign changes of the first
level filters..
Figure 1: First and second layer filters from faces
Figure 2: a contrast normalized face, and its reconstruction from 40 filter
responses.
### 4.2 Caltech motorcycles
We also train on the motorbikes-side dataset, available at
http://www.vision.caltech.edu/html-files/archive.html which consists of color
images of various motorcycles. The motorcycles are centered in each image. We
convert each image to gray level, resize to $64\times 64$, and contrast
normalize. We train $8$ $16\times 16$ filters. As before, we then train a new
16 element dictionary on the subsampled absolute value rectified responses of
the first level. In figure 3 we display the first level filters, and the
second level filters up to shifts of size 8 and sign changes of the first
level filters..
Figure 3: First and second layer filters from motorcycles
Figure 4: A contrast normalized motorcycle, and its reconstruction from 40
filter responses.
### 4.3 Images from PASCAL VOC
We also show results trained on “unclassified” natural images from the PASCAL
visual object challenge dataset available at
http://pascallin.ecs.soton.ac.uk/challenges/VOC/. We randomly subsample $5000$
grayscaled images by a factor of 1 to 4, and then pick from each image a
$64\times 64$ patch, and then contrast normalize. We train $8$ $8\times 8$
filters. We then train a new $4\times 4$ 64 element dictionary on the
subsampled absolute value rectified responses of the first level. In figure 5
we display the first level filters, and the second level filters up to shifts
of size 8 and sign changes of the first level filters.
Figure 5: First and second layer filters from natural images
In order to show the dependence of the filters on the number of filters used,
in figure 6 we display an $8$, $16$, and $64$ element $16\times 16$ dictionary
trained on the same set as above.
Figure 6: Dictionaries with varying numbers of elements trained on natural
images.
## References
* [1] M. Aharon, M. Elad, and A. Bruckstein. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11):4311–4322, 2006\.
* [2] Francis R. Bach. Consistency of the group lasso and multiple kernel learning. Journal of Machine Learning Research, 9:1179–1225, 2008.
* [3] Sven Behnke. Discovering hierarchical speech features using convolutional non-negative matrix factorization. In IJCNN, pages 7–12, 2008.
* [4] Alfred M. Bruckstein, David L. Donoho, and Michael Elad. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Review, 51(1):34–81, 2009.
* [5] Y. Boureau K. Gregor M. Mathieu Y. LeCun K. kavukcuoglu, P. Sermanet. Learning convolutional feature hierarchies for visual recognition. Advances in NIPS, 2010.
* [6] Stephane Mallat and Zhifeng Zhang. Matching pursuit with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41:3397–3415, 1993.
* [7] Graham Taylor Matthew Zeiler, Dilip Krishnan and Rob Fergus. Hierarchical convolutional sparse image decomposition. In The Twenty-Third IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, CA, June 2010.
* [8] B. Olshausen and D. Field. Sparse coding with an overcomplete basis set: A strategy employed by v1?, 1997.
* [9] Ming Yuan, Ming Yuan, Yi Lin, and Yi Lin. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, Series B, 68:49–67, 2006\.
|
arxiv-papers
| 2010-10-03T16:55:56 |
2024-09-04T02:49:13.415080
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arthur Szlam, Koray Kavukcuoglu, and Yann LeCun",
"submitter": "Arthur Szlam",
"url": "https://arxiv.org/abs/1010.0422"
}
|
1010.0584
|
Wigner function evolution in self-Kerr Medium derived by Entangled
state representationWork supported by a grant from the Key Programs Foundation of Ministry of
Education of China (No. 210115) and the Research Foundation of the Education
Department of Jiangxi Province of China (No. GJJ10097).
Li-yun Hu
Corresponding author. E-mail addresses: hlyun2008@126.com,
hlyun2008@gmail.com), Zheng-lu Duan, Xue-xiang Xu, and Zi-sheng Wang
College of Physics & Communication Electronics, Jiangxi Normal
University, Nanchang 330022, China
By introducing the thermo entangled state representation, we convert
the calculation of Wigner function (WF) of density operator to an overlap
between "two pure" states in a two-mode enlarged Fock space. Furthermore, we
derive a new WF evolution formula of any initial state in self-Kerr Medium
with photon loss and find that the photon number distribution for any
initial state is independent of the coupling factor with Kerr Medium, where
the number state is not affected by the Kerr nonlinearity and evolves into a
density operator of binomial distribution.
Keywords: Wigner function, Kerr Medium, entangled state
§ INTRODUCTION
Nonclassicality of optical fields has been a topic of great interest in
quantum optics and quantum information processing [1], which is
usually associated with quantum interference and entanglement. The phase
space Wigner function (WF) [2, 3] of quantum states of light is a
powerful tool for investigating such nonclassical effects. The WF was first
introduced by Wigner in 1932 to calculate quantum corrections to a classical
distribution function of a quantum-mechanical system. The partial negativity
of the WF is indeed a good indication of the highly nonclassical character
of the state [4] and monitors a decoherence process of a quantum state,
e. g. the excited coherent state in both photon-loss and thermal channels
[5, 6], the single-photon subtracted squeezed vacuum state in both
amplitude decay and phase damping channels [7], and so on <cit.>.
Nonlinear interaction of light in a medium provides a very useful framework
to study various nonclassical properties of quantum states of radiation. The
Kerr medium is one of the simplest nonlinearity, which shall allow us to
investigate the full time-dependent WF dynamics with or without a quantum
noise. Recently, a Fokker-Planck equation for the WF evolution in a noisy
Kerr medium ($\chi^{(3)}$ nonlinearity) is presented [13]. Then the
authors numerically solved this equation assuming coherent state as an
initial condition and discussed its dissipation effects. However, for any
initial condition, as far as we are concerned, there is no report about the
WF evolution. On the other hand, by using the thermo entangled state
representation (TESR) we solved various master equations to obtain density
operators with an infinite operator-sum representation [14] and then
revealed that the WF of density operator can be expressed as an overlap
between two pure states (see Eq. (<ref>) below) [15]. This brings
much convenience to calculate time evolution of WFs when quantum decoherence
happens. Thus the TESR is beneficial to quantum decoherence theory.
In this paper, we shall appeal the TESR $\left\vert \eta\right\rangle $ to
treat the WF evolution at any initial condition in self-Kerr Medium with
photon loss and present a new formula to calculate time evolution of the WF
for quantum decoherence. In addition, based on the derived WF evolution
formula, we shall deduce the photon number distribution for any initial
state in presence of Kerr interaction, where the photon number distribution
is independent of the coupling factor $\chi$ that is relative to the Kerr
medium. As examples, the WF formula is applied to the cases of initial
coherent state, and number state, respectively. Conclusions are involved in
the last section.
§ BRIEF REVIEW OF THERMO ENTANGLED STATE REPRESENTATION
We begin with briefly reviewing the thermo entangled state representation
(TESR). On the basis of Umezawa-Takahash thermo field dynamics (TFD) <cit.> we constructed the TESR in the doubled Fock space [19, 20, 21],
\begin{align}
\left\vert \eta\right\rangle & =D\left( \eta\right) \left\vert
\eta=0\right\rangle \notag \\
& =\exp\left[ -\frac{1}{2}|\eta|^{2}+\eta a^{\dagger}-\eta^{\ast}\tilde {a}%
^{\dagger}+a^{\dagger}\tilde{a}^{\dagger}\right] \left\vert 0,\tilde {0}%
\right\rangle , \label{2.1}
\end{align}
where $D\left( \eta\right) =e^{\eta a^{\dagger}-\eta^{\ast}a}$ is the
displacement operator, $\tilde{a}^{\dagger}$ is a fictitious mode
accompanying the real photon creation operator $a^{\dagger},$ $\left\vert 0,%
\tilde {0}\right\rangle =\left\vert 0\right\rangle \left\vert \tilde{0}%
\right\rangle ,$ and $\left\vert \tilde{0}\right\rangle $ is annihilated by $%
\tilde{a}$ with the relations $\left[ \tilde{a},\tilde{a}^{\dagger}\right]
=1 $ and $\left[ a,\tilde{a}^{\dagger}\right] =0$. The structure of $%
\left\vert \eta \right\rangle $ is similar to that of the EPR eigenstateshown in Ref. [19]. Operating $a$ and $\tilde{a}$ on $\left\vert
\eta\right\rangle $ in Eq.(<ref>) we can obtain the eigen-equations of $%
\left\vert \eta \right\rangle $,
\begin{align}
(a-\tilde{a}^{\dagger})\left\vert \eta\right\rangle & =\eta\left\vert
\eta\right\rangle ,\;(a^{\dagger}-\tilde{a})\left\vert \eta\right\rangle
=\eta^{\ast}\left\vert \eta\right\rangle , \notag \\
\left\langle \eta\right\vert (a^{\dagger}-\tilde{a}) & =\eta^{\ast
}\left\langle \eta\right\vert ,\ \left\langle \eta\right\vert (a-\tilde {a}%
^{\dagger})=\eta\left\langle \eta\right\vert . \label{2.2}
\end{align}
Note that $\left[ (a-\tilde{a}^{\dagger}),(a^{\dagger}-\tilde{a})\right] =0,$
thus $\left\vert \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 $\left\vert 0,\tilde{0}\right\rangle \left\langle 0,%
\tilde{0}\right\vert =\colon\exp\left( -a^{\dagger}a-\tilde {a}^{\dagger}%
\tilde{a}\right) \colon$ and the technique of integration within an ordered
product (IWOP) of operators [22, 23, 24], we can easily prove that $%
\left\vert \eta\right\rangle $ is complete and orthonormal,
\begin{equation}
\int\frac{\mathtt{d}^{2}\eta}{\pi}\left\vert \eta\right\rangle \left\langle
\eta\right\vert =1,\text{ }\left\langle \eta^{\prime}\right. \left\vert
\eta\right\rangle =\pi\delta\left( \eta^{\prime}-\eta\right) \delta\left(
\eta^{\prime\ast}-\eta^{\ast}\right) . \label{2.3}
\end{equation}
It is easily seen that $\left\vert \eta=0\right\rangle $ has the properties
\begin{equation}
\text{ \ }\left\vert \eta=0\right\rangle =e^{a^{\dagger}\tilde{a}%
^{\dagger}}\left\vert 0,\tilde{0}\right\rangle
=\sum_{n=0}^{\infty}\left\vert n,\tilde{n}\right\rangle , \label{2.4}
\end{equation}
(where $n=\tilde{n}$, and $\tilde{n}$ denotes the number in the fictitious
Hilbert space) and
\begin{align}
a\text{\ }\left\vert \eta=0\right\rangle & =\tilde{a}^{\dagger}\left\vert
\eta=0\right\rangle , \notag \\
a^{\dagger}\left\vert \eta=0\right\rangle & =\tilde{a}\left\vert
\eta=0\right\rangle , \label{2.5} \\
\left( a^{\dagger}a\right) ^{n}\left\vert \eta=0\right\rangle & =\left(
\tilde{a}^{\dagger}\tilde{a}\right) ^{n}\left\vert \eta=0\right\rangle .
\notag
\end{align}
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.
In a similar way, we can introduce the state vector $\left\vert \xi
\right\rangle $ conjugated to $\left\vert \eta\right\rangle $, defined as
\begin{align}
\left\vert \xi\right\rangle & =D\left( \xi\right) e^{-a^{\dagger}\tilde {a}%
^{\dagger}}\left\vert 0,\tilde{0}\right\rangle \notag \\
& =\exp\left( -\frac{1}{2}|\xi|^{2}+\xi a^{\dagger}+\xi^{\ast}\tilde {a}%
^{\dagger}-a^{\dagger}\tilde{a}^{\dagger}\right) \left\vert 0,\tilde {0}%
\right\rangle \notag \\
& =(-1)^{a^{\dagger}a}\left\vert \eta=-\xi\right\rangle , \label{2.9}
\end{align}
which also possesses orthonormal and complete properties
\begin{equation}
\int\frac{\mathtt{d}^{2}\xi}{\pi}\left\vert \xi\right\rangle \left\langle
\xi\right\vert =1,\text{ }\left\langle \xi^{\prime}\right. \left\vert
\xi\right\rangle =\pi\delta\left( \xi^{\prime}-\xi\right) \delta\left(
\xi^{\prime\ast}-\xi^{\ast}\right) . \label{2.10}
\end{equation}
§ MASTER EQUATION FOR A SELF-KERR INTERACTION
In the Markov approximation and interaction picture the master equation for
a dissipative cavity with Kerr medium has the form [25, 26]
\begin{equation}
\frac{d\rho }{dt}=-i\chi \left[ \left( a^{\dagger }a\right) ^{2},\rho \right]
+\gamma \left( 2a\rho a^{\dagger }-a^{\dagger }a\rho -\rho a^{\dagger
}a\right) , \label{3.5}
\end{equation}
where $\gamma $ is decaying parameter of the dissipative cavity, $\chi $ is
coupling factor depending on the Kerr medium. Milburn and Holmes [27]
solved this equation by changing it to a partial differential equation for
the Q-function and for an initial coherent state. Here we will solve the
master equation by virtue of the entangled state representation and present
the infinite sum representation of density operator.
Operating the both sides of Eq.(<ref>) on the state $\left\vert \eta
=0\right\rangle ,$ letting $\left\vert \rho \right\rangle =\rho \left\vert
\eta =0\right\rangle $ (Here one should understand the single-mode density
operator $\rho $ in the left of Eq.(<ref>) as the direct product $\rho
\otimes \tilde{I}$ when $\rho $ acts onto the two-mode state $\left\vert
\eta =0\right\rangle =e^{a^{\dagger }\tilde{a}^{\dagger }}\left\vert 0,%
\tilde{0}\right\rangle $, where $\tilde{I}$ is the identity operator in
the auxiliary mode.), and using Eq.(<ref>) we can convert the master
equation in Eq. (<ref>) into the following form,
\begin{eqnarray}
\frac{d}{dt}\left\vert \rho \right\rangle &=&\left\{ -i\chi \left[ \left(
a^{\dagger }a\right) ^{2}-\left( \tilde{a}^{\dagger }\tilde{a}\right) ^{2}%
\right] \right. \notag \\
&&+\left. \gamma \left( 2a\tilde{a}-a^{\dagger }a-\tilde{a}^{\dagger }\tilde{%
a}\right) \right\} \left\vert \rho \right\rangle , \label{3.6}
\end{eqnarray}
i.e., an evolution equation of state vector $\left\vert \rho \right\rangle $
. Its solution is then of the form
\begin{equation}
\left\vert \rho \right\rangle =e^{-i\chi t\left[ \left( a^{\dagger }a\right)
^{2}-\left( \tilde{a}^{\dagger }\tilde{a}\right) ^{2}\right] +\gamma t\left(
2a\tilde{a}-a^{\dagger }a-\tilde{a}^{\dagger }\tilde{a}\right) }\left\vert
\rho _{0}\right\rangle , \label{3.7}
\end{equation}
where $\left\vert \rho _{0}\right\rangle =\rho _{0}\left\vert \eta
=0\right\rangle ,$ $\rho _{0}$ is an initial density operator. The advantage
of using thermo field notation over more traditional algebraic manipulation
with superoperators is that in many situations (and, particularly, ones of
our interest) it enables to simplify, make more illustrative and less
cumbersome finding the solution (<ref>) and estimation of time-dependent
matrix elements. In particular, it allows to represent in a simple form a
factorization of the superoperator exp{$\cdots $} into multipliers with
easily estimated actions on the number states [28].
By introducing the following operators,
\begin{equation}
K_{0}=a^{\dagger }a-\tilde{a}^{\dagger }\tilde{a},\text{\ }K_{z}=\frac{%
a^{\dagger }a+\tilde{a}^{\dagger }\tilde{a}+1}{2},\text{\ }K_{-}=a\tilde{a},
\label{3.8}
\end{equation}
which satisfy $\left[ K_{0},K_{z}\right] =\left[ K_{0},K_{-}\right] =0,$ we
can rewrite Eq.(<ref>) as
\begin{align}
\left\vert \rho \right\rangle & =e^{\left\{ -i\chi t\left[ K_{0}(2K_{z}-1)%
\right] +\gamma t\left( 2K_{-}-2K_{z}+1\right) \right\} }\left\vert \rho
_{0}\right\rangle \notag \\
& =\exp \left[ i\chi tK_{0}+\gamma t\right] \notag \\
& \times \exp \left\{ -2t\left( \gamma +i\chi K_{0}\right) \left[ K_{z}+%
\frac{-\gamma }{\gamma +i\chi K_{0}}K_{-}\right] \right\} \left\vert \rho
_{0}\right\rangle . \label{3.9}
\end{align}
With the aid of the operator identity [30]
\begin{eqnarray}
e^{\lambda \left( A+\sigma B\right) } &=&e^{\lambda A}\exp \left[ \sigma
B\left( 1-e^{-\lambda \tau }\right) /\tau \right] \notag \\
&=&\exp \left[ \sigma B\left( e^{\lambda \tau }-1\right) /\tau \right]
e^{\lambda A}, \label{3.10}
\end{eqnarray}
which is valid for $\left[ A,B\right] =\tau B,$ and noticing $\left[
K_{z},K_{-}\right] =-K_{-},$ we can reform Eq.(<ref>) as
\begin{equation}
\left\vert \rho \right\rangle =\exp \left[ i\chi tK_{0}+\gamma t\right] \exp %
\left[ \Gamma _{z}K_{z}\right] \exp \left[ \Gamma _{-}K_{-}\right]
\left\vert \rho _{0}\right\rangle , \label{2.12}
\end{equation}
\begin{equation}
\Gamma _{z}=-2t\left( \gamma +i\chi K_{0}\right) ,\text{ }\Gamma _{-}=\frac{%
\gamma (1-e^{-2t\left( \gamma +i\chi K_{0}\right) })}{\gamma +i\chi K_{0}}.
\label{2.13}
\end{equation}
From Eq.(<ref>) we can obtain the infinite operator-sum form of $\rho
\left( t\right) $, (see Appendix A)
\begin{equation}
\rho \left( t\right) =\sum_{m,n,l=0}^{\infty }M_{m,n,l}\rho _{0}\mathcal{M}%
_{m,n,l}^{\dagger }, \label{2.26}
\end{equation}
where the two operators $M_{m,n,l}$ and $\mathcal{M}_{m,n,l}^{\dagger }$ are
respectively defined as
\begin{align}
M_{m,n,l}& \equiv \sqrt{\frac{\Lambda _{m,n}^{l}}{l!}}e^{-i\chi
tm^{2}-\gamma tm}\left\vert m\right\rangle \left\langle m\right\vert a^{l},%
\text{ } \notag \\
\mathcal{M}_{m,n,l}^{\dagger }& \equiv \left\{ \sqrt{\frac{\Lambda _{n,m}^{l}%
}{l!}}e^{-i\chi tn^{2}-\gamma tn}\left\vert n\right\rangle \left\langle
n\right\vert a^{l}\right\} ^{\dag }. \label{2.27}
\end{align}
Although $M_{m,n,l}$ is not hermite conjugate to $\mathcal{M}%
_{m,n,l}^{\dagger }$, the normalization still holds, ($\sum_{m,n,l=0}^{%
\infty }\mathcal{M}_{m,n,l}^{\dagger }M_{m,n,l}=1,$ see Appendix B <cit.>) i.e., they are trace-preserving in a general sense, so $M_{m,n,l}$
and $\mathcal{M}_{m,n,l}^{\dagger }$ may be named the generalized Kraus
§ EVOLUTION OF WIGNER FUNCTION FOR SELF-KERR CHANNEL
In this section, we consider Wigner function's time evolution in the
self-Kerr medium channel. For this purpose, we shall derive a new expression
of Wigner function in the TESR. According to the definition of Wigner
function of density operator $\rho,$
\begin{equation}
W\left( \alpha,\alpha^{\ast}\right) =\text{Tr}\left[ \Delta\left(
\alpha,\alpha^{\ast}\right) \rho\right] , \label{2.6}
\end{equation}
where $\Delta\left( \alpha,\alpha^{\ast}\right) $ is the single-mode Wigner
operator [2, 30], whose explicit normally ordered form is [31]
\begin{equation}
\Delta\left( \alpha,\alpha^{\ast}\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}. \label{2.7}
\end{equation}
By using $\left\langle \tilde{n}\right\vert \left. \tilde{m}\right\rangle
=\delta_{n,m}$ ($n=\tilde{n},m=\tilde{m}$) and noticing (<ref>) as well
as $\left\vert \rho\right\rangle =\rho\left\vert \eta=0\right\rangle ,$
we can reform Eq.(<ref>) as
\begin{align}
W\left( \alpha,\alpha^{\ast}\right) & =\sum_{m,n}^{\infty}\left\langle n,%
\tilde{n}\right\vert \Delta\left( \alpha,\alpha^{\ast}\right) \rho\left\vert
m,\tilde{m}\right\rangle \notag \\
& =\frac{1}{\pi}\left\langle \eta=0\right\vert D\left( 2\alpha\right)
(-1)^{a^{\dagger}a}\left\vert \rho\right\rangle \notag \\
& =\frac{1}{\pi}\left\langle \eta=-2\alpha\right\vert
(-1)^{a^{\dagger}a}\left\vert \rho\right\rangle \notag \\
& =\frac{1}{\pi}\left\langle \xi=2\alpha\right\vert \left. \rho\right\rangle
, \label{2.8}
\end{align}
where Eq. (<ref>) is the Wigner function formula in thermo entangled
state representation, with which the Wigner function of density operator is
simplified as an overlap between two “pure
states” in enlarged Fock space, rather than using ensemble
average in the system-mode space. This will brings much convenience to
calculate the time evolution of Wigner functions when quantum decoherence
Projecting (<ref>) on the entangled state representation $\frac{1}{\pi }%
\left\langle \xi _{=2\alpha }\right\vert ,$ and inserting the completeness
relation (<ref>), we find
\begin{equation}
W\left( \alpha ,\alpha ^{\ast },t\right) =4\int \frac{\mathtt{d}^{2}\beta }{%
\pi }G\left( \alpha ,\beta ,t\right) W\left( \beta ,\beta ^{\ast },0\right) ,
\label{3.1}
\end{equation}
where $W\left( \alpha ,\alpha ^{\ast },t\right) $ and $W\left( \beta ,\beta
^{\ast },0\right) $ are the Wigner functions at the evolving time $t$ and
initial time, respectively, and
\begin{eqnarray}
G\left( \alpha ,\beta ,t\right) &=&\left\langle \xi _{=2\alpha }\right\vert
\exp \left[ i\chi tK_{0}+\gamma t\right] \notag \\
&&\times \exp \left[ \Gamma _{z}K_{z}\right] \exp \left[ \Gamma _{-}K_{-}%
\right] \left\vert \xi _{=2\beta }^{\prime }\right\rangle . \label{3.2}
\end{eqnarray}
It is convenient to the matrix element in (<ref>) according to the
two-mode Fock space. Thus the $\left\langle \xi _{=2\alpha }\right\vert $ is
expanded as
\begin{equation}
\left\langle \xi \right\vert =\left\langle 0,\tilde{0}\right\vert
\sum_{m,n=0}^{\infty }\frac{a^{m}\tilde{a}^{n}}{m!n!}H_{m,n}\left( \xi
^{\ast },\xi \right) e^{-\left\vert \xi \right\vert ^{2}/2}. \label{3.3}
\end{equation}
By using the two-mode Fock state $\left\vert m,\tilde{n}\right\rangle
=a^{\dag m}\tilde{a}^{\dag n}/\sqrt{m!n!}\left\vert 0,\tilde{0}\right\rangle
$, we get
\begin{equation}
\left\langle \xi \right\vert \left. m,\tilde{n}\right\rangle =H_{m,n}\left(
\xi ^{\ast },\xi \right) e^{-\left\vert \xi \right\vert ^{2}/2}/\sqrt{m!n!},
\label{3.4}
\end{equation}
where $H_{m,n}\left( \xi ^{\ast },\xi \right) $ is the two-variable Hermite
polynomials [32, 33]. Inserting the complete relation $%
\sum_{m,n=0}^{\infty }\left\vert m,\tilde{n}\right\rangle \left\langle m,%
\tilde{n}\right\vert =1,$ after a long but straight calculation, then the
Wigner function's evolution is given by (see Appendix C)
\begin{equation}
W\left( \alpha ,\alpha ^{\ast },t\right) =\sum_{m,n=0}^{\infty
}C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right) E_{m,n}, \label{3.11}
\end{equation}
where $\Lambda _{m,n}\equiv \frac{\gamma (1-e^{-2t\left( \gamma +i\chi
\left( m-n\right) \right) })}{\gamma +i\chi \left( m-n\right) },$
\begin{eqnarray}
&&C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
&\equiv &\frac{e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left(
m+n\right) }e^{-2\left\vert \alpha \right\vert ^{2}}}{m!n!\left( \Lambda
_{m,n}+1\right) ^{(m+n+2)/2}}H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right) ,
\label{3.12}
\end{eqnarray}
\begin{eqnarray}
E_{m,n} &=&4\int \frac{\mathtt{d}^{2}\beta }{\pi }W\left( \beta ,\beta
^{\ast },0\right) e^{\frac{2\left( \Lambda _{m,n}-1\right) }{\Lambda _{m,n}+1%
}\left\vert \beta \right\vert ^{2}} \notag \\
&&\times H_{m,n}\left( \frac{2\beta }{\sqrt{\Lambda _{m,n}+1}},\frac{2\beta
^{\ast }}{\sqrt{\Lambda _{m,n}+1}}\right) . \label{3.13}
\end{eqnarray}
It is obvious that, when $\chi =0,$ the case of photon loss, $\Lambda
_{m,n}\rightarrow (1-e^{-2\gamma t})=T$ and Eq.(<ref>) just does reduce
to (see Appendix E)
\begin{equation}
W\left( \alpha ,\alpha ^{\ast },t\right) =\frac{2}{T}\int \frac{\mathtt{d}%
^{2}\beta }{\pi }e^{-\allowbreak \frac{2}{T}\left\vert \alpha -\beta
e^{-\gamma t}\right\vert ^{2}}W\left( \beta ,\beta ^{\ast },0\right) ,
\label{3.14}
\end{equation}
which is just the evolving formula of Wigner function for amplitude-damping
channel. While for $\gamma =0,$ without photon-loss, Eq.(<ref>) reduces
\begin{align}
& W\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
& =\sum_{m,n=0}^{\infty }\frac{\exp \left[ -i\chi t\left( m^{2}-n^{2}\right) %
\right] }{m!n!e^{2\left\vert \alpha \right\vert ^{2}}}H_{m,n}\left( 2\alpha
^{\ast },2\alpha \right) \notag \\
& \times 4\int \frac{\mathtt{d}^{2}\beta }{\pi }e^{-2\left\vert \beta
\right\vert ^{2}}H_{m,n}\left( 2\beta ,2\beta ^{\ast }\right) W\left( \beta
,\beta ^{\ast },0\right) . \label{3.15}
\end{align}
§ PHOTON NUMBER DISTRIBUTION IN PRESENCE OF KERR INTERACTION
Now we consider photon number (PN) distribution in presence of Kerr medium.
According to the TFD, we can reform the PN $p\left( n\right) =\mathtt{tr}%
\left[ \rho \left\vert n\right\rangle \left\langle n\right\vert \right] $ as
\begin{eqnarray}
p\left( n\right) &=&\left\langle n\right\vert \rho \left\vert n\right\rangle
=\sum_{m=0}^{\infty }\left\langle n,\tilde{n}\right\vert \rho \left\vert m,%
\tilde{m}\right\rangle \notag \\
&=&\left\langle n,\tilde{n}\right\vert \rho \left\vert \eta =0\right\rangle
=\left\langle n,\tilde{n}\right\vert \left. \rho \right\rangle , \label{5.1}
\end{eqnarray}
which is converted to the matrix element $\left\langle n,\tilde{n}%
\right\vert \left. \rho \right\rangle $ in the context of thermo dynamics.
Then using the completeness of $\left\langle \xi \right\vert $ and Eq.(<ref>) as well as Eq.(<ref>), we have
\begin{align}
p\left( n\right) & =\int \frac{\mathtt{d}^{2}\xi }{\pi }\left\langle n,%
\tilde{n}\right\vert \left. \xi \right\rangle \left\langle \xi \right\vert
\left. \rho \right\rangle \notag \\
& =\int \mathtt{d}^{2}\xi \left\langle n,\tilde{n}\right\vert \left. \xi
\right\rangle W\left( \alpha =\xi /2,\alpha ^{\ast }=\xi ^{\ast }/2\right)
\notag \\
& =4\pi \int \mathtt{d}^{2}\alpha W_{\left\vert n\right\rangle \left\langle
n\right\vert }\left( \alpha ,\alpha ^{\ast }\right) W\left( \alpha ,\alpha
^{\ast }\right) , \label{5.2}
\end{align}
where $W_{\left\vert n\right\rangle \left\langle n\right\vert }\left( \alpha
,\alpha ^{\ast }\right) =\frac{(-1)^{n}}{\pi }e^{-2\left\vert \alpha
\right\vert ^{2}}L_{n}(4\left\vert \alpha \right\vert ^{2})$ is the Wigner
function of number state $\left\vert n\right\rangle \left\langle
n\right\vert $ as shown in [34, 35]. Thus one can calculate the PN by
combining Eqs.(<ref>) and (<ref>).
Next we evaluate the PN for the above decoherence model in Eq.(<ref>).
Substituting Eq.(<ref>) into Eq.(<ref>), we have
\begin{align}
p\left( s\right) & =4\pi \sum_{m,n=0}^{\infty }\int \mathtt{d}^{2}\alpha
W_{\left\vert s\right\rangle \left\langle s\right\vert }\left( \alpha
,\alpha ^{\ast }\right) C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right)
E_{m,n} \notag \\
& =\sum_{m,n=0}^{\infty }\frac{4\pi e^{-i\chi t\left( m^{2}-n^{2}\right)
-\gamma t\left( m+n\right) }}{m!n!\left( \Lambda _{m,n}+1\right) ^{(m+n+2)/2}%
}E_{m,n}\cdot F_{m,n}, \label{5.3}
\end{align}
\begin{eqnarray}
F_{m,n} &\equiv &\int \mathtt{d}^{2}\alpha e^{-2\left\vert \alpha
\right\vert ^{2}}W_{\left\vert s\right\rangle \left\langle s\right\vert
}\left( \alpha ,\alpha ^{\ast }\right) H_{m,n}\left( 2\alpha ^{\ast
},2\alpha \right) \notag \\
&=&\frac{s!}{4}\delta _{m,s}\delta _{n,s}. \label{5.4}
\end{eqnarray}
Then substituting Eqs.(<ref>) and (<ref>) into (<ref>) yields
\begin{eqnarray}
p\left( s\right) &=&\frac{4(-1)^{s}e^{2\gamma t}}{\left( 2e^{2\gamma
t}-1\right) ^{s+1}}\int \mathtt{d}^{2}\beta \exp \left\{ -\frac{2\left\vert
\beta \right\vert ^{2}}{2e^{2\gamma t}-1}\right\} \notag \\
&&\text{ \ \ \ \ \ \ \ \ \ }\times L_{s}\left( \frac{4e^{2\gamma
t}\left\vert \beta \right\vert ^{2}}{2e^{2\gamma t}-1}\right) W\left( \beta
,\beta ^{\ast },0\right) , \label{5.5}
\end{eqnarray}
which corresponds to the photon number of density operator in the
amplitude-damping quantum channel [15]. From Eq.(<ref>), it is
easily to see that, for any initial state, the photon number distribution $%
p\left( s\right) $ is independent of the coupling factor $\chi $ that is
relative to the Kerr medium, as respected in [36].
§ EVOLUTION OF QUANTUM STATES
The phase space Wigner distribution function description of quantum states
of light is a powerful tool to investigate nonclassical effects, such as
quantum interference and entanglement. In this section, as the applications
of WF evolution formula, we take two special initial states as examples.
(1) When the initial state is the coherent state $\left\vert z\right\rangle $
, whose WF is given $W\left( \beta ,\beta ^{\ast }\right) =\frac{1}{\pi }%
e^{-2\left\vert \beta -z\right\vert ^{2}},$ thus substituting into Eq.(<ref>) yields (see Appendix G)
\begin{equation}
E_{m,n}=\frac{1}{\pi }\left( \Lambda _{m,n}+1\right) ^{\frac{m+n+2}{2}%
}e^{\left( \allowbreak \Lambda _{m,n}-1\right) \left\vert z\right\vert
^{2}}z^{m}z^{\ast n}, \label{5.6}
\end{equation}
\begin{eqnarray}
&&W\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
&=&\frac{e^{-2\left\vert \alpha \right\vert ^{2}}}{\pi }\sum_{m,n=0}^{\infty
}\frac{z^{m}z^{\ast n}}{m!n!}e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma
t\left( m+n\right) } \notag \\
&&\times e^{\left( \allowbreak \Lambda _{m,n}-1\right) \left\vert
z\right\vert ^{2}}H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right) ,
\label{5.7}
\end{eqnarray}
which is a new expression of the evolution of WF for any initial state. In
particular, when $\gamma =0,$ without the dissipation, Eq.(<ref>)
reduces to
\begin{eqnarray}
&&W\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
&=&\frac{e^{-\left\vert z\right\vert ^{2}}}{\pi e^{2\left\vert \alpha
\right\vert ^{2}}}\sum_{m,n=0}^{\infty }\frac{z^{m}z^{\ast n}}{m!n!}%
e^{-i\chi t(m^{2}-n^{2})}H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right) ,
\label{5.8}
\end{eqnarray}
which is identical to Eq.(7) in Ref.[13], where Eq.(7) is used to make
the numerical calculation since it is much more rapid than the other
expression Eq.(6). In addition, from Eq.(<ref>) one can see that the WF
can be obtained very quickly when the dissipation cannot be negligible.
Further when $\chi t=2\pi $, Eq.(<ref>) just returns to the WF of the
initial coherent state.
Fig.1 presents the plots of the WF for different parameters $\chi t$ and $%
\alpha=2\mathtt{.}$ From Fig.1, one can see that the WF turns into an
ellipse and squeezing appears in an appropriate direction. Then the ellipse
changes into a banana shape and a tailor of the interference fringes appears
where the distribution takes the negative values.
(Color online) The WF for different parameters $\protect\chi t$ and
$\protect\alpha =2\mathtt{,(a)}$ $\protect\chi t=0,\mathtt{(b)}$ $\protect%
\chi t=0.04,\mathtt{(c)}$ $\protect\chi t=0.06,\mathtt{(d)}$ $\protect\chi %
t=0.08,\mathtt{(e)}$ $\protect\chi t=0.1,\mathtt{(f)}$ $\protect\chi t=0.2.$
(2) Another example is number state, where the WF of number state $%
\left\vert s\right\rangle $ is given by
\begin{eqnarray}
W_{s}\left( \beta ,\beta ^{\ast },0\right) &=&\frac{1/s!}{\pi }%
e^{-2\left\vert \beta \right\vert ^{2}}H_{s,s}\left( 2\beta ,2\beta ^{\ast
}\right) \notag \\
&=&\frac{(-1)^{s}}{\pi }e^{-2\left\vert \beta \right\vert
^{2}}L_{s}(4\left\vert \beta \right\vert ^{2}), \label{5.9}
\end{eqnarray}
substituting it into (<ref>) and using the generating function of
two-variable Hermite polynomials [see below Eq.(F3)], we have
\begin{equation}
E_{m,n}=\frac{s!}{\pi }\frac{\Lambda _{m,n}^{s-m}\left( \Lambda
_{m,n}+1\right) ^{m+1}}{\left( s-m\right) !}\delta _{m,n}, \label{5.10}
\end{equation}
thus the evolution of WF for $\left\vert s\right\rangle $ is
\begin{eqnarray}
&&W_{s}\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
&=&\sum_{m=0}^{s}\binom{s}{m}e^{-2m\gamma t}\left( 1-e^{-2\gamma t}\right)
^{s-m}W_{m}\left( \alpha ,\alpha ^{\ast },0\right) , \label{5.11}
\end{eqnarray}
which is the WF of number state in the photon-loss channel and indicates
that the number state is not affected by the Kerr nonlinearity. In
particular, when $\gamma =0,$ or $t=0,$ Eq.(<ref>) just reduces to the
WF of number state. From Eq.(<ref>), on the other hand, it is found
that the number state evolves into a density operator of binomial
distribution (a mixed state) if $e^{-2\gamma t}$ is a binomial parameter.
§ CONCLUSIONS
In summary, by converting Wigner function for quantum state into an overlap
between two "pure states" in a two-mode enlarged Fock space, we investigate
the WF evolution of any initial condition in self-Kerr Medium with photon
loss and present a new formula for calculating time evolution of the WF for
quantum decoherence. Based on the derived WF evolution formula, in addition,
we discuss the photon number distribution for any initial state in presence
of Kerr interaction. It is found that the photon number distribution is
independent of the coupling factor $\chi $ in correlation with the Kerr
medium, as expected by people. As applications, furthermore, the two cases
of initial coherent state and number state are considered. It is shown that
the coherent state can be squeezed due to the presence of Kerr medium, while
the number state is not affected by the Kerr nonlinearity and evolves into a
density operator of binomial distribution (a mixed state) with $e^{-2\gamma
t}$ being a binomial parameter.
ACKNOWLEDGEMENTS Work supported by a grant from the Key Programs
Foundation of Ministry of Education of China (No. 210115) and the Research
Foundation of the Education Department of Jiangxi Province of China (No.
Appendix A: Derivation of Eq.(<ref>)
In order to obtain the infinite operator-sum form of $\rho \left( t\right) $
from Eq.(<ref>), using the completeness relation of Fock state in the
enlarged space $\sum_{m,n=0}^{\infty }\left\vert m,\tilde{n}\right\rangle
\left\langle m,\tilde{n}\right\vert =1$ and noticing $a^{\dagger
l}\left\vert n\right\rangle =\sqrt{\frac{\left( l+n\right) !}{n!}}\left\vert
n+l\right\rangle $, we have
\begin{align}
\left\vert \rho \right\rangle & =e^{i\chi tK_{0}+\gamma t}e^{\Gamma
_{z}K_{z}}e^{\Gamma _{-}K_{-}}\left\vert \rho _{0}\right\rangle \notag \\
& =e^{i\chi tK_{0}+\gamma t}e^{\Gamma _{z}K_{z}}\sum_{m,n=0}^{\infty
}\left\vert m,\tilde{n}\right\rangle \left\langle m,\tilde{n}\right\vert
e^{\Gamma _{-}K_{-}}\left\vert \rho _{0}\right\rangle \notag \\
& =\sum_{m,n=0}^{\infty }e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma
t\left( m+n\right) } \notag \\
& \times \left\vert m,\tilde{n}\right\rangle \left\langle m,\tilde{n}%
\right\vert e^{\Lambda _{m,n}a\tilde{a}}\left\vert \rho _{0}\right\rangle ,
\tag{A1}
\end{align}
\begin{equation}
\Lambda _{m,n}=\frac{\gamma (1-e^{-2t\left( \gamma +i\chi \left( m-n\right)
\right) })}{\gamma +i\chi \left( m-n\right) }. \tag{A2}
\end{equation}
Furthermore, using the relations
\begin{equation}
\left\langle n\right\vert \left. \eta =0\right\rangle =\left\vert \tilde{n}%
\right\rangle ,\text{ }\left\vert m,\tilde{n}\right\rangle =\left\vert
m\right\rangle \left\langle n\right\vert \left. \eta =0\right\rangle ,
\tag{A3}
\end{equation}
we find
\begin{align}
\left\langle m,\tilde{n}\right\vert a^{l}\rho _{0}a^{\dagger l}\left\vert
\eta =0\right\rangle & =\left\langle m\right\vert \left\langle \tilde{n}%
\right\vert a^{l}\rho _{0}a^{\dagger l}\left\vert \eta =0\right\rangle
\notag \\
& =\left\langle m\right\vert a^{l}\rho _{0}a^{\dagger l}\left( \left\langle
\tilde{n}\right\vert \left. \eta =0\right\rangle \right) \notag \\
& =\left\langle m\right\vert a^{l}\rho _{0}a^{\dagger l}\left\vert
n\right\rangle . \tag{A4}
\end{align}
Thus Eq.(A1) becomes
\begin{align}
\left\vert \rho \right\rangle & =\sum_{m,n,l=0}^{\infty }\frac{\Lambda
_{m,n}^{l}}{l!}e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left(
m+n\right) } \notag \\
& \times \left\vert m,\tilde{n}\right\rangle \left\langle m\right\vert
a^{l}\rho _{0}a^{\dagger l}\left\vert n\right\rangle \notag \\
& =\sum_{m,n,l=0}^{\infty }\frac{\sqrt{\left( n+l\right) !\left( m+l\right) !%
}}{\sqrt{n!m!}l!\Lambda _{m,n}^{-l}} \notag \\
& \times e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left( m+n\right)
}\left\vert m,\tilde{n}\right\rangle \rho _{0,m+l,n+l}, \tag{A5}
\end{align}
where $\rho _{0,m+l,n+l}\equiv \left\langle m+l\right\vert \rho
_{0}\left\vert n+l\right\rangle .$ Using Eq.(A3) again, we see
\begin{align}
\left\vert \rho \right\rangle & =\sum_{m,n,l=0}^{\infty }\frac{\sqrt{\left(
n+l\right) !\left( m+l\right) !}}{\sqrt{n!m!}l!\Lambda _{m,n}^{-l}} \notag
\\
& \times e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left( m+n\right)
}\rho _{0,m+l,n+l}\left\vert m\right\rangle \left\langle n\right\vert \left.
\eta =0\right\rangle . \tag{A6}
\end{align}
After depriving $\left\vert \eta =0\right\rangle $ from the both sides of
Eq.(A6), the solution of master equation (<ref>) appears as an infinite
operator-sum form
\begin{align}
\rho \left( t\right) & =\sum_{m,n,l=0}^{\infty }\sqrt{\frac{\left(
n+l\right) !\left( m+l\right) !}{n!m!}}\frac{\Lambda _{m,n}^{l}}{l!}%
e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left( m+n\right) }\left\vert
m\right\rangle \left\langle m+l\right\vert \rho _{0}\left\vert
n+l\right\rangle \left\langle n\right\vert \notag \\
& =\sum_{m,n,l=0}^{\infty }\frac{\Lambda _{m,n}^{l}}{l!}e^{-i\chi t\left(
m^{2}-n^{2}\right) -\gamma t\left( m+n\right) }\left\vert m\right\rangle
\left\langle m\right\vert a^{l}\rho _{0}a^{\dagger l}\left\vert
n\right\rangle \left\langle n\right\vert . \tag{A7}
\end{align}
Note that the factor $\left( m-n\right) $ appears in the
denominator of $\Lambda _{m,n}$ (see Eq.(A2)), (this is originated from the
nonlinear term of $\left( a^{\dagger }a\right) ^{2}$) so that it is
impossible to move all $n-$dependent$\ $terms to the right of $a^{l}\rho
_{0}a^{\dagger l}$. Fortunately, we can formally express Eq.(A7) as Eq.(<ref>).
Appendix B: Proof of normalization for the generalized Kraus
Using the operator identity $e^{\lambda a^{\dagger}a}=\colon\exp\left[
\left( e^{\lambda}-1\right) a^{\dagger}a\right] \colon$ and the IWOP
technique, we can prove that
\begin{align}
& \ \sum_{m,n,l=0}^{\infty}\mathcal{M}_{m,n,l}^{\dagger}M_{m,n,l} \notag \\
& =\sum_{n,l=0}^{\infty}\frac{\left( n+l\right) !}{n!}\frac{(1-e^{-2t\gamma
})^{l}}{l!}e^{-2n\gamma t}\left\vert n+l\right\rangle \left\langle
n+l\right\vert \notag \\
& =\sum_{n,l=0}^{\infty}\frac{(1-e^{-2t\gamma})^{l}}{l!}a^{\dag
l}e^{-2\gamma ta^{\dagger}a}\left\vert n\right\rangle \left\langle
n\right\vert a^{l} \notag \\
& =\sum_{l=0}^{\infty}\frac{(1-e^{-2t\gamma})^{l}}{l!}\colon\exp\left[
\left( e^{-2\gamma t}-1\right) a^{\dagger}a\right] \left( a^{\dagger
}a\right) ^{l}\colon=1, \tag{B1}
\end{align}
from which one can see that the normalization still holds, i.e., they are
trace-preserving in a general sense, so $M_{m,n,l}$ and $\mathcal{M}%
_{m,n,l}^{\dagger}$ may be named the generalized Kraus operators.
Appendix C: Derivation of Eq.(<ref>)
Using Eq.(<ref>) and (A1) as well as (A2), Eq.(<ref>) can be
rewritten as
\begin{align}
G\left( \alpha ,\beta ,t\right) & =\sum_{m,n=0}^{\infty }e^{-i\chi t\left(
m^{2}-n^{2}\right) -\gamma t\left( m+n\right) }\left\langle \xi _{=2\alpha
}\right. \left\vert m,\tilde{n}\right\rangle \left\langle m,\tilde{n}%
\right\vert e^{\Lambda _{m,n}a\tilde{a}}\left\vert \xi _{=2\beta }^{\prime
}\right\rangle \notag \\
& =\sum_{m,n=0}^{\infty }\frac{e^{-2\left\vert \alpha \right\vert ^{2}}}{%
\sqrt{m!n!}}e^{-i\chi t\left( m^{2}-n^{2}\right) -\gamma t\left( m+n\right)
}H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right) \sum_{l=0}^{\infty }\frac{%
\Lambda _{m,n}^{l}}{l!}\left\langle m,\tilde{n}\right\vert a^{l}\tilde{a}%
^{l}\left\vert \xi _{=2\beta }^{\prime }\right\rangle \notag \\
& =e^{-2\left\vert \beta \right\vert ^{2}-2\left\vert \alpha \right\vert
^{2}}\sum_{m,n=0}^{\infty }\frac{e^{-i\chi t\left( m^{2}-n^{2}\right)
-\gamma t\left( m+n\right) }}{m!n!}H_{m,n}\left( 2\alpha ^{\ast },2\alpha
\right) \sum_{l=0}^{\infty }\frac{\Lambda _{m,n}^{l}}{l!}H_{m+l,n+l}\left(
2\beta ,2\beta ^{\ast }\right) . \tag{C1}
\end{align}
Further using a new sum formula (see appendix D)
\begin{align}
& \sum_{l=0}^{\infty }\frac{z^{l}}{l!}H_{m+l,n+l}\left( x,y\right) \notag \\
& =\frac{e^{\frac{z\allowbreak xy}{z+1}}}{\left( z+1\right) ^{(m+n+2)/2}}%
H_{m,n}\left( \frac{x}{\sqrt{z+1}},\frac{y}{\sqrt{z+1}}\right) , \tag{C2}
\end{align}
thus Eq.(C1) can be recast into the following form
\begin{align}
G\left( \alpha ,\beta ,t\right) & =\sum_{m,n=0}^{\infty }C_{m,n}\left(
\alpha ,\alpha ^{\ast },t\right) e^{2\frac{\Lambda _{m,n}-1}{\Lambda _{m,n}+1%
}\left\vert \beta \right\vert ^{2}} \notag \\
& \times H_{m,n}\left( \frac{2\beta }{\sqrt{\Lambda _{m,n}+1}},\frac{2\beta
^{\ast }}{\sqrt{\Lambda _{m,n}+1}}\right) , \tag{C3}
\end{align}
where $C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right) $ is defined in (<ref>). Substituting Eq.(C3) into (<ref>) yields (<ref>) and (<ref>).
Appendix D: Derivation of Eq.(C2)
Using the integratal expression of two-mode Hermite polynomials,
\begin{equation}
H_{m,n}\left( \xi ,\eta \right) =(-1)^{n}e^{\xi \eta }\int \frac{d^{2}z}{\pi
}z^{n}z^{\ast m}e^{-\left\vert z\right\vert ^{2}+\xi z-\eta z^{\ast }},
\tag{D1}
\end{equation}
we have
\begin{align}
\sum_{l=0}^{\infty }\frac{\alpha ^{l}}{l!}H_{m+l,n+l}\left( x,y\right) &
=\sum_{l=0}^{\infty }\frac{\alpha ^{l}}{l!}(-1)^{n+l}e^{xy}\int \frac{d^{2}z%
}{\pi }z^{n+l}z^{\ast m+l}\exp \left[ -\left\vert z\right\vert
^{2}+xz-yz^{\ast }\right] \notag \\
& =e^{xy}(-1)^{n}\int \frac{d^{2}z}{\pi }\sum_{l=0}^{\infty }\frac{\left(
-\alpha \left\vert z\right\vert ^{2}\right) ^{l}}{l!}z^{n}z^{\ast m}\exp %
\left[ -\left\vert z\right\vert ^{2}+xz-yz^{\ast }\right] \notag \\
& =e^{xy}(-1)^{n}\int \frac{d^{2}z}{\pi }z^{n}z^{\ast m}\exp \left[ -\left(
\alpha +1\right) \left\vert z\right\vert ^{2}+xz-yz^{\ast }\right] \notag
\\
& =\frac{e^{\frac{\alpha \allowbreak xy}{\alpha +1}}}{\left( \alpha
+1\right) ^{(m+n+2)/2}}(-1)^{n}e^{xy/\left( \alpha +1\right) }\int \frac{%
d^{2}z}{\pi }z^{n}z^{\ast m}\exp \left[ -\left\vert z\right\vert ^{2}+\frac{%
xz-yz^{\ast }}{\sqrt{\alpha +1}}\right] \notag \\
& =\frac{e^{\frac{\alpha \allowbreak xy}{\alpha +1}}}{\left( \alpha
+1\right) ^{(m+n+2)/2}}H_{m,n}\left( \frac{x}{\sqrt{\alpha +1}},\frac{y}{%
\sqrt{\alpha +1}}\right) , \tag{D2}
\end{align}
thus we have completed the proof of (C4).
Appendix E: Derivation of Eq.(<ref>)
When $\chi =0,$ $\Lambda _{m,n}\rightarrow (1-e^{-2\gamma t})=T,$ and
\begin{align}
& C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right) \notag \\
& \equiv \frac{\exp \left[ -\gamma t\left( m+n\right) \right] }{m!n!\left(
T+1\right) ^{(m+n+2)/2}}H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right)
e^{-2\left\vert \alpha \right\vert ^{2}}, \tag{E1}
\end{align}
then we have
\begin{align}
& \sum_{m,n=0}^{\infty }C_{m,n}\left( \alpha ,\alpha ^{\ast },t\right)
H_{m,n}\left( 2\beta ,2\beta ^{\ast }\right) \notag \\
& =\frac{e^{-2\left\vert \alpha \right\vert ^{2}}}{T+1}\sum_{m,n=0}^{\infty }%
\frac{\left( \frac{e^{-\gamma t}}{\sqrt{T+1}}\right) ^{m+n}}{m!n!} \notag \\
& \times H_{m,n}\left( 2\alpha ^{\ast },2\alpha \right) H_{m,n}\left( 2\beta
,2\beta ^{\ast }\right) . \tag{E2}
\end{align}
Using the following formula
\begin{align}
& \sum_{m,n=0}^{\infty }\frac{s^{m}t^{n}}{m!n!}H_{m,n}\left( x,y\right)
H_{m,n}\left( \alpha ,\beta \right) \notag \\
& =\frac{1}{1-st}\exp \left[ \allowbreak \frac{sx\alpha +ty\beta -\left(
xy+\alpha \beta \right) st}{1-st}\right] , \tag{E3}
\end{align}
Eq.(E2) can be rewritten as
\begin{equation}
\text{(E2)}=\frac{e^{-2\left\vert \alpha \right\vert ^{2}}}{2T}e^{\frac{%
\frac{4e^{-\gamma t}}{\sqrt{T+1}}\left( \alpha ^{\ast }\beta +\alpha \beta
^{\ast }\right) -\left( \alpha ^{\ast }\alpha +\beta \beta ^{\ast }\right)
\frac{4e^{-2\gamma t}}{T+1}}{1-\frac{e^{-2\gamma t}}{T+1}}}. \tag{E4}
\end{equation}
Thus Eq.(<ref>) becomes
\begin{align}
W\left( \alpha,\alpha^{\ast},t\right) & =4\int\frac{\mathtt{d}^{2}\beta }{\pi%
}e^{2\frac{T-1}{T+1}\left\vert \beta\right\vert
^{2}}\sum_{m,n=0}^{\infty}C_{m,n}\left( \alpha,\alpha^{\ast},t\right)
H_{m,n}\left( \frac{2\beta}{\sqrt{T+1}},\frac{2\beta^{\ast}}{\sqrt{T+1}}%
\right) W\left( \beta,\beta^{\ast},0\right) \notag \\
& =\frac{2}{T}e^{-2\left\vert \alpha\right\vert ^{2}}\int\frac{\mathtt{d}%
^{2}\beta}{\pi}\exp\left[ 2\frac{T-1}{T+1}\left\vert \beta\right\vert ^{2}+%
\frac{2e^{-\gamma t}\left( \alpha^{\ast}\beta+\alpha\beta^{\ast}\right)
-2e^{-2\gamma t}\left( \left\vert \alpha\right\vert ^{2}+\frac{\left\vert
\beta\right\vert ^{2}}{T+1}\right) }{T}\right] W\left( \beta,\beta^{\ast
},0\right) \notag \\
& =R.H.S.\text{ of Eq.(\ref{3.14}).} \tag{E5}
\end{align}
Appendix F: Derivation of Eq.(<ref>)
Using the relation between Hermite polynomial and Lagurre polynomial,
\begin{equation}
L_{m}\left( xy\right) =\frac{(-1)^{m}}{m!}H_{m,m}\left( x,y\right) ,
\tag{F1}
\end{equation}
we can recast the left of Eq.(<ref>) into the following form
\begin{align}
F_{m,n} & =\frac{1}{s!}\int\frac{\mathtt{d}^{2}\alpha}{\pi}e^{-4\left\vert
\alpha\right\vert ^{2}}(-1)^{s}s!L_{s}\left( 4\left\vert \alpha\right\vert
^{2}\right) H_{m,n}\left( 2\alpha^{\ast},2\alpha\right) \notag \\
& =\frac{1}{s!}\int\frac{\mathtt{d}^{2}\alpha}{\pi}e^{-4\left\vert
\alpha\right\vert ^{2}}H_{s,s}\left( 2\alpha^{\ast},2\alpha\right)
H_{m,n}\left( 2\alpha^{\ast},2\alpha\right) \notag \\
& =\frac{1/4}{s!}\int\frac{\mathtt{d}^{2}\alpha}{\pi}e^{-\left\vert
\alpha\right\vert ^{2}}H_{s,s}\left( \alpha^{\ast},\alpha\right)
H_{m,n}\left( \alpha^{\ast},\alpha\right) . \tag{F2}
\end{align}
Further using the generating function of $H_{m,n}\left( \epsilon
,\varepsilon\right) $,
\begin{equation}
H_{m,n}\left( \epsilon,\varepsilon\right) =\frac{\partial^{m+n}}{\partial
t^{m}\partial t^{\prime n}}\left. \exp\left[ -tt^{\prime}+\epsilon
t+\varepsilon t^{\prime}\right] \right\vert _{t=t^{\prime}=0}, \tag{F3}
\end{equation}
and the integration formula,
\begin{equation}
\int\frac{d^{2}z}{\pi}e^{\zeta\left\vert z\right\vert ^{2}+\xi z+\eta
z^{\ast }}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(
\zeta\right) <0, \tag{F4}
\end{equation}
we have
\begin{align}
& \int\frac{\mathtt{d}^{2}\alpha}{\pi}e^{-\left\vert \alpha\right\vert
^{2}}H_{m^{\prime},n^{\prime}}\left( \alpha^{\ast},\alpha\right)
H_{m,n}\left( \alpha^{\ast},\alpha\right) \notag \\
& =\frac{\partial^{m+n}}{\partial t^{m}\partial t^{\prime n}}\frac {%
\partial^{m+n}}{\partial\tau^{m}\partial\tau^{\prime n}}\exp\left[
-tt^{\prime}-\tau\tau^{\prime}\right] \notag \\
& \times\int\frac{\mathtt{d}^{2}\alpha}{\pi}\left. \exp\left[ -\left\vert
\alpha\right\vert ^{2}+\left( t+\tau\right) \alpha^{\ast}+\left( t^{\prime
}+\tau^{\prime}\right) \alpha\right] \right\vert _{t=t^{\prime}=\tau
=\tau^{\prime}=0} \notag \\
& =\frac{\partial^{m^{\prime}+n^{\prime}}}{\partial t^{m^{\prime}}\partial
t^{\prime n^{\prime}}}\frac{\partial^{m+n}}{\partial\tau^{m}\partial
\tau^{\prime n}}\exp\left[ t\tau^{\prime}+\tau t^{\prime}\right]
_{t=t^{\prime}=\tau=\tau^{\prime}=0} \notag \\
& =m!n!\delta_{m^{\prime},n}\delta_{n^{\prime},m}, \tag{F5}
\end{align}
thus Eq.(F2) becomes the right hand side of Eq.(<ref>).
Appendix G: Derivation of Eq.(<ref>)
For this purpose, from Eq.(<ref>) we have
\begin{equation}
E_{m,n}=4\int \frac{\mathtt{d}^{2}\beta }{\pi ^{2}}e^{-2y\left\vert \beta
\right\vert ^{2}}H_{m,n}\left( 2x\beta ,2x\beta ^{\ast }\right)
e^{-2\left\vert \beta -z\right\vert ^{2}}, \tag{G1}
\end{equation}
where we have set
\begin{equation}
y=\frac{1-\Lambda _{m,n}}{1+\Lambda _{m,n}},x=\frac{1}{\sqrt{\Lambda _{m,n}+1%
}}. \tag{G2}
\end{equation}
Using Eqs.(F3) and (F4), Eq.(G1) can be recast into the following form,
\begin{align}
E_{m,n} & =4e^{-2\left\vert z\right\vert ^{2}}\frac{\partial^{m+n}}{\partial
t^{m}\partial t^{\prime n}}e^{-tt^{\prime}}\int\frac{\mathtt{d}^{2}\beta}{%
\pi^{2}}\left. \exp\left[ -2\left( y+1\right) \left\vert \beta\right\vert
^{2}+2\beta\left( xt+z^{\ast}\right) +2\beta^{\ast}\left( xt^{\prime
}+z\right) \right] \right\vert _{t=t^{\prime}=0} \notag \\
& =e^{-2\left\vert z\right\vert ^{2}}\frac{\partial^{m+n}}{\partial
t^{m}\partial t^{\prime n}}e^{-tt^{\prime}}\int\frac{\mathtt{d}^{2}\beta}{%
\pi^{2}}\left. \exp\left[ -\frac{y+1}{2}\left\vert \beta\right\vert
^{2}+\beta\left( xt+z^{\ast}\right) +\beta^{\ast}\left( xt^{\prime
}+z\right) \right] \right\vert _{t=t^{\prime}=0} \notag \\
& =\frac{1}{\pi}\frac{2}{y+1}e^{-\frac{2y}{y+1}\left\vert z\right\vert ^{2}}%
\frac{\partial^{m+n}}{\partial t^{m}\partial t^{\prime n}}\exp\left[ -\left(
1-\frac{2x^{2}}{y+1}\right) tt^{\prime}+\frac{2xz}{y+1}t+\frac{2xz^{\ast}}{%
y+1}t^{\prime}\right] _{t=t^{\prime}=0}. \tag{G3}
\end{align}
Note that $1-\frac{2x^{2}}{y+1}=\allowbreak 0,y+1=\frac{%
1-\Lambda _{m,n}}{1+\Lambda _{m,n}}+1=\allowbreak \frac{2}{\Lambda _{m,n}+1}%
,-\frac{2y}{y+1}=\allowbreak \Lambda _{m,n}-1,$ then we find
\begin{align}
E_{m,n}& =\frac{1}{\pi }\frac{2}{y+1}e^{-\frac{2y}{y+1}\left\vert
z\right\vert ^{2}}\left( \frac{2xz}{y+1}\right) ^{m}\left( \frac{2xz^{\ast }%
}{y+1}\right) ^{n} \notag \\
& =\frac{1}{\pi }\left( \Lambda _{m,n}+1\right) ^{\frac{m+n+2}{2}}e^{\left(
\allowbreak \Lambda _{m,n}-1\right) \left\vert z\right\vert
^{2}}z^{m}z^{\ast n}. \tag{G4}
\end{align}
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|
arxiv-papers
| 2010-09-25T11:19:26 |
2024-09-04T02:49:13.438168
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu, Zheng-lu Duan, Xue-xiang Xu, and Zi-sheng Wang",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1010.0584"
}
|
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